Modelling and Forecasting Stock Index Volatility –a comparison between GARCH models and the Stochastic Volatility mode

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: k=3 • 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

12. TestedHypotheses 1. Normality Histogram of the BET-C returnsBET-C return quantile plotted against the Normal quantile

13. BET-C return series 2.Homoscedasticity BET-C squared return series

14. 3. Stationarity Unit root tests for BET-C return series

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

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

18. Residual tests • Normality test • Autocorrelation tests • ARCH-LM test and White Heteroscedasticity Test

19. GARCH (1,1) – Normal Distribution – QML parameter estimates 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 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 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

21. In-sample model evaluationa) Residual tests • Autocorrelation of the residuals • Autocorrelation of the squared residuals • Kurtosis explanation

22. b) In-sample forecast evaluation 1Benchmark model - Random Walk

23. 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

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 1Benchmark model - Random Walk

26. 5-step ahead forecast evaluation 1Benchmark model - Random Walk

27. 10-step ahead forecast evaluation 1Benchmark model - Random Walk

28. Comparison between the statistical features of the two sample periods

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;

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33. Appendix – GARCH mean equation 1. The AR(1) model with intercept

34. 2.The AR(1) model without intercept

35. Appendix – Residual Tests Correlogram of Residuals

36. Correlogram of Squared Residuals

37. ARCH-LM test

38. White Heteroskedasticity Test