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Supervisor: Professor Moisa Altar

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

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Supervisor: Professor Moisa Altar

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  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;

  31. 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;

  32. 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

  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

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