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

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

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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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