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Modelling and Forecasting StockIndex Volatility
a comparison between GARCH models and theStochastic Volatility model
Supervisor:Professor Moisa Altar
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Table of Contents
Competing volatility models
Data description
Model estimates and forecasting
performances Concluding remarks
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The Stylized Facts
The distribution of financial time series has heaviertails 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 managementpolicy issues
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Competing Volatility ModelsARCH/GARCH class of models
Engle (1982)
Bollerslev (1986)
Nelson (1991)
Glosten, Jaganathan, and Runkle (1993)
Stochastic Volatility (Variance) model
Taylor (1986)
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The GARCH model
p
1j jtj
q
1i
2
iti0t
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
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Error distribution1. Normal
The density function:
Implied kurtosis:
k=3
The log-likelihood function:
t
t
t
thh
f2
21exp
21
T
t t
t
tNormalh
hL1
2
ln2ln2
1
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2. Student-t
Bollerslev (1987) The density function:
Implied kurtosis:
The log-likelihood function:
2,2
svar;
s12
s21f tt21
t
2
t
21
21
tt
4,
4
23
k
T
t t
t
tStudentTL
1
2
2
2
21ln1ln
2
12ln
2
1
2ln
2
1ln
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3. Generalized Error Distribution (GED)
Nelson (1991) The density function:
Implied kurtosis:
The log-likelihood function:
3
21
;12
2
1exp
f
2
1
t
t
2351
k
T
t
t
GEDL
1
1ln2ln
1
2
1ln
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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
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Forecast Evaluation Measures Root Mean Square Error (RMSE)
Mean Absolute Error (MAE)
Theil-U Statistics
LINEX loss function
I
i
iiI
RMSE1
222 )(1
I
i
iiI
MAE1
22
1
I
i ii
I
i iiUTheil
1
222
1
1
222
)(
)(
I
i
iiii aa
I
LINEX1
2222 1)())(exp(1
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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 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
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
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TestedHypotheses
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
.
.
.
.
. . .
.
.
.
.-4
-3
-2
-1
0
1
2
3
4
-.10 -.05 .00 .05 .10 .15
R
NormalQuantile
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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
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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.
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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)
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-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
1 5 913
17
21
25
29
33
AC
PA
Autocorrelation coefficients for squared returns (lags 1 to 36)
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Model estimates and forecastingperformances
Constant Y(-1) R-squared
Mean equation with in tercept -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
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Lagnumber
Correlogram of
residuals
Correlogram of
squared residualsQ-stat Prob Q-stat Prob
1 0.0085 0.927 103.60 0.000
5 3.3598 0.645 162.76 0.000
10 5.7904 0.833 165.21 0.000
15 8.0496 0.922 167.21 0.0000
40
80
120
160
200
-0.05 0.00 0.05
Series: Residuals
Sample 3 1004
Observations 1002
Mean -0.000355Median -0.000463
Maximum 0.093143
Minimum -0.077582
Std. Dev. 0.014613
Skewness -0.022081
Kurtosis 8.209193
Jarque-Bera 1132.997
Probab ilit y 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 estimates
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GARCH (1,1)Normal DistributionQML parameter estimates
Coefficient 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 DistributionQML 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 0.41479 0.67830Negative 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 Prob
Sign 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
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SVQML parameter estimates
Coefficient Std. Err or 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 DistributionQML parameter estimates
Coefficient Std.Er ror t-value Probabil ity
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 0.0000GARCH(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 0.63592Negative Size Bias t-Test 0.82446 0.40968
Positive Size Bias t-Test 0.14047 0.88829Joint Test for the Three Effects 0.74931 0.86155
SV modelTo estimate the SV model, the return series was first filtered in order
to eliminate the first order autocorrelation of the returns
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In-sample model evaluationa) Residual tests
Autocorrelation of the residualsLag 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-value
1 1.131 0.287 2.289 0.130 2.014 0.156 0.506 0.477
5 3.286 0.511 4.755 0.313 4.408 0354 2.802 0.591
10 5.654 0.774 7.046 0.632 6.720 0.667 6.237 0.716
15 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 SV
Q-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 0.443
5 3.198 0.362 3.606 0.307 3.499 0.321 2.681 0.613
10 6.033 0.644 6.235 0.621 6.180 0.627 6.539 0.685
15 6.782 0.913 6.936 0.905 6.895 0.907 8.824 0.842
Autocorrelation of the squared residuals
Kurtosis explanationUnexplained
kurtosis
GARCH (1,1) Normal 4.28
GARCH (1,1) Student-t -7.21
GARCH (1,1) GED 2.56
SV -2.05
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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
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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
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Forecast output
0
0,0005
0,001
0,0015
0,002
0,0025
0,003
0,0035
0,004
0,0045
0,005
1
1
9
3
7
5
5
7
3
9
1
10
9
12
7
14
5
16
3
18
1
19
9
21
7
23
5
25
3
1day
5days
10 days
0
0,001
0,002
0,003
0,004
0,005
0,006
119
37
55
73
91
1
09
1
27
1
45
1
63
1
81
1
99
2
17
2
35
2
53
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 96115
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
119
37
55
73
91
1
09
1
27
1
45
1
63
1
81
1
99
2
17
2
35
2
53
1 day
5 days
10 days
a) GARCH (1,1) Normal c) GARCH (1,1) GED
b) GARCH (1,1) Student-t d) SV
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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
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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 0.846648GARCH 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
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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
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
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Comparison between the statistical featuresof the two sample periods
In-sample Out-of-sample
Number of observations 1004 252
Mean -0.000468 0.002371
Median -0.000378 0.001137Maximum 0.093332 0.103860
Minimum -0.097570 -0.065731
Standard Deviati on 0.015209 0.015531
Skewness -0.116772 0.925148
Kurtosis 8.666434 11.94869
Jarque-Bera 1344.146 880.2563
Probability 0 0
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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;
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Concluding remarks
Sample construction problems;
Further research:
- allowing for switching regimes;
- allowing for leptokurtotic distributions in the SV
- a better proxy for realized volatility;
Bibli h
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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
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(4), 1978, 549-564; Bluhm, H.H.W. and J. Yu (2000) - Forecasting volatility: Evidence from the German stock market, Working
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Hol, E. and S. J. Koopman (2000) - Forecasting the Variability of Stock Index Returns with Stochastic VolatilityModels and Implied Volatility, Tinbergen Institute Discussion Paper;
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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 -0.000487Adjusted 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
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2.The AR(1) model without interceptDependent 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
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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
7/30/2019 Stock Index Volatility
36/38
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
7/30/2019 Stock Index Volatility
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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 -12.19544Sum 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
7/30/2019 Stock Index Volatility
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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