The likelihood ratio test for a change-point in simple linear regression

We consider likelihood ratio tests to detect a change-point in simple linear regression (a) when the alternative specifies that only the intercept changes and (b) when the alternative permits the intercept and the slope to change. Approximations for the significance level are obtained under reasonably general assumptions about the empirical distribution of the independent variable. The approximations are compared with simulations in order to assess their accuracy. For the model in which only the intercept is allowed to change, a confidence region for the change-point and an approximate joint confidence region for the change-point, the difference in intercepts, and the slope are obtained by inversion of the appropriate likelihood ratio tests.