Multiscale inference and long-run variance estimation in non-parametric regression with time series errors

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2020 · Volume 82 · Issue 1

Abstract

We develop new multiscale methods to test qualitative hypotheses about the functionmin the non-parametric regression modelYt,T=m(t/T)+ɛtwith time series errorsɛt. In time series applications,mrepresents a non-parametric time trend. Practitioners are often interested in whether the trendmhas certain shape properties. For example, they would like to know whethermis constant or whether it is increasing or decreasing in certain time intervals. Our multiscale methods enable us to test for such shape properties of the trendm. To perform the methods, we require an estimator of the long-run error variance. We propose a new difference-based estimator ofσ2for the case that {ɛt} belongs to the class of auto-regressive AR(∞) processes. In the technical part of the paper, we derive asymptotic theory for the proposed multiscale test and the estimator of the long-run error variance. The theory is complemented by a simulation study and an empirical application to climate data.

Keywords

Anticoncentration bounds, Long-run variance, Multiscale statistics, Non-parametric regression, Shape constraints, Strong approximations, Time series errors

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