Detection of Stationary Errors in Multiple Regressions with Integrated Regressors and Cointegration

This article studies the problem of detecting sequentially stationary error terms in a multiple regression model with a difference-stationary multivariate I(1)-regressor. The detection of cointegration is covered as a special case. We provide the asymptotic distribution theory for a monitoring procedure that is related to a well-known nonparametric unit root test statistic calculated from sequentially updated least squares residuals. Functional limit theorems for the corresponding sequential processes and central limit theorems for the detectors used to raise an alarm are established under the no-change null hypothesis as well as under change-point models covering a change to I(0)-errors and a change of the regression coefficients as well. We also discuss extensions to the case that continuous time processes are discretely sampled to obtain the data allowing to apply the procedures to high-frequency data as well. Our results show that we can handle the infill asymptotics assuming that nonstationary continuous-time processes such as semimartingales are discretely observed, by virtue of the general assumptions that we impose. The finite sample properties are investigated by a simulation study.