Asymptotic Theory of Test Statistic for Sphericity of High†Dimensional Time Series

We consider the testing problem for the sphericity hypothesis regarding the covariance matrix based on high†dimensional time series, under the assumption that the sample size n and the dimension p satisfy limn,p→∞p/n=c∈(0,∞). Recently, several studies on test statistics for sphericity of independent and identically distributed p†dimensional random variables have been carried out under the assumption that both n and p diverge to infinity. A test statistic for sphericity has been proved to be well behaved even when p>n. We investigate the test statistic under situations of high†dimensional time series. The asymptotic null distribution of the test statistic is shown to be standard normal distribution when the observations come from Gaussian stationary processes. In the simulation study, we illustrate the properties of the test statistic for several time series models. We apply the test to a problem of portfolio selection in our empirical study.