Methods for Nonparametric and Semiparametric Regressions with Endogeneity: A Gentle Guide

Abstract

This paper reviews recent advances in estimation and inference for nonparametric and semiparametric models with endogeneity. It first describes methods of sieves and penalization for estimating unknown functions identified via conditional moment restrictions. Examples include nonparametric instrumental variables regression (NPIV), nonparametric quantile IV regression and many more semi-nonparametric structural models. Asymptotic properties of the sieve estimators and the sieve Wald, quasi-likelihood ratio (QLR) hypothesis tests of functionals with nonparametric endogeneity are presented. For sieve NPIV estimation, the rate-adaptive datadriven choices of sieve regularization parameters and the sieve score bootstrap uniform confidence bands are described. Finally, simple sieve variance estimation and over-identification test for semiparametric two-step GMM are reviewed. Monte Carlo examples are included.

A Unified Framework of Policy Learning for Contextual Bandit with Confounding Bias and Missing Observations

“…Combining (i) and (iii) we see that the IV is also outcome-independent, i.e., Z K K Y | pA, U, Xq. We remark that assumptions (i)-(iii) are standard in the IV literature (Baiocchi et al, 2014;Newey and Powell, 2003;Singh et al, 2019;Chen and Qiu, 2016;Chen et al, 2011), and an IV satisfying these assumptions is referred to as a valid IV. Assumption (iv) shows that the missingness is unconfounded and outcome-independent, since R X and R Z are neither caused nor have a direct effect on U and Y .…”

“…Causal inference. A series of previous works in causal inference address adjusting the confounding bias via use of side observations like instrumental variables (Chen et al, 2011Chen and Qiu, 2016;Bennett et al, 2019Bennett et al, , 2023Wong, 2021;Athey and Wager, 2019) or proxy variables (Bennett and Kallus, 2021;Pearl, 2009;Miao et al, 2018a;Lee and Bareinboim, 2021). Among these works, Pearl (2009); Miao et al (2018a); Lee and Bareinboim (2021) are most relevant to this paper since the our framework covers side observations proposed in them.…”