Asymptotic distribution of the least squares estimator in the first-order autoregressive process

This study is about the asymptotic distribution of the least squares estimator in nonstationary first-order autoregressive processes. These processes are commonly used to model economic time series and the desired distribution is important in finding the size of the so-called unit root tests.

Our approach is based on the asymptotic characterization of the distribution in terms of a functional of the standard Wiener process. We use the Karhunen- Loeve expansion for the Wiener process and obtain the solution using characteristic functions and the Fourier inversion theorem. As compared to the previous studies, our method provides a conceptually simple framework in which one can investigate more complicated models.