  Generalized Method of Moments (GMM) estimators in their various forms,
including the popular Maximum Likelihood (ML) estimator, are frequently applied
for the evaluation of complex econometric models with not analytically
computable moment or likelihood functions. As the objective functions of GMM-
and ML-estimators themselves constitute the approximation of an integral, more
precisely of the expected value over the real world data space, the question
arises whether the approximation of the moment function and the simulation of
the entire objective function can be combined. Motivated by the popular Probit
and Mixed Logit models, we consider double integrals with a linking function
which stems from the considered estimator, e.g. the logarithm for Maximum
Likelihood, and apply a sparse tensor product quadrature to reduce the
computational effort for the approximation of the combined integral. Given
H\"older continuity of the linking function, we prove that this approach can
improve the order of the convergence rate of the classical GMM- and
ML-estimator by a factor of two, even for integrands of low regularity or high
dimensionality. This result is illustrated by numerical simulations of Mixed
Logit and Multinomial Probit integrals which are estimated by ML- and
GMM-estimators, respectively.
