Multiscale Inference and Long-Run Variance Estimation in Non-Parametric Regression with Time Series Errors

Abstract

Summary We develop new multiscale methods to test qualitative hypotheses about the function m in the non‐parametric regression model Yt,T=m(t/T)+ɛt with time series errors ɛt. In time series applications, m represents a non‐parametric time trend. Practitioners are often interested in whether the trend m has certain shape properties. For example, they would like to know whether m is 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 trend m. To perform the methods, we require an estimator of the long‐run error variance σ2=normalΣl=−∞∞covfalse(ε0,εlfalse). We propose a new difference‐based estimator of σ2 for 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.

Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

“…Lipschitz continuous, as are sinusoids. Such an assumption is commonly made in the literature, see for example Vogt (2012) and Khismatullina and Vogt (2020). Furthermore, in Section 3.6…”

“…Other methods, such as Dahlhaus and Neumann (2001) which focuses on the semi-parametric setting, consider results where the mean function is time-varying and/or estimated. Khismatullina and Vogt (2020) test for increases and decreases in trend in the presence of stationary time series errors, while Dette and Wu (2020) consider the problem of prediction in locally stationary time series.…”

Modelling time-varying first and second-order structure of time series via wavelets and differencing

“…In particular, polynomials (restricted to the interval [0, 1]) are Lipschitz continuous, as are sinusoids. Such an assumption is commonly made in the literature, see for example [31] and [13]. Furthermore, in Section 3.6 we will discuss the case when the trend is not Lipschitz.…”

“…Other methods, such as [6] which focuses on the semi-parametric setting, consider results where the mean function is time-varying and/or estimated. [13] test for increases and decreases in trend in the presence of stationary time series errors, while [9] consider the problem of prediction in locally stationary time series.…”

Detecting changes in the trend function of heteroscedastic time series

“…The observations are standardized to yield a theoretical long run variance κ 2 Y = 1. [19] or Zhang and Wu [39] and the references therein). We test for the stationarity of the mean of the annual average temperatures from 1659 to 2020, such that n = 362.…”