Transcript Slide 1

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
Why model and forecast volatility?
investment
security valuation
risk management
policy issues
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
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
 mean eq. : rt  h t  t
 variance eq. : h t   0  i 1 r  j1  jh t  j
q
p
2
i t i
Parameter constraints:


 0  0,
ensuring variance to be positive  i  0  i  1,
  0  j  1
 i
stationarity condition:

q
i 1
 i   j 1  j  1
p
Error distribution
1. Normal

The density function:
f  t  


 1  t2 
 exp    
2ht
 2 ht 
1
Implied kurtosis:
k=3
The log-likelihood function:
LNormal
 t2 
1 T 
   ln2   lnht   
2 t 1 
ht 
2. Student-t

Bollerslev (1987)

The density function:
  1 2 s t 1 2
s t
f  t  
,  2
 1 2 ; var t  
12
2
 2
   21   t s t 



Implied kurtosis:
3  2
k
,  4
 4

LStudent
The log-likelihood function:
     1 
 1 T 

 t2 
    1
2

 T  ln 
  ln     ln   2   ln  t  1    ln1  2

2
2
2
2





2




t

1





t





 
3. Generalized Error Distribution (GED)

Nelson (1991)

The density function:


2
  1  t  

1
  
  exp  


2


2

  

 
f  t  
;


 1
3
1

 
  2  
 
 
Implied kurtosis:
2
1 5
3
k        
   
 
The log-likelihood function:
   1 
t
  ln  
2 
t 1    

T
LGED

  1
 1 

 ln2  ln  
  
  
The SV model
 mean eq. : rt   t  t ,  t ~ N(0,1)
1
 t  exp( h t )
2
 volat ilit y eq. : h t    h t 1  v t , v t ~ N 0, v2
Parameter constraints:
 stationarity condition: |  |  1

Linearized form:
 y t  ln(rt 2 )  ht  ln( t2 )  1.27  ht   t

ht    ht 1  vt
E  t   0, Var t  
2
2

Forecast Evaluation Measures

Root Mean Square Error (RMSE)
RMSE 

1 I
(ˆ i2   i2 ) 2

I i 1
Mean Absolute Error (MAE)
1 I
MAE   ˆ i2   i2
I i 1

Theil-U Statistics

Theil  U 

2
2 2
ˆ
(



i
i )
i 1
I
2
2 2
(



)
i

1
i
i 1
I

LINEX loss function
1 I
LINEX   exp(a(ˆ i2   i2 ))  a(ˆ i2   i2 )  1
I i 1
Data Description
Daily closing prices of BET-C index

data series: BET-C stock index
1300

time length:
April 17, 1998 - April 21, 2003
1100
1200
1000
900

1255 daily returns
rt  lnP t   lnPt 1 
Pt – daily closing value of BET-C

Software: Eviews, Ox
800
700
600
500
400
250
500
750
1000
BETC
Descriptive statistics for BET-C return series
Mean
Median Maximum Minimum Std. Dev. Skewness Kurtosis Jarque-Bera Prob.
0.000102 -0.0000519 0.1038602 -0.0975698 0.0153105 0.106634
9.423705
2160.141
0.000
1250
Tested Hypotheses
1. Normality
500
4
Series: R100
Sample 2 1257
3
Observations 1256
2
Mean
0.010406
1
Median
-0.004572
Maximum 0 10.38602
Minimum
-9.756982
Std. Dev. -1 1.531051
Skewness
0.106367
-2
Kurtosis
9.430919
Normal Quantile
400
300
200
100
-3
Jarque-Bera
Probability-4
0
-10
-5
0
5
10
2166.704
0.000000
-.10
-.05
.00
.05
.10
R
Histogram of the BET-C returns
BET-C return quantile plotted
against the Normal quantile
.15
2.Homoscedasticity
.12
BET-C return series
.08
.04
.00
-.04
-.08
-.12
250
500
750
RETURN
1000
1250
.012
.010
.008
.006
.004
.002
BET-C squared return series
.000
250
500
750
1000
SQUARED_RETURN
1250
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
Autocorrelation coefficients for returns (lags 1 to 36)
0.3
0.25
0.2
0.15
AC
0.1
PAC
0.05
0
1
-0.05
-0.1
4
7
10 13 16 19 22 25 28 31 34
Autocorrelation coefficients for squared returns (lags 1 to 36)
0.3
0.25
0.2
0.15
AC
0.1
PA
C
0.05
-0.1
33
29
25
21
17
9
13
-0.05
5
1
0
Model estimates and forecasting
performances
Methodology:
- two sets: 1004 observations for model estimation
252 observations for out-of-sample forecast evaluation
 GARCH models
Mean equation specification
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)
-
Residual tests
 Normality test
200
 Autocorrelation tests
Lag
number
1
5
10
15
Correlogram of
residuals
Q-stat
0.0085
3.3598
5.7904
8.0496
Prob
0.927
0.645
0.833
0.922
Correlogram of
squared residuals
Q-stat
103.60
162.76
165.21
167.21
Prob
0.000
0.000
0.000
0.000
Series: Residuals
Sample 3 1004
Observations 1002
160
120
80
40
0
-0.05
0.00
0.05
 ARCH-LM test and White Heteroscedasticity Test
ARCH Test:
F-statistic
114.8229
Probability
0.000000
Obs*R-squared
103.1921
Probability
0.000000
White Heteroskedasticity Test:
F-statistic
63.32189
Probability
0.000000
Obs*R-squared
112.7329
Probability
0.000000
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.000355
-0.000463
0.093143
-0.077582
0.014613
-0.022081
8.209193
Jarque-Bera
Probability
1132.997
0.000000
GARCH (1,1) – Normal Distribution – QML parameter estimates
Coefficient
Std.Error
t-value
Probability
0.302055
0.045561
6.630
0.0000
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
AR (1)
Constant (V)
Diagnostic test based on the news impact curve (EGARCH vs. GARCH)
Test
Prob
Sign Bias t-Test
0.41479 0.67830
Negative Size Bias t-Test
0.66864
0.50373
Positive Size Bias t-Test
0.02906 0.97682
Joint Test for the Three Effects
0.47585
0.92416
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
Prob
Sign Bias t-Test
0.38456 0.70056
Negative Size Bias t-Test
0.81038 0.41772
Positive Size Bias t-Test
0.21808 0.82736
Joint Test for the Three Effects
0.73189 0.86568
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
0.0000
Diagnostic test based on the news impact curve (EGARCH vs. GARCH)
Test
Prob
Sign Bias t-Test
0.47340
0.63592
Negative Size Bias t-Test
0.82446
0.40968
Positive Size Bias t-Test
0.14047
0.88829
Joint 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
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
In-sample model evaluation
a) Residual tests

Autocorrelation of the residuals
Lag
GARCH(1,1) Nomal
1
5
10
15
Q-stat.
1.131
3.286
5.654
8.679

Lag
p-value
0.287
0.511
0.774
0.851

Q-stat.
2.289
4.755
7.046
10.144
GARCH(1,1) GED
p-value
0.130
0.313
0.632
0.752
Q-stat.
2.014
4.408
6.720
9.796
p-value
0.156
0354
0.667
0.777
SV
Q-stat.
0.506
2.802
6.237
7.571
p-value
0.477
0.591
0.716
0.910
Autocorrelation of the squared residuals
GARCH(1,1) Nomal
Q-stat.
p-value
0.127
1
3.198
0.362
6.033
0.644
6.782
0.913
1
5
10
15
GARCH(1,1) Student-T
GARCH(1,1) Student-T
Q-stat.
p-value
0.204
1
3.606
0.307
6.235
0.621
6.936
0.905
Kurtosis explanation
GARCH (1,1) Normal
GARCH (1,1) Student-t
GARCH (1,1) GED
SV
Unexplained
kurtosis
4.28
-7.21
2.56
-2.05
GARCH(1,1) GED
Q-stat.
p-value
0.186
1
3.499
0.321
6.180
0.627
6.895
0.907
SV
Q-stat.
0.589
2.681
6.539
8.824
p-value
0.443
0.613
0.685
0.842
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
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
LINEX
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
.012
.010
.008
.006
.004
.002
.000
1050
1100
1150
RR
1200
1250
Forecast output
a) GARCH (1,1) Normal
c) GARCH (1,1) GED
0,005
0,005
0,0045
0,0045
0,004
0,004
0,0035
0,0035
0,003
1day
0,003
1day
0,0025
5days
0,0025
5days
b) GARCH (1,1) Student-t
248
229
210
191
172
153
134
115
96
77
58
39
1
253
235
217
199
181
163
145
0
127
0
109
0,0005
91
0,0005
73
0,001
55
0,001
37
0,0015
19
0,0015
1
10 days
0,002
20
10 days
0,002
d) SV
0,006
0,0008
0,0007
0,005
0,0006
0,004
1day
0,003
5days
10 days
0,002
0,0005
1 day
0,0004
5 days
10 days
0,0003
0,0002
0,001
0,0001
253
235
217
199
181
163
145
127
109
91
73
55
37
19
253
235
217
199
181
163
145
127
109
91
73
55
37
1
19
1
0
0
Evaluation Measures
1-step ahead forecast evaluation

RMSE
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
1
Benchmark model - Random Walk
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
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 evaluation

RMSE
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
1
Benchmark model - Random Walk
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
Comparison between the statistical features
of the two sample periods
In-sample
Out-of-sample
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
0
0
Number of observations
Probability
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 Lancaster,
Working paper;
Ruiz, E. (1994) - Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models, Journal of
Econometrics, 63, 289-306;
Ruiz, Esther, Angeles Carnero and Daniel Pena (2001) – Is Stochastic Volatility More Flexible than Garch? ,
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,
Financial Markets Group, London School of Economics, Discussion Paper 248;
Shephard, H. (1993) – Fitting Nonlinear Time-series Models with Applications to stochastic Variance models,
Journal of Applied Econometrics, Vol. 8, S135-S152;
Shephard, Neil, S. Kim and S. Chib (1998) – Stochastic Volatility: Likelihood Inference and Comparison with
ARCH Models, Review of Economic Studies 65, 361-393;
Taylor, S.J. (1986) - Modelling Financial Time Series, John Wiley;
Terasvirta, T. (1996) - Two Stylized Facts and the GARCH(1,1) Model, W.P. Series in Finance and Economics 96,
Stockholm School of Economics;
Walsh, D. and G. Tsou (1998) - Forecasting Index Volatility: Sampling Interval and Non-Trading Effects, Applied
Financial Economics, 8, 477-485
Appendix – GARCH mean equation
1. The AR(1) model with intercept
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
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 Tests
Date: 06/23/03 Time: 00:48
Correlogram of Residuals
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
Date: 06/23/03 Time: 00:49
Correlogram of Squared Residuals
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
ARCH-LM test
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
White Heteroskedasticity Test
White Heteroskedasticity Test:
F-statistic
63.32189
Probability
0.000000
Obs*R-squared
112.7329
Probability
0.000000
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
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Date: 06/23/03 Time: 00:53
Sample: 3 1004
Included observations: 1002
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