Testing for Breaks in Regression Models with Dependent Data
The paper examines a test for smoothness/breaks in a nonparametric regression model with dependent data. The test is based on the supremum of the difference between the one-sided kernel regression estimates. When the errors of the model exhibit strong dep
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Abstract The paper examines a test for smoothness/breaks in a nonparametric regression model with dependent data. The test is based on the supremum of the difference between the one-sided kernel regression estimates. When the errors of the model exhibit strong dependence, we have that the normalization constants to obtain the asymptotic Gumbel distribution are data dependent and the critical values are difficult to obtain, if possible. This motivates, together with the fact that the rate of convergence to the Gumbel distribution is only logarithmic, the use of a bootstrap analogue of the test. We describe a valid bootstrap algorithm and show its asymptotic validity. It is interesting to remark that neither subsampling nor the sieve bootstrap will lead to asymptotic valid inferences in our scenario. Finally, we indicate how to perform a test for k breaks against the alternative of k + k0 breaks for some k0 . Keywords Nonparametric regression · Breaks/smoothness · Strong dependence · Extreme-values distribution · Frequency domain bootstrap algorithms
1 Introduction The literature on breaks/continuity on parametric regression models is both extensive and exhaustive in both econometric and statistical literature, see [23] for a survey. Because as in many other situations an incorrect specification of the model can lead to misleading conclusions, see for instance [14], it is of interest to develop tests which do not rely on any functional specification of the regression model. Although some work has been done in the nonparametric setup, the literature appears to focus mostly on the estimation of the break point, see for instance [22], Chu and Wu (1992) and [8], rather than on the testing of its existence. With this view, the purpose of this paper is to fill this gap by looking at testing for the hypothesis of continuity against the J. Hidalgo (B) London School of Economics, Houghton Street, London WC2A 2AE, UK e-mail: [email protected] V. Dalla National and Kapodistrian University of Athens, Sofokleous 1, 10559 Athens, Greece © Springer International Publishing Switzerland 2016 R. Cao et al. (eds.), Nonparametric Statistics, Springer Proceedings in Mathematics & Statistics 175, DOI 10.1007/978-3-319-41582-6_3
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J. Hidalgo and V. Dalla
alternative of the existence of (at least) one discontinuity point in a nonparametric regression model, although we shall indicate how to perform a test for k breaks against the alternative of k + k0 breaks for some k0 . More specifically, we consider the regression model yt = r (xt ) + u t ; t = 1, . . . , n,
(1.1)
where we assume that the homoscedastic errors {u t }t∈Z follow a covariance stationary linear process exhibiting possibly strong dependence, to be more precise in Condition C1 below. We shall assume that xt is deterministic, say a time trend. A classical example of interest in time series is a polynomial trend, that is xt = t, t 2 , . . . , t p , and/or when regressors are of the type “cos tλ0 ” and/or “sin tλ0 ”, where λ0 = 0. The latter type of regressors
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