Least absolute deviation estimation for AR(1) processes with roots close to unity

We establish the asymptotic theory of least absolute deviation estimators for AR(1) processes with autoregressive parameter satisfying $n({\rho }_n-1)\to \gamma$ for some fixed $\gamma$ as $n\to \mathrm{\infty }$, which is parallel to the results of ordinary least squares estimators developed by Andrews and Guggenberger (2008) in the case $\gamma =0$ or Chan and Wei (1987) and Phillips (1987) in the case $\gamma \mathrm{\ne }0$. Simulation experiments are conducted to confirm the theoretical results and to demonstrate the robustness of the least absolute deviation estimation.