A bootstrapped test of covariance stationarity based on orthonormal transformations

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 893-922), who exploit Walsh (Amer. J. Math. 45 (1923) 5-24) functions 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 in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions, and we allow for linear or nonlinear 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 basis generated sub-sample counter k (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter HT and KT. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.