A time-domain test for some types of non-linearity

The bispectrum and third-order moment can be viewed as equivalent tools for testing for the presence of nonlinearity in stationary time series. This is because the bispectrum is the Fourier transform of the third order moment. An advantage of the bispectrum is that its estimator comprises terms which are asymptotically independent at distinct bifrequencies under the null hypothesis of linearity. An advantage of the third order moment is that its values in any subset of joint lags can be used in the test, whereas when using the bispectrum the entire (or truncated) third order moment is required to construct the Fourier transform. In this paper we propose a test for nonlinearity based upon the estimated third order moment. We use the phase scrambling bootstrap method to give a non-parametric estimate of the variance of our test statistic under the null hypothesis. Using a simulation study we demonstrate that the test obtains its target significance level, with large power, when compared to an existing standard parametric test that uses the bispectrum. Further we show how the proposed test can be used to identify the source of non-linearity due to interactions at specific frequencies. We also investigate implications for heuristic diagnosis of non-stationarity.