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Detecting at-most-\(\mathfrak{m}\) changes in linear regression models. (English) Zbl 1368.62239

Summary: In this article, we provide a new procedure to test for at-most-\(\mathfrak{m}\) changes in the time-dependent regression model \(y_t=\mathbf{x}_t^\top\beta_t+e_t\), \(1\leq t\leq T\), that is, \(\beta_1=\beta_2=\cdots=\beta_T\) under the no-change null hypothesis against the alternative \(y_t=x_t^\top\beta^{(i)}+e_t\), if \(k_{i-1}^\ast<t<k^\ast_i\), \(1\leq i\leq\mathfrak{m}+1\) and \(\beta^{(j)}\neq\beta^{(\ell)}\) for some \(1\leq j\), \(\ell\leq\mathfrak{m}+1\) with \(k_0^\ast=0\), \(1<k_1^\ast<k_2^\ast<\cdots<k_{\mathfrak{m}}^\ast<T\), \(k_{\mathfrak{m}+1}^\ast=T\). Our procedure is based on weighted sums of the residuals, incorporating the possibility of math formulam changes. The weak limit of the proposed test statistic is the sum of two double-exponential random variables. A small Monte Carlo simulation illustrates the applicability of the limit results in case of small and moderate sample sizes. We compare the new method to the cumulative sum control chart (CUSUM) and standardized (weighted) CUSUM procedures and obtain the power curves of the test statistics under the alternative. We apply our method to find changes in the unconditional four-factor capital asset pricing model.

MSC:

62M07 Non-Markovian processes: hypothesis testing
62J05 Linear regression; mixed models
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
62P05 Applications of statistics to actuarial sciences and financial mathematics
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