A Global Wavelet Based Bootstrapped Test of Covariance Stationarity

TL;DR

This paper introduces a wavelet-based bootstrap test for covariance stationarity in complex time series, capable of detecting subtle deviations and variance breaks without relying on strict assumptions or covariance matrix inversion.

Contribution

It extends previous methods by using a general orthonormal basis and bootstrap techniques, allowing for nonlinear processes and non-iid innovations in testing stationarity.

Findings

Effective detection of variance breaks and mild deviations from stationarity.

Applicable to nonlinear and non-iid processes in macroeconomics and finance.

Avoids complex covariance matrix estimation and inversion.

Abstract

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 (2015), who exploit Walsh (1923) 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 $\mathcal{H}_{T}$ and $\mathcal{K}_{T}$. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.