Simple Tests for Models of Dependence between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rates

Evidence that asset returns are more highly correlated during volatile markets and during market downturns (see Longin and Solnik, 2001, and Ang and Chen, 2002) has lead some researchers to propose alternative models of dependence. In this paper we develop two simple goodness-of-fit tests for such models. We use these tests to determine whether the multivariate Normal or the Student’s t copula models are compatible with U.S. equity return and exchange rate data. Both tests are robust to specifications of marginal distributions, and are based on the multivariate probability integral transform and kernel density estimation. The first test is consistent but requires the estimation of a multivariate density function and is recommended for testing the dependence structure between a small number of assets. The second test may not be consistent against all alternatives but it requires kernel estimation of only a univariate density function, and hence is useful for testing the dependence structure between a large number of assets. We justify our tests for both observable multivariate strictly stationary time series and for standardized innovations of GARCH models. A simulation study demonstrates the efficacy of both tests. When applied to equity return data and exchange rate return data, we find strong evidence against the normal copula, but little evidence against the more flexible Student’s t copula.