Valid edgeworth expansions of M-estimators in regression models with weakly dependent residuals

Consider a linear regression model y_t = x_tβ + 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 β̂_n of β corresponding to a smooth score function. Then, the second-order Edgeworth expansion for β̂_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(β̂_n) of β̂_n. Then, a sufficient condition for T to extinguish the second-order terms is given. The results are applicable to many statistical problems.