Tests for Volatility Dynamics in Bivariate Stochastic Volatility Models

Abstract

Recently Chiba and Kobayashi (2013) have proposed the Lagrange multiplier (LM) test for the null hypothesis that volatilities of two asset return processes are driven by only one stochastic volatility (SV) process in a bivariate SV model. They apply their LM test to Asian stock market index returns, and find that the null hypothesis is not rejected for several pairs of those returns. They, however, derive the test statistic under an unconventional assumption that the conditional distribution of log squared return is normally distributed conditional on the SV, which seems inappropriate for a model of financial assets returns. This paper develops their analysis in two aspects. First, by a method of simulation, we examine the performance of their LM test under a conventional (bivariate) SV model in that each return (not log squared return) follows a normal distribution conditional on the SV. Second, because the null hypothesis examined in Chiba and Kobayashi (2013) is restrictive for many financial time series, we propose two LM tests for less restrictive but plausible null hypotheses. Our simulation study demonstrates that even for the conventional SV model, the actual size of their LM test is reasonably close to the nominal sizes and its power is high enough so that their LM test is useful for practical applications. Moreover, our empirical analysis with newly proposed LM tests finds that for many Asian stock market index returns, although their volatility processes themselves are not identical, they are likely to be driven by a common (or perfectly positively correlated) shock.