A New Class of Change Point Test Statistics of Rényi Type

Abstract

A new class of change point test statistics is proposed that utilizes a weighting and trimming scheme for the cumulative sum (CUSUM) process inspired by Rényi (1953). A thorough asymptotic analysis and simulations both demonstrate that this new class of statistics possess superior power compared to traditional change point statistics based on the CUSUM process when the change point is near the beginning or end of the sample. Generalizations of these "Rényi" statistics are also developed to test for changes in the parameters in linear and non-linear regression models, and in generalized method of moments estimation. In these contexts we applied the proposed statistics, as well as several others, to test for changes in the coefficients of Fama-French factor models. We observed that the Rényi statistic was the most effective in terms of retrospectively detecting change points that occur near the endpoints of the sample.

Detecting early or late changes in linear models with heteroscedastic errors

“…More recently, Horváth et al (2020) use sequential averages of the sample residuals to develop efficient tests to detect early or late changes in the linear model parameters. For a survey on the change point problem from a time series point of view, we refer to Aue and Horváth (2013).…”

“…Detecting such changes is often of interest when applying change point detection procedures retrospectively to a sample where a change is suspected to have occurred recently. In this paper, we aim to extend the residual based test of Horváth et al (2020), which is based on a novel trimming scheme for the CUSUM process that is effective for end of sample change point detection, to the setting of heteroscedastic linear models.…”

“…The residual-based approaches require additional conditions on the change (1) − (T) . For example, the Rényi-type statistic in Horváth et al (2020) will not detect a change if (1) − (T) and ∑ T t=1 x t are orthogonal vectors. Further these results hold under the above general heteroscedasticity assumptions.…”

“…The tests that we consider include the proposed test based onẐ T , along with the Rényi-type statistic of Horváth et al (2020) with trimming parameter a T = b T = T 1/2 computed from the residuals, the CUSUM statistic of Ploberger and Krämer (1992), Andrews (1993) Wald-type statistic, and the Hidalgo and Seo (2013) statistic. The Rényi-type statistic of Horváth et al (2020) and the CUSUM statistic of Ploberger and Krämer (1992) are the univariate statistics computed from the estimated residuals of the regression model. The Andrews and Hidalgo-Seo statistics were tailored to the regression context as described in their papers, and are based on functionals of the process of estimated regression coefficientŝt ,1 −̂t ,2 , similar to the proposed test.…”

“…The Hidalgo-Seo statistic uses the periodogram matrix for LRV estimation. The statistics of the present paper, the Rényi-type statistics of Horváth et al (2020) and the CUSUM statistic of Ploberger and Ploberger and Krämer (1992) use either the LRV estimation with kernel methods (correlated errors) or the sample variance (uncorrelated data). If LRV estimation is needed we use the quadratic spectral kernel, and the bandwidth was selected using the procedure advocated by Andrews (1991) and implemented in Aschersleben and Wagner (2016).…”