Modelling Volatility using Markov-Switching Garch Models

Abstract

Modelling of financial time-series has for decades been subject to extensive research and gradually evolved with numerous developments and applications. A common subject of interest is the volatility of such time series. As investments are inherently risky, modelling and forecasting volatility accurately is an important but non-trivial task. Engle (1982) introduced the ARCH model, which Bollerslev (1986) later generalized, with the Generalized Autoregressive Conditional Heteroscedasticity, GARCH model, to account for the timevarying and clustering properties of volatility. Although widely used and shown to generally provide good volatility forecasts (Bollerslev, 1998), Krämer et al. (2012) suggest modelling volatility with regime-dependent parameters instead, due to the persistence issue of the single-regime GARCH model. In this thesis, we model log returns of the S&P 500 with Markov switching GARCH models, with normal and t-distributed conditional innovations, to compare the relative performance to their singleregime counterparts and examine if regime-switching provides better in-sample estimation results and out-of-sample risk forecast. We find that the Markov switching models can identify the structural changes in the variance process, taking place in the estimation period, and shift accordingly. They generate higher likelihood values from the estimation, although the tGARCH exhibits the best BIC score, indicating possible overfitting of data for the Markov switching models. We observe that the regime switching models, can reduce persistence when a period of high volatility is followed by a period of low volatility. All models capture autocorrelation in the squared standardized residuals to a satisfying degree. We find strong evidence, of the t-distributed models, being able to better model the tail-distribution of the returns. For out-of-sample performance, several tests have been conducted, to examine the model’s ability to forecast Value at Risk and Expected Shortfall. In this regard, our results show, that the regimeswitching models are unable to outperform the single-regime models. Instead, our finding points towards the importance of the chosen distribution assumptions, as the tGARCH exhibits the best performance, and the normal distributed models are unable to forecast Expected Shortfall, as they fail to capture the leptokurtic property of financial data.