Testing for a Two-Phase Multiple Regression

Maximum likelihood methods are used to estimate the parameters of two separate multiple regressions that switch at an unknown point in the data. Normal errors with constant variance are assumed and likelihood ratio statistics are used to test for the presence of two separate regressions. Our main result is a conservative bound on the null distribution function of the test statistic. This bound is based on an improved Bonferroni inequality, and a simple power-series approximation is provided. Similar bounds are given for likelihood ratio statistics that test for a shift in the constant term of the regression only. The accuracies of the bounds and approximations are evaluated on a number of examples.