Structural change in AR(1) models

This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR(1) model with a single structural break of unknown timing. Let beta (1) and beta (2) be the preshift and postshift AR parameter, respectively. Three cases are considered: (i) \ beta (1)\ < 1 and <vertical bar>beta (2)\ < 1; (ii) <vertical bar>beta (1)\ < 1 and <beta>(2) = 1; and (iii) beta (1) = 1 and \ beta (2)\ < 1. Cases (ii) and (iii) are of particular interest but are rarely discussed in the literature. Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be consistently estimated and the residual sum of squares divided by the sample size converges to a discontinuous function of the change point. In case (iii), <(beta )over cap>(2) does not converge to beta (2) whenever the change-point estimate is lower than the true change point. Further, the limiting distribution of the break-point estimator for shrinking break is asymmetric for case (ii), whereas those for cases (i) and (iii) are symmetric. The appropriate shrinking rate is found to be different in all cases.