Nonlinear dynamics: evidence for Bucharest Stock Exchange

1. Nonlinear dynamics: evidence for Bucharest Stock Exchange Dissertation paper: Anca Svoronos(Merdescu)

2. Goals • To analyse a good volatility model by its ability to capture “stylized facts” • To analyse changes in models behavior with respect to temporal aggregation • To perform an empirical evidence for Bucharest Stock Exchange using its reference index BET

3. Introduction • The finding of nonlinear dynamics in financial time series dates back to the works of Mandelbrot and Fama in the 1960’s: - Mandelbrot first noted in 1963 that “large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes” - Fama developed the efficient-market hypothesis (EMH) – which asserts that financial markets are “informationally efficient”

4. Volatility Models • GARCH models Engle (1982) Bollerslev (1986) Nelson (1991) Glosten, Jagannathan and Runkle (1993) • Markov regime switching model Hamilton (1989)

5. GARCH models • GARCH (p,q) • TARCH (p,q) • EGARCH (p,q)

6. Markov switching model The model assumes the existence of an unobserved variable denoted: State where , is i.i.d N(0,1). The conditional mean and variance are defined: The transition (=conditional) probabilities are : The maximum likelihood will estimate the following vector containing six parameters:

7. Data Description • Data series: BET stock index • Time length: Jan 3rd, 2001 – March 4th, 2009 • 2131 daily returns:

8. Statistical properties of the returns • Non-normal distribution Histogram of BET returns

9. Statistical properties of the returns • Heteroscedasticity

10. Statistical properties of the returns • Autocorrelation • High serial dependence in returns • The Ljung-Box statistic for 20 lags is 85,75 (0.000) • The Ljung-Box statistic for 20 lags is 1442,6 (0.000) • LM (1): 260,61 => BET index returns exhibit ARCH effects

11. Statistical properties of the returns • BDS independence test (Brocht, Dechert, Scheinkman) • of the null hypotheses that time series is independently and identically distributed, is a general test for • identifying nonlinear dependence • (m=5, ε=0,7) • The results presented above show a rejection of the independence hypothesis for all embedding dimensions m

12. Statistical properties of the returns • Stationarity: Unit root tests for BET return series

13. Models specification (daily data) Model 1:TARCH (1, 1) Model 2: GARCH (1,1) Model 3: EGARCH (1,1) Model 4: Markov Switching (MS)

14. BDS test Model Estimates Model 1 – TGARCH(1,1) - Null hypothesis of BDS is not rejected at any significance level - The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level - Volatility persistence given by is 0,874179 < 1, implying a half life volatility of about 8 days - > 0 therefore we could stress that a leverage effect exists but testing the null hypothesis of = 0 at 1% level of significance we find that the shock is symmetric => a symmetric model specification should be tested *Denotes significance at the 1% level of significance **Denotes significance at the 5% level of significance

15. BDS test Model Estimates Model 2 – GARCH(1,1) • Null hypothesis of BDS is accepted at any significance level for all 5 dimensions; • The standardized squared residuals are serially uncorrelated at both significance level of 5% and 1% • Volatility persistence is 0,8772 < 1, implying a half life volatility of about 8 days, similar to the one implied by Model 1 *Denotes significance at the 1% level of significance **Denotes significance at the 5% level of significance

16. BDS test Model Estimates Model 3 – EGARCH(1,1) - Null hypothesis of BDS is being rejected by dimension m=5 and m=4 if using a significance level of 5%(1,64) and by m=5 for 1%(2,33); - The standardized squared residuals are serially uncorrelated both at 5% and 1% significance level - Volatility persistence given by is 0,857612 < 1, implying a half life volatility of about 8 days - < 0 therefore we can stress a leverage effect exists although testing the null hypothesis of = 0 at 1% level of significance we find that the shock is still symmetric *Denotes significance at the 1% level of significance **Denotes significance at the 5% level of significance

17. Model estimates • Both probabilities are quite small which means neither regime is too persistent – there is no evidence for “long swings” hypothesis • We find slight asymmetry in the persistence of the regimes – upward moves are short and sharp (a01 is positive and p11 is small) and downwards moves could be gradual and drawn out (a02 negative and p22 larger) • The ML estimates associate state 1 with a 0,16% daily increase while in state 2 the stock index falls by -0,2% with considerably more variability in state 2 than in state 1 • SIC value is significantly higher than the values estimated with GARCH models Model 3 – Markov Switching

18. Evidence for lower frequenciesMonthly data (99 observations) Monthly returns for BET Monthly closing prices for BET => There are no significant evidence of dynamics

19. Model estimation • GARCH Models failed to converge (see Appendix 3) • Markov Switching models • Two states are again high mean/lower volatility and low mean/higher volatility • p22 is larger than p11 which means regime 2 should be slightly more persistent – again there is no evidence for “long swings” hypothesis • again we find asymmetry in the persistence of the regimes • The ML estimates associate state 1 with an approx 3% monthly increase while in state 2 the stock index falls by -3,5% with considerably more variability in state 2 than in state 1 • In general, the characteristics of the regimes are still present at a monthly frequency in contrast with GARCH

20. Concluding remarks • If judging from the behavior of residuals, out of the GARCH models, GARCH (1,1) is the model of choice. • Compared with Markov Switching by SIC value we find GARCH(1,1) superior • Considering temporal aggregation, we find that GARCH models fail to converge while Markov Switching model still shows power • Further research: -forecast ability of both models

21. Bibliography • Akgiray, V (1989) - Conditional Heteroscedasticity, Journal of Business 62: 55 - 80 • 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; • Baillie, T. R. And DeGennaro, R.P. (1990) – Stock Returns and Volatility, Journal of Financial and Quantitative Analysis, vol.25, no.2, June 1990; • Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of Econometrics, Volume 4, Chapter 49, North Holland; • Brooks, C (2002) – Introductory econometrics for finance – Cambridge University Press 2002; • Byström, H. (2001) - Managing Extreme Risks in Tranquil and Volatile Markets Using Conditional Extreme Value Theory, Department of Economics, Lund University; • DeFusco, A. R., McLeavey D. W., Pinto E. J., Runkle E. D. – Quantitative Methods for Investment Analysis, United Book Press, Inc., Baltimore, MD, August 2001; • 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; • Engle, C and Hamilton, J (1990) – Long Swings in the Dollar:Are they in the Data and Do Markets know it?, American Economic Review 80:689-712; • 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; • Kanzler L.(1999) – Very fast and correctly sized estimation of the BDS statistic, Department of Economics, University of Oxford;

22. Bibliography • Kaufmann, S and Scheicher, M (1996) – Markov Regime Switching in Economic Variables:Part I. Modelling, Estimating and Testing.- Part II. A selective survey, Institute for Advanced Studies, Vienna, Economic Series, no.38, Nov. 1996 • Kim, D and Kon, S (1994) – Alternative Models for the Conditional Heteroskedasticity of Stock Returns, Journal of Business 67: 563-98; • Nelson, Daniel B. (1991) – Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59, 347-370; • Pagan, A. R. and Schwert, G. W. (1990) – Alternative models for conditional stock volatility, Journal of Econometrics 45, 1990, pag. 267-290, North-Holland; • 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; • 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; • Rockinger, M. (1994) – Switching Regressions of Unexpected Macroeconomic Events: Explaining the French Stock Index; • Scheicher, M (1994) – Nonlinear Dynamics: Evidence for a small Stock Exchange, Department of Economics, BWZ University of Viena, Working Paper No. 9607 • Sola, M. and Timmerman, A. (1994) – Fitting the moments: A Comparison of ARCH and Regime Switching Models for Daily Stock Returns • 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; • Van Norden, S and Schaller, H (1993) – Regime Switching in Stock Market Returns – Working paper, November 18th, 1993;

23. Appendix 1 BDS test for TARCH (1,1)

24. Appendix 1 BDS test for GARCH (1,1)

25. Appendix 1 BDS test for EGARCH (1,1)

26. Appendix 2 Residuals histogram following GARCH(1,1)

27. Appendix 2 Residuals histogram following GARCH(1,1)

28. Appendix 2 Residuals histogram following EGARCH(1,1)

29. Appendix 3 GARCH(1,1) on monthly data