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Modelling and Forecasting Stock Index Volatility –a comparison between GARCH models and the Stochastic Volatility model–. Supervisor: Professor Moisa Altar. Table of Contents. Competing volatility models Data description - PowerPoint PPT Presentation
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Modelling and Forecasting Stock Index Volatility
–a comparison between GARCH models and the Stochastic Volatility model–
Supervisor: Professor Moisa Altar
Table of Contents
Competing volatility models Data description Model estimates and forecasting
performances Concluding remarks
The Stylized Facts The distribution of financial time series has
heavier tails than the normal distribution
Highly correlated values for the squared returns
Changes in the returns tend to cluster
Why model and forecast volatility?
investment
security valuation
risk management
policy issues
Competing Volatility Models
ARCH/GARCH class of models Engle (1982) Bollerslev (1986) Nelson (1991) Glosten, Jaganathan, and Runkle
(1993) Stochastic Volatility (Variance) model
Taylor (1986)
The GARCH model
p
1j jtj
q
1i
2iti0t
ttt
hrh:.eqiancevar
hr:.eqmean
Parameter constraints: ensuring variance to be positive
stationarity condition:
1j0
,1i0
,0
i
i
0
p
j j
q
i i 111
Error distribution1. Normal
The density function:
Implied kurtosis: k=3
The log-likelihood function:
t
t
t
t hhf
2
2
1exp
2
1
T
t t
ttNormal h
hL1
2
ln2ln2
1
2. Student-t Bollerslev (1987)
The density function:
Implied kurtosis:
The log-likelihood function:
2,
2
svar;
s12
s21f t
t21
t2t
21
21t
t
4,
4
23
k
T
t t
ttStudent TL
12
22
21ln1ln
2
12ln
2
1
2ln
2
1ln
3. Generalized Error Distribution (GED) Nelson (1991)
The density function:
Implied kurtosis:
The log-likelihood function:
3
21
;1
2
21
exp
f
2
1
t
t
2351
k
T
t
tGEDL
1
1ln2ln
1
2
1ln
The SV model
2vtt1tt
tt
tttt
0,N~v,vhh:.eqvolatility
)h2
1exp(
)1,0(N~,r:.eqmean
Parameter constraints: stationarity condition:
Linearized form:
1||
ttt
tttttt
vhh
hhry
1
22 27.1)ln()ln(
2
,02 tt VarE
Forecast Evaluation Measures Root Mean Square Error (RMSE)
Mean Absolute Error (MAE)
Theil-U Statistics
LINEX loss function
I
iiiI
RMSE1
222 )ˆ(1
I
iiiI
MAE1
22ˆ1
I
i ii
I
i iiUTheil
1
2221
1
222
)(
)ˆ(
I
iiiii aa
ILINEX
1
2222 1)ˆ())ˆ(exp(1
Data Description
data series: BET-C stock index
time length: April 17, 1998 - April 21, 2003
1255 daily returns
Pt – daily closing value of BET-C
Software: Eviews, Ox
Descriptive statistics for BET-C return seriesMean Median Maximu
mMinimum Std.
Dev.Skewnes
sKurtosis Jarque-
BeraProb.
0.000102
-0.0000519
0.1038602
-0.0975698
0.0153105
0.106634 9.423705 2160.141 0.000
1ttt PlnPlnr
400
500
600
700
800
900
1000
1100
1200
1300
250 500 750 1000 1250
BETC
Daily closing prices of BET-C index
Tested Hypotheses 1. Normality
Histogram of the BET-C returns BET-C return quantile plotted
against the Normal quantile
0
100
200
300
400
500
-10 -5 0 5 10
Series: R100Sample 2 1257Observations 1256
Mean 0.010406Median -0.004572Maximum 10.38602Minimum -9.756982Std. Dev. 1.531051Skewness 0.106367Kurtosis 9.430919
Jarque-Bera 2166.704Probability 0.000000-4
-3
-2
-1
0
1
2
3
4
-.10 -.05 .00 .05 .10 .15
R
Norm
al Q
uantil
e
2.Homoscedasticity
-.12
-.08
-.04
.00
.04
.08
.12
250 500 750 1000 1250
RETURN
.000
.002
.004
.006
.008
.010
.012
250 500 750 1000 1250
SQUARED_RETURN
BET-C return series
BET-C squared return series
3. Stationarity
Unit root tests for BET-C return series
ADF Test Statistic -13.53269 1% Critical Value* -3.4384
5% Critical Value -2.8643
10% Critical Value -2.5683
*MacKinnon critical values for rejection of hypothesis of a unit root.
PP Test Statistic -28.07887 1% Critical Value* -3.4384
5% Critical Value -2.8643
10% Critical Value -2.5682
*MacKinnon critical values for rejection of hypothesis of a unit root.
4. Serial independence
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1 4 7 10 13 16 19 22 25 28 31 34
AC
PAC
Autocorrelation coefficients for returns (lags 1 to 36)
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.31 5 9 13
17
21
25
29
33
AC
PAC
Autocorrelation coefficients for squared returns (lags 1 to 36)
Model estimates and forecasting performances
Constant Y(-1) R-squared
Mean equation with intercept -0.000355 0.276034 0.076278
t-statistic (probability that the coefficient equals 0)
-0.768264 (0.4425)
9.087175(0.000)
-
Mean equation without intercept - 0.276769 0.075733
t-statistic (probability that the coefficient equals 0)
- 9.117758(0.000)
-
Mean equation specification
GARCH models
Methodology: - two sets: 1004 observations for model estimation 252 observations for out-of-sample forecast evaluation
Lagnumber
Correlogram of residuals
Correlogram ofsquared residuals
Q-stat Prob Q-stat Prob
1 0.0085 0.927 103.60 0.0005 3.3598 0.645 162.76 0.000
10 5.7904 0.833 165.21 0.00015 8.0496 0.922 167.21 0.000 0
40
80
120
160
200
-0.05 0.00 0.05
Series: ResidualsSample 3 1004Observations 1002
Mean -0.000355Median -0.000463Maximum 0.093143Minimum -0.077582Std. Dev. 0.014613Skewness -0.022081Kurtosis 8.209193
Jarque-Bera 1132.997Probability 0.000000
ARCH Test:
F-statistic 114.8229 Probability 0.000000
Obs*R-squared 103.1921 Probability 0.000000
Residual tests
White Heteroskedasticity Test:
F-statistic 63.32189 Probability 0.000000
Obs*R-squared 112.7329 Probability 0.000000
ARCH-LM test and White Heteroscedasticity Test
Autocorrelation tests
Normality test
GARCH (1,1) – Normal Distribution – QML parameter estimatesCoefficient Std.Error t-value Probability
AR (1)
0.302055 0.045561 6.630 0.0000
Constant (V) 0.0000472947 0.141153 3.351 0.0008
ARCH(Alpha1) 0.320832 0.065118 4.927 0.0000
GARCH(Beta1) 0.483147 0.102838 4.698 0.0000
GARCH (1,1) – Student-T Distribution – QML parameter estimates
Coefficient Std.Error t-value Probability
AR(1)
0.280817 0.037364 7.516 0.0000
Constant(V)
0.0000527251 0.144746
3.643 0.0003
ARCH(Alpha1) 0.350230 0.067874
5.160 0.0000
GARCH(Beta1) 0.439533 0.091994
4.778 0.0000
Student(DF)
4.512539 0.656110
6.878 0.0000
Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.41479 0.67830 Negative Size Bias t-Test 0.66864 0.50373Positive Size Bias t-Test 0.02906 0.97682Joint Test for the Three Effects 0.47585 0.92416
Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.38456 0.70056Negative Size Bias t-Test 0.81038 0.41772Positive Size Bias t-Test 0.21808 0.82736Joint Test for the Three Effects 0.73189 0.86568
SV– QML parameter estimates
Coefficient Std. Error z-Statistic Probability
C(1) -1.269102 0.450023 -2.820081 0.0048
C(2) 0.858869 0.050340 17.06149 0.0000
C(3) -1.486221 0.456019 -3.259119 0.0011
GARCH (1,1) –GED Distribution – QML parameter estimates
Coefficient Std.Error t-value Probability
AR(1)
0.285181 0.057321 4.975 0.0000
Constant(V)
0.0000496321
0.130000 3.818 0.0001
ARCH(Alpha1) 0.333678 0.062854 5.309 0.0000
GARCH(Beta1) 0.450807 0.091152 4.946 0.0000
Student(DF)
1.172517 0.081401 14.40 0.0000Diagnostic test based on the news impact curve (EGARCH vs. GARCH) Test ProbSign Bias t-Test 0.47340 0.63592Negative Size Bias t-Test 0.82446 0.40968Positive Size Bias t-Test 0.14047 0.88829Joint Test for the Three Effects 0.74931 0.86155
SV model To estimate the SV model, the return series was first filtered in order to eliminate the first order autocorrelation of the returns
In-sample model evaluationa) Residual tests Autocorrelation of the residuals
Lag GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) GED SV
Q-stat. p-value Q-stat. p-value Q-stat. p-value Q-stat. p-value1 1.131 0.287 2.289 0.130 2.014 0.156 0.506 0.4775 3.286 0.511 4.755 0.313 4.408 0354 2.802 0.59110 5.654 0.774 7.046 0.632 6.720 0.667 6.237 0.71615 8.679 0.851 10.144 0.752 9.796 0.777 7.571 0.910
Lag GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) GED SVQ-stat. p-value Q-stat. p-value Q-stat. p-value Q-stat. p-value
1 0.127 1 0.204 1 0.186 1 0.589 0.4435 3.198 0.362 3.606 0.307 3.499 0.321 2.681 0.61310 6.033 0.644 6.235 0.621 6.180 0.627 6.539 0.68515 6.782 0.913 6.936 0.905 6.895 0.907 8.824 0.842
Autocorrelation of the squared residuals
Kurtosis explanationUnexplained
kurtosisGARCH (1,1) Normal 4.28
GARCH (1,1) Student-t -7.21GARCH (1,1) GED 2.56SV -2.05
b) In-sample forecast evaluation
RMSE MAE THEIL-U1
GARCH 11 Normal 0.0000196062 0.000257336 0.646352
GARCH 11 T 0.0000195026 0.000256516 0.639539
GARCH 11 GED 0.0000194814 0.000253146 0.638149
SV 0.0000186253 0.000231101 0.583293
LINEX a=-20 a=-10 a= 10 a= 20
GARCH 11 Normal 7,70895E-09 1,92751E-09 1,92806E-09 7,71335E-09
GARCH 11 T 7,62777E-09 1,9072E-09 1,90773E-09 7,63198E-09
GARCH 11 GED 7,61114E-09 1,90305E-09 1,90359E-09 7,61545E-09
SV 6,95655E-09 1,73942E-09 1,73999E-09 6,96113E-09
1 Benchmark model - Random Walk
Out-of-sample Forecast Evaluation
Forecast methodology - rolling sample window: 1004 observations - at each step, the n-step ahead forecast is stored - n=1, 5, 10 Benchmark: realized volatility = squared returns
.000
.002
.004
.006
.008
.010
.012
1050 1100 1150 1200 1250
RR
Forecast output
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
0,004
0,0045
0,005
1
19 37 55 73 91
109
127
145
163
181
199
217
235
253
1day
5days
10 days
0
0,001
0,002
0,003
0,004
0,005
0,006
1
19 37 55 73 91
109
127
145
163
181
199
217
235
253
1day
5days
10 days
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
0,004
0,0045
0,005
1 20 39 58 77 96 115
134
153
172
191
210
229
248
1day
5days
10 days
0
0,0001
0,0002
0,0003
0,0004
0,0005
0,0006
0,0007
0,0008
1 19 37 55 73 91 109
127
145
163
181
199
217
235
253
1 day
5 days
10 days
a) GARCH (1,1) Normal c) GARCH (1,1) GED
b) GARCH (1,1) Student-t d) SV
Evaluation Measures
1-step ahead forecast evaluationRMSE MAE THEIL-U1
GARCH 11 Normal 0,000035300 0,00022591 0,583721
GARCH 11 T 0,000035111 0,000204242 0,580597
GARCH 11 GED 0,000035760 0,000203486 0,591337
SV 0,000048823 0,000253071 0,807336
LINEX a=-20 a=-10 a= 10 a= 20
GARCH 11 Normal 6,30398E-09 1,57614E-09 1,57644E-09 6,30638E-09
GARCH 11 T 6,23593E-09 1,55923E-09 1,55971E-09 6,2398E-09
GARCH 11 GED 6,46868E-09 1,61743E-09 1,61795E-09 6,47286E-09
SV 1,2055E-08 3,01454E-09 3,01612E-09 1,20676E-08
1 Benchmark model - Random Walk
5-step ahead forecast evaluation
RMSE MAE THEIL-U1
GARCH 11 Normal 0.0000512767 0.0003042315 0.847915
GARCH 11 T 0.0000512001 0.0003077174 0.846648
GARCH 11 GED 0.0000511668 0.0002983467 0.846097
SV 0.0000511653 0.0002851430 0.846073
1 Benchmark model - Random Walk
LINEX a=-20 a=-10 a= 10 a= 20
GARCH 11 Normal 1.3297E-08 3.325E-09 3.3268E-09 1.33108E-08
GARCH 11 T 1.3257E-08 3.315E-09 3.3169E-09 1.32711E-08
GARCH 11 GED 1.3241E-08 3.311E-09 3.3126E-09 1.32539E-08
SV 1.3239E-08 3.310E-09 3.3125E-09 1.32534E-08
10-step ahead forecast evaluationRMSE MAE THEIL-U1
GARCH 11 Normal 0.0000513675 0.0003060239 0.849416
GARCH 11 T 0.0000513716 0.0003107481 0.849484
GARCH 11 GED 0.0000513779 0.000300542 0.849588
SV 0.0000514735 0.0002870131 0.851169
LINEX a=-20 a=-10 a= 10 a= 20
GARCH 11 Normal 1,33445E-08 3,33699E-09 3,33871E-09 1,33583E-08
GARCH 11 T 1,33467E-08 3,33753E-09 3,33925E-09 1,33604E-08
GARCH 11 GED 1,33499E-08 3,33834E-09 3,34007E-09 1,33637E-08
SV 1,33996E-08 3,35077E-09 3,35251E-09 1,34135E-08
1 Benchmark model - Random Walk
Comparison between the statistical features of the two sample periods
In-sample Out-of-sample
Number of observations 1004 252
Mean -0.000468 0.002371
Median -0.000378 0.001137
Maximum 0.093332 0.103860
Minimum -0.097570 -0.065731
Standard Deviation 0.015209 0.015531
Skewness -0.116772 0.925148
Kurtosis 8.666434 11.94869
Jarque-Bera 1344.146 880.2563
Probability 0 0
Concluding remarks
In-sample analysis: a) residual tests: all models may be appropriate; b) evaluation measures: SV model is the best
performer;
Out-of-sample analysis: - for a 1-day forecast horizon GARCH models
outperform SV; - for the 5-day and 10-day forecast horizon, model
performances seem to converge; - the best model changes with forecast horizon and
with forecast evaluation measure; - there is no clear winner;
Concluding remarks
Sample construction problems;
Further research: - allowing for switching regimes; - allowing for leptokurtotic distributions in
the SV - a better proxy for realized volatility;
Bibliography Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, John Wiley &Sons, Ltd.; Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility Models Do
Provide Accurate Forecasts, International Economic Review; Armstrong, J.S. (1995) - On the Selection of Error Measures for Comparisons Among Forecasting
Methods, Journal of Forecasting; Armstrong, J.S (1978) – Forecasting with Econometric Methods: Folklore versus Fact, Journal of
Business, 51 (4), 1978, 549-564; Bluhm, H.H.W. and J. Yu (2000) - Forecasting volatility: Evidence from the German stock market,
Working paper, University of Auckland; Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of
Econometrics, Volume 4, Chapter 49, North Holland; Byström, H. (2001) - Managing Extreme Risks in Tranquil and Volatile Markets Using Conditional
Extreme Value Theory, Department of Economics, Lund University; Christodoulakis, G.A. and Stephen E. Satchell (2002) – Forecasting Using Log Volatility Models, Cass
Business School, Research Paper; Christoffersen, P. F and F. X. Diebold. (1997) - How Relevant is Volatility Forecasting for Financial Risk
Management?, The Wharton School, University of Pennsylvania; Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the variance of
UK inflation, Econometrica, 50, pp. 987-1008; Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on Volatility, The
Journal of Fiance, Vol. XLVIII, No. 5; Engle, R. (2001) – Garch 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of
Economic Perspectives – Volume 15, Number 4 – Fall 2001 – Pages 157-168; Engle, R. and A. J. Patton (2001) – What good is a volatility model?, Research Paper, Quantitative
Finance, Volume 1, 237-245; Engle, R. (2001) – New Frontiers for ARCH Models, prepared for Conference on Volatility Modelling
and Forecasting, Perth, Australia, September 2001; Glosten, L. R., R. Jaganathan, and D. Runkle (1993) – On the Relation between the Expected Value
and the Volatility of the Normal Excess Return on Stocks, Journal of Finance, 48, 1779-1801; Hamilton, J.D. (1994) – Time Series Analysis, Princeton University Press; Hamilton J.D. (1994) – State – Space Models, Handbook of Econometrics, Volume 4, Chapter 50, North
Holland;
Hol, E. and S. J. Koopman (2000) - Forecasting the Variability of Stock Index Returns with Stochastic Volatility Models and Implied Volatility, Tinbergen Institute Discussion Paper;
Koopman, S.J. and Eugenie Hol Uspenski (2001) –The Stochastic volatility in Mean model: Empirical evidence from international stock markets,
Liesenfeld, R. and R.C. Jung (2000) Stochastic Volatility Models: Conditional Normality versus Heavy-Tailed Distributions, Journal of Applied Econometrics, 15, 137-160;
Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models, Economic Research Deparment, Federal Reserve Bank of San Francisco;
Nelson, Daniel B. (1991) – Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59, 347-370;
Ozaki, T. and P.J. Thomson (1998) – Transformation and Seasonal Adjustment, Technical Report, Institute of Statistics and Operations Research, New Zealand
Peters, J. (2001) - Estimating and Forecasting Volatility of Stock Indices Using Asymmetric GARCH Models and (Skewed) Student-T Densities, Ecole d’Administration des Affaires, University of Liege;
Peters, J. and S. Laurent (2002) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models;
Pindyck, R.S and D.L. Rubinfeld (1998) – Econometric Models and Economic Forecasts, Irwin/McGraw-Hill; Poon, S.H. and C. Granger (2001) - Forecasting Financial Market Volatility - A Review, University of
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, Universidad Carlos III de Madrid, Statistics and Econometrics Series, Working Paper 01-08; Sandmann, G. and S.J. Koopman (1997)– Maximum Likelihood Estimation of Stochastic Volatility Models,
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Appendix – GARCH mean equation
Dependent Variable: Y
Method: Least Squares
Date: 06/23/03 Time: 00:45
Sample(adjusted): 3 1004
Included observations: 1002 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C -0.000355 0.000462 -0.768264 0.4425
Y(-1) 0.276034 0.030376 9.087175 0.0000
R-squared 0.076278 Mean dependent var -0.000487
Adjusted R-squared 0.075354 S.D. dependent var 0.015204
S.E. of regression 0.014620 Akaike info criterion -5.610880
Sum squared resid 0.213740 Schwarz criterion -5.601080
Log likelihood 2813.051 F-statistic 82.57675
Durbin-Watson stat 2.002722 Prob(F-statistic) 0.000000
1. The AR(1) model with intercept
2.The AR(1) model without intercept
Dependent Variable: Y
Method: Least Squares
Date: 06/23/03 Time: 00:46
Sample(adjusted): 3 1004
Included observations: 1002 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
Y(-1) 0.276769 0.030355 9.117758 0.0000
R-squared 0.075733 Mean dependent var -0.000487
Adjusted R-squared 0.075733 S.D. dependent var 0.015204
S.E. of regression 0.014617 Akaike info criterion -5.612286
Sum squared resid 0.213866 Schwarz criterion -5.607386
Log likelihood 2812.755 Durbin-Watson stat 2.003016
Appendix – Residual TestsDate: 06/23/03 Time: 00:48
Sample: 3 1004
Included observations: 1002
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.| | .| | 1 -0.003 -0.003 0.0085 0.927
.| | .| | 2 -0.011 -0.011 0.1228 0.940
.| | .| | 3 0.041 0.041 1.8102 0.613
.| | .| | 4 0.004 0.004 1.8256 0.768
.| | .| | 5 0.039 0.040 3.3598 0.645
.| | .| | 6 0.030 0.028 4.2395 0.644
.| | .| | 7 0.013 0.014 4.4124 0.731
.| | .| | 8 0.027 0.025 5.1482 0.742
.| | .| | 9 -0.025 -0.027 5.7834 0.761
.| | .| | 10 -0.003 -0.005 5.7904 0.833
.| | .| | 11 0.034 0.029 6.9812 0.801
.| | .| | 12 0.008 0.008 7.0442 0.855
.| | .| | 13 0.030 0.029 7.9561 0.846
.| | .| | 14 -0.007 -0.009 8.0088 0.889
.| | .| | 15 0.006 0.007 8.0496 0.922
.| | .| | 16 -0.049 -0.055 10.543 0.837
.| | .| | 17 0.021 0.020 10.994 0.857
.| | .| | 18 -0.002 -0.008 10.998 0.894
.| | .| | 19 0.007 0.009 11.051 0.922
.| | .| | 20 0.023 0.023 11.599 0.929
Correlogram of Residuals
Date: 06/23/03 Time: 00:49
Sample: 3 1004
Included observations: 1002
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|** | .|** | 1 0.321 0.321 103.60 0.000
.|* | .|* | 2 0.194 0.101 141.44 0.000
.|* | .| | 3 0.125 0.041 157.05 0.000
.|* | .| | 4 0.075 0.010 162.73 0.000
.| | .| | 5 0.005 -0.043 162.76 0.000
.| | .| | 6 0.008 0.005 162.82 0.000
.| | .| | 7 0.042 0.045 164.59 0.000
.| | .| | 8 0.024 0.003 165.18 0.000
.| | .| | 9 0.005 -0.012 165.21 0.000
.| | .| | 10 -0.027 -0.040 165.97 0.000
.| | .| | 11 -0.004 0.012 165.98 0.000
.| | .| | 12 -0.009 0.000 166.06 0.000
.| | .| | 13 -0.028 -0.022 166.84 0.000
.| | .| | 14 -0.011 0.005 166.96 0.000
.| | .| | 15 -0.016 -0.012 167.21 0.000
.| | .| | 16 0.007 0.020 167.26 0.000
.| | .| | 17 -0.019 -0.020 167.61 0.000
.| | .| | 18 -0.004 0.005 167.62 0.000
.| | .| | 19 0.000 0.003 167.62 0.000
.| | .| | 20 -0.017 -0.019 167.91 0.000
Correlogram of Squared Residuals
ARCH Test:
F-statistic 114.8229 Probability 0.000000
Obs*R-squared 103.1921 Probability 0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 06/23/03 Time: 00:52
Sample(adjusted): 4 1004
Included observations: 1001 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000145 1.83E-05 7.903650 0.0000
RESID^2(-1) 0.321081 0.029964 10.71555 0.0000
R-squared 0.103089 Mean dependent var 0.000213
Adjusted R-squared 0.102191 S.D. dependent var 0.000573
S.E. of regression 0.000543 Akaike info criterion -12.19544
Sum squared resid 0.000295 Schwarz criterion -12.18564
Log likelihood 6105.819 F-statistic 114.8229
Durbin-Watson stat 2.064939 Prob(F-statistic) 0.000000
ARCH-LM test
White Heteroskedasticity Test:
F-statistic 63.32189 Probability 0.000000
Obs*R-squared 112.7329 Probability 0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 06/23/03 Time: 00:53
Sample: 3 1004
Included observations: 1002
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000144 1.82E-05 7.933013 0.0000
Y(-1) -0.000222 0.001125 -0.197479 0.8435
Y(-1)^2 0.299471 0.026700 11.21598 0.0000
R-squared 0.112508 Mean dependent var 0.000213
Adjusted R-squared 0.110731 S.D. dependent var 0.000573
S.E. of regression 0.000541 Akaike info criterion -12.20501
Sum squared resid 0.000292 Schwarz criterion -12.19031
Log likelihood 6117.708 F-statistic 63.32189
Durbin-Watson stat 2.075790 Prob(F-statistic) 0.000000
White Heteroskedasticity Test