The stochastic volatility in mean model: empirical evidence from international stock markets

Abstract

In this paper we present an exact maximum likelihood treatment for the estimation of a Stochastic Volatility in Mean (SVM) model based on Monte Carlo simulation methods. The SVM model incorporates the unobserved volatility as an explanatory variable in the mean equation. The same extension is developed elsewhere for Autoregressive Conditional Heteroskedastic (ARCH) models, known as the ARCH in Mean (ARCH-M) model. The estimation of ARCH models is relatively easy compared with that of the Stochastic Volatility ( S V ) m o d e l . However, e cient Monte Carlo simulation methods for SV models have been developed to overcome some of these problems. The details of modi cations required for estimating the volatility-in-mean e ect are presented in this paper together with a Monte Carlo study to investigate the small-sample properties of the SVM estimators. Taking these developments of estimation methods into account, we regard SV and SVM models as practical alternatives to their ARCH counterparts and therefore it is of interest to study and compare the two classes of volatility m o d e l s . We present an empirical study about the intertemporal relationship between stock index returns and their volatility for the United Kingdom, United States and Japan. This phenomenon has been discussed in the nancial literature but has proved hard to nd empirically we nd evidence of a negative but weak relationship.

The inflation and inflation uncertainty relationship for Turkey: a dynamic framework

“…Note that parallel to Koopman and Uspensky (2002), Cov(η t , ε t ) was taken as zero. Thus, there is neither the contemporaneous effect of inflation shock to inflation variability nor the contemporaneous effect of inflation volatility shock to inflation.…”

“…Moreover, since both mean and variance equations have a latent variable h t , the construction likelihood for the SVM model is complicated. Koopman and Uspensky (2002) adopt the Monte Carlo likelihood approach developed by Shephard and Pitt (1997) and Durbin and Koopman (1997). This simulation method of computing the likelihood function can be derived as…”

“…In order to capture the dynamics of inflation, we adopt the Stochastic Volatility in Mean model (SVM) proposed by Koopman and Uspensky (2002). The mean equation for the SVM model is…”

Estimation methods for stochastic volatility models: a survey

“…The incorporation of the unobserved volatility as an explanatory variable in the mean equation is analyzed by Koopman and Uspensky (2002) who t the SV-M model to three di erent nancial series. In this empirical application, estimates of the volatility are obtained by means of the particle ltering technique of Pitt and Shephard (1999).…”

The Stochastic Volatility Model, Regime Switching and Value-at-Risk (VaR) in International Equity Markets

“…The volatility of daily stock index returns has been estimated with stochastic volatility models but usually results have relied on extensive pre-modeling of these series, thus avoiding the problem of simultaneous estimation of the mean and variance. Koopman and Hol Uspensky [14] proposed the Stochastic Volatility in Mean model (SVM) that incorporates volatility as one of the determinants of the mean. This modification makes the model suitable for empirical applications between the mean and variance of returns.…”