The Performance of GARCH Models with Short-memory, Long-memory, or Jump Dynamics: Evidence from Global Financial Markets Yaw-Huei Wang National Central University Co-authored with Chih-Chiang Hsu National Central University

Motivation Volatility is a key input for both risk management and derivative pricing. ARCH-type models are the most successful and popular framework for describing volatility. In the past two decades, the model has been developed to be more realistic, but more complicated unfortunately.

Research Questions Would the more complicated model performs any better for a particular purpose?  Conditional distribution: skewed, fat-tailed  Memory: short or long If so, can such improved performance be globally valid for different markets?

Objectives Develop a nested volatility model based on the EGARCH framework to investigate the performance of (1)short-memory, (2)long-memory, and (3)jump models in terms of (1)model fitting, (2)volatility forecasting, and (3)VaR prediction for 8 relatively large stock markets.

The Model

Estimation: MLE  Maximize the log likelihood function

The Model Short-memory: EGARCH (Nelson, 1991)  λ t = d = 0. Long-memory: FIEGARCH (Bollerslev & Mikkelsen, 1996)  λ t = 0. Jump dynamics: EGARCH-jump (Maheu & McCurdy, 2004)  d = 0. EGARCH-skewed-t (Hansen, 1994)

Measures of Performance Model fitting:  Likelihood ratio test:  Akaike information criteria (AIC) Volatility forecasting:  Mean squared errors (MSEs)

Measures of Performance VaR prediction: Likelihood ratio test In practice, a preferred model should have a violation rate which is no greater than the threshold.

Data The Datastream stock market indices of US, Japan, the UK, Germany, France, Canada, Italy and Spain. From July 1990 to June Excluding holiday, there are, on average, about 3,785 observations. Preliminary tests for the absolute values of returns support the existence of long memory in volatility, particularly clear for US and Canada.

Empirical Results Model fitting:  The EGARCH-jump model has the smallest AIC and the highest LR statistic globally.  The EGARCH-skewed-t model, with two more parameters, can also provide substantial improvements.  The FIEGARCH model does not necessarily result in any significant improvement.

Empirical Results Volatility forecasting  The EGARCH-jump performs best for some countries, particularly good for German and Canadian markets.  FIEGARCH has fairly satisfactory improvement as well (but with higher variation), although the performance in model fitting is bad.  By contrast, the EGARCH-skewed-t model does not provide any improvement, although the performance in model fitting is good.

Empirical Results VaR prediction  Almost of all models pass the LR tests for all countries at the 5% significance level.  Both the EGARCH-jump model and the EGARCH- skewed-t model pass the LR tests for all countries and at all significance levels.  However, for the EGARCH-jump model, 87.5% of all violation rates are lower than the corresponding significance levels, while 62.5% for the EGARCH-skewed-t model.

Conclusions The EGARCH-jump model outperforms all other models in all aspects, with the single exception of volatility forecasting for some indices. However, the computation load is substantially increased.

Conclusions If less-expensive volatility are preferred, alternative models include  the use of the EGARCH-skew-t model for model fitting and VaR prediction.  the FIEGARCH model for volatility forecasting, since these models also demonstrate fairly good performance for these particular purposes. Interestingly, the FIEGARCH model performs relatively satisfactory for the US market only.