Identification and Estimation with Deconfounded Instruments

Abstract

The primary contribution of this research is the introduction of a novel methodology, called common confounding (CC), for identifying and estimating the causal effects of endogenous (treatment) variables on an outcome variable with partially endogenous instrumental variables. A crucial estimation step called deconfounding recovers variation in the instruments, which is unassociated with some observed variables called proxies, and consequently with any unobserved variables that explain the association between the instruments and proxies. These unobserved variables are called common confounders of the instruments and proxies. If the instruments are excluded and exogenous conditional on the unobserved common confounders, the deconfounding step recovers excluded and exogenous instrument variation, where conditioning on the proxies as covariates would generally not. In this sense, the deconfounding step discards more instrument variation than conditioning on the same proxies as covariates would. While this discarding of instrument variation naturally incurs a cost in terms of estimation precision, deconfounding may permit the identification and estimation of causal effects with instruments that violate exclusion or exogeneity due to such common confounders. The deconfounding step is at the heart of the identification proofs and estimation theory developed in this research. Although the linear model, with its simplicity, is often used for illustrative purposes, all identification frameworks in this thesis are semiparameric, featuring a low-dimensional causal effect or parameter of interest and nonparametric nuisance functions. This research explores three principal frameworks for instrument deconfounding: moment-based in Chapter 1, nonparametric with index sufficiency in Chapter 2, and nonparametric bridge functions in Chapter 3. The moment-based framework allows for probabilistic deconfounding, where the deconfounding step can be interpreted as a probabilistic construction of hypothetical exogenous instruments from partially endogenous observed categorical instruments. In the index-sufficient and bridge function frameworks, point identification results for structural causal quantities are provided under two sets of standard parametric assumptions in instrumental variable (IV) approaches: the linear separability of a disturbance in the outcome model [Newey and Powell, 2003] and first-stage strict monotonicity [Imbens and Newey, 2009]. In Chapter 3, no further parametric assumptions are imposed compared to traditional IV, such that unlike in Chapters 1 and 2, a standard exclusion and exogeneity assumption is required for the proxies with respect to the treatment variable. The setting in Chapter 3 illustrates transparently how the CC approach bridges the identification assumptions of nonparametric IV and proximal learning (PL) [Cui et al., 2020]. The CC approach was in part inspired by recent advances in the proximal learning literature [Miao et al., 2018]. Proximal learning extends the literature on nonclassical measurement error models with mismeasured confounders [Hu, 2008, Kuroki and Pearl, 2014], which, similar to mixture models [Hall et al., 2003, Allman et al., 2009, Bonhomme et al., 2016], feature independence conditions between some observed variables conditional on the unobserved variable. Unlike all identification results in mixture models, identification in the CC approach does not require the conditional independence of three observed variables. Instead, under minimal exogeneity and exclusion requirements, the crucial deconfounding step extracts excluded and exogenous variation in the instruments, which is then used to identify some average structural effects of an endogenous treatment on an outcome variable. In Chapters 1 and 2, the identification theorems are complemented by comprehensive estimation theory. In the moment-based framework with nonparametric nuisances in Chapter 1, under some regularity conditions, a double robust and Neyman orthogonal score can be constructed and utilised for debiased machine learning. The estimation of typical causal quantities in the index-sufficient framework in Chapter 2 motivates novel semiparametric estimation theory, combining debiasing with respect to sequentially dependent nuisance functions [Singh, 2021, Chernozhukov et al., 2022] and strong identification subject to nuisance functions, which are defined as solutions to possibly ill-posed inverse problems [Bennett et al., 2022]. An extensive simulation in Chapter 1 sheds light on some expected behaviours of estimators based on deconfounded instruments, while two empirical applications in Chapters 1 and 2 demonstrate the appeal of this approach in practical settings with partially endogenous instruments. In the empirical application of Chapter 1, the CC approach is used with local linear moments to estimate the average treatment effect of substance use on antisocial behaviour in adolescents with peer behaviour instruments. In the empirical application of Chapter 2, pre-college GPA measures are deconfounded with respect to the unobserved common confounder ability and used to infer the causal effect of obtaining a BA degree on net worth later in life in a linear model. This thesis equips applied researchers with comprehensive identification and estimation frameworks for the more nuanced CC approach to instrument exclusion and exogeneity. Deconfounding enables the use of partially endogenous instruments to identify new causal effects and can be employed to challenge instrument exogeneity assumptions in previous applied research. The novel identification and estimation approach developed in this thesis disrupts the conventional dichotomy of exogenous versus endogenous instruments, a well-known limiting factor for instrumental variable methods.