Weak consistency of estimators in linear regression model

A linear regression model and M-estimator of its regression coefficients are considered. We present a derivation of a weak consistency of the M-estimator together with a rate. Derivation is made under general conditions set on the error term, say “asymptotic stationarity” property. The results are proved by means of L2-convergence and cover the cases as the error term is ARMA, ARCH, GARCH process or it is attracted by an ARMA, ARCH, GARCH process. We do not separate random and deterministic covariates. Both cases are treated in one general setting.