A Unified Approach to Structural Change Tests Based on ML Scores,FStatistics, and OLS Residuals

ABSTRACT Three classes of structural change tests (or tests for parameter instability) that have been receiving much attention in both the statistics and the econometrics communities but have been developed in rather loosely connected lines of research are unified by embedding them into the framework of generalized M-fluctuation tests (Zeileis and Hornik, 2003 Zeileis , A. , Hornik , K. ( 2003 ). Generalized M-fluctuation tests for parameter instability . Report 80, SFB “Adaptive Information Systems and Modelling in Economics and Management Science”. URL http://www.wu-wien.ac.at/am/reports.htm#80 . Google Scholar). These classes are tests based on maximum likelihood scores (including the Nyblom–Hansen test), on F statistics (sup F, ave F, exp F tests), and on OLS residuals (OLS-based CUSUM and MOSUM tests). We show that (representatives from) these classes are special cases of the generalized M-fluctuation tests, based on the same functional central limit theorem but employing different functionals for capturing excessive fluctuations. After embedding these tests into the same framework and thus understanding the relationship between these procedures for testing in historical samples, it is shown how the tests can also be extended to a monitoring situation. This is achieved by establishing a general M-fluctuation monitoring procedure and then applying the different functionals corresponding to monitoring with ML scores, F statistics, and OLS residuals. In particular, an extension of the sup F test to a monitoring scenario is suggested and illustrated on a real-world data set.