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 \(n(\rho _n-1)\rightarrow \gamma\) for some fixed \(\gamma\) as \(n\rightarrow \infty\), which is parallel to the results of ordinary least squares estimators developed by Andrews and Guggenberger (Journal of Time Series Analysis, 29, 203–212, 2008) in the case \(\gamma = 0\) or Chan and Wei (Annals of Statistics, 15, 1050–1063, 1987) and Phillips (Biometrika, 74, 535–574, 1987) in the case \(\gamma \ne 0\). Simulation experiments are conducted to confirm the theoretical results and to demonstrate the robustness of the least absolute deviation estimation.