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

Abstract

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,ldots ,X_n, we shall study the least squares estimator widehat{t}_0 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, widehat{t}_0/n is a consistent estimator for tau _0, where t_0 =lfloor ntau _0rfloor , with 0<tau _0<1, i.e., widehat{t}_0/n ,{mathop {rightarrow }limits ^{P}},tau _0(nrightarrow infty ). 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.