Testing for changes in linear models using weighted residuals

We study methods for detecting change points in linear regression models. Motivated by statistics arising from maximally selected likelihood ratio tests, we provide an asymptotic theory for weighted functionals of the cumulative sum (CUSUM) process of linear model residuals. Special attention is given to standardized quadratic form statistics, leading to Darling–Erdős type limit results, as well as novel heavily weighted CUSUM statistics that increase the power of the tests to detect changes that occur early or late in the sample. We discuss improved finite-sample approaches to estimate the critical values for the proposed statistics, which are shown to work well in a Monte Carlo simulation study. The proposed tests are applied to the environmental Kuznets curve, and a COVID-19 dataset.