Valid Edgeworth expansions of M-Estimators in regression models with weakly dependent residuals

Consider a linear regression model y _t = x _t β+u _t where the u _t 's are weakly dependent random variables, the x _t ,'s are known design nonrandom variables, and β is an unknown parameter. We define an M-estimator β _n of β corresponding to a smooth score function. Then, the second-order Edgeworth expansion for β _n is derived. Here we do not assume the normality of (u _t ), and (u _t ) includes the usual ARMA processes. Second, we give the second-order Edgeworth expansion for a transformation T(β _n ) of β _n . Then, a sufficient condition for T to extinguish the second-order terms is given. The results are applicable to many statistical problems.