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Abstract
A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives. Our proposal is based on a modified CUSUM type test procedure, which uses a sequential marked empirical process of residuals. We show weak convergence of the considered process to a centered Gaussian process under the null hypothesis of no change in the mean function and a stationarity assumption. This requires some sophisticated arguments for sequential empirical processes of weakly dependent variables. As a consequence we obtain convergence of Kolmogorov-Smirnov and Cramér-von Mises type test statistics. The proposed procedure acquires a very simple limiting distribution and nice consistency properties, features from which related tests are lacking. We moreover suggest a bootstrap version of the procedure and discuss its applicability in the case of unstable variances.
Highlights
Assume a finite sequence (Xt, Yt), t = 1, . . . , n, of a weakly dependent Rd × Rvalued time series has been observed
A weakly dependent time series regression model with multivariate covariates and univariate observations is considered, for which we develop a procedure to detect whether the nonparametric conditional mean function is stable in time against change point alternatives
We suggested a new test for structural breaks in the regression function in nonparametric time seriesregression
Summary
N (for some not further specified function m) should be tested against structural changes over time such as change point alternatives Literature on such tests for nonparametric regression functions is rare in the time series context. Su and Xiao (2008) extended these tests to strongly mixing and not necessarily stationary processes, allowing for heteroscedasticity, while Su and White (2010) proposed change point tests in partially linear time series models. Marked empirical processes have been suggested in a seminal paper by Stute (1997) for lack-of-fit testing in nonparametric regression models with i.i.d. data. We suggest a wild bootstrap version of our test that can be applied to detect changes in the mean function in the case of stable variances (as alternative to using the asymptotic distribution, e.g. for multivariate covariates) as well as in the case of non-stable variances.
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