On nonparametric and semiparametric testing for multivariate linear time series

TL;DR

This paper introduces new nonparametric and semiparametric tests for multivariate stationary linear time series, based on spectral density estimators, with applications to various hypotheses including independence, equality of autocovariances, and nonlinear constraints.

Contribution

It unifies hypothesis testing for multivariate linear time series and proposes new test statistics with known asymptotic distributions, enhancing practical applicability and flexibility.

Findings

Test statistics have normal limiting distributions under null hypotheses.

Tests are consistent and diverge under false null hypotheses as sample size increases.

Applicable to a wide range of hypotheses, including nonlinear constraints.

Abstract

We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the independence between the component series, the equality of the autocovariance functions or the autocorrelation functions of the component series, the separability of the covariance matrix function and the time reversibility. Furthermore, a null hypothesis with a nonlinear constraint like the conditional independence between the two series can be tested in the same way.