Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point t(0) at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations X-1,..., X-n, we shall study the least squares estimator (t(0)) over cap for t(0), which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, (t(0)) over cap /n is a consistent estimator for t(0), where t(0) = left perpendicularn tau(0)right perpendicular, with 0 < tau(0) < 1, i.e., (t(0)) over cap /n P ->(P) tau(0) (n ->infinity). Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator.