GMM weighted estimation

Jan Ámos Víšek

Charles University in Prague, Faculty of Social Sciences

The point estimation was from the very beginning of the statistics (and econometrics) one of the key topics. In the early days, the unbiasedness was assumed to play the crucial role but later the (weak) consistency overtook the governance.

The (classical and/or robust) statistics developed a bunch of ``principles'', heuristics of which promised to yield the estimators being not only consistent but also premiant in a widely considered competition (achieving e. g. efficiency). Some of them worked, some not. Typically, the estimator was given as a solution of a (vector) equation (normal equations) - interpretable as $p$-tuple of orthogonality conditions (of residuals to the columns of design matrix, e. g.).

The consistency requires orthogonality of residuals to the (estimated) model and hence the estimators are defined as solution of a $q$-tuple of orthogonality conditions ($p\leq q$). It allows for direct employment of additional information about the parameter in question. In such a case we speak about Generalized Method of Moments estimation.

Despite the forty years of robust studies, the econometrics haven't taken seriously (possible) fatal consequences of a slight deviation of the assumed model from the underlying one as Fisher did already in 1922 or of a few contaminating observations as considered by Hampel or Huber much later on.

Since the weighting down the order statistics of squared residuals appeared to be powerful tool for influential-points-recognition, the present paper offers an idea of the generalized method of moments weighted estimators and shows that the least weighted squares are special case of them.