Testing error distribution in various multivariate time series models

Abstract

Knowing the error distribution is essential in many multivariate time series applications. To alleviate the risk of error distribution misspecification, testing methodologies are needed to detect whether the chosen error distribution is correct. However, the majority of existing tests only deal with the multivariate normal distribution for some special multivariate time series models, and thus cannot be used for testing the often observed heavy-tailed and skewed error distributions in applications. In the first part of this thesis, a new consistent test is constructed for error distributions in general multivariate time series models, based on the kernelized Stein discrepancy (KSD). Tractable formulas of the KSD-based test statistic are derived for various multivariate distributions, and they are computationally easy regardless of the dimension of data. To account for the estimation uncertainty and unobserved initial values, asymptotics of the KSD-based test statistic are studied and a parametric bootstrap method is provided to calculate the critical values of the test. This new test is easy-to-implement for a large scope of multivariate error distributions, including multivariate normal, t, skewed-normal, and skewed-t distributions. Extensive simulation studies show that the KSD-based test has the power advantage over the existing tests. A real financial example illustrates the importance of this test by forecasting portfolio Value-at-Risk (VaR). The second part of this thesis extends the kernelized Stein discrepancy test to error distributions in copula-based multivariate time series models. Tractable formulas of the KSD-based test statistic are computed for Gaussian and Student's t copulas. A similar resampling scheme is proposed to compute the critical values of the test. Simulation studies and a real data application of exchange rate returns illustrate the performance of this test. The last part of this thesis further extends the KSD-based test to non-stationary multivariate time series models. A copula--exponentially weighted moving average (copula--EWMA) model is proposed, to account for the nonstationarity and volatility spillover effects in time series data. Explicit forms of the test statistics are derived, and bootstrap methods are introduced to compute the critical values of the test. Simulations and real applications demonstrate the usefulness of the KSD-based test in non-stationary scenarios.