Testing structural change in time-series nonparametric regression models

Abstract

We propose a CUSUM type of test for structural change in dynamic nonparametric regression models. It is based upon the cumulative sums of weighted residuals from a single nonparametric regression and complements the conventional parameter instability tests in parametric models. We derive the limiting distributions of the test under both the null hypothesis and sequences of local alternatives. A bootstrap procedure is also proposed and its validity is justified. Finally, simulation experiments are conducted to investigate the finite sample properties of our test.

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

“…In particular, we generalize some of Hansen's () results in Lemma and Proposition 1 in the online supplement. Su & Xiao () also prove a generalization such as Proposition 1 under slightly different conditions. Instead of stating assumptions on the density averages, they state their assumptions on for every j . Theorem Under the assumptions (K), (C’), (W’), (F’), (E’), (X’), and (Z’) and (M) under the fixed alternative H 1 with a change point in ⌊ nθ 0 ⌋, we have Corollary Under the assumptions of Theorem , the Kolmogorov–Smirnov type test based on the process is consistent against fixed alternatives H 1 .…”

“…This possibility should be investigated, see Delgado & Hidalgo (), Chen et al . () or Su & Xiao () as well as Hidalgo () or Honda (). Remark It is possible to generalize the test statistic by introducing a bounded weight function ω that can reflect a prior information about when the change is likely to occur and replacing sup s ∈ [0,1] by . The weight function could be of the form , 0 < s 1 < s 2 < 1, to examine the stability of the innovations on a subregion.…”

Testing Structural Change in Partially Linear Models

“…If m nt (·) is absent from the DGP in (2.1), we obtain the conventional time-varying linear regression DGP. If x 0 nt γ nt is absent, however, the DGP in (2.1) becomes time-varying nonparametric (see, e.g., Su and Xiao, 2008).…”

“…Indeed, following Su and Xiao (2008), we permit time-varying behavior in the conditional variance process and a nonstationary distribution for {x nt , z nt , ε nt } under both the null and alternative hypotheses. Nevertheless, to facilitate the presentation we will assume that some aspects of the process {x nt , z nt , ε nt } are asymptotically stationary in a sense to be defined precisely below.…”

“…Nonparametric models allow for much greater flexibility and thus may have certain advantages in applications. For this reason, Delgado and Hidalgo (2000) advocate conducting nonparametric inference when testing for structural breaks; and Su and Xiao (2008) propose a nonparametric test of structural change in dynamic nonparametric regression models. Nevertheless, as Robinson (1988) has remarked, a correctly specified parametric model affords precise inferences, a badly misspecified one, possibly seriously misleading inference; whereas nonparametric modeling is associated with both greater robustness and lesser precision.…”