Two-stage generalized moment method with applications to regressions with heteroscedasticity of unknown form

We describe in a general framework a two-stage generalized moment method. More precisely we explain how to partition the set of estimating constraints and to unfold the set of parameters in order to derive a two-stage approach which is asymptotically equivalent to GMM. Then we consider a linear model Y_t = X_tb + u_t, and the GMM estimator based on the constraints: EX′_tu_t = ,..., EX′_tu_t^2p+1 = 0. We explain how to find an asymptotically equivalent estimator by replacing some of the errors u_t by their corresponding OLS residuals. Finally, we focus on constraints based on first and third conditional moments. The estimator thus obtained is asymptotically more efficient than the OLS estimator. The efficiency gain is linked with a marginal kurtosis of the u_t's distribution or with the conditional heteroscedasticity.