# A test for second order stationarity of a time series based on the Discrete Fourier Transform (Technical Report)

## TL;DR

This paper introduces a new statistical test based on the Discrete Fourier Transform to determine if a time series is second order stationary, with proven theoretical properties and demonstrated effectiveness through simulations and real data examples.

## Contribution

It develops a Portmanteau type test for second order stationarity using DFT, including its asymptotic distribution under null and alternative hypotheses.

## Key findings

- Test statistic follows chi-square distribution under null hypothesis.
- The test has good power against local alternatives.
- Effective in real-world data examples.

## Abstract

We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical frequencies. It is well known that the discrete Fourier transform is asymptotically uncorrelated at the canonical frequencies if and if only the time series is second order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic is approximately a chi square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a type of noncentral chi-square, where the noncentrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power. Some real examples are also included to illustrate the test.