A Global Wavelet Based Bootstrapped Test of Covariance Stationarity

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in the new setting of Jin, Wang and Wang (2015) who exploit Walsh (1923) functions (global square waves) in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis, and exploit linearity in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix. Conversely, we allow for linear or linear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag h and Walsh function generated sub-sample counter k (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter H and K, and in the supplemental material we present a data driven method for selecting H and K. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.