Efficient Estimation of Average Derivatives in NPIV Models: Simulation Comparisons of Neural Network EstimatorsAuthor(s):
Publication Date: December 2021
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.
Artiﬁcial neural networks, Relu, Sigmoid, Nonparametric instrumental variables, Weighted average derivatives, Optimal sieve minimum distance, Eﬀicient influence, Semiparametric eﬀiciency, Endogenous demand
JEL Classification Codes: C14, C22