Time series analysis of

Daily Closing Stock Price Indices of

Standard amp Poorrsquos 500 (SampP 500)

Course Instructor Prof Samarjit Das

Report by Mohar Sen QE1201

Data

Data analyzed ndash Unadjusted closing price of SampP 500

Vintage ndash 03011950 ndash 09122013

Source ndash Yahoo Finance

Number of Observations ndash 16088

Nature ndash Non Seasonal (Daily data)

Variables

Level data ndash series P

First difference ndash Series DP

Data

Data analyzed ndash Unadjusted closing price of SampP 500

Vintage ndash 03011950 ndash 09122013

Source ndash Yahoo Finance

Number of Observations ndash 16088

Nature ndash Non Seasonal (Daily data)

Variables

Level data ndash series P

First difference ndash Series DP

Methodology

Stationarity Analysis

bull Raw data plot

bull Unit root test

bull Removal of non-stationarity

bull Confirmation of stationarity

Conditional mean

bull Box-Jenkins analysis

bull Estimation

bull Information criteria ndash finalizing specification

bull Residual diagnostics

Volatility analysis

bull ARCH test

bull ARCHGARCH model fitting

bull Information criteria ndash finalizing specification

bull Residual diagnostics

Conclusion

bull Final specification of the model

bull Recommendation if any

Plot of Raw Data

0

400

800

1200

1600

2000

55 60 65 70 75 80 85 90 95 00 05 10

P

Observations

bull Uneven trendbull Non-Stationary data

Checking for Trend

Observations

Both trend and intercept terms are significant

Therefore we proceed to test for unit root using ADF test with trend and intercept

Dependent Variable P

Method Least Squares

Date 121113 Time 1132

Sample 1031950 12092013

Included observations 16088

Variable Coefficient Std Error t-Statistic Prob

C -3072818 3871625 -7936765 00000

TREND 0092369 0000417 2215926 00000

R-squared 0753242 Mean dependent var 4356903

Adjusted R-squared 0753226 SD dependent var 4942939

SE of regression 2455470 Akaike info criterion 1384498

Sum squared resid 970E+08 Schwarz criterion 1384593

Log likelihood -1113670 Hannan-Quinn criter 1384529

F-statistic 4910329 Durbin-Watson stat 0000960

Prob(F-statistic) 0000000

ADF test (series p)

Observations

P-Valuegt005

Null hypothesis rejected Unit root present (as expected)

First difference calculated dp=p-p(-1)

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 21 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -0874981 09572

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

P(-1) -0000213 0000243 -0874981 03816

D(P(-1)) -0065128 0007889 -8255719 00000

D(P(-2)) -0042691 0007907 -5399168 00000

D(P(-3)) -0017087 0007914 -2159080 00309

D(P(-4)) -0009753 0007906 -1233615 02174

D(P(-5)) -0038168 0007906 -4827940 00000

D(P(-6)) -0012548 0007909 -1586680 01126

D(P(-7)) -0030904 0007907 -3908456 00001

D(P(-8)) 0007349 0007909 0929223 03528

D(P(-9)) -0004196 0007908 -0530685 05956

D(P(-10)) 0032448 0007898 4108179 00000

D(P(-11)) -0011639 0007902 -1472910 01408

D(P(-12)) 0048758 0007900 6172055 00000

D(P(-13)) 0021209 0007909 2681615 00073

D(P(-14)) -0019073 0007911 -2410950 00159

D(P(-15)) -0024191 0007909 -3058820 00022

D(P(-16)) 0027656 0007911 3496115 00005

D(P(-17)) 0018294 0007908 2313352 00207

D(P(-18)) -0047570 0007910 -6014067 00000

D(P(-19)) 0008477 0007918 1070654 02843

D(P(-20)) 817E-05 0007911 0010329 09918

D(P(-21)) -0046335 0007896 -5868435 00000

C -0145949 0140840 -1036276 03001

TREND(1031950) 463E-05 259E-05 1790536 00734

R-squared 0020029 Mean dependent var 0111498

Adjusted R-squared 0018624 SD dependent var 7611029

SE of regression 7539821 Akaike info criterion 6879767

Sum squared resid 9119701 Schwarz criterion 6891246

Log likelihood -5524117 Hannan-Quinn criter 6883563

F-statistic 1425548 Durbin-Watson stat 1999061

Prob(F-statistic) 0000000

Plot of Series dp (after first differencing of p)

Observations

bull Apparently stationarybull Conditionally

heteroskedasticbull Unit Root test required

for confirmation of stationarity

-120

-80

-40

0

40

80

120

55 60 65 70 75 80 85 90 95 00 05 10

DP

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Plot of Raw Data

0

400

800

1200

1600

2000

55 60 65 70 75 80 85 90 95 00 05 10

P

Observations

bull Uneven trendbull Non-Stationary data

Checking for Trend

Observations

Both trend and intercept terms are significant

Therefore we proceed to test for unit root using ADF test with trend and intercept

Dependent Variable P

Method Least Squares

Date 121113 Time 1132

Sample 1031950 12092013

Included observations 16088

Variable Coefficient Std Error t-Statistic Prob

C -3072818 3871625 -7936765 00000

TREND 0092369 0000417 2215926 00000

R-squared 0753242 Mean dependent var 4356903

Adjusted R-squared 0753226 SD dependent var 4942939

SE of regression 2455470 Akaike info criterion 1384498

Sum squared resid 970E+08 Schwarz criterion 1384593

Log likelihood -1113670 Hannan-Quinn criter 1384529

F-statistic 4910329 Durbin-Watson stat 0000960

Prob(F-statistic) 0000000

ADF test (series p)

Observations

P-Valuegt005

Null hypothesis rejected Unit root present (as expected)

First difference calculated dp=p-p(-1)

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 21 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -0874981 09572

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

P(-1) -0000213 0000243 -0874981 03816

D(P(-1)) -0065128 0007889 -8255719 00000

D(P(-2)) -0042691 0007907 -5399168 00000

D(P(-3)) -0017087 0007914 -2159080 00309

D(P(-4)) -0009753 0007906 -1233615 02174

D(P(-5)) -0038168 0007906 -4827940 00000

D(P(-6)) -0012548 0007909 -1586680 01126

D(P(-7)) -0030904 0007907 -3908456 00001

D(P(-8)) 0007349 0007909 0929223 03528

D(P(-9)) -0004196 0007908 -0530685 05956

D(P(-10)) 0032448 0007898 4108179 00000

D(P(-11)) -0011639 0007902 -1472910 01408

D(P(-12)) 0048758 0007900 6172055 00000

D(P(-13)) 0021209 0007909 2681615 00073

D(P(-14)) -0019073 0007911 -2410950 00159

D(P(-15)) -0024191 0007909 -3058820 00022

D(P(-16)) 0027656 0007911 3496115 00005

D(P(-17)) 0018294 0007908 2313352 00207

D(P(-18)) -0047570 0007910 -6014067 00000

D(P(-19)) 0008477 0007918 1070654 02843

D(P(-20)) 817E-05 0007911 0010329 09918

D(P(-21)) -0046335 0007896 -5868435 00000

C -0145949 0140840 -1036276 03001

TREND(1031950) 463E-05 259E-05 1790536 00734

R-squared 0020029 Mean dependent var 0111498

Adjusted R-squared 0018624 SD dependent var 7611029

SE of regression 7539821 Akaike info criterion 6879767

Sum squared resid 9119701 Schwarz criterion 6891246

Log likelihood -5524117 Hannan-Quinn criter 6883563

F-statistic 1425548 Durbin-Watson stat 1999061

Prob(F-statistic) 0000000

Plot of Series dp (after first differencing of p)

Observations

bull Apparently stationarybull Conditionally

heteroskedasticbull Unit Root test required

for confirmation of stationarity

-120

-80

-40

0

40

80

120

55 60 65 70 75 80 85 90 95 00 05 10

DP

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Checking for Trend

Observations

Both trend and intercept terms are significant

Therefore we proceed to test for unit root using ADF test with trend and intercept

Dependent Variable P

Method Least Squares

Date 121113 Time 1132

Sample 1031950 12092013

Included observations 16088

Variable Coefficient Std Error t-Statistic Prob

C -3072818 3871625 -7936765 00000

TREND 0092369 0000417 2215926 00000

R-squared 0753242 Mean dependent var 4356903

Adjusted R-squared 0753226 SD dependent var 4942939

SE of regression 2455470 Akaike info criterion 1384498

Sum squared resid 970E+08 Schwarz criterion 1384593

Log likelihood -1113670 Hannan-Quinn criter 1384529

F-statistic 4910329 Durbin-Watson stat 0000960

Prob(F-statistic) 0000000

ADF test (series p)

Observations

P-Valuegt005

Null hypothesis rejected Unit root present (as expected)

First difference calculated dp=p-p(-1)

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 21 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -0874981 09572

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

P(-1) -0000213 0000243 -0874981 03816

D(P(-1)) -0065128 0007889 -8255719 00000

D(P(-2)) -0042691 0007907 -5399168 00000

D(P(-3)) -0017087 0007914 -2159080 00309

D(P(-4)) -0009753 0007906 -1233615 02174

D(P(-5)) -0038168 0007906 -4827940 00000

D(P(-6)) -0012548 0007909 -1586680 01126

D(P(-7)) -0030904 0007907 -3908456 00001

D(P(-8)) 0007349 0007909 0929223 03528

D(P(-9)) -0004196 0007908 -0530685 05956

D(P(-10)) 0032448 0007898 4108179 00000

D(P(-11)) -0011639 0007902 -1472910 01408

D(P(-12)) 0048758 0007900 6172055 00000

D(P(-13)) 0021209 0007909 2681615 00073

D(P(-14)) -0019073 0007911 -2410950 00159

D(P(-15)) -0024191 0007909 -3058820 00022

D(P(-16)) 0027656 0007911 3496115 00005

D(P(-17)) 0018294 0007908 2313352 00207

D(P(-18)) -0047570 0007910 -6014067 00000

D(P(-19)) 0008477 0007918 1070654 02843

D(P(-20)) 817E-05 0007911 0010329 09918

D(P(-21)) -0046335 0007896 -5868435 00000

C -0145949 0140840 -1036276 03001

TREND(1031950) 463E-05 259E-05 1790536 00734

R-squared 0020029 Mean dependent var 0111498

Adjusted R-squared 0018624 SD dependent var 7611029

SE of regression 7539821 Akaike info criterion 6879767

Sum squared resid 9119701 Schwarz criterion 6891246

Log likelihood -5524117 Hannan-Quinn criter 6883563

F-statistic 1425548 Durbin-Watson stat 1999061

Prob(F-statistic) 0000000

Plot of Series dp (after first differencing of p)

Observations

bull Apparently stationarybull Conditionally

heteroskedasticbull Unit Root test required

for confirmation of stationarity

-120

-80

-40

0

40

80

120

55 60 65 70 75 80 85 90 95 00 05 10

DP

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

ADF test (series p)

Observations

P-Valuegt005

Null hypothesis rejected Unit root present (as expected)

First difference calculated dp=p-p(-1)

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 21 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -0874981 09572

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

P(-1) -0000213 0000243 -0874981 03816

D(P(-1)) -0065128 0007889 -8255719 00000

D(P(-2)) -0042691 0007907 -5399168 00000

D(P(-3)) -0017087 0007914 -2159080 00309

D(P(-4)) -0009753 0007906 -1233615 02174

D(P(-5)) -0038168 0007906 -4827940 00000

D(P(-6)) -0012548 0007909 -1586680 01126

D(P(-7)) -0030904 0007907 -3908456 00001

D(P(-8)) 0007349 0007909 0929223 03528

D(P(-9)) -0004196 0007908 -0530685 05956

D(P(-10)) 0032448 0007898 4108179 00000

D(P(-11)) -0011639 0007902 -1472910 01408

D(P(-12)) 0048758 0007900 6172055 00000

D(P(-13)) 0021209 0007909 2681615 00073

D(P(-14)) -0019073 0007911 -2410950 00159

D(P(-15)) -0024191 0007909 -3058820 00022

D(P(-16)) 0027656 0007911 3496115 00005

D(P(-17)) 0018294 0007908 2313352 00207

D(P(-18)) -0047570 0007910 -6014067 00000

D(P(-19)) 0008477 0007918 1070654 02843

D(P(-20)) 817E-05 0007911 0010329 09918

D(P(-21)) -0046335 0007896 -5868435 00000

C -0145949 0140840 -1036276 03001

TREND(1031950) 463E-05 259E-05 1790536 00734

R-squared 0020029 Mean dependent var 0111498

Adjusted R-squared 0018624 SD dependent var 7611029

SE of regression 7539821 Akaike info criterion 6879767

Sum squared resid 9119701 Schwarz criterion 6891246

Log likelihood -5524117 Hannan-Quinn criter 6883563

F-statistic 1425548 Durbin-Watson stat 1999061

Prob(F-statistic) 0000000

Plot of Series dp (after first differencing of p)

Observations

bull Apparently stationarybull Conditionally

heteroskedasticbull Unit Root test required

for confirmation of stationarity

-120

-80

-40

0

40

80

120

55 60 65 70 75 80 85 90 95 00 05 10

DP

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Plot of Series dp (after first differencing of p)

Observations

bull Apparently stationarybull Conditionally

heteroskedasticbull Unit Root test required

for confirmation of stationarity

-120

-80

-40

0

40

80

120

55 60 65 70 75 80 85 90 95 00 05 10

DP

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

ADF test (series dp)

Observations

P-Valueltlt005 =gt Null hypothesis accepted ndash no unit root in the series dp

It is established that series dp is stationary We try to model it as an ARMA(pq) process by Box Jenkinrsquosmethod

Null Hypothesis P has a unit root

Exogenous Constant Linear Trend

Lag Length 20 (Automatic - based on SIC maxlag=42)

t-Statistic Prob

Augmented Dickey-Fuller test statistic -2906799 00000

Test critical values 1 level -3958605

5 level -3410082

10 level -3126769

MacKinnon (1996) one-sided p-values

Variable Coefficient Std Error t-Statistic Prob

DP(-1) -1208127 0041562 -2906799 00000

D(DP(-1)) 0142840 0040571 3520791 00004

D(DP(-2)) 0099996 0039534 2529330 00114

D(DP(-3)) 0082756 0038448 2152420 00314

D(DP(-4)) 0072855 0037418 1947064 00515

D(DP(-5)) 0034538 0036323 0950846 03417

D(DP(-6)) 0021837 0035136 0621521 05343

D(DP(-7)) -0009215 0033957 -0271356 07861

D(DP(-8)) -0002010 0032778 -0061307 09511

D(DP(-9)) -0006353 0031506 -0201653 08402

D(DP(-10)) 0025942 0030062 0862927 03882

D(DP(-11)) 0014150 0028576 0495183 06205

D(DP(-12)) 0062754 0026917 2331407 00197

D(DP(-13)) 0083808 0025169 3329787 00009

D(DP(-14)) 0064577 0023271 2774993 00055

D(DP(-15)) 0040232 0021310 1887948 00591

D(DP(-16)) 0067737 0019191 3529544 00004

D(DP(-17)) 0085886 0016976 5059121 00000

D(DP(-18)) 0038172 0014478 2636591 00084

D(DP(-19)) 0046510 0011535 4032085 00001

D(DP(-20)) 0046458 0007894 5884990 00000

C -0080367 0119239 -0674003 05003

TREND(1031950) 267E-05 129E-05 2074122 00381

R-squared 0539061 Mean dependent var 0000204

Adjusted R-squared 0538429 SD dependent var 1109785

SE of regression 7539766 Akaike info criterion 6879690

Sum squared resid 9120137 Schwarz criterion 6890691

Log likelihood -5524155 Hannan-Quinn criter 6883328

F-statistic 8528214 Durbin-Watson stat 1999071

Prob(F-statistic) 0000000

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Correlogram analysis (series dp)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

This hints at ARMA(pq) model and not AR(p) or MA(q) model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0063 -0063 -799058 64008 0

2 -0039 -0043 -545389 87923 0

3 -0007 -0012 -152201 8871 0

4 -0001 -0004 -050734 88726 0

5 -004 -0041 -520022 11425 0

6 -0006 -0012 -152201 11482 0

7 -0026 -0031 -393187 1255 0

8 0009 0004 0507338 12688 0

9 -0006 -0008 -101468 12738 0

10 0034 0032 4058705 14622 0

11 -002 -0017 -215619 15252 0

12 005 0048 6088058 19218 0

13 0019 0025 3170863 19781 0

14 -0023 -0017 -215619 20648 0

15 -0029 -0026 -32977 2198 0

16 0035 003 3805036 23985 0

17 0014 0022 279036 24304 0

18 -0052 -0048 -608806 28721 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0013 001 1268345 28988 0

20 0008 0003 0380504 29104 0

21 -0048 -0046 -583439 32879 0

22 0018 001 1268345 3343 0

23 0007 0004 0507338 33519 0

24 0003 0004 0507338 33536 0

25 -001 -0013 -164885 33703 0

26 -0009 -0014 -177568 33843 0

27 0033 0035 4439209 35586 0

28 0007 0009 1141511 35656 0

29 0019 0019 2409856 36249 0

30 0005 0013 1648849 36284 0

31 -0003 0006 0761007 36299 0

32 0001 -0002 -025367 363 0

33 -0016 -0014 -177568 36714 0

34 -0066 -0061 -773691 4372 0

35 0006 -0004 -050734 4377 0

36 0018 0009 1141511 44317 0

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

ARMA Estimation ndash Information Criteria

Information Criteria(SIC)

MA(0) MA(1) MA(2)

AR(0) AIC 6895772

6896728

6891609

6892565

6890118

6891551SIC

AR(1) AIC 6892031

6892986

6889474

6890907

6889569

6891480SIC

AR(2) AIC 6890391

6891824

6889632

6891544

6889077

6891466SIC

Observations

ARMA(11) appears to be the best fitted model

The suitability has been in line with both AIC and SIC

Sufficient data present in the model so SIC is the best criteria

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Estimation of the model

Observations

As expected all the terms are significant in the ARMA(11) model

The Durbin Watson Statistic is asymp 2 This suggests that the autocorrelation among residuals may be small

This is in fact confirmed by the correlogramQ-statistics of the residuals

Dependent Variable DP

Method Least Squares

Date 121113 Time 0144

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 8 iterations

MA Backcast 1041950

Variable Coefficient Std Error t-Statistic Prob

C 0111277 0049883 2230768 00257

AR(1) 0607851 0056980 1066776 00000

MA(1) -0672801 0053089 -1267311 00000

R-squared 0006645 Mean dependent var 0111371

Adjusted R-squared 0006522 SD dependent var 7606298

SE of regression 7581454 Akaike info criterion 6889474

Sum squared resid 9244258 Schwarz criterion 6890907

Log likelihood -5540904 Hannan-Quinn criter 6889948

F-statistic 5379525 Durbin-Watson stat 2004706

Prob(F-statistic) 0000000

Inverted AR Roots 61

Inverted MA Roots 67

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Correlogram Q statistic (of residuals)

Observations

All ACF values from 5th lag onwards significant

No definite threshold found such that PACF vanishes above that threshold

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 -0002 -0002 -025366 00897

2 -0001 -0001 -012683 01221

3 0015 0015 1902459 35484 006

4 0011 0011 1395137 55919 0061

5 -0031 -0031 -393175 20833 0

6 -0002 -0003 -038049 20928 0

7 -0022 -0022 -279027 28685 0

8 0011 0011 1395137 30563 0

9 -0002 -0001 -012683 30637 0

10 0036 0035 4439071 50953 0

11 -0014 -0014 -177563 54261 0

12 005 0049 6214699 95031 0

13 002 002 2536612 10169 0

14 -0021 -0021 -266344 1086 0

15 -0027 -0025 -317076 12008 0

16 0033 003 3804918 13763 0

17 0012 0017 215612 14007 0

18 -0051 -005 -634153 18163 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 001 0011 1395137 18319 0

20 0006 0001 0126831 18372 0

21 -0047 -0045 -570738 21954 0

22 0016 0014 1775628 22378 0

23 0007 0006 0760984 22456 0

24 0003 0004 0507322 22474 0

25 -0009 -0011 -139514 226 0

26 -0007 -001 -126831 22681 0

27 0033 0037 4692732 24466 0

28 0009 0009 1141475 24598 0

29 002 0017 215612 25252 0

30 0005 0009 1141475 25299 0

31 -0003 0002 0253661 25318 0

32 -0002 -0006 -076098 25322 0

33 -0019 -0018 -228295 25902 0

34 -0066 -006 -760984 32879 0

35 0003 0001 0126831 32889 0

36 0017 0013 1648798 33334 0

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Correlogram Q statistic (of squared residuals)

Observations

All ACF values significant

No definite threshold found such that PACF vanishes above that threshold

However square of the lag values are larger (in ACF)

This suggests ARCH type modelling is more appropriate

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0236 0236 2993202 89426

2 0387 035 4439071 32984

3 0243 0123 1560016 42517 0

4 0297 0139 1762945 56715 0

5 0323 0189 2397098 73461 0

6 0311 0137 1737579 88992 0

7 0303 0098 124294 10377 0

8 0295 0089 1128792 11777 0

9 0283 0067 849765 13068 0

10 0311 0094 1192208 14621 0

11 0307 0089 1128792 16138 0

12 0287 0045 5707377 17464 0

13 0235 -0027 -342443 18356 0

14 022 -0039 -494639 19138 0

15 0237 0001 0126831 20041 0

16 0271 0051 646836 21225 0

17 0269 0045 5707377 22387 0

18 0272 0047 5961038 23577 0

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0209 -0026 -32976 24280 0

20 0253 003 3804918 25307 0

21 0289 0105 1331721 26653 0

22 0223 -0013 -16488 27456 0

23 0242 -0004 -050732 28402 0

24 0191 -002 -253661 28991 0

25 0209 -0012 -152197 29696 0

26 0205 -0013 -16488 30372 0

27 0276 0079 1001962 31600 0

28 0233 0023 2917104 32474 0

29 0214 -002 -253661 33215 0

30 0185 -0022 -279027 33768 0

31 0194 -0006 -076098 34373 0

32 0251 0067 849765 35388 0

33 02 -0005 -063415 36032 0

34 0232 003 3804918 36902 0

35 0169 -0018 -228295 37360 0

36 0208 0013 1648798 38055 0

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

ARCH Heteroskedasticity test

Observations

We reaffirm the ARCH effect using the Heteroskedasticity ndash ARCH test

Clearly all residue lags seem significant (uptolag 9)

This hints that we may have to opt for a GARCH(pq) model

Hence we have not tested for pure ARCH models any further

Heteroskedasticity Test ARCH

F-statistic 6246957 Prob F(916067) 00000

ObsR-squared 4167458 Prob Chi-Square(9) 00000

Variable Coefficient Std Error t-Statistic Prob

C 1160923 2045328 5675975 00000

RESID^2(-1) 0013088 0007872 1662714 00964

RESID^2(-2) 0213112 0007842 2717408 00000

RESID^2(-3) 0016465 0007996 2059229 00395

RESID^2(-4) 0062821 0007948 7903630 00000

RESID^2(-5) 0147233 0007879 1868761 00000

RESID^2(-6) 0111116 0007948 1397978 00000

RESID^2(-7) 0080432 0007996 1005965 00000

RESID^2(-8) 0087330 0007842 1113558 00000

RESID^2(-9) 0066907 0007872 8499328 00000

R-squared 0259219 Mean dependent var 5749988

Adjusted R-squared 0258804 SD dependent var 2862004

SE of regression 2463978 Akaike info criterion 1385239

Sum squared resid 975E+08 Schwarz criterion 1385717

Log likelihood -1113425 Hannan-Quinn criter 1385397

F-statistic 6246957 Durbin-Watson stat 2012623

Prob(F-statistic) 0000000

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH(11)

Observations

P-Value of Coefficient of GARCH(-1) is more less than 005

Thus we conclude volatility is of GARCH kind

Thus we do not check for ARCH We check for the best form of GARCH that fits the data

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0305

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 32 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0018850 0003353 5622112 00000

AR(1) -0141277 0075434 -1872843 00611

MA(1) 0242258 0073980 3274633 00011

Variance Equation

C 0000213 329E-05 6465720 00000

RESID(-1)^2 0066426 0001506 4410476 00000

GARCH(-1) 0939441 0001222 7689159 00000

R-squared -0022203 Mean dependent var 0111371

Adjusted R-squared -0022330 SD dependent var 7606298

SE of regression 7690752 Akaike info criterion 3800477

Sum squared resid 9512719 Schwarz criterion 3803344

Log likelihood -3056124 Hannan-Quinn criter 3801425

Durbin-Watson stat 2322505

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH(12)

Observations

the coefficients of RESID(-1)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0312

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 33 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1)

Variable Coefficient Std Error z-Statistic Prob

C 0019100 0003404 5610811 00000

AR(1) -0137936 0072289 -1908117 00564

MA(1) 0242221 0070576 3432082 00006

Variance Equation

C 0000161 262E-05 6145381 00000

RESID(-1)^2 0107198 0004442 2413017 00000

RESID(-2)^2 -0050822 0005119 -9928459 00000

GARCH(-1) 0948322 0001527 6212227 00000

R-squared -0023299 Mean dependent var 0111371

Adjusted R-squared -0023427 SD dependent var 7606298

SE of regression 7694877 Akaike info criterion 3798666

Sum squared resid 9522926 Schwarz criterion 3802010

Log likelihood -3054567 Hannan-Quinn criter 3799772

Durbin-Watson stat 2328892

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH(22)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 0319

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 36 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)GARCH(-1) + C(8)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

Inverted AR Roots -14

Inverted MA Roots -24

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH(23)

Observations

the coefficients of RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 GARCH(-1) and GARCH(-2) are all significant at 5 level of significance

we cannot draw any inference talking about the best fitted volatility model to the data

Likewise all models upto GARCH(33) have been estimated though they have not been pasted in this presentation for brevity

To see which of these models best fit the data we consider the minimum AIC and BIC values

Dependent Variable DP

Method ML - ARCH (Marquardt) - Normal distribution

Date 121113 Time 1223

Sample (adjusted) 1051950 12092013

Included observations 16086 after adjustments

Convergence achieved after 179 iterations

MA Backcast 1041950

Presample variance backcast (parameter = 07)

GARCH = C(4) + C(5)RESID(-1)^2 + C(6)RESID(-2)^2 + C(7)RESID(-3)^2+ + C(8)GARCH(-1) + C(9)GARCH(-2)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

Inverted AR Roots -14

Inverted MA Roots -25

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Volatility Estimation ndash Information Criteria

Information Criteria(SIC)

AIC BIC

GARCH(11) 3800477 3803344

GARCH(12) 3798666 3802010

GARCH(13) 3798186 3802009

GARCH(21) 3800477 3802872

GARCH(22) 3794703 3798525

GARCH(23) 3794321 3798621

GARCH (31) 3798906 3802729

GARCH(32) 3799317 3803617

GARCH(33) 3799790 3804568

Observations

GARCH (22) and GARCH (23) appear to be the best fitted models

We proceed to estimate both models

The final model selection will depend upon the residual diagnostics

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0019824 0003325 5962788 00000

AR(1) -0142799 0071582 -1994902 00461

MA(1) 0248253 0069856 3553797 00004

Variance Equation

C 195E-06 427E-07 4569525 00000

RESID(-1)^2 0103399 0004377 2362315 00000

RESID(-2)^2 -0133792 0009751 -1372090 00000

RESID(-3)^2 0031082 0005770 5386980 00000

GARCH(-1) 1904230 0005893 3231577 00000

GARCH(-2) -0904878 0005823 -1554076 00000

R-squared -0023694 Mean dependent var 0111371

Adjusted R-squared -0023822 SD dependent var 7606298

SE of regression 7696361 Akaike info criterion 3794321

Sum squared resid 9526601 Schwarz criterion 3798621

Log likelihood -3050872 Hannan-Quinn criter 3795743

Durbin-Watson stat 2331385

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15477

2 0002 0002 0253661 15945

3 0 0 0 15952 0207

4 0015 0015 1902459 52146 0074

5 -0004 -0004 -050732 54646 0141

6 -0014 -0014 -177563 8698 0069

7 -0003 -0002 -025366 88176 0117

8 0008 0008 1014645 99152 0128

9 -0003 -0003 -038049 10031 0187

10 0018 0019 2409781 15317 0053

11 -0007 -0008 -101464 16206 0063

12 0013 0013 1648798 19079 0039

13 0005 0005 0634153 19468 0053

14 -0004 -0005 -063415 19747 0072

15 0 0 0 19748 0102

16 0011 0011 1395137 21735 0084

17 -001 -0011 -139514 23501 0074

18 -001 -0009 -114148 25015 007

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0003 0003 0380492 25131 0092

20 0005 0004 0507322 25486 0112

21 -0006 -0006 -076098 26126 0127

22 -001 -001 -126831 27786 0115

23 -0003 -0004 -050732 2797 0141

24 0013 0012 1521967 30601 0105

25 -0013 -0013 -16488 3345 0074

26 -0016 -0015 -190246 37473 0039

27 0006 0007 0887814 38039 0046

28 0005 0004 0507322 38424 0055

29 0008 0009 1141475 3955 0056

30 0004 0005 0634153 3977 0069

31 -0013 -0013 -16488 42344 0052

32 0005 0004 0507322 42673 0063

33 0011 0011 1395137 44582 0054

34 -0012 -0012 -152197 46866 0044

35 -0007 -0006 -076098 47736 0047

36 0001 0001 0126831 4777 0059

Estimation Correlogram Q statistics of Residuals

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

GARCH (23)

Variable Coefficient Std Error z-Statistic Prob

C 0020273 0003412 5940926 00000

AR(1) -0137986 0072567 -1901495 00572

MA(1) 0242984 0070989 3422855 00006

Variance Equation

C 430E-06 860E-07 5000378 00000

RESID(-1)^2 0085055 0002502 3400018 00000

RESID(-2)^2 -0083594 0002442 -3423010 00000

GARCH(-1) 1871332 0006706 2790638 00000

GARCH(-2) -0872703 0006578 -1326706 00000

R-squared -0023536 Mean dependent var 0111371

Adjusted R-squared -0023663 SD dependent var 7606298

SE of regression 7695767 Akaike info criterion 3794703

Sum squared resid 9525129 Schwarz criterion 3798525

Log likelihood -3051279 Hannan-Quinn criter 3795967

Durbin-Watson stat 2330303

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 001 001 1268306 15963

2 0001 0001 0126831 16214

3 0 0 0 16214 0203

4 0014 0014 1775628 49305 0085

5 -0004 -0004 -050732 51411 0162

6 -0014 -0014 -177563 85074 0075

7 -0003 -0003 -038049 86544 0124

8 0008 0008 1014645 96626 014

9 -0002 -0002 -025366 97411 0204

10 0018 0018 2282951 14693 0065

11 -0007 -0008 -101464 15589 0076

12 0013 0013 1648798 18335 005

13 0005 0005 0634153 18735 0066

14 -0004 -0005 -063415 19059 0087

15 0 0 0 19059 0121

16 0011 0011 1395137 21107 0099

17 -0011 -0011 -139514 22955 0085

18 -001 -0009 -114148 24507 0079

Lags AC PACSQRT(N)PAC Q-Stat Prob

19 0002 0003 0380492 24605 0104

20 0005 0004 0507322 24978 0126

21 -0006 -0006 -076098 25604 0142

22 -001 -001 -126831 27214 0129

23 -0003 -0004 -050732 27395 0158

24 0013 0013 1648798 30322 0111

25 -0014 -0014 -177563 33438 0074

26 -0016 -0016 -202929 37617 0038

27 0006 0007 0887814 38194 0044

28 0005 0004 0507322 38651 0053

29 0008 0008 1014645 39671 0055

30 0004 0005 0634153 39933 0067

31 -0013 -0013 -16488 42486 0051

32 0005 0004 0507322 42841 0061

33 0011 0011 1395137 44682 0053

34 -0012 -0012 -152197 46952 0043

35 -0007 -0006 -076098 47832 0046

36 0001 0001 0126831 4786 0058

Estimation Correlogram Q statistics of Residuals

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

The best fit model

Observations

The Residual diagnostics (previous slides) hint that GARCH(23) is a better fit compared to GARCH(22)

We check the squared residuals of GARCH(23) to affirm the goodness of fit

The correlogram affirms our result that GARCH (23) is indeed the best fit model

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

1 0011 0011 1395137 20998

2 0002 0002 0253661 21909

3 0001 0001 0126831 22065 0137

4 -0002 -0002 -025366 22434 0326

5 0 0 0 22444 0523

6 -001 -001 -126831 37929 0435

7 -0012 -0012 -152197 6194 0288

8 -0001 -0001 -012683 62132 04

9 001 001 1268306 77643 0354

10 0001 0 0 7769 0456

11 -0002 -0002 -025366 7816 0553

12 -0006 -0006 -076098 83879 0591

13 -0004 -0004 -050732 8619 0657

14 0002 0002 0253661 86817 073

15 -0005 -0005 -063415 91452 0762

16 -0004 -0004 -050732 94626 08

17 -0009 -0009 -114148 10793 0767

18 -0005 -0005 -063415 11272 0792

lags AC PAC

SQRT(N)

PAC Q-Stat Prob

19 0002 0002 0253661 11368 0837

20 -0002 -0002 -025366 11424 0876

21 -0002 -0001 -012683 1146 0907

22 -0005 -0005 -063415 11849 0921

23 0 0 0 11849 0944

24 0002 0001 0126831 11894 096

25 -0001 -0001 -012683 11921 0972

26 -0022 -0021 -266344 19371 0732

27 -0005 -0004 -050732 19765 0759

28 -0008 -0008 -101464 20862 0749

29 0001 0001 0126831 20871 0792

30 -0008 -0008 -101464 21971 0783

31 -0005 -0005 -063415 22344 0806

32 -0007 -0008 -101464 23234 0806

33 0009 0009 1141475 24675 0782

34 0004 0003 0380492 24903 081

35 0008 0008 1014645 25907 0805

36 0002 0002 0253661 25965 0837

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

Conclusion

The data has significant stochastic trend which got removed after first differencing

No presence of seasonality in the data

The best fitted conditional mean model is ARMA(11)

Heteroskedasticity exists in the data

The best fitted volatility model is GARCH(23)

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das

DATA SOURCE Yahoo FinanceSOFTWARE USEDEviews 7Microsoft Office 2013

Reference ARCHGARCH Models by Prof Samarjit Das