The Stochastic Volatility in Mean Model

In this paper we present an exact maximum likelihood treatment forthe estimation of a Stochastic Volatility in Mean(SVM) model based on Monte Carlo simulation methods. The SVM modelincorporates the unobserved volatility as anexplanatory variable in the mean equation. The same extension isdeveloped elsewhere for Autoregressive ConditionalHeteroskedastic (ARCH) models, known as the ARCH in Mean (ARCH-M)model. The estimation of ARCH models isrelatively easy compared with that of the Stochastic Volatility (SV)model. However, efficient Monte Carlo simulationmethods for SV models have been developed to overcome some of theseproblems. The details of modificationsrequired for estimating the volatility-in-mean effect are presentedin this paper together with a Monte Carlo study toinvestigate the small-sample properties of the SVM estimators. Takingthese developments of estimation methods intoaccount, we regard SV and SVM models as practical alternatives totheir ARCH counterparts and therefore it is ofinterest to study and compare the two classes of volatility models.We present an empirical study about theintertemporal relationship between stock index returns and theirvolatility for the United Kingdom, United States andJapan. This phenomenon has been discussed in the financial literaturebut has proved hard to find empirically; we findevidence of a negative but weak relationship.