Long-run variance estimation for linear processes under possible degeneracy

We analyze the asymptotic behavior of the long-run variance estimator for linear processes under degeneracy, where the spectral density function near the origin equals to zero. Given degeneracy which typically arises from over-differencing, standard results in the literature of heteroskedasticity and autocorrelation consistent (HAC) estimation are invalid. We provide asymptotic distribution of the long-run variance estimator from long term trends in linear processes. Further, we propose a test statistic to testing degeneracy, which achieves asymptotic normality. Our test is directly applied to testing for trend stationarity. Under the null of trend stationarity, the spectrum near the origin for the differenced process becomes zero. On the other hand, under the alternative of difference stationarity, the spectrum becomes strictly positive at the zero frequency. It is found that, depending on the signal-to-ratio, our test has significant power advantages over the KPSS test. Thus, the proposed test becomes an useful complement to the KPSS test.