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
β1 and β2 be the preshift and
postshift AR parameter, respectively. Three cases are considered:
(i) |β1| < 1 and
|β2| < 1; (ii)
|β1| < 1 and β2 = 1;
and (iii) β1 = 1 and |β2|
< 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), [circumflex over beta]2
does not converge to β2 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.
Moment approximation for least‐squares estimators in dynamic regression models with a unit root