Autocovariance function estimation via difference schemes for a semiparametric change point model with m $$ m $$ -dependent errors

Abstract

We discuss a broad class of difference-based estimators of the autocovariance function in a semiparametric regression model where the signal consists of the sum of a smooth function and another stepwise function whose number of jumps and locations are unknown (change points) while the errors are stationary and $m$-dependent. We establish that the influence of the smooth part of the signal over the bias of our estimators is negligible; this is a general result as it does not depend on the distribution of the errors. We show that the influence of the unknown smooth function is negligible also in the mean squared error (MSE) of our estimators. Although we assumed Gaussian errors to derive the latter result, our finite sample studies suggest that the class of proposed estimators still show small MSE when the errors are not Gaussian. Our simulation study also demonstrates that, when the error process is mis-specified as an AR$\left(1\right)$ instead of an $m$-dependent process, our proposed method can estimate autocovariances about as well as some methods specifically designed for the AR(1) case, and sometimes even better than them. We also allow both the number of change points and the magnitude of the largest jump grow with the sample size $n$. In this case, we provide conditions on the interplay between the growth rate of these two quantities as well as the vanishing rate of the modulus of continuity (of the signal's smooth part) that ensure $\sqrt{n}$ consistency of our autocovariance estimators. As an application, we use our approach to provide a better understanding of the possible autocovariance structure of a time series of global averaged annual temperature anomalies. Finally, the R package dbacf complements this article.