Valid Edgeworth Expansions of M-Estimators in Regression Models with Weakly Dependent Resfduals

Consider a linear regression model y₁ = x₁β + u₁, where the u₁'S afe weakly dependent random variables, the x₁'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 (u₁), and (u₁) includes the usual ARMA processes. Second, we give the second-order Edgeworth expansion for a transformation T(βn) of β_n. Then, a sufficient condition for <italic>T</italic> to extinguish the second-order terms is given. The results are applicable to many statistical problems.