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MR4060975 62G08 62M10
Khismatullina, Marina (D-BONN-NDM) ;Vogt, Michael [Vogt, Michael2 ] (D-BONN-EC)
Multiscale inference and long-run variance estimation in non-parametric
regression with time series errors. (English summary)
J. R. Stat. Soc. Ser. B. Stat. Methodol. 82 (2020), no. 1, 5–37.

This article is concerned with a testing problem for certain shape properties of a
nonparametric time trend. For T observations of a univariate time series {Yt,T }Tt=1 , thetest is built on the nonparametric regression model tYt,T = m+ εt ,Twhere m: [0, 1] → R is an unknown nonparametric function, and {εt } is a zero-mean
stationary error process.
The null hypothesis for this test can be formulated as follows. Throughout it is assumed that m is continuously differentiable on [0, 1]. Also let H0 (u, h) be the hypothesisthat m is constant on the interval [u − h, u + h] for some location u and bandwidth h.
It follows that H0 (u, h) : m0 (w) = 0 for all w ∈ [u − h, u + h], where m0 is the first-orderderivative of m. The overall null hypothesis considered in this article is designed to test
for constancy of m on many such intervals simultaneously. It is in the form of
H0 : the hypothesis H0 (u, h) holds true for all (u, h) ∈ GT ,where GT is some large set of points (u, h) that expands with the sample size T ingeneral.
Accordingly, the test statistic for the overall null hypothesis H0 is constructed fromthe one for H0 (u, h). For the latter, the following rescaled local linear estimator of thederivative m0 (u) is considered:bT (u, h) =ψTX

wt,T (u, h) Yt,T ,t=1

where
Λt,T (u, h)wt,T (u, h) = qP,T
2 (u, h)Λt=1 t,Tt/T − ut/T − uΛt,T (u, h) = K
ST,0 (u, h)− ST,1 (u, h) ,hh`T
1 Xt/T − ut/T − uST,` (u, h) =KT h t=1hhfor ` = 0, 1, 2, and K is a nonnegative symmetric kernel function with support onbT (u, h)/b[−1, 1]. The test statistic for H0 (u, h) is then given byPψ
σ , where σb2 is some∞2
consistent estimator of the long-run variance of σ = j=−∞ E(εt εt−j ) of the error
process {εt }. In particular, instead of leaving the dependent structure of the processunspecified, this article assumes that it obeys the class of AR(∞) processes and proposesa difference-based long-run variance estimator.

In the end, the test statistic for H0 is defined as()bT (u, h)ψb T = maxΨ− λ (h) ,σb(u,h)∈GTpwhere λ(h) = 2 log{1/(2h)} is the additive correction term that is implied by thedistribution of the maximum of standard normal random variables. Because the null
distribution of this test statistic is non-standard, critical values should be computedby Monte Carlo simulation. Specifically, qT (α), the critical value for a given levelof significance α ∈ (0, 1), can be obtained as the (1 − α)-quantile of the empirical
distribution ofφT (u, h)− λ (h) ,ΦT = maxσ(u,h)∈GTPTiidwhere φT (u, h) = t=1 wt,T (u, h)σZt for {Zt }Tt=1 ∼ N (0, 1). For such qT (α), we rejectb T > qT (α). It is also demonstrated that the test has
the overall null hypothesis H0 if Ψ
asymptotic power 1 against some local alternatives.
Masayuki Hirukawa
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