Specification Testing for Multivariate Time Series Volatility Models

Volatility models have been playing an important role in economics and finance. Using a multivariate generalized spectral approach, we propose a new class of generally applicable omnibus tests for univariate and multivariate volatility models. Both GARCH models and stochastic volatility models are covered. Our tests have a convenient asymptotic null N(0,1) distribution, and can detect a wide range of misspecifications for volatility dynamics. Distinct from the existing tests for volatility models, our tests are robust to higher order time-varying moments of unknown form (e.g., time-varying skewness and kurtosis). Our tests check a large number of lags and are therefore expected to be powerful against neglected volatility dynamics that occurs at higher order lags or display long memory properties. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally discounts higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by the recent past events than by the remote past events. No specific estimation method is required, and parameter estimation uncertainty has no impact on the limit distribution of the test statistics. Moreover, there is no need to formulate an alternative volatility model, and only estimated standardized residuals are needed to implement our tests. We do not have to calculate tedious score functions or derivatives of volatility models with respect to estimated parameters, which are model-specific and are required in some existing popular tests for volatility models. We examine the finite sample performance of the proposed tests. An empirical application to some popular GARCH models for stock returns illustrates our approach.