We contribute to the asymptotic theory of M-estimators for multiple linear regression by analysing absolutely continuous objective functions under weak assumptions. Particular cases are the Huber-skip and quantile estimation. We allow for a variety of regressors, both deterministic and stochastic, where the stochastic regressors can be either stationary or random walks. We prove consistency and find a stochastic expansion of the M-estimator, from which one can derive limit distributions under weak assumptions on the objective function and regressors. The results are obtained using some recent martingale results, see Johansen and Nielsen (2016), which build on an iterated version of the exponential inequality for martingales by Bercu and Touati (2008).
