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

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

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”

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

GARCH models GARCH (p,q) TARCH (p,q) EGARCH (p,q)

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

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

Statistical properties of the returns Non-normal distribution Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability Observations 2130 Histogram of BET returns

Statistical properties of the returns Heteroscedasticity

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

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) DimensionBDS StatisticStd. Errorz-StatisticProb The results presented above show a rejection of the independence hypothesis for all embedding dimensions m

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

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

Model Estimates VariableCoeffT-statSignif 1. Intercept (b 1 ) AR(1) (b 2 ) Constant (a 0 ) ARCH (a 1 ) Asymmetric coeff (a 4 ) ** GARCH (a 3 ) Dummy(π) Q(20) st res28.953** Q(20) st sq res19.120** SIC Model 1 – TGARCH(1,1) *Denotes significance at the 1% level of significance **Denotes significance at the 5% level of significance - 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, < 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 DimensionBDS StatisticStd. Errorz-StatisticProbab / , BDS test

Model Estimates VariableCoeffT-statSignif 1. Intercept (b 1 ) AR(1) (b 2 ) AR(11) (b 3 ) AR(14) (b 4 ) AR(19) (b 5 ) Constant (a 0 ) ARCH (a 1 ) GARCH (a 2 ) Dummy (π) Q(20) st res19.044** Q(20) st sq res20.531** SIC 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 DimensionBDS StatisticStd. Errorz-StatisticProb BDS test

Model Estimates VariableCoeffT-statSignif 1. Intercept (b 1 ) AR(1) (b 2 ) AR(11) (b 3 ) AR(14) (b 4 ) AR(19) (b 5 ) Constant (a 0 ) ARCH (a 1 ) Asymmetric coeff (a 2 ) ** GARCH (a 3 ) Dummy coeff (π) Q(20) st res18.731**0.539 Q(20) st sq res13.912**0.835 SIC Model 3 – EGARCH(1,1) DimensionBDS StatisticStd. Errorz-StatisticProb BDS test - 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, < 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

Model estimates Model 3 – Markov Switching VariableCoeffT-statSignif 1. Mean State 1(a01) Variance State 1 (σ 1 ) Mean State 2 (a02) Variance State 2 (σ 2 ) Matrix of Markov transition probabilities SIC 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

Evidence for lower frequencies Monthly data (99 observations) Mean Median Maximum Minimum Std. Dev. SkewnessKurtosis Jarque-Bera Prob , Monthly closing prices for BET Autocorrelation at lag Signif Q(20)4.7020Signif LM(1) Signif Q(20) for squares12.434Signif Monthly returns for BET => There are no significant evidence of dynamics

Model estimation GARCH Models failed to converge (see Appendix 3) Markov Switching models VariableCoeffT-statSignif 1. Mean State 1(a01) Variance State 1 (σ 1 ) Mean State 2 (a02) Variance State 2 (σ 2 ) Matrix of Markov transition probabilities SIC978 -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

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

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Appendix 1 BDS test for TARCH (1,1)

Appendix 1 BDS test for GARCH (1,1)

Appendix 1 BDS test for EGARCH (1,1)

Appendix 2 Residuals histogram following GARCH(1,1)

Appendix 2 Residuals histogram following GARCH(1,1)

Appendix 2 Residuals histogram following EGARCH(1,1)

Appendix 3 GARCH(1,1) on monthly data