A new bootstrap assisted test for checking second order stationarity

A new computationally driven method is proposed to check if a possibly nonlinear time series is second order stationary or not, which is important in time series modeling. The new test relies on blocks of blocks bootstrap covariance matrix estimates and Walsh transformations in order to capture the nonlinearity features of time series. The asymptotic normality of the Walsh coefficients and their asymptotic covariance matrix under the null hypothesis are derived for nonlinear processes. In addition, the asymptotic covariance matrix of an increasing dimension is shown to be consistently estimated by a blocks of blocks bootstrap procedure. In the framework of locally stationary nonlinear processes, it is shown that the proposed test is consistent under a sequence of local alternatives. A simulation study is conducted to examine the finite sample performance of the procedure. In many nonlinear time series settings, the proposed test works well while existing methods may have highly inflated type I error rates. The proposed test is applied to an analysis of a financial data set.