Microsoft - Volatility modeling and analysis
201
Microsoft (MSFT) Microsoft (MSFT) Augusto Pucci Augusto Pucci
-
Upload
augusto-pucci-phd -
Category
Business
-
view
1.311 -
download
1
description
A ga
Transcript of Microsoft - Volatility modeling and analysis
Microsoft Microsoft (MSFT)(MSFT)Augusto PucciAugusto Pucci
OverviewOverview MSFT – Company OverviewMSFT – Company Overview MSFT – Return AnalysisMSFT – Return Analysis RT – AR(2) modelRT – AR(2) model RT – AR(2) – ARCH(1) modelRT – AR(2) – ARCH(1) model RT – AR(2) – ARCH(2) modelRT – AR(2) – ARCH(2) model RT – AR(2) – GARCH(1,1) modelRT – AR(2) – GARCH(1,1) model RT – AR(2) – TGARCH(1,1) modelRT – AR(2) – TGARCH(1,1) model Range model → Range2 modelRange model → Range2 model abs(RT) model → RT2 modelabs(RT) model → RT2 model RT – GARCH(1,1) model, Extended…RT – GARCH(1,1) model, Extended… RT – GARCH(1,1) model, Extended 2…RT – GARCH(1,1) model, Extended 2… RT – AR(2) – TGARCH(1,1) ShortFallRT – AR(2) – TGARCH(1,1) ShortFall Volatility Forecasting from TGARCH(1,1) modelVolatility Forecasting from TGARCH(1,1) model Volatility Forecasting from GARCH(1,1) eXt. modelVolatility Forecasting from GARCH(1,1) eXt. model Extra Stuff…Extra Stuff…
Microsoft CampusMicrosoft Campus
Microsoft: Company Microsoft: Company OverviewOverview
Financial HighlightsFinancial Highlights Beta: Beta: 1.081.08 Fiscal Year Ends: Fiscal Year Ends: 30-June30-June Profitability Profit Margin: Profitability Profit Margin:
27.80%27.80% Operating Margin:Operating Margin:38.06%38.06% Return on Assets (ttm):Return on Assets (ttm):22.15%22.15% Return on Equity (ttm):Return on Equity (ttm):50.01%50.01%
Income StatementIncome Statement Revenue: Revenue: 61.98B61.98B Revenue Per Share: Revenue Per Share: 6.7816.781 Qtrly Revenue Growth:Qtrly Revenue Growth:1.60%1.60% Gross Profit:Gross Profit:48.82B48.82B EBITDA:EBITDA:25.94B25.94B Net Income Avl to Net Income Avl to
Common:Common:17.23B17.23B Diluted EPS:Diluted EPS:1.871.87 Qtrly Earnings Growth:Qtrly Earnings Growth:--
11.30%11.30%William Henry Gates III
(Seattle, 10/28/1955)
Financial HighlightsFinancial Highlights
Balance SheetBalance Sheet Total Cash: Total Cash: 20.30B20.30B Total Cash Per Share:Total Cash Per Share:2.2832.283 Total Debt:Total Debt:2.00B2.00B Total Debt/Equity:Total Debt/Equity:N/AN/A Current Ratio:Current Ratio:1.5911.591 Book Value Per Share:Book Value Per Share:3.8793.879
Cash Flow StatementCash Flow Statement Operating Cash Operating Cash
Flow:Flow:20.32B20.32B Levered Free Cash Levered Free Cash
Flow:Flow:14.40B14.40B
Steven Anthony Ballmer (Detroit, 03/24/1956)
Important DatesImportant Dates 1975 1975 Microsoft foundedMicrosoft founded Jan. 1, 1979Jan. 1, 1979 Microsoft moves from Albuquerque, New Mexico to Bellevue, Microsoft moves from Albuquerque, New Mexico to Bellevue,
WashingtonJuneWashingtonJune 25, 1981 Microsoft incorporates25, 1981 Microsoft incorporates Aug. 12, 1981Aug. 12, 1981 IBM introduces its personal computer with Microsoft's 16-bit IBM introduces its personal computer with Microsoft's 16-bit
operating system, MS-DOS 1.0operating system, MS-DOS 1.0 Feb. 26, 1986Feb. 26, 1986 Microsoft moves to corporate campus in Redmond, Washington Microsoft moves to corporate campus in Redmond, Washington March 13, 1986March 13, 1986 Microsoft stock goes public Microsoft stock goes public Aug. 1, 1989Aug. 1, 1989 Microsoft introduces earliest version of Office suite of productivity Microsoft introduces earliest version of Office suite of productivity
applicationsapplications May 22, 1990May 22, 1990 Microsoft launches Windows 3.0 Microsoft launches Windows 3.0 Aug. 24, 1995Aug. 24, 1995 Microsoft launches Windows 95Microsoft launches Windows 95 Dec. 7, 1995Dec. 7, 1995 Bill Gates outlines Microsoft's commitment to supporting and Bill Gates outlines Microsoft's commitment to supporting and
enhancing the Internetenhancing the Internet June 25, 1998June 25, 1998 Microsoft launches Windows 98Microsoft launches Windows 98 Jan. 13, 2000Jan. 13, 2000 Steve Ballmer named president and chief executive officer for Steve Ballmer named president and chief executive officer for
MicrosoftMicrosoft Feb. 17, 2000Feb. 17, 2000 Microsoft launches Windows 2000 Microsoft launches Windows 2000 Apr. 3, 2000Apr. 3, 2000 Microsoft accused of abusive monopolyMicrosoft accused of abusive monopoly June 22, 2000June 22, 2000 Bill Gates and Steve Ballmer outline Microsoft's .NET strategy for Bill Gates and Steve Ballmer outline Microsoft's .NET strategy for
Web servicesWeb services May 31, 2001May 31, 2001 Microsoft launches Office XP Microsoft launches Office XP
Important Dates [2]Important Dates [2] Oct. 25, 2001Oct. 25, 2001 Microsoft launches Windows XP Microsoft launches Windows XP Jan. 15, 2002Jan. 15, 2002 Bill Gates outlines Microsoft's commitment to Trustworthy Bill Gates outlines Microsoft's commitment to Trustworthy
ComputingComputing Nov. 7, 2002Nov. 7, 2002 Microsoft and partners launch Tablet PC Microsoft and partners launch Tablet PC Jan. 16, 2003Jan. 16, 2003 Microsoft declares annual dividend Microsoft declares annual dividend April 24, 2003April 24, 2003 Microsoft launches Windows Server 2003 Microsoft launches Windows Server 2003 Oct. 21, 2003Oct. 21, 2003 Microsoft launches Microsoft Office System Microsoft launches Microsoft Office System March, 2004March, 2004 European antitrust legal action against MicrosoftEuropean antitrust legal action against Microsoft July 20, 2004July 20, 2004 Microsoft announces plans to return up to $75 billion to shareholders Microsoft announces plans to return up to $75 billion to shareholders
in dividends and stock buybacksin dividends and stock buybacks June 15, 2006June 15, 2006 Microsoft announces that Bill Gates will transition out of a day-to- Microsoft announces that Bill Gates will transition out of a day-to-
day role in the company in July 2008, Ray Ozzie is named chief software architect day role in the company in July 2008, Ray Ozzie is named chief software architect and Craig Mundie chief research and strategy officerand Craig Mundie chief research and strategy officer
July 20, 2006July 20, 2006 Microsoft announces a new $20 billion tender offer and authorizes an Microsoft announces a new $20 billion tender offer and authorizes an additional share-repurchase program of up to $20 billion over five yearsadditional share-repurchase program of up to $20 billion over five years
Jan. 30, 2007Jan. 30, 2007 Microsoft launches Windows Vista and the 2007 Microsoft Office Microsoft launches Windows Vista and the 2007 Microsoft Office System to consumers worldwideSystem to consumers worldwide
Feb. 27, 2008Feb. 27, 2008 Microsoft launches Windows Server 2008, SQL Server 2008 and Microsoft launches Windows Server 2008, SQL Server 2008 and Visual Studio 2008Visual Studio 2008
June 27, 2008June 27, 2008 Bill Gates transitions from his day-to-day role at Microsoft to spend Bill Gates transitions from his day-to-day role at Microsoft to spend more time on his work at The Bill & Melinda Gates Foundationmore time on his work at The Bill & Melinda Gates Foundation
Jan. 2009Jan. 2009 Microsoft announces layoffs of up to 5,000 employeesMicrosoft announces layoffs of up to 5,000 employees
MSFT – Return MSFT – Return AnalysisAnalysis
Adj_CloseAdj_Closefrom 03/13/1986 to from 03/13/1986 to
02/05/200902/05/2009
0
10
20
30
40
50
60
86 88 90 92 94 96 98 00 02 04 06 08
ADJ CLOSE
9/11Win95Win98monopol
yaccuse
European antitrust
action
5,000 emp.
layoffs
RT from 03/13/1986 to RT from 03/13/1986 to 02/05/200902/05/2009
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT
9/11Win95Win98monopol
yaccuse
European antitrust
action
5,000 emp.
layoffs
Windows 95 & Windows Windows 95 & Windows 9898
0
10
20
30
40
50
60
1995 1996 1997 1998 1999
ADJ CLOSE
Win95 Win98
Windows 95 & Windows Windows 95 & Windows 9898
-160
-120
-80
-40
0
40
80
120
160
1995 1996 1997 1998 1999
RT
Win95 Win98
Dot.Com Bubble & 9/11Dot.Com Bubble & 9/11
10
20
30
40
50
60
1998 1999 2000 2001
ADJ CLOSE
9/11monopol
yaccuse
Dot.Com Bubble & 9/11Dot.Com Bubble & 9/11
-300
-200
-100
0
100
200
300
1998 1999 2000 2001
RT
9/11monopol
yaccuse
European antitrust accuse & European antitrust accuse & massive layoffsmassive layoffs
16
20
24
28
32
36
40
2003 2004 2005 2006 2007 2008
ADJ CLOSE
European antitrust
action
5,000 emp.
layoffs
European antitrust accuse & European antitrust accuse & massive layoffsmassive layoffs
-300
-200
-100
0
100
200
300
2003 2004 2005 2006 2007 2008
RT
European antitrust
action
5,000 emp.
layoffs
RT - HistogramRT - Histogram
0
400
800
1200
1600
2000
2400
-625 -500 -375 -250 -125 0 125 250
Series: RTSample 3/13/1986 2/05/2009Observations 5776
Mean 1.503975Median 0.000000Maximum 283.3044Minimum -602.4211Std. Dev. 39.78974Skewness -0.619675Kurtosis 17.56243
Jarque-Bera 51406.45Probability 0.000000
Windows 95 & Windows Windows 95 & Windows 9898
0
40
80
120
160
200
-120 -80 -40 0 40 80 120
Series: RTSample 1/03/1995 12/31/1999Observations 1263
Mean 3.425266Median 1.458384Maximum 149.6213Minimum -147.2664Std. Dev. 34.82695Skewness 0.096270Kurtosis 3.856238
Jarque-Bera 40.53260Probability 0.000000
Dot.Com Bubble & 9/11Dot.Com Bubble & 9/11
0
50
100
150
200
250
300
-200 -100 0 100 200 300
Series: RTSample 1/02/1998 12/31/2001Observations 1004
Mean 1.134653Median 0.960695Maximum 283.3044Minimum -269.1723Std. Dev. 44.87066Skewness -0.220573Kurtosis 7.322283
Jarque-Bera 789.6769Probability 0.000000
RT Synth - HistogramRT Synth - Histogram
0
100
200
300
400
500
600
700
-160 -120 -80 -40 0 40 80 120
Series: RT_SYNTHSample 3/13/1986 2/05/2009Observations 5777
Mean 1.418457Median 1.724196Maximum 143.1277Minimum -154.1308Std. Dev. 39.77694Skewness -0.064970Kurtosis 3.075513
Jarque-Bera 5.436769Probability 0.065981
RT Vs. RT SynthRT Vs. RT SynthRT_SYNTH RT
Mean 1.410508 1.503975
Median 1.712924 0.000000
Maximum 143.1277 283.3044
Minimum -154.1308 -602.4211
Std. Dev. 39.77580 39.78974
Skewness -0.064653 -0.619675
Kurtosis 3.076041 17.56243
Jarque-Bera 5.415586 51406.45
Probability 0.066684 0.000000
Sum 8147.096 8686.960
Sum Sq. Dev. 9136709. 9143113.
Observations 5776 5776
RT SynthRT Synth
-160
-120
-80
-40
0
40
80
120
160
86 88 90 92 94 96 98 00 02 04 06 08
RT_SYNTH
RT Vs. RT Synth [2]RT Vs. RT Synth [2]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT RT_SYNTH
RT Vs. RT Synth [3]RT Vs. RT Synth [3]
-160
-120
-80
-40
0
40
80
120
160
86 88 90 92 94 96 98 00 02 04 06 08
RT_SYNTH
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT
RT - CorrelogramRT - Correlogram
Sign. Level (5%) = ± 0.025
RTRT22 - Correlogram - Correlogram
Sign. Level (5%) = ± 0.025
abs(RT) - Correlogramabs(RT) - Correlogram
Sign. Level (5%) = ± 0.025
RTRT22
0
100000
200000
300000
400000
86 88 90 92 94 96 98 00 02 04 06 08
RT2
RTRT22 - Histogram - Histogram
0
1000
2000
3000
4000
5000
6000
0 100000 200000 300000
Series: RT2Sample 3/13/1986 2/05/2009Observations 5776
Mean 1585.211Median 326.5124Maximum 362911.2Minimum 0.000000Std. Dev. 6425.544Skewness 33.63031Kurtosis 1758.877
Jarque-Bera 7.43e+08Probability 0.000000
abs(RT)abs(RT)
0
100
200
300
400
500
600
700
86 88 90 92 94 96 98 00 02 04 06 08
RT_ABS
abs(RT) - Histogramabs(RT) - Histogram
0
400
800
1200
1600
2000
2400
2800
3200
3600
0 100 200 300 400 500 600
Series: RT_ABSSample 3/13/1986 2/05/2009Observations 5776
Mean 26.22082Median 18.06964Maximum 602.4211Minimum 0.000000Std. Dev. 29.96389Skewness 3.571797Kurtosis 36.66024
Jarque-Bera 284959.5Probability 0.000000
RT – AR(2) modelRT – AR(2) model
RTF - AR(2) Static RTF - AR(2) Static ForecastForecast
-120
-80
-40
0
40
80
120
86 88 90 92 94 96 98 00 02 04 06 08
RTF
Forecast: RTFActual: RTForecast sample: 3/13/1986 2/05/2009Adjusted sample: 3/18/1986 2/05/2009Included observations: 5774
Root Mean Squared Error 39.65853Mean Absolute Error 26.44798Mean Abs. Percent Error 88.89025Theil Inequality Coefficient 0.935928 Bias Proportion 0.000000 Variance Proportion 0.896076 Covariance Proportion 0.103924
RT Vs. RTF AR(2) Static RT Vs. RTF AR(2) Static ForecastForecast
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT
-20
-10
0
10
20
30
40
86 88 90 92 94 96 98 00 02 04 06 08
RTF
RTF - AR(2) Dynamic RTF - AR(2) Dynamic ForecastForecast
-100
-50
0
50
100
86 88 90 92 94 96 98 00 02 04 06 08
RTF
Forecast: RTFActual: RTForecast sample: 3/13/1986 2/05/2009Adjusted sample: 3/18/1986 2/05/2009Included observations: 5774
Root Mean Squared Error 39.71565Mean Absolute Error 26.38541Mean Abs. Percent Error 87.99759Theil Inequality Coefficient 0.963430 Bias Proportion 0.000000 Variance Proportion 0.993422 Covariance Proportion 0.006578
RT AR(2) – Residual PlotRT AR(2) – Residual Plot
-800
-600
-400
-200
0
200
400
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
Residual Actual Fitted
RT AR(2) – Residual Plot RT AR(2) – Residual Plot [2][2]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT Residuals
RT AR(2) – Residual RT AR(2) – Residual HistogramHistogram
0
400
800
1200
1600
2000
2400
-625 -500 -375 -250 -125 0 125 250
Series: ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean -3.06e-10Median -1.577343Maximum 282.5983Minimum -606.2254Std. Dev. 39.66197Skewness -0.674704Kurtosis 17.79372
Jarque-Bera 53090.73Probability 0.000000
RT AR(2) – Residual RT AR(2) – Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT AR(2) – Residual RT AR(2) – Residual ARCH TestARCH Test
RT – AR(2) – ARCH(1) RT – AR(2) – ARCH(1) modelmodel
RT – AR(2) – ARCH(1) RT – AR(2) – ARCH(1) modelmodel
σ2 = 1,618.1026σ = 40.225647
RT – ARCH(1) Residual PlotRT – ARCH(1) Residual Plot
RT – ARCH(1) Conditional RT – ARCH(1) Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
20000
40000
60000
80000
100000
120000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – ARCH(1) Residual Vs. Conditional RT – ARCH(1) Residual Vs. Conditional Variance PlotVariance Plot
RT – ARCH(1) Std. Residual RT – ARCH(1) Std. Residual PlotPlot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – ARCH(1) Residuals Vs. Std. Residuals RT – ARCH(1) Residuals Vs. Std. Residuals PlotPlot
RT – ARCH(1) Std. Residuals Vs. RT – ARCH(1) Std. Residuals Vs. ResidualsResiduals
RT – ARCH(1) Conditional Variance Vs. Std. RT – ARCH(1) Conditional Variance Vs. Std. ResidualsResiduals
RT – ARCH(1) Residual RT – ARCH(1) Residual HistogramHistogram
0
400
800
1200
1600
2000
-8 -6 -4 -2 0 2 4 6
Series: Standardized ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean 0.002564Median -0.036891Maximum 7.251408Minimum -8.116231Std. Dev. 1.000086Skewness -0.098849Kurtosis 9.585103
Jarque-Bera 10441.96Probability 0.000000
RT – ARCH(1) Std. Residual RT – ARCH(1) Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – ARCH(1) Squared Std. Residual RT – ARCH(1) Squared Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT ARCH(1) – Residual RT ARCH(1) – Residual ARCH TestARCH Test
RT – AR(2) – ARCH(2) RT – AR(2) – ARCH(2) modelmodel
RT – AR(2) – ARCH(2) RT – AR(2) – ARCH(2) modelmodel
σ2 = 1,635.1865σ = 40.437440
RT – ARCH(2) Residual PlotRT – ARCH(2) Residual Plot
RT – ARCH(2) Conditional RT – ARCH(2) Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – ARCH(2) Residual Vs. Conditional RT – ARCH(2) Residual Vs. Conditional Variance PlotVariance Plot
RT – ARCH(2) Std. Residual RT – ARCH(2) Std. Residual PlotPlot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – ARCH(2) Residuals Vs. Std. Residuals RT – ARCH(2) Residuals Vs. Std. Residuals PlotPlot
RT – ARCH(2) Std. Residuals Vs. RT – ARCH(2) Std. Residuals Vs. ResidualsResiduals
RT – ARCH(2) Conditional Variance Vs. Std. RT – ARCH(2) Conditional Variance Vs. Std. ResidualsResiduals
RT – ARCH(2) Residual RT – ARCH(2) Residual HistogramHistogram
0
400
800
1200
1600
2000
-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5
Series: Standardized ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean -0.004841Median -0.047257Maximum 7.802617Minimum -9.038780Std. Dev. 1.000086Skewness -0.064868Kurtosis 10.17817
Jarque-Bera 12400.39Probability 0.000000
RT – ARCH(2) Std. Residual RT – ARCH(2) Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – ARCH(2) Squared Std. Residual RT – ARCH(2) Squared Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT ARCH(2) – Residual RT ARCH(2) – Residual ARCH TestARCH Test
RT – AR(2) – GARCH(1,1) RT – AR(2) – GARCH(1,1) modelmodel
RT – AR(2) – GARCH(1,1) RT – AR(2) – GARCH(1,1) modelmodel
σ2 = 2,391.1118σ = 48.898996
RT – GARCH(1,1) Residual PlotRT – GARCH(1,1) Residual Plot
RT – GARCH(1,1) Conditional RT – GARCH(1,1) Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
4000
8000
12000
16000
20000
24000
28000
32000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – GARCH(1,1) Residual Vs. Conditional RT – GARCH(1,1) Residual Vs. Conditional Variance PlotVariance Plot
RT – GARCH(1,1) Std. Residual RT – GARCH(1,1) Std. Residual PlotPlot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – GARCH(1,1) Residuals Vs. Std. Residuals RT – GARCH(1,1) Residuals Vs. Std. Residuals PlotPlot
RT – GARCH(1,1) Std. Residuals Vs. RT – GARCH(1,1) Std. Residuals Vs. ResidualsResiduals
RT – GARCH(1,1) Conditional Variance Vs. RT – GARCH(1,1) Conditional Variance Vs. Std. ResidualsStd. Residuals
RT – GARCH(1,1) Residual RT – GARCH(1,1) Residual HistogramHistogram
0
400
800
1200
1600
2000
-10 -5 0 5
Series: Standardized ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean -0.002391Median -0.030605Maximum 6.955800Minimum -11.64137Std. Dev. 0.999853Skewness -0.334206Kurtosis 9.932213
Jarque-Bera 11668.86Probability 0.000000
RT – GARCH(1,1) Std. Residual RT – GARCH(1,1) Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – GARCH(1,1) Squared Std. Residual RT – GARCH(1,1) Squared Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT GARCH(1,1) – Residual RT GARCH(1,1) – Residual ARCH TestARCH Test
RT GARCH(1,1) - Sign RT GARCH(1,1) - Sign Bias TestBias Test
RT GARCH(1,1) – Negative Size RT GARCH(1,1) – Negative Size Bias TestBias Test
RT – AR(2) – TGARCH(1,1) RT – AR(2) – TGARCH(1,1) modelmodel
RT – AR(2) – TGARCH(1,1) RT – AR(2) – TGARCH(1,1) modelmodel
σ2 = 2,656.5854σ = 51.542074
RT – TGARCH(1,1) Residual RT – TGARCH(1,1) Residual PlotPlot
RT – TGARCH(1,1) Conditional RT – TGARCH(1,1) Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
4000
8000
12000
16000
20000
24000
28000
32000
36000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – TGARCH(1,1) Residual Vs. Conditional RT – TGARCH(1,1) Residual Vs. Conditional Variance PlotVariance Plot
RT – TGARCH(1,1) Std. RT – TGARCH(1,1) Std. Residual PlotResidual Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – TGARCH(1,1) Residuals Vs. Std. RT – TGARCH(1,1) Residuals Vs. Std. Residuals PlotResiduals Plot
RT – TGARCH(1,1) Std. Residuals Vs. RT – TGARCH(1,1) Std. Residuals Vs. ResidualsResiduals
RT – TGARCH(1,1) Conditional Variance Vs. RT – TGARCH(1,1) Conditional Variance Vs. Std. ResidualsStd. Residuals
0
400
800
1200
1600
2000
-10 -5 0 5
Series: Standardized ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean 0.004466Median -0.025805Maximum 6.494105Minimum -11.52324Std. Dev. 0.999918Skewness -0.334445Kurtosis 9.651680
Jarque-Bera 10752.21Probability 0.000000
RT – TGARCH(1,1) Residual RT – TGARCH(1,1) Residual HistogramHistogram
RT – TGARCH(1,1) Std. Residual RT – TGARCH(1,1) Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – TGARCH(1,1) Squared Std. Residual RT – TGARCH(1,1) Squared Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT TGARCH(1,1) – Residual ARCH RT TGARCH(1,1) – Residual ARCH TestTest
Range & RangeRange & Range22
range = range = log(high/low)*sqr(252/(4*log(2)))*10log(high/low)*sqr(252/(4*log(2)))*10
00
Range model → Range model → RangeRange22 modelmodel
RangeRange22 model model
E[ RangeE[ Range22t t | I| I(t-1) (t-1) ] (from Range ] (from Range
MEM)MEM)
0
4000
8000
12000
16000
20000
24000
28000
32000
86 88 90 92 94 96 98 00 02 04 06 08
RANGE2_HAT
RangeRange22tt Vs. E[ Range Vs. E[ Range22
t t | | II(t-1) (t-1) ]]
0
20000
40000
60000
80000
100000
120000
86 88 90 92 94 96 98 00 02 04 06 08
RANGE2 RANGE2_HAT
abs(RT) model → RTabs(RT) model → RT22 modelmodel
RTRT22 model model
E[ RTE[ RT22t t | I| I(t-1) (t-1) ] (from abs(RT) ] (from abs(RT)
MEM)MEM)
0
10000
20000
30000
40000
86 88 90 92 94 96 98 00 02 04 06 08
RT2_HAT
RTRT22tt Vs. E[ RT Vs. E[ RT22
t t | I| I(t-1) (t-1) ]]
0
100000
200000
300000
400000
86 88 90 92 94 96 98 00 02 04 06 08
RT2 RT2_HAT
RT – GARCH(1,1) modelRT – GARCH(1,1) modelExtended…Extended…
RT – GARCH(1,1) eXt. RT – GARCH(1,1) eXt. modelmodel
RT – GARCH(1,1) eXt.RT – GARCH(1,1) eXt. Residual Residual PlotPlot
RT – GARCH(1,1) eXt. Conditional RT – GARCH(1,1) eXt. Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
10000
20000
30000
40000
50000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – GARCH(1,1) eXt. Residual Vs. Conditional Variance RT – GARCH(1,1) eXt. Residual Vs. Conditional Variance PlotPlot
RT – GARCH(1,1) eXt. Std. RT – GARCH(1,1) eXt. Std. Residual PlotResidual Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – GARCH(1,1) eXt. Residuals Vs. Std. RT – GARCH(1,1) eXt. Residuals Vs. Std. Residuals PlotResiduals Plot
RT – GARCH(1,1) eXt. Std. Residuals Vs. RT – GARCH(1,1) eXt. Std. Residuals Vs. ResidualsResiduals
RT – GARCH(1,1) eXt. Conditional Variance RT – GARCH(1,1) eXt. Conditional Variance Vs. Std. ResidualsVs. Std. Residuals
RT – GARCH(1,1) eXt. Residual RT – GARCH(1,1) eXt. Residual HistogramHistogram
0
400
800
1200
1600
2000
-8 -6 -4 -2 0 2 4
Series: Standardized ResidualsSample 3/17/1986 2/05/2009Observations 5775
Mean 0.040886Median 0.000000Maximum 5.427238Minimum -9.489069Std. Dev. 0.999128Skewness -0.221303Kurtosis 7.435210
Jarque-Bera 4780.495Probability 0.000000
RT – GARCH(1,1) eXt. Std. Residual RT – GARCH(1,1) eXt. Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – GARCH(1,1) eXt. Squared Std. RT – GARCH(1,1) eXt. Squared Std. Residual CorrelogramResidual Correlogram
Sign. Level (5%) = ± 0.025
RT - GARCH(1,1) eXt – Residual RT - GARCH(1,1) eXt – Residual ARCH TestARCH Test
RT – GARCH(1,1) modelRT – GARCH(1,1) modelExtended 2…Extended 2…
RT – GARCH(1,1) eXt.2 RT – GARCH(1,1) eXt.2 modelmodel
RT – GARCH(1,1) eXt.2RT – GARCH(1,1) eXt.2 Residual PlotResidual Plot
RT – GARCH(1,1) eXt.2 Conditional RT – GARCH(1,1) eXt.2 Conditional Variance PlotVariance Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
10000
20000
30000
40000
50000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RT – GARCH(1,1) eXt.2 Residual Vs. Conditional RT – GARCH(1,1) eXt.2 Residual Vs. Conditional Variance PlotVariance Plot
RT – GARCH(1,1) eXt.2 Std. RT – GARCH(1,1) eXt.2 Std. Residual PlotResidual Plot
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RT – GARCH(1,1) eXt.2 Residuals Vs. Std. RT – GARCH(1,1) eXt.2 Residuals Vs. Std. Residuals PlotResiduals Plot
RT – GARCH(1,1) eXt.2 Std. Residuals RT – GARCH(1,1) eXt.2 Std. Residuals Vs. ResidualsVs. Residuals
RT – GARCH(1,1) eXt.2 Conditional Variance RT – GARCH(1,1) eXt.2 Conditional Variance Vs. Std. ResidualsVs. Std. Residuals
RT – GARCH(1,1) eXt.2 Residual RT – GARCH(1,1) eXt.2 Residual HistogramHistogram
0
400
800
1200
1600
2000
-10 -5 0 5
Series: Standardized ResidualsSample 3/14/1986 2/05/2009Observations 5776
Mean 0.041525Median 0.000000Maximum 5.839957Minimum -10.81130Std. Dev. 0.999256Skewness -0.272238Kurtosis 8.627599
Jarque-Bera 7693.228Probability 0.000000
RT – GARCH(1,1) eXt.2 Std. Residual RT – GARCH(1,1) eXt.2 Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT – GARCH(1,1) eXt.2 Squared Std. Residual RT – GARCH(1,1) eXt.2 Squared Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RT - GARCH(1,1) eXt.2 – Residual RT - GARCH(1,1) eXt.2 – Residual ARCH TestARCH Test
RT – AR(2) – TGARCH(1,1) RT – AR(2) – TGARCH(1,1) ShortFallShortFall
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT LOSS_HAT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -1.000*sqr(GARCH) ]1.000*sqr(GARCH) ]
Zα = 1.000
-600
-500
-400
-300
-200
-100
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
Zα = 1.000
0
100
200
300
400
500
600
-500 -400 -300 -200 -100 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 701
Mean -22.68773Median -13.04273Maximum -0.013943Minimum -538.5431Std. Dev. 34.00902Skewness -6.561950Kurtosis 81.88871
Jarque-Bera 186806.7Probability 0.000000
Shortfall Histogram Shortfall Histogram [12.1406 %]
Zα = 1.000
[12.1406 %]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
LOSS_HAT RT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -2.000*sqr(GARCH) ]2.000*sqr(GARCH) ]
Zα = 2.000
-500
-400
-300
-200
-100
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
Zα = 2.000
0
10
20
30
40
50
60
70
-400 -300 -200 -100 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 110
Mean -34.69664Median -21.39431Maximum -0.045751Minimum -474.6651Std. Dev. 54.15019Skewness -5.307540Kurtosis 41.24680
Jarque-Bera 7221.030Probability 0.000000
Shortfall Histogram Shortfall Histogram [1.9050 %]
Zα = 2.000
[1.9050 %]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
LOSS_HAT RT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -2.250*sqr(GARCH) ]2.250*sqr(GARCH) ]
Zα = 2.250
-500
-400
-300
-200
-100
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
Zα = 2.250
0
10
20
30
40
50
-400 -300 -200 -100 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 78
Mean -37.84129Median -21.40151Maximum -0.397317Minimum -458.6956Std. Dev. 58.78183Skewness -5.043842Kurtosis 35.14740
Jarque-Bera 3689.453Probability 0.000000
Shortfall Histogram Shortfall Histogram [1.3508 %]
Zα = 2.250
[1.3508 %]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
LOSS_HAT RT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -2.250*sqr(GARCH) ]2.250*sqr(GARCH) ]
Zα = 2.426
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
-500
-400
-300
-200
-100
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL Zα = 2.426
0
5
10
15
20
25
30
35
-400 -300 -200 -100 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 62
Mean -39.82862Median -22.99443Maximum -0.788588Minimum -447.4530Std. Dev. 62.32909Skewness -4.809761Kurtosis 30.90806
Jarque-Bera 2251.104Probability 0.000000
Shortfall Histogram Shortfall Histogram [1.0737 %]
Zα = 2.426
[1.0737 %]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
LOSS_HAT RT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -3.000*sqr(GARCH) ]3.000*sqr(GARCH) ]
Zα = 3.000
-500
-400
-300
-200
-100
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
Zα = 3.000
0
5
10
15
20
25
30
-400 -300 -200 -100 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 32
Mean -44.26316Median -27.63546Maximum -0.542937Minimum -410.7870Std. Dev. 75.18691Skewness -3.863235Kurtosis 18.99085
Jarque-Bera 420.5407Probability 0.000000
Shortfall Histogram Shortfall Histogram [0.5542 %]
Zα = 3.000
0.5542 %]
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
LOSS_HAT RT
RT Vs. Expected Loss [ -RT Vs. Expected Loss [ -4.000*sqr(GARCH) ]4.000*sqr(GARCH) ]
Zα = 4.000
-350
-300
-250
-200
-150
-100
-50
0
86 88 90 92 94 96 98 00 02 04 06 08
SHORTFALL
Shortfall [ min{rt-Shortfall [ min{rt-loss_hat,0}]loss_hat,0}]
Zα = 4.000
0
1
2
3
4
5
6
-350 -300 -250 -200 -150 -100 -50 0
Series: SHORTFALLSample 3/13/1986 2/05/2009 IF SHORTFALL<0Observations 8
Mean -84.82702Median -31.48301Maximum -1.965430Minimum -346.9090Std. Dev. 113.7577Skewness -1.734299Kurtosis 4.692967
Jarque-Bera 4.965769Probability 0.083502
Shortfall Histogram Shortfall Histogram [0.1383 %]
Zα = 4.000
[0.1383 %]
Volatility ForecastingVolatility Forecasting
from: TGARCH(1,1) modelfrom: TGARCH(1,1) model
TGARCH(1,1) - Plot RT TGARCH(1,1) - Plot RT ±±2 2 σσ
TGARCH(1,1) – Variance Dynamic TGARCH(1,1) – Variance Dynamic ForecastForecast
(out of the sample)(out of the sample)02/06/2009 - 02/06/201002/06/2009 - 02/06/2010
-150
-100
-50
0
50
100
150
2009M04 2009M07 2009M10 2010M01
RTF
3000
3100
3200
3300
3400
3500
3600
3700
3800
2009M04 2009M07 2009M10 2010M01
Forecast of Variance
TGARCH(1,1) - Plot RT TGARCH(1,1) - Plot RT ±±2 2 σσ
Variance Dynamic Forecast (out of the Variance Dynamic Forecast (out of the sample)sample)
TGARCH(1,1) – Variance Dynamic TGARCH(1,1) – Variance Dynamic ForecastForecast
(in the sample)(in the sample)
-100
-50
0
50
100
2008M04 2008M07 2008M10 2009M01
RTF
Forecast: RTFActual: RTForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 49.70124Mean Absolute Error 35.52234Mean Abs. Percent Error 103.4389Theil Inequality Coefficient 0.974449 Bias Proportion 0.009734 Variance Proportion 0.988903 Covariance Proportion 0.001363
600
800
1000
1200
1400
1600
1800
2000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
TGARCH(1,1) - Plot RT TGARCH(1,1) - Plot RT ±±2 2 σσ
Variance Dynamic Forecast (in the Variance Dynamic Forecast (in the sample)sample)
-200
-100
0
100
200
2008M04 2008M07 2008M10 2009M01
RTF
Forecast: RTFActual: RTForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 49.46354Mean Absolute Error 35.39842Mean Abs. Percent Error 103.4706Theil Inequality Coefficient 0.959512 Bias Proportion 0.010551 Variance Proportion 0.953923 Covariance Proportion 0.035525
0
1000
2000
3000
4000
5000
6000
7000
8000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
TGARCH(1,1) – Variance Static TGARCH(1,1) – Variance Static ForecastForecast
(in the sample)(in the sample)
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
TGARCH(1,1) - Plot RT TGARCH(1,1) - Plot RT ±±2 2 σσ
Variance Static Forecast (in the Variance Static Forecast (in the sample)sample)
Volatility ForecastingVolatility Forecasting
from: Rangefrom: Range22 model model
RangeRange22 - Plot RT - Plot RT ±±2 2 σσ
-120
-80
-40
0
40
80
120
2008M04 2008M07 2008M10 2009M01
RANGEF
Forecast: RANGEFActual: RANGEForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 39.61914Mean Absolute Error 33.85970Mean Abs. Percent Error 100.0000Theil Inequality Coefficient 1.000000 Bias Proportion 0.730392 Variance Proportion 0.269608 Covariance Proportion 0.000000
0
500
1000
1500
2000
2500
3000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
RangeRange22 – Variance Dynamic – Variance Dynamic ForecastForecast
(in the sample)(in the sample)
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
RangeRange22 - Plot RT - Plot RT ±±2 2 σσ Variance Dynamic Forecast (in the Variance Dynamic Forecast (in the
sample)sample)
-200
-150
-100
-50
0
50
100
150
200
2008M04 2008M07 2008M10 2009M01
RANGEF
Forecast: RANGEFActual: RANGEForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 39.61914Mean Absolute Error 33.85970Mean Abs. Percent Error 100.0000Theil Inequality Coefficient 1.000000 Bias Proportion 0.730392 Variance Proportion 0.269608 Covariance Proportion 0.000000
0
1000
2000
3000
4000
5000
6000
7000
8000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
RangeRange22 – Variance Static Forecast – Variance Static Forecast(in the sample)(in the sample)
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
RangeRange22 - Plot RT - Plot RT ±±2 2 σσ Variance Static Forecast (in the Variance Static Forecast (in the
sample)sample)
Volatility ForecastingVolatility Forecasting
from: GARCH(1,1) eXt. from: GARCH(1,1) eXt. modelmodel
GARCH(1,1) eXt.2GARCH(1,1) eXt.2 - Plot - Plot RT RT ±±2 2 σσ
-200
-100
0
100
200
2008M04 2008M07 2008M10 2009M01
RTF
Forecast: RTFActual: RTForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 49.58125Mean Absolute Error 35.33305Mean Abs. Percent Error 99.27798Theil Inequality Coefficient 1.000000 Bias Proportion 0.004929 Variance Proportion 0.995071 Covariance Proportion 0.000000
0
2000
4000
6000
8000
10000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
GARCH(1,1) eXt.2 – Variance Dynamic GARCH(1,1) eXt.2 – Variance Dynamic ForecastForecast
(in the sample)(in the sample)
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
GARCH(1,1) eXt.2 - Plot RT GARCH(1,1) eXt.2 - Plot RT ±±2 2 σσ
Variance Dynamic Forecast (in the Variance Dynamic Forecast (in the sample)sample)
GARCH(1,1) eXt.2 –GARCH(1,1) eXt.2 – Variance Static Variance Static ForecastForecast
(in the sample)(in the sample)
Training Set: 03/13/1986 - 12/31/2007
Test Set: 01/01/2008 - 02/05/2009
-300
-200
-100
0
100
200
300
2008M04 2008M07 2008M10 2009M01
RTF
Forecast: RTFActual: RTForecast sample: 1/02/2008 2/05/2009Included observations: 277
Root Mean Squared Error 49.58125Mean Absolute Error 35.33305Mean Abs. Percent Error 99.27798Theil Inequality Coefficient 1.000000 Bias Proportion 0.004929 Variance Proportion 0.995071 Covariance Proportion 0.000000
0
2000
4000
6000
8000
10000
12000
2008M04 2008M07 2008M10 2009M01
Forecast of Variance
GARCH(1,1) eXt.2 - Plot RT GARCH(1,1) eXt.2 - Plot RT ±±2 2 σσ
Variance Static Forecast (in the Variance Static Forecast (in the sample)sample)
Conditional Variance Conditional Variance ComparisonsComparisons
Extra Stuff…Extra Stuff…
S&P 500S&P 500
-400
-300
-200
-100
0
100
200
86 88 90 92 94 96 98 00 02 04 06 08
RM
RT MSFT Vs. RM S&P500RT MSFT Vs. RM S&P500
-800
-600
-400
-200
0
200
400
86 88 90 92 94 96 98 00 02 04 06 08
RT MSFT
-400
-300
-200
-100
0
100
200
86 88 90 92 94 96 98 00 02 04 06 08
RM S&P500
RX = RT - RMRX = RT - RM
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
RX = RT - RM
9/11Win95Win98monopoly
accuse
European antitrust
action
5,000 emp.
layoffs
RX - HistogramRX - Histogram
0
400
800
1200
1600
2000
2400
-200 -100 0 100 200
Series: RXSample 3/13/1986 2/05/2009Observations 5776
Mean 1.149853Median 0.045307Maximum 229.0877Minimum -263.9850Std. Dev. 32.61286Skewness -0.192156Kurtosis 11.08033
Jarque-Bera 15749.09Probability 0.000000
RX - CorrelogramRX - Correlogram
Sign. Level (5%) = ± 0.025
RXRX22 - Correlogram - Correlogram
Sign. Level (5%) = ± 0.025
RX – AR(2) modelRX – AR(2) model
RXF - AR(2) Static RXF - AR(2) Static ForecastForecast
-80
-40
0
40
80
86 88 90 92 94 96 98 00 02 04 06 08
RXF
Forecast: RXFActual: RXForecast sample: 3/13/1986 2/05/2009Adjusted sample: 3/18/1986 2/05/2009Included observations: 5774
Root Mean Squared Error 32.49066Mean Absolute Error 21.72832Mean Abs. Percent Error 146.6416Theil Inequality Coefficient 0.952797 Bias Proportion 0.000000 Variance Proportion 0.934372 Covariance Proportion 0.065628
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
RX
-8
-4
0
4
8
12
86 88 90 92 94 96 98 00 02 04 06 08
RXF
RX Vs. RXF AR(2) Static RX Vs. RXF AR(2) Static ForecastForecast
RXF - AR(2) Dynamic RXF - AR(2) Dynamic ForecastForecast
-80
-60
-40
-20
0
20
40
60
80
86 88 90 92 94 96 98 00 02 04 06 08
RXF
Forecast: RXFActual: RXForecast sample: 3/13/1986 2/05/2009Adjusted sample: 3/18/1986 2/05/2009Included observations: 5774
Root Mean Squared Error 32.50845Mean Absolute Error 21.74181Mean Abs. Percent Error 141.7840Theil Inequality Coefficient 0.966035 Bias Proportion 0.000000 Variance Proportion 0.994750 Covariance Proportion 0.005250
RX AR(2) – Residual PlotRX AR(2) – Residual Plot
-300
-200
-100
0
100
200
300
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
Residual Actual Fitted
RX AR(2) – Residual Plot RX AR(2) – Residual Plot [2][2]
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
RX Residuals
RX AR(2) – Residual RX AR(2) – Residual HistogramHistogram
0
400
800
1200
1600
2000
2400
-200 -100 0 100 200
Series: ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean -1.34e-10Median -1.075838Maximum 229.3795Minimum -265.8949Std. Dev. 32.49348Skewness -0.222536Kurtosis 10.99945
Jarque-Bera 15442.88Probability 0.000000
RX AR(2) – Residual RX AR(2) – Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RX AR(2) – Squared RX AR(2) – Squared Residual CorrelogramResidual Correlogram
Sign. Level (5%) = ± 0.025
RX AR(2) – Residual RX AR(2) – Residual ARCH TestARCH Test
RX – AR(2) – GARCH(1,1) RX – AR(2) – GARCH(1,1) modelmodel
RX – AR(2) – GARCH(1,1) RX – AR(2) – GARCH(1,1) modelmodel
σ2 = 1,055.5790σ = 32.489675
RX – AR(2) - GARCH(1,1) RX – AR(2) - GARCH(1,1) Residual PlotResidual Plot
RX – AR(2) - GARCH(1,1) RX – AR(2) - GARCH(1,1) Conditional Variance PlotConditional Variance Plot
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
RESID
0
2000
4000
6000
8000
10000
12000
14000
86 88 90 92 94 96 98 00 02 04 06 08
GARCH
RX – AR(2) – GARCH(1,1) Residual Vs. Conditional RX – AR(2) – GARCH(1,1) Residual Vs. Conditional Variance PlotVariance Plot
RX – AR(2) -GARCH(1,1) Std. RX – AR(2) -GARCH(1,1) Std. Residual PlotResidual Plot
-300
-200
-100
0
100
200
300
86 88 90 92 94 96 98 00 02 04 06 08
RESID
-16
-12
-8
-4
0
4
8
86 88 90 92 94 96 98 00 02 04 06 08
STD_RESID
RX – AR(2) - GARCH(1,1) Residuals Vs. Std. RX – AR(2) - GARCH(1,1) Residuals Vs. Std. Residuals PlotResiduals Plot
RX – AR(2) - GARCH(1,1) Std. Residuals Vs. RX – AR(2) - GARCH(1,1) Std. Residuals Vs. ResidualsResiduals
RX – AR(2) - GARCH(1,1) Conditional Variance Vs. Std. RX – AR(2) - GARCH(1,1) Conditional Variance Vs. Std. ResidualsResiduals
RX – AR(2) - GARCH(1,1) Residual RX – AR(2) - GARCH(1,1) Residual HistogramHistogram
0
400
800
1200
1600
2000
2400
-10 -5 0 5
Series: Standardized ResidualsSample 3/18/1986 2/05/2009Observations 5774
Mean 0.010620Median -0.019988Maximum 6.570933Minimum -13.00676Std. Dev. 0.999150Skewness -0.432622Kurtosis 11.72297
Jarque-Bera 18486.16Probability 0.000000
RX – AR(2) - GARCH(1,1) Std. Residual RX – AR(2) - GARCH(1,1) Std. Residual CorrelogramCorrelogram
Sign. Level (5%) = ± 0.025
RX – AR(2) - GARCH(1,1) Squared Std. RX – AR(2) - GARCH(1,1) Squared Std. Residual CorrelogramResidual Correlogram
Sign. Level (5%) = ± 0.025
RX - AR(2) - GARCH(1,1) – Residual RX - AR(2) - GARCH(1,1) – Residual ARCH TestARCH Test
RX - AR(2) - GARCH(1,1) – RX - AR(2) - GARCH(1,1) – Variance Dynamic ForecastVariance Dynamic Forecast
-80
-40
0
40
80
2009M04 2009M07 2009M10 2010M01
RXF
1150
1200
1250
1300
1350
1400
1450
1500
2009M04 2009M07 2009M10 2010M01
Forecast of Variance
Grazie dell’Attenzione !!!Grazie dell’Attenzione !!!
