Efficient Estimation of Average Derivatives in NPIV Models: Simulation Comparisons of Neural Network Estimators

Artificial Neural Networks (ANNs) can be viewed as {nonlinear sieves} that can approximate complex functions of high dimensional variables more effectively than linear sieves. We investigate the computational performance of various ANNs in nonparametric instrumental variables (NPIV) models of moderately high dimensional covariates that are relevant to empirical economics. We present two efficient procedures for estimation and inference on a weighted average derivative (WAD): an orthogonalized plug-in with optimally-weighted sieve minimum distance (OP-OSMD) procedure and a sieve efficient score (ES) procedure. Both estimators for WAD use ANN sieves to approximate the unknown NPIV function and are root-n asymptotically normal and first-order equivalent. We provide a detailed practitioner's recipe for implementing both efficient procedures. This involves the choice of tuning parameters for the unknown NPIV, the conditional expectations and the optimal weighting function that are present in both procedures but also the choice of tuning parameters for the unknown Riesz representer in the ES procedure. We compare their finite-sample performances in various simulation designs that involve smooth NPIV function of up to 13 continuous covariates, different nonlinearities and covariate correlations. Some Monte Carlo findings include: 1) tuning and optimization are more delicate in ANN estimation; 2) given proper tuning, both ANN estimators with various architectures can perform well; 3) easier to tune ANN OP-OSMD estimators than ANN ES estimators; 4) stable inferences are more difficult to achieve with ANN (than spline) estimators; 5) there are gaps between current implementations and approximation theories. Finally, we apply ANN NPIV to estimate average partial derivatives in two empirical demand examples with multivariate covariates.