Estimation of the Long‐Run Variance of Nonlinear Time Series With an Application to Change Point Analysis

ABSTRACT For a broad class of nonlinear time series known as Bernoulli shifts, we establish the asymptotic normality of the smoothed periodogram estimator of the long‐run variance. This estimator uses only a narrow band of Fourier frequencies around the origin and so has been extensively used in local Whittle estimation. Existing asymptotic normality results apply only to linear time series, so our work substantially extends the scope of the applicability of the smoothed periodogram estimator. As an illustration, we apply it to a test of changes in mean against long‐range dependence. A simulation study is also conducted to illustrate the performance of the test for nonlinear time series.