Regression Arima Garch Var Model

7/28/2019 Regression Arima Garch Var Model

1/19

Forecasting using Regression,ARIMA and VAR Models

Submitted to Prof. Sajal Ghosh for partial fulfillment of Part TimePGPM Oct 2011 batch

Modeling and Forecasting in Energy and Financial

Markets

Submitted by:

Manish Jindal (11PT2-36)


2/19

1. RegressionInput Data Used (from E-Views Database):

Table 7.3

Real Gross Product, Labor Days and Real Capital Input in theAgricultural Sector, Taiwan, 1958-1972

YEAR = YearY = Real Gross Product, Millions of NT $X2 = Labor Days, Millions of DaysX3 = Real Capital Input, Millions of NT $

YEAR Y X2 X31958 16607.7 275.5 17803.71959 17511.3 274.4 18096.81960 20171.2 269.7 18271.81961 20932.9 267.0 19167.3

1962 20406.0 267.8 19647.61963 20831.6 275.0 20803.51964 24806.3 283.0 22076.61965 26465.8 300.7 23445.21966 27403.0 307.5 24939.01967 28628.7 303.7 26713.71968 29904.5 304.7 29957.81969 27508.2 298.6 31585.91970 29035.5 295.5 33474.51971 29281.5 299.0 34821.81972 31535.8 288.1 41794.3

E-views Output for multiple linear regression model:

ls y c x2 x3

Dependent Variable: Y

Method: Least Squares

Date: 04/24/13 Time: 17:39

Sample: 1958 1972

Included observations: 15

Variable Coefficient Std. Error t-Statistic Prob.

C -28067.17 9432.066 -2.975718 0.0116

X2 147.9362 36.44344 4.059338 0.0016

X3 0.403563 0.073561 5.486131 0.0001

R-squared 0.909559 Mean dependent var 24735.33

Adjusted R-squared 0.894485 S.D. dependent var 4874.173

S.E. of regression 1583.279 Akaike info criterion 17.74924

Sum squared resid 30081287 Schwarz criterion 17.89085

Management Development Institute Page 2 October, 2011 PT-PGPM


3/19

Log likelihood -130.1193 F-statistic 60.34143

Durbin-Watson stat 1.039019 Prob(F-statistic) 0.000001

Analysis:

High R-squared and Adjusted R-squared indicate a good explanatory power of the

model.

t-test (Probability values < 0.05) indicates that both Labour Days (X2) and Real Capital

Input (X3) are significant in explaining the variability of the explained variable, Real

GDP.

Also, F-test (probability value < 0.05) indicates that the model as a whole is significant.

However, the Durbin-Watson stat value of 1.039 is in the doubtful range and therefore,

serial correlation LM test is performed to make sure no auto correlation exists in the

model. Lag order of 1 is used for LM test. The outcome is given below:

Serial Correlation LM Test:

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 1.237281 Probability 0.289714

Obs*R-squared 1.516612 Probability 0.218133

Test Equation:

Dependent Variable: RESID

Method: Least SquaresDate: 04/24/13 Time: 18:00

Presample missing value lagged residuals set to zero.


C 3243.336 9784.721 0.331469 0.7465

X2 -12.76956 37.87033 -0.337192 0.7423

X3 0.016801 0.074393 0.225839 0.8255

RESID(-1) 0.333779 0.300071 1.112331 0.2897

R-squared 0.101107 Mean dependent var -1.36E-11

Adjusted R-squared -0.144045 S.D. dependent var 1465.832







4/19

Here, the Null hypothesis is that no auto correlation exists in the model. A probability of

0.21 and 0.28 are greater than the threshold of 0.05, therefore, Null Hypothesis of No

Auto Correlation existing is accepted.

Check for Multicollinearity:

Correlation Matrix:

X2 X3

X2 1.000000 0.620523

X3 0.620523 1.000000

Correlation between X2 and X3 is found to be 0.620523 which is not high enough to

indicate a multicollinearity problem existing in the model. Therefore, the model can now

be used for forecasting.

The final regression model used for forecasting is as follows:Y = -28067.16664 + 147.9362123*X2 + 0.4035625699*X3

Forecasting using the model:

12000

16000

20000

24000

28000

32000

36000

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

YF

Forecast: YF

Actual: Y

Forecast sample: 1958 1972


Root Mean Squared Error 1416.128

Mean Absolute Error 1100.348

Mean Abs. Percent Error 5.169292

Theil Inequality Coefficient 0.028143

Bias Proportion 0.000000Variance Proportion 0.023694

Covariance Proportion 0.976306

Theil Inequality Coefficient of 0.023 is quite close to zero, which indicates a near perfect

fit of the model for forecasting purpose.



5/19

2. ARIMAInput Data Used peak demand.xls:

peak demand.xls

E-views Output for correlogram:

Date: 06/23/13 Time: 14:59

Sample: 2000M04 2008M07


Autocorrelation Partial Correlation AC PAC Q-Stat Prob

. |*******| . |*******| 1 0.970 0.970 96.941 0.000

. |*******| . | . | 2 0.940 -0.015 188.91 0.000

. |*******| . | . | 3 0.910 -0.015 275.99 0.000

. |*******| . | . | 4 0.880 -0.015 358.29 0.000

. |*******| . | . | 5 0.850 -0.015 435.90 0.000

. |****** | . | . | 6 0.820 -0.015 508.94 0.000

. |****** | . | . | 7 0.791 -0.015 577.50 0.000

. |****** | . | . | 8 0.761 -0.015 641.71 0.000

. |****** | . | . | 9 0.731 -0.015 701.67 0.000

. |***** | . | . | 10 0.702 -0.015 757.52 0.000

. |***** | . | . | 11 0.673 -0.015 809.38 0.000

. |***** | . | . | 12 0.643 -0.016 857.38 0.000

. |***** | . | . | 13 0.614 -0.015 901.64 0.000

. |**** | . | . | 14 0.586 -0.016 942.30 0.000

. |**** | . | . | 15 0.557 -0.015 979.50 0.000

. |**** | . | . | 16 0.528 -0.016 1013.4 0.000

. |**** | . | . | 17 0.500 -0.015 1044.1 0.000

. |**** | . | . | 18 0.472 -0.016 1071.8 0.000

. |*** | . | . | 19 0.444 -0.015 1096.6 0.000

. |*** | . | . | 20 0.416 -0.015 1118.6 0.000

. |*** | . | . | 21 0.389 -0.016 1138.1 0.000

. |*** | . | . | 22 0.361 -0.015 1155.2 0.000

. |*** | . | . | 23 0.334 -0.015 1170.0 0.000

. |** | . | . | 24 0.308 -0.016 1182.7 0.000

. |** | . | . | 25 0.281 -0.015 1193.5 0.000

. |** | . | . | 26 0.255 -0.016 1202.4 0.000

. |** | . | . | 27 0.229 -0.015 1209.8 0.000

. |** | . | . | 28 0.204 -0.016 1215.7 0.000

. |*. | . | . | 29 0.179 -0.015 1220.3 0.000

. |*. | . | . | 30 0.154 -0.016 1223.7 0.000



6/19

. |*. | . | . | 31 0.130 -0.015 1226.2 0.000

. |*. | . | . | 32 0.106 -0.015 1227.9 0.000

. |*. | . | . | 33 0.082 -0.016 1228.9 0.000

. | . | . | . | 34 0.059 -0.015 1229.4 0.000

. | . | . | . | 35 0.036 -0.015 1229.6 0.000

. | . | . | . | 36 0.013 -0.016 1229.7 0.000

The correlogram shows signature of AR(1) process, however, it could be an illusion untilstationarity of the series is confirmed.

Performing the ADF Unit Root test for check of

stationarity:

ADF unit root test output at level:

Null Hypothesis: DEMAND has a unit root

Exogenous: Constant, Linear Trend

Lag Length: 0 (Automatic based on SIC, MAXLAG=11)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -3.370152 0.0626

Test critical values: 1% level -4.073859

5% level -3.465548

10% level -3.159372

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(DEMAND)


Date: 06/23/13 Time: 15:03

Sample (adjusted): 2000M05 2007M02

Included observations: 82 after adjustments


DEMAND(-1) -0.249099 0.073913 -3.370152 0.0012

C 4861.437 1432.497 3.393681 0.0011

@TREND(2000M04) 29.69826 9.602382 3.092802 0.0027









7/19

The probability value of >5% indicates that H0 of unit root present is accepted at 5%

critical level and therefore, ADF Unit root test is done again on first difference with the

following outcome:

Null Hypothesis: D(DEMAND) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.902358

10% level -2.588587



Dependent Variable: D(DEMAND,2)


Date: 06/23/13 Time: 15:08




D(DEMAND(-1)) -4.811954 0.635495 -7.571975 0.0000

D(DEMAND(-1),2) 3.506694 0.574176 6.107346 0.0000

D(DEMAND(-2),2) 3.219195 0.511100 6.298564 0.0000

D(DEMAND(-3),2) 2.809151 0.456585 6.152531 0.0000

D(DEMAND(-4),2) 2.374129 0.408596 5.810456 0.0000

D(DEMAND(-5),2) 2.039335 0.353954 5.761583 0.0000

D(DEMAND(-6),2) 1.665176 0.295530 5.634549 0.0000

D(DEMAND(-7),2) 1.228536 0.244074 5.033460 0.0000

D(DEMAND(-8),2) 0.815078 0.190846 4.270875 0.0001

D(DEMAND(-9),2) 0.463414 0.121058 3.828023 0.0003

C 579.9569 122.9400 4.717396 0.0000

R-squared 0.692708 Mean dependent var -1.263889








8/19

The H0 of unit root is now rejected and therefore, the series is considered I(1)

Checking the correlogram with first difference series for

trend and seasonality:

Trend component is added in the first difference series series dd=d(demand, 1,0) and

correlogram checked with following outcome:

Date: 06/23/13 Time: 15:14

Sample: 2000M04 2008M07



. | . | . | . | 1 0.030 0.030 0.0747 0.785

. | . | . | . | 2 0.016 0.015 0.0967 0.953

.*| . | .*| . | 3 -0.111 -0.112 1.1721 0.760

**| . | **| . | 4 -0.199 -0.195 4.6597 0.324

.*| . | .*| . | 5 -0.112 -0.106 5.7834 0.328

.*| . | .*| . | 6 -0.134 -0.147 7.4030 0.285

.*| . | **| . | 7 -0.165 -0.229 9.9096 0.194

.*| . | **| . | 8 -0.131 -0.243 11.508 0.175

. | . | **| . | 9 -0.042 -0.200 11.675 0.232

.*| . | ***| . | 10 -0.094 -0.353 12.528 0.251

. |*** | . |*. | 11 0.352 0.085 24.557 0.011

. |** | . |*. | 12 0.252 0.079 30.822 0.002

. |** | . |*. | 13 0.260 0.167 37.577 0.000

. | . | . | . | 14 0.003 -0.016 37.578 0.001

.*| . | . | . | 15 -0.123 -0.046 39.137 0.001

.*| . | .*| . | 16 -0.151 -0.090 41.512 0.000

.*| . | . | . | 17 -0.099 -0.020 42.558 0.001

.*| . | . | . | 18 -0.090 -0.005 43.427 0.001

. | . | . |*. | 19 0.006 0.169 43.431 0.001

.*| . | . | . | 20 -0.129 -0.036 45.292 0.001

**| . | .*| . | 21 -0.195 -0.146 49.592 0.000

. |*. | . | . | 22 0.103 -0.018 50.814 0.000

. |*. | . | . | 23 0.098 -0.053 51.932 0.001

. |** | . | . | 24 0.308 0.057 63.234 0.000

. |*. | . | . | 25 0.156 0.035 66.175 0.000

. | . | . | . | 26 -0.022 -0.049 66.234 0.000

.*| . | . | . | 27 -0.084 -0.038 67.111 0.000

. | . | . | . | 28 -0.044 0.062 67.359 0.000

.*| . | . |*. | 29 -0.074 0.075 68.075 0.000

.*| . | .*| . | 30 -0.083 -0.096 68.979 0.000



9/19

. | . | . | . | 31 -0.050 -0.055 69.315 0.000

.*| . | . | . | 32 -0.110 0.009 70.994 0.000

.*| . | . | . | 33 -0.077 -0.039 71.830 0.000

. |*. | . |*. | 34 0.079 0.173 72.732 0.000

. | . | .*| . | 35 0.061 -0.061 73.273 0.000

. |** | . | . | 36 0.221 0.034 80.603 0.000

Nothing can be derived from the correlogram regarding AR and MA components so

trying with the seasonal component (with 12 as seasonality): Series dd=d(demand,0,12)

Date: 06/23/13 Time: 15:15

Sample: 2000M04 2008M07



. |**** | . |**** | 1 0.486 0.486 17.479 0.000

. |**** | . |** | 2 0.475 0.313 34.414 0.000

. |*** | . |*. | 3 0.380 0.101 45.423 0.000

. |*** | . | . | 4 0.340 0.063 54.342 0.000

. |** | . | . | 5 0.303 0.050 61.572 0.000

. |** | . | . | 6 0.222 -0.041 65.492 0.000

. |*. | .*| . | 7 0.088 -0.166 66.118 0.000

. |*. | . | . | 8 0.132 0.054 67.557 0.000

. |** | . |** | 9 0.246 0.266 72.615 0.000

. |*. | .*| . | 10 0.072 -0.152 73.050 0.000

. |*. | . |*. | 11 0.174 0.085 75.652 0.000.*| . | ***| . | 12 -0.142 -0.398 77.429 0.000

. | . | . |** | 13 0.062 0.201 77.767 0.000

. | . | . | . | 14 0.043 0.058 77.939 0.000

. | . | . |*. | 15 0.051 0.093 78.178 0.000

. | . | . |*. | 16 0.031 0.072 78.271 0.000

. |*. | . |*. | 17 0.096 0.069 79.151 0.000

. |*. | . |*. | 18 0.141 0.072 81.100 0.000

. |*. | . | . | 19 0.195 -0.022 84.903 0.000

. |*. | . | . | 20 0.177 -0.053 88.092 0.000

. | . | . | . | 21 0.008 -0.047 88.098 0.000

. |*. | . | . | 22 0.145 0.001 90.326 0.000

. |*. | . |*. | 23 0.071 0.124 90.864 0.000

. |*. | .*| . | 24 0.125 -0.146 92.584 0.000

. | . | . |*. | 25 0.063 0.095 93.038 0.000

. |*. | . | . | 26 0.092 0.020 94.015 0.000

. | . | .*| . | 27 0.058 -0.062 94.406 0.000

. |*. | . | . | 28 0.106 -0.028 95.760 0.000

. | . | .*| . | 29 0.021 -0.084 95.812 0.000

.*| . | .*| . | 30 -0.123 -0.158 97.721 0.000



10/19

.*| . | .*| . | 31 -0.151 -0.118 100.68 0.000

**| . | .*| . | 32 -0.237 -0.112 108.16 0.000

This correlogram shows an indication of AR(2), SMA(12) process. Confirming the

significance of ar(1), ar(2) and sma(12) using regression

Confirming AR and MA components from the

correlogram:

ls d(demand,0,12) c ar(1) ar(2) sma(12)

Dependent Variable: D(DEMAND,0,12)


Date: 06/23/13 Time: 15:18


Included observations: 69 after adjustmentsConvergence achieved after 12 iterations

Backcast: 2000M06 2001M05


C 1701.522 214.0576 7.948896 0.0000

AR(1) 0.471909 0.118750 3.973966 0.0002

AR(2) 0.287270 0.117415 2.446625 0.0171

MA(12) -0.854022 0.041146 -20.75597 0.0000







Inverted AR Roots .82 -.35

Inverted MA Roots .99 .85+.49i .85-.49i .49+.85i

.49-.85i -.00-.99i -.00+.99i -.49-.85i

-.49+.85i -.85+.49i -.85-.49i -.99

All 3 processes are showing significance at 5% confidence level. Confirming whether the

Residual is white Noise of not.

Date: 06/23/13 Time: 15:20

Sample: 2001M06 2007M02




11/19

Q-statisticprobabilities

adjusted for 3 ARMAterm(s)


. | . | . | . | 1 -0.043 -0.043 0.1322

.*| . | .*| . | 2 -0.075 -0.077 0.5457

. | . | . | . | 3 0.052 0.046 0.7499

. | . | . | . | 4 -0.022 -0.024 0.7863 0.375

. | . | . | . | 5 0.006 0.012 0.7892 0.674

. | . | . | . | 6 -0.029 -0.035 0.8565 0.836

**| . | **| . | 7 -0.203 -0.205 4.1276 0.389

. | . | . | . | 8 -0.011 -0.038 4.1382 0.530

. |*. | . |*. | 9 0.138 0.114 5.6902 0.459

.*| . | .*| . | 10 -0.097 -0.075 6.4719 0.486

. |*. | . |** | 11 0.182 0.199 9.2745 0.320

.*| . | .*| . | 12 -0.119 -0.145 10.488 0.312

. | . | . |*. | 13 0.043 0.070 10.650 0.385

. | . | . | . | 14 0.037 -0.051 10.772 0.463

. | . | . |*. | 15 0.035 0.073 10.885 0.539

. | . | . | . | 16 -0.047 -0.016 11.092 0.603

.*| . | .*| . | 17 -0.066 -0.078 11.508 0.646

. | . | . | . | 18 -0.029 0.004 11.587 0.710

. |*. | . |*. | 19 0.133 0.125 13.319 0.649

. | . | . | . | 20 0.000 -0.044 13.319 0.715

**| . | .*| . | 21 -0.227 -0.150 18.556 0.420

. | . | .*| . | 22 0.016 -0.075 18.583 0.484

.*| . | .*| . | 23 -0.109 -0.108 19.848 0.467

. | . | .*| . | 24 -0.007 -0.079 19.854 0.530

. | . | . |*. | 25 0.047 0.076 20.096 0.577

. | . | . |*. | 26 0.025 0.080 20.169 0.632

. | . | . | . | 27 -0.017 -0.006 20.202 0.685

. |*. | . | . | 28 0.136 0.050 22.401 0.612

Confirmed that the residuals components are white noise; so proceeding with forecasting:

Forecasting using the model:

Outcome of the static forecasting:



12/19

18000

20000

22000

24000

26000

28000

30000

32000

34000

2001 2002 2003 2004 2005 2006

DEMANDF

Forecast: DEMANDF

Actual: DEMAND

Forecast sample: 2000M04 2008M07

Adjusted sample: 2001M06 2007M03






Bias Proportion 0.013140

Variance Proportion 0.020256


Outcome of dynamic forecasting:

26000

28000

30000

32000

34000

36000

07M01 07M04 07M07 07M10 08M01 08M04 08M07

DEMANDF

Forecast: DEMANDF

Actual: DEMAND

Forecast sample: 2007M01 2008M07






Bias Proportion 0.999701

Variance Proportion 0.000299


MAPE is between 1% and 3%, which is good enough so model can be used for forecasting.



13/19

3. VARInput Data Used sensex_ex_oil_VAR.xls:

sensex_ex_oil_VAR.xls

Stationarity test for all the three series:

ADF Unit Root Test for Sensex:

Null Hypothesis: SEN has a unit root

Exogenous: Constant, Linear Trend


t-Statistic Prob.*



5% level -3.416877

10% level -3.130796



Dependent Variable: D(SEN)Method: Least Squares

Date: 06/23/13 Time: 15:54

Sample (adjusted): 2 639



SEN(-1) -0.008067 0.005254 -1.535520 0.1252

C 134.9426 66.52498 2.028450 0.0429

@TREND(1) 0.037049 0.087721 0.422351 0.6729









14/19


15/19

Outcome with lag 2

Vector Autoregression Estimates

Date: 06/23/13 Time: 15:59



Standard errors in ( ) & t-statistics in [ ]

DLOG(SEN) DLOG(OIL) DLOG(EX)

DLOG(SEN(-1)) 0.062916 0.041426 -0.007926

(0.03938) (0.05878) (0.01270)

[ 1.59786] [ 0.70476] [-0.62406]

DLOG(SEN(-2)) -0.047235 0.069754 -0.032615

(0.03926) (0.05861) (0.01266)

[-1.20309] [ 1.19016] [-2.57540]

DLOG(OIL(-1)) 0.027833 -0.046923 0.000759

(0.02660) (0.03970) (0.00858)

[ 1.04652] [-1.18187] [ 0.08852]

DLOG(OIL(-2)) -0.046258 -0.019528 -0.012440

(0.02550) (0.03807) (0.00823)

[-1.81390] [-0.51295] [-1.51229]

DLOG(EX(-1)) -0.043623 -0.018486 -0.032198

(0.12249) (0.18286) (0.03951)

[-0.35613] [-0.10110] [-0.81489]

DLOG(EX(-2)) -0.461301 0.051077 -0.061351

(0.12216) (0.18236) (0.03940)

[-3.77614] [ 0.28008] [-1.55695]

C 0.000873 0.001302 -4.56E-05

(0.00063) (0.00094) (0.00020)

[ 1.39264] [ 1.39148] [-0.22567]

R-squared 0.033549 0.005954 0.018689

Adj. R-squared 0.024330 -0.003529 0.009328

Sum sq. resids 0.154639 0.344605 0.016089

S.E. equation 0.015680 0.023406 0.005058

F-statistic 3.639180 0.627866 1.996506

Log likelihood 1743.907 1489.092 2463.518

Akaike AIC -5.461973 -4.660667 -7.724899

Schwarz SC -5.412938 -4.611632 -7.675864

Mean dependent 0.000889 0.001300 -9.69E-05



16/19

S.D. dependent 0.015874 0.023365 0.005081

Determinant resid covariance (dof adj.) 3.43E-12

Determinant resid covariance 3.32E-12

Log likelihood 5698.165

Akaike information criterion -17.85272Schwarz criterion -17.70561

The lag length criteria output is as follows:

VAR Lag Order Selection CriteriaEndogenous variables: DLOG(SEN) DLOG(OIL)DLOG(EX)

Exogenous variables: C

Date: 06/23/13 Time: 16:01

Sample: 1 639Included observations: 630

Lag LogL LR FPE AIC SC HQ

0 5655.428 NA 3.23e-12 -17.94421 -17.92304* -17.93599*

1 5658.995 7.088918 3.29e-12 -17.92697 -17.84229 -17.89408

2 5675.220 32.09070 3.21e-12 -17.94991 -17.80172 -17.89234

3 5697.734 44.31307* 3.08e-12* -17.99281* -17.78111 -17.91058

4 5703.470 11.23404 3.11e-12 -17.98244 -17.70723 -17.87554

5 5711.659 15.96324 3.12e-12 -17.97987 -17.64115 -17.84830

6 5715.384 7.225424 3.17e-12 -17.96312 -17.56089 -17.80689

7 5721.169 11.16626 3.20e-12 -17.95292 -17.48718 -17.77201

8 5727.525 12.20586 3.23e-12 -17.94452 -17.41527 -17.73895

* indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level)

FPE: Final prediction error

AIC: Akaike information criterion

SC: Schwarz information criterion

HQ: Hannan-Quinn information criterion

The lag length 3 is selected by the criterion and therefore, it will be used for furtheranalysis.

Cointegration Test: The Johansen and Jeselius cointegration test is performed to validate the existence of

any cointegration between variables with the following outcome:



17/19

Date: 06/23/13 Time: 16:06



Trend assumption: Linear deterministic trend

Series: DLOG(SEN) DLOG(OIL) DLOG(EX)

Lags interval (in first differences): 1 to 3

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.272322 472.1336 29.79707 0.0001

At most 1 * 0.234106 270.5869 15.49471 0.0001

At most 2 * 0.147925 101.4914 3.841466 0.0000

Trace test indicates 3 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

Hypothesized Max-Eigen 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None * 0.272322 201.5467 21.13162 0.0001

At most 1 * 0.234106 169.0955 14.26460 0.0001

At most 2 * 0.147925 101.4914 3.841466 0.0000

Max-eigenvalue test indicates 3 cointegrating eqn(s) at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level

**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):


-106.7013 16.58454 -290.3165

12.35526 88.88217 94.37331

69.51591 11.45247 -287.3705

Unrestricted Adjustment Coefficients (alpha):

D(DLOG(SEN)) 0.007827 -0.001086 -0.003048

D(DLOG(OIL)) -0.003808 -0.012067 -0.001340

D(DLOG(EX)) 0.001309 -0.000759 0.001783



18/19

1 Cointegrating Equation(s): Log likelihood 5590.689

Normalized cointegrating coefficients (standard error in parentheses)


1.000000 -0.155430 2.720834

(0.05594) (0.25062)

Adjustment coefficients (standard error in parentheses)

D(DLOG(SEN)) -0.835138

(0.06505)

D(DLOG(OIL)) 0.406332

(0.11089)

D(DLOG(EX)) -0.139619

(0.02283)

2 Cointegrating Equation(s): Log likelihood 5675.236

Normalized cointegrating coefficients (standard error in parentheses)


1.000000 0.000000 2.824833

(0.25049)

0.000000 1.000000 0.669108

(0.32764)

Adjustment coefficients (standard error in parentheses)

D(DLOG(SEN)) -0.848561 0.033247

(0.06532) (0.05498)

D(DLOG(OIL)) 0.257236 -1.135738(0.09882) (0.08318)

D(DLOG(EX)) -0.149002 -0.045799

(0.02275) (0.01915)

The H0 of Cointegration is rejected for all 3 possible pairs for existence of cointegration.

The unrestricted VAR can, therefore, be applied now.

Granger Causality Test:

VAR Granger Causality/Block Exogeneity Wald Tests

Date: 06/23/13 Time: 16:12

Sample: 1 639


Dependent variable: DLOG(SEN)



19/19

Excluded Chi-sq df Prob.

DLOG(OIL) 5.287301 3 0.1519

DLOG(EX) 49.19361 3 0.0000

All 54.27170 6 0.0000

Dependent variable: DLOG(OIL)


DLOG(SEN) 1.856538 3 0.6027

DLOG(EX) 0.518338 3 0.9148

All 2.386966 6 0.8809

Dependent variable: DLOG(EX)


DLOG(SEN) 9.790470 3 0.0204

DLOG(OIL) 4.567472 3 0.2064

All 13.33178 6 0.0381

The granger causality test with H0 of non-causality running from the independent

variable DLOG(EX) to the dependent variable DLOG(SEN) has been rejected and

therefore, it is confirmed that Exchange Rate movement Granger causes Sensex

movement.

Similarly, the H0 of non-causality running from the independent variable DLOG(SEN) to

the dependent variable DLOG(EX) has been rejected and therefore, it is confirmed that

Sensex movement Granger causes Exchange rate movement.

In other words, there is a bidirectional causality that exists between the sensex and the

exchange rate movement.

Forecasting can be done now using the above VAR model with lag structure of 3 between

Sensex and Exchange Rate.

Management Development Institute Page 19 October 2011 PT PGPM

Regression Arima Garch Var Model

Documents

Transcript of Regression Arima Garch Var Model