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

We discuss the estimation of a change-point $$t_0$$ t 0 at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations $$X_1,\ldots ,X_n$$ X 1 , … , X n , we shall study the least squares estimator $$\widehat{t}_0$$ t ^ 0 for $$t_0$$ 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, $$\widehat{t}_0/n$$ t ^ 0 / n is a consistent estimator for $$\tau _0$$ τ 0 , where $$t_0 =\lfloor n\tau _0\rfloor $$ t 0 = ⌊ n τ 0 ⌋ , with $$0