Modelling and Forecasting Stock Index Volatility a comparison

Modelling and Forecasting Stock Index Volatility –a comparison between GARCH models and the Stochastic Volatility model– Supervisor: Professor Moisa Altar

Table of Contents n Competing volatility models n Data description n n Model estimates and forecasting performances Concluding remarks

Why model and forecast volatility? Øinvestment Øsecurity valuation Ørisk management Øpolicy issues The Stylized Facts n The distribution of financial time series has heavier tails than the normal distribution n Highly correlated values for the squared returns n Changes in the returns tend to cluster

Competing Volatility Models n ARCH/GARCH class of models Engle (1982) Ø Bollerslev (1986) Ø Nelson (1991) Ø Glosten, Jaganathan, and Runkle (1993) Ø n Stochastic Volatility (Variance) model Ø Taylor (1986)

The GARCH model Parameter constraints: ü ensuring variance to be positive ü stationarity condition:

Error distribution 1. Normal Ø Ø Ø The density function: Implied kurtosis: k=3 The log-likelihood function:

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

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

The SV model Parameter constraints: ü stationarity condition: Linearized form:

Forecast Evaluation Measures n Root Mean Square Error (RMSE) n Mean Absolute Error (MAE) n Theil-U Statistics n LINEX loss function

Data Description Daily closing prices of BET-C index ü data series: BET-C stock index ü time length: April 17, 1998 - April 21, 2003 ü 1255 daily returns Pt – daily closing value of BET-C ü 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

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

BET-C return series 2. Homoscedasticity 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 *Mac. Kinnon 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 *Mac. Kinnon critical values for rejection of hypothesis of a unit root.

4. Serial independence Autocorrelation coefficients for returns (lags 1 to 36)

Autocorrelation coefficients for squared returns (lags 1 to 36)

Model estimates and forecasting performances Methodology: - two sets: 1004 observations for model estimation 252 observations for out-of-sample forecast evaluation Ø 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 v Normality test v Autocorrelation tests Lag number 1 5 10 15 Correlogram of residuals Correlogram of squared residuals Q-stat 0. 0085 3. 3598 5. 7904 8. 0496 Q-stat 103. 60 162. 76 165. 21 167. 21 Prob 0. 927 0. 645 0. 833 0. 922 Prob 0. 000 v 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

GARCH (1, 1) – Normal Distribution – QML parameter estimates Coefficient Std. Error t-value Probability 0. 302055 0. 045561 6. 630 0. 0000472947 0. 141153 3. 351 0. 0008 ARCH(Alpha 1) 0. 320832 0. 065118 4. 927 0. 0000 GARCH(Beta 1) 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(Alpha 1) 0. 350230 0. 067874 5. 160 0. 0000 GARCH(Beta 1) 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(Alpha 1) 0. 333678 0. 062854 5. 309 0. 0000 GARCH(Beta 1) 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 Ø 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 Ø Autocorrelation of the residuals Lag GARCH(1, 1) Nomal 1 5 10 15 Q-stat. 1. 131 3. 286 5. 654 8. 679 Ø Lag p-value 0. 287 0. 511 0. 774 0. 851 Ø 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 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-U 1 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= 20 GARCH 11 Normal 7, 70895 E-09 1, 92751 E-09 1, 92806 E-09 7, 71335 E-09 GARCH 11 T 7, 62777 E-09 1, 9072 E-09 1, 90773 E-09 7, 63198 E-09 GARCH 11 GED 7, 61114 E-09 1, 90305 E-09 1, 90359 E-09 7, 61545 E-09 SV 6, 95655 E-09 1, 73942 E-09 1, 73999 E-09 6, 96113 E-09 1 Benchmark model - Random Walk LINEX

Out-of-sample Forecast Evaluation n n 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

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

Evaluation Measures 1 -step ahead forecast evaluation n RMSE MAE THEIL-U 1 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= 20 GARCH 11 Normal 6, 30398 E-09 1, 57614 E-09 1, 57644 E-09 6, 30638 E-09 GARCH 11 T 6, 23593 E-09 1, 55923 E-09 1, 55971 E-09 6, 2398 E-09 GARCH 11 GED 6, 46868 E-09 1, 61743 E-09 1, 61795 E-09 6, 47286 E-09 SV 1, 2055 E-08 3, 01454 E-09 3, 01612 E-09 1, 20676 E-08

5 -step ahead forecast evaluation n RMSE MAE THEIL-U 1 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= 20 GARCH 11 Normal 1. 3297 E-08 3. 325 E-09 3. 3268 E-09 1. 33108 E-08 GARCH 11 T 1. 3257 E-08 3. 315 E-09 3. 3169 E-09 1. 32711 E-08 GARCH 11 GED 1. 3241 E-08 3. 311 E-09 3. 3126 E-09 1. 32539 E-08 SV 1. 3239 E-08 3. 310 E-09 3. 3125 E-09 1. 32534 E-08

10 -step ahead forecast evaluation n RMSE MAE THEIL-U 1 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= 20 GARCH 11 Normal 1, 33445 E-08 3, 33699 E-09 3, 33871 E-09 1, 33583 E-08 GARCH 11 T 1, 33467 E-08 3, 33753 E-09 3, 33925 E-09 1, 33604 E-08 GARCH 11 GED 1, 33499 E-08 3, 33834 E-09 3, 34007 E-09 1, 33637 E-08 SV 1, 33996 E-08 3, 35077 E-09 3, 35251 E-09 1, 34135 E-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 n In-sample analysis: a) residual tests: all models may be appropriate; b) evaluation measures: SV model is the best performer; n 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 n n Sample construction problems; Further research: - allowing for switching regimes; - allowing for leptokurtotic distributions in the SV - a better proxy for realized volatility;

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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. 0085 0. 927 . | | 2 -0. 011 0. 1228 0. 940 . | | 3 0. 041 1. 8102 0. 613 . | | 4 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 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 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. 321 103. 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. 83 E-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: 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. 82 E-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