Three essays on multivariate volatility modelling and estimation

Abstract

Many financial time series, such as asset returns or exchange rates, exhibit common regularities like time varying volatilities or comovements. These similarities are known in the literature as stylized facts. A desirable time series model is the one that explains one or more of these regularities. Some of these stylized facts are the following: ¿ Volatility clustering It is observed that periods of high (low) volatility are followed by periods of high (low) volatility. This kind of persistent behavior suggests that there might be an autoregressive structure governing the dynamics of the volatilities. The ARCH models introduced by Engle (1982) and SV models introduced by Taylor (1986) and their extensions referred to throughout this thesis are developed to mimic this volatility clustering. ¿ Thick tails It is documented in Mandelbrot (1963) and Fama (1963, 1965) among others that asset returns tend to present a heavy tailed or leptokurtic distribution. To match this stylized fact different distributions are used in the literature such as Student-t distribution (see for example Fiorentini et al. 2003, Sandmann and Koopman (1998)). The first chapter of this thesis works with Student-t distribution as well. ¿ Leverage effects Leverage effect refers to the negative correlation between the returns and volatilities: i.e. a negative return is expected to increase volatility more than a positive return. The intuition is that a decrease in the stock prices implies higher leverage of the firms, which increases the risks and uncertainty, hence causes high volatility. Examples can be found in Nelson (1991) or Jungbacker and Koopman (2005). In the second chapter of this thesis, two multivariate models are proposed to capture leverage effects. ¿ Comovements The volatility spillovers and correlations between returns have been increasingly of interest in the literature. Among many others Jeantheau (1998), Longin and Solnik (1995), Bae and Karolyi (1994) are examples analyzing theoretically or empirically the volatility spillovers. Bollerslev (1990), Engle (2002), Tse (2000), Pelletier (2006) are some of the papers studying the correlations between returns of stock markets. The first chapter of this thesis refers to the model of Jeantheau (1998) and the third chapter proposes a model to capture volatility spillovers. In all chapters of this thesis constant and/or time varying correlation models are considered. To explain the time varying volatilities, Engle (1982) and Bollerslev (1996) proposed generalized autoregressive conditional heteroskedasticity (GARCH) models. In GARCH set up, the volatilities follow a deterministic equation of the squared previous day returns and volatilities. Therefore the dynamics of the volatilities in this model is observation driven. Later, the GARCH models have been extended to multivariate settings (MGARCH) to capture the volatility spillovers and correlations between series, see for example, Bauwens et al. (2006) and Silvennoinen and Teräsvirta (2009) for a survey. Among others, for example EGARCH proposed by Nelson (1991) is developed to explain the leverage effects, ECCC-GARCH of Jeantheau (1998) is developed for capturing the volatility spillovers, Bollerslev (1990) proposes the constant conditional correlation GARCH (CCC-GARCH) model capturing the correlation between two series and Engle (2002) proposes the DCC-GARCH model to allow these correlations between returns to vary over time. Alternatively, the stochastic volatility (SV) literature started by Taylor (1986, 1994) and Hull and White (1987) suggests to model the time varying volatility as an unobserved component and lets its logarithm follow an autoregressive process. In this set up, the volatilities are parameter driven. Starting with the constant correlations multivariate stochastic volatility (CC-MSV) of Harvey et al. (1994), the multivariate extensions of the SV method have been developed. Among others, Asai and McAleer (2006) proposes the MSV with leverage model to explain the leverage effects in a time series data, Harvey et al. (1994) proposes the CC-MSV model to capture the correlation between the returns of a time series data while Jungbacker and Koopman (2006) proposes the time varying correlations MSV model to allow these correlations between returns to change over time. The SV approach is attractive in the sense that it is closer to the models used in the financial theory to describe the behavior of prices; see Shephard and Andersen (2008). Moreover it has been shown that the SV models describe the behavior of volatilities more accurately compared to GARCH models; see Danielsson (1994), Kim et al. (1998) and Carnero et al. (2004). Although statistically more attractive than the GARCH models, SV models have the disadvantage in terms of estimation because the exact likelihood functions of these models are difficult to evaluate. The main objective of this thesis is to compare the generalized autoregressive conditional heteroscedasticity (GARCH) models and stochastic volatility (SV) models in explaining one or more of the stylized facts in time series literature and analyze the small sample performances of several estimation methods in estimating the parameters of these models. In particular, one of the objectives of this thesis is to analyze the small simple performance of the multiple steps estimators of the parameters of multivariate GARCH models in the case of Gaussian and Student-t errors. When we would like to fit a multivariate GARCH model to data with high number of series or observations, estimating mean, variance and correlation parameters in separate steps makes the estimation process much easier than estimating these parameters all at once because in the latter case the convergence might be too slow. On the other hand, we lose from efficiency when we estimate the parameters in multiple steps. Therefore one question we seek to answer is: can we estimate the parameters of the multivariate GARCH models in multiple steps without losing much from efficiency? Another objective of this thesis is to compare the performance of the quasi-maximum likelihood (QML) and Monte Carlo likelihood (MCL) methods in estimating several multivariate stochastic volatility models. The QML method is relatively easier to implement and is much more flexible to admit samples with high number of series or observations. However it is based on approximations and therefore is inefficient. MCL is asymptotically efficient but is harder to implement and requires much more time to converge. Moreover it requires derivatives with respect to high dimensional state vectors, which makes the use of MCL method very hard in practice for with samples of large number of data points. Therefore in this thesis we also seek an answer for the question if there are some multivariate stochastic volatility models and parameter values for which the QML method performs closer to the MCL method. On the other hand, the leverage effects, as defined before, refers to the correlation between the returns and future volatilities of one series. In this work, we develop a new MSV model with leverage which allows the returns of one series to be correlated with the future volatilities of the other series. The third and last objective of this thesis is to develop a new multivariate GARCH model to capture volatility spillovers. In the literature, popular models like the BEKK and ECCC-GARCH require estimation of high number of parameters. For this reason, the estimation of these models is difficult for the samples with high number of series. With this model, our objective is to produce a model that could make use of the correlation dynamics to explain partially the effect of volatility spillovers and therefore requiring estimation of less number of parameters, so that it is flexible to admit samples with high number of series In all the chapters of the thesis, the plan of study consisted of explaining theoretical and econometric backgrounds of the models and estimation methods and performing simulation experiments to analyze the small sample performance of the models or methods of question. The results of the simulation experiments are reported in tables via means, standard errors and root mean squared errors. Moreover, we include some figures to plotting the kernel density estimates for the differences between the estimated and the true values of the volatilities and correlations. Finally in each chapter, an empirical estimation is included for illustration purposes using the data from stock markets. All the programming and implementation of the models and estimation methods are done in MATLAB. To fulfill the objectives, this thesis consists of three independent chapters. The hypothesis of the first chapter is that when the errors are Gaussian, it is a reasonable alternative to estimate the mean, variance and correlation parameters of the conditional correlation GARCH models in multiple steps and we wouldn¿t be losing much from efficiency in small samples in comparison to the one step estimation. When the errors follow a Student-t distribution, we expect that this hypothesis may not hold. The hypothesis of the second chapter is that there are models and parameter values for which the quasi-maximum likelihood (QML) method performs close to the Monte Carlo likelihood (MCL) method. The other hypothesis is that the multivariate stochastic volatility models with leverage effects proposed in this chapter can be estimated by the MCL method and can explain the leverage behavior in the data. The hypothesis of the third chapter is that the multivariate GARCH model we develop in this chapter can capture the volatility spillovers between series with less number of parameters and therefore is preferable when samples with high number of series are considered. This thesis is a collection of three independent articles. These articles will be sent to internationally acknowledged journals in the areas of Statistics and Econometrics.