Modelling and Forecasting Stock Index Volatility

1
Modelling and Forecasting Stock Index Volatility
a comparison between GARCH models and the
Stochastic Volatility model

• Supervisor
  Professor Moisa Altar

2
Table of Contents

• Competing volatility models
• Data description
• Model estimates and forecasting performances
• Concluding remarks

3
The Stylized Facts
Why model and forecast volatility?

• investment
• security valuation
• risk management
• policy issues

• 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

4
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)

5
The GARCH model

• Parameter constraints
• ensuring variance to be positive
• stationarity condition

6

• Error distribution
• 1. Normal
• The density function
• Implied kurtosis
• k3
• The log-likelihood function

7

• 2. Student-t
• Bollerslev (1987)
• The density function
• Implied kurtosis
• The log-likelihood function

8

• 3. Generalized Error Distribution (GED)
• Nelson (1991)
• The density function
• Implied kurtosis
• The log-likelihood function

9
The SV model

• Parameter constraints
• stationarity condition
• Linearized form

10
Forecast Evaluation Measures

• Root Mean Square Error (RMSE)
• Mean Absolute Error (MAE)
• Theil-U Statistics
• LINEX loss function

11
Data Description
Daily closing prices of BET-C index

• 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
12
Tested Hypotheses

• 1. Normality
• Histogram of the BET-C returns
  BET-C return quantile plotted
• 
  against the Normal
  quantile

13

• 2.Homoscedasticity

BET-C return series
BET-C squared return series
14

• 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. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. 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. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root.
15

• 4. Serial independence

Autocorrelation coefficients for returns (lags 1
to 36)
16
Autocorrelation coefficients for squared returns
(lags 1 to 36)
17
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) -
18
Residual tests

• Normality test

• Autocorrelation tests

Lag number Correlogram of residuals Correlogram of residuals Correlogram of squared residuals Correlogram of squared residuals
Lag number Q-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 0.000

• ARCH-LM test and White Heteroscedasticity Test

ARCH Test ARCH Test ARCH Test ARCH Test
F-statistic 114.8229 Probability 0.000000
ObsR-squared 103.1921 Probability 0.000000
White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test
F-statistic 63.32189 Probability 0.000000
ObsR-squared 112.7329 Probability 0.000000
19
GARCH (1,1) Normal Distribution QML 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
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
20
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
21
In-sample model evaluationa) Residual tests

• Autocorrelation of the residuals

Lag GARCH(1,1) Nomal GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) Student-T GARCH(1,1) GED GARCH(1,1) GED SV SV
Lag 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

• Autocorrelation of the squared residuals

Lag GARCH(1,1) Nomal GARCH(1,1) Nomal GARCH(1,1) Student-T GARCH(1,1) Student-T GARCH(1,1) GED GARCH(1,1) GED SV SV
Lag 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

• Kurtosis explanation

Unexplained kurtosis
GARCH (1,1) Normal 4.28
GARCH (1,1) Student-t -7.21
GARCH (1,1) GED 2.56
SV -2.05
22
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
1 Benchmark model - Random Walk
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
23
Out-of-sample Forecast Evaluation

• Forecast methodology
• - rolling sample window 1004 observations
• - at each step, the n-step ahead forecast
  is stored
• - n1, 5, 10
• Benchmark realized volatility squared returns

24
Forecast output
a) GARCH (1,1) Normal
c) GARCH (1,1) GED
b) GARCH (1,1) Student-t
d) SV
25
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
26

• 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
27

• 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
28
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
29
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

30
Concluding remarks

• Sample construction problems
• Further research
• - allowing for switching regimes
• - allowing for leptokurtotic
  distributions in the SV
• - a better proxy for realized volatility

31
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33
Appendix GARCH mean equation
1. The AR(1) model with intercept
Dependent Variable Y Dependent Variable Y Dependent Variable Y Dependent Variable Y Dependent Variable Y
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Date 06/23/03 Time 0045 Date 06/23/03 Time 0045 Date 06/23/03 Time 0045 Date 06/23/03 Time 0045 Date 06/23/03 Time 0045
Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004
Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints 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 Mean dependent var -0.000487
Adjusted R-squared 0.075354 S.D. dependent var S.D. dependent var 0.015204
S.E. of regression 0.014620 Akaike info criterion Akaike info criterion -5.610880
Sum squared resid 0.213740 Schwarz criterion Schwarz criterion -5.601080
Log likelihood 2813.051 F-statistic F-statistic 82.57675
Durbin-Watson stat 2.002722 Prob(F-statistic) Prob(F-statistic) 0.000000
34
2.The AR(1) model without intercept
Dependent Variable Y Dependent Variable Y Dependent Variable Y Dependent Variable Y Dependent Variable Y
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Date 06/23/03 Time 0046 Date 06/23/03 Time 0046 Date 06/23/03 Time 0046 Date 06/23/03 Time 0046 Date 06/23/03 Time 0046
Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004 Sample(adjusted) 3 1004
Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints Included observations 1002 after adjusting endpoints 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 Mean dependent var -0.000487
Adjusted R-squared 0.075733 S.D. dependent var S.D. dependent var 0.015204
S.E. of regression 0.014617 Akaike info criterion Akaike info criterion -5.612286
Sum squared resid 0.213866 Schwarz criterion Schwarz criterion -5.607386
Log likelihood 2812.755 Durbin-Watson stat Durbin-Watson stat 2.003016
35
Appendix Residual Tests
Date 06/23/03 Time 0048 Date 06/23/03 Time 0048 Date 06/23/03 Time 0048 Date 06/23/03 Time 0048 Date 06/23/03 Time 0048 Date 06/23/03 Time 0048 Date 06/23/03 Time 0048
Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004
Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 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
36
Date 06/23/03 Time 0049 Date 06/23/03 Time 0049 Date 06/23/03 Time 0049 Date 06/23/03 Time 0049 Date 06/23/03 Time 0049 Date 06/23/03 Time 0049 Date 06/23/03 Time 0049
Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004
Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 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
37
ARCH-LM test
ARCH Test ARCH Test ARCH Test ARCH Test ARCH Test
F-statistic 114.8229 Probability Probability 0.000000
ObsR-squared 103.1921 Probability Probability 0.000000

Test Equation Test Equation Test Equation Test Equation Test Equation
Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Date 06/23/03 Time 0052 Date 06/23/03 Time 0052 Date 06/23/03 Time 0052 Date 06/23/03 Time 0052 Date 06/23/03 Time 0052
Sample(adjusted) 4 1004 Sample(adjusted) 4 1004 Sample(adjusted) 4 1004 Sample(adjusted) 4 1004 Sample(adjusted) 4 1004
Included observations 1001 after adjusting endpoints Included observations 1001 after adjusting endpoints Included observations 1001 after adjusting endpoints Included observations 1001 after adjusting endpoints Included observations 1001 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000145 1.83E-05 7.903650 0.0000
RESID2(-1) 0.321081 0.029964 10.71555 0.0000
R-squared 0.103089 Mean dependent var Mean dependent var 0.000213
Adjusted R-squared 0.102191 S.D. dependent var S.D. dependent var 0.000573
S.E. of regression 0.000543 Akaike info criterion Akaike info criterion -12.19544
Sum squared resid 0.000295 Schwarz criterion Schwarz criterion -12.18564
Log likelihood 6105.819 F-statistic F-statistic 114.8229
Durbin-Watson stat 2.064939 Prob(F-statistic) Prob(F-statistic) 0.000000
38
White Heteroskedasticity Test
White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test White Heteroskedasticity Test
F-statistic 63.32189 Probability Probability Probability 0.000000
ObsR-squared 112.7329 Probability Probability Probability 0.000000

Test Equation Test Equation Test Equation Test Equation Test Equation Test Equation
Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2 Dependent Variable RESID2
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Date 06/23/03 Time 0053 Date 06/23/03 Time 0053 Date 06/23/03 Time 0053 Date 06/23/03 Time 0053 Date 06/23/03 Time 0053 Date 06/23/03 Time 0053
Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004 Sample 3 1004
Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002 Included observations 1002
Variable Coefficient Std. Error t-Statistic t-Statistic Prob.
C 0.000144 1.82E-05 7.933013 7.933013 0.0000
Y(-1) -0.000222 0.001125 -0.197479 -0.197479 0.8435
Y(-1)2 0.299471 0.026700 11.21598 11.21598 0.0000
R-squared 0.112508 Mean dependent var Mean dependent var Mean dependent var 0.000213
Adjusted R-squared 0.110731 S.D. dependent var S.D. dependent var S.D. dependent var 0.000573
S.E. of regression 0.000541 Akaike info criterion Akaike info criterion Akaike info criterion -12.20501
Sum squared resid 0.000292 Schwarz criterion Schwarz criterion Schwarz criterion -12.19031
Log likelihood 6117.708 F-statistic F-statistic F-statistic 63.32189
Durbin-Watson stat 2.075790 Prob(F-statistic) Prob(F-statistic) Prob(F-statistic) 0.000000