Detecting relevant changes in time series models

Most of the literature on change-point analysis by means of hypothesis testing considers hypotheses of the form H0: θ1 = θ2 vs. H1: θ1 6 = θ2, where θ1 and θ2 denote parameters of the process before and after a change point. This paper takes a different perspective and investigates the null hypotheses of no relevant changes, i.e. H0: ||θ1 − θ2| | ≤ ∆, where ‖ · ‖ is an appropriate norm. This formulation of the testing problem is motivated by the fact that in many applications a modification of the statistical analysis might not be necessary, if the difference between the parameters before and after the change-point is small. A general approach to problems of this type is developed which is based on the CUSUM principle. For the asymptotic analysis weak convergence of the sequential empirical process has to be established under the alternative of non-stationarity, and it is shown that the resulting test statistic is asymptotically normal distributed. The results can also be used to establish similarity of the parameters, i.e. H1: ||θ1−θ2| | ≤ ∆, at a controlled type one error and to estimate the magnitude ||θ1 − θ2| | of the change with a corresponding confidence interval. Several applications of the methodology are given including tests for relevant changes in the mean, variance, parameter in a linear regression model and distribution function among others. The finite sample properties of the new tests are investigated by means of a simulation study and illustrated by analyzing a data example from portfolio management