ACF estimation via difference schemes for a semiparametric model with $m$-dependent errors

Abstract

We discuss a class of difference-based estimators of the autocovariance function in a semiparametric regression model where the signal consists of the sum of an identifiable smooth function and another function with jumps (change points) and the errors are m-dependent. We establish that the influence of the smooth part of the signal over the bias of our estimators is negligible; this result does not depend on any distributional assumption. Under Gaussianity of the errors, we show that the mean squared error of the proposed estimators does not depend on either the unknown smooth function or on the values of the difference sequence coefficients; our finite sample studies suggest that the latter feature holds true regardless of the Gaussianity assumption. We also allow that both the number of change points and the magnitude of the largest jump grow with the sample size. In this case, we provide conditions on the interplay between the growth rate of these two quantities in order to ensure √ n consistency of our estimators.

Optimal difference-based variance estimators in time series: A general framework

“…Without the nuisance structure 2 below, this task was studied by, for example, Hall, Kay and Titterington (1990). Similar and extended results include Anderson (1971), Rice (1984) and Levine and Tecuapetla-Gómez (2019).…”

“…Such v is just an estimator of the marginal variance γ 0 but not v = k∈Z γ k . Recently, Tecuapetla-Gómez and Munk (2017) and Levine and Tecuapetla-Gómez (2019) extended it to estimation of γ k for M-dependent time series, that is,…”