We considered parametric estimation and error estimation in two classical regression models. First, a heteroscedastic linear regression model is considered where responses are allowed to be missing at random and with conditional variance modeled as a function of the mean response. Maximum empirical likelihood estimation is studied for an empirical likelihood with an increasing number of estimated constraints. The resulting estimator is shown to be asymptotically normal and can perform outperform the ordinary least squares estimator. Second, we proved a stochastic expansion for a residual-based estimator of the error distribution in semi-parametric model. It implies a functional central limit theorem.