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Conditional variance model checking. (English) Zbl 1181.62053

Summary: This paper discusses the problem of fitting a parametric model to the conditional variance function in a class of heteroscedastic regression models. The proposed test is based on the supremum of the E. V. Khmaladze type martingale transformation [see Ann. Stat. 21, No. 2, 798-829 (1993; Zbl 0801.62043); ibid. 16, No. 4, 1503–1516 (1998; Zbl 0671.62048)] of a certain partial sum process of calibrated squared residuals. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. The test is shown to be consistent against a large class of fixed alternatives and to have nontrivial asymptotic power against a class of nonparametric \(n^{-1/2}\)-local alternatives, where \(n\) is the sample size. Simulation studies are conducted to assess the finite sample performance of the proposed test and to make a finite sample comparison with an existing test.

MSC:

62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)
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