Testing for Trend Stationarity Using Spectral Density Estimators

Testing for Trend Stationarity Using Spectral Density Estimators

We propose a new test statistic for trend stationarity against di¤erence stationarity using spectral density estimators. The spectral density of the …rst di¤erenced process equals to zero at the zero frequency under the null of trend stationarity, whereas di¤erence stationarity yields positive spectrum near zero frequency. With this one-sided nature of the spectrum, we construct valid testing procedures based on kernel-based spectral density estimators. Note that the spectral density estimator becomes degenerate under the null, where one do not simply apply standard results in the literature of heteroskedasticity and autocorrelation consistent (HAC) estimation. We provide new results on asymptotic distribution of the spectral density estimator under degeneracy. It is found that the convergence rates ensuring nondegenerating asymptotic variance of the estimator are much faster than the rate required for conventional HAC estimators. Consistency of the proposed test is also discussed. Simulation studies show that our spectrum-based test is competitive in terms of power in comparison with well-known KPSS test. Applications to some US macroeconomic series are presented.