IV with heterogeneous treatment effects and causal mechanisms

The paper develops a new fully Bayesian method (IVBMA) for incorporating model uncertainty into instrumental variable regression systems by extending an existing Gibbs sampler.

“We develop a method to perform model averaging in two-stage linear regression systems subject to endogeneity. Our method extends an existing Gibbs sampler for instrumental variables to incorporate a component of model uncertainty.”◌ not checked against the paper’s text as it now stands

It introduces conditional Bayes factors (CBFs) to make model comparison tractable within the instrumental variable framework, since direct evaluation of model probabilities is intractable.

“Direct evaluation of model probabilities is intractable in this setting. We show that by nesting model moves inside the Gibbs sampler, model comparison can be performed via conditional Bayes factors, leading to straightforward calculations.”◌ not checked against the paper’s text as it now stands

The paper formally defines the instrumental variable model as a two-stage system with an outcome equation and an instrument equation linking endogenous regressors to instruments and covariates.

“We consider the classic, two-stage endogenous variable model: Y =Xβ+Wγ+ϵ X =Zδ+Wτ+η”◌ not checked against the paper’s text as it now stands

The paper applies its IV model averaging method to real datasets, instrumenting institutions and integration in a macroeconomic growth model and instrumenting wholesale on retail prices in a demand estimation, showing computational and inferential advantages over prior IV-BMA approaches.

“We conclude with a study of two different modeling challenges: incorporating uncertainty into the determinants of macroeconomic growth, and estimating a demand function by instrumenting wholesale on retail prices.”◌ not checked against the paper’s text as it now stands

The paper develops a new identification approach for a correlated random coefficients model that allows binary and discrete instrumental variables, unlike prior approaches requiring continuous instruments.

“First, while […] require a continuous instrument (see the discussion on page […]florens_cts_req_remark), we can achieve identification with binary and discrete instruments in many cases.”◌ not checked against the paper’s text as it now stands

The paper builds a control function using the instrument to show that the endogenous regressors are independent of the unobservable coefficients conditional on this control function, enabling identification via the instrument.

“[…] and […] imply that (R, B) ⊥ Z. If […] also holds, then W ⊥ B | R.”◌ not checked against the paper’s text as it now stands

The paper imposes an instrument exogeneity assumption requiring the unobservables and first-stage heterogeneity to be independent of the instrument.

“* (Instrument exogeneity) (B, V) ⊥ Z.”◌ not checked against the paper’s text as it now stands

The paper develops a computationally straightforward estimator based on kernel-weighted linear regressions using the instrument's first-stage quantile regression, and applies it empirically using a binary instrument derived from the 1970 Clean Air Act Amendments.

The paper introduces a new stochastic monotonicity condition as a weaker alternative to deterministic monotonicity for instrumental variable analysis.

“We introduce a stochastic monotonicity condition which relaxes deterministic monotonicity in that it does not require that a monotonic increasing relationship hold within subjects between the levels of the IV and the level of the treatment that the subject would take if given a level of the IV, but only that a monotonic increasing relationship hold across subjects between the IV and the treatment in a certain manner.”◌ not checked against the paper’s text as it now stands

The paper shows that under stochastic monotonicity, the instrumental variable method identifies a strength-of-IV weighted average treatment effect (SIVWATE) rather than the standard local average treatment effect.

“We show that under stochastic monotonicity, the IV method identifies a weighted average of treatment effects with greater weight on subgroups of subjects on whom the IV has a stronger effect.”◌ not checked against the paper’s text as it now stands

The paper provides bounds on the global average treatment effect and a sensitivity analysis for violations of the stochastic monotonicity assumption used in IV estimation.

“We provide bounds on the global average treatment effect under stochastic monotonicity and a sensitivity analysis for violations of the stochastic monotonicity assumption.”

The paper introduces a new instrumental constraint on the joint distribution of potential outcomes called the stochastically monotone instrumental variable (SMIV) to derive sharp bounds and testable implications for the Roy model.

“We characterize sharp bounds on the joint distribution of potential outcomes and testable implications of the Roy self-selection model under an instrumental constraint on the joint distribution of potential outcomes we call stochastically monotone instrumental variable (SMIV).”◌ not checked against the paper’s text as it now stands

The authors define stochastically monotone instrumental variables as selection shifters restricted to affect potential outcomes only monotonically, resolving drawbacks of standard independent instruments.

“To resolve both of these issues, we introduce stochastically monotone instrumental variables. They are selection shifters that are restricted to affect potential outcomes monotonically.”◌ not checked against the paper’s text as it now stands

They show their SMIV assumption is stronger than Manski's classical monotone instrumental variable assumption because it restricts the whole conditional distribution rather than just the conditional mean.

“Our stochastically monotone instrumental variable assumption is stronger than the […] monotone instrumental variable assumption, which only requires mean potential outcomes to be monotonic in the instrument, rather than the whole distribution.”

The paper extends instrumental variable estimation methods to competing risk time-to-event data, allowing arbitrary types of exposure and instrument.

“In particular, we show how to infer the effect of an arbitrary exposure on cause-specific hazard functions under a semi-parametric model that imposes relatively weak restrictions on the observed data distribution. The proposed approach is flexible accommodating exposures and instrumental variables of arbitrary type, and enables covariate adjustment.”◌ not checked against the paper’s text as it now stands

The paper formally defines the instrumental variable assumptions required (association with exposure, no direct effect on outcome except through exposure, and no confounding of the instrument-outcome association).

“This is a variable which is (a) associated with the exposure, (b) has no direct effect on the outcome other than through the exposure, and (c) whose association with the outcome is not confounded by unmeasured variables (see e.g. Hernán and Robins, 2006). Condition (a) is empirically verifiable, but conditions (b) and (c) are not.”◌ not checked against the paper’s text as it now stands

The paper derives an unbiased estimating equation using the instrumental variable (conditional on covariates) under the structural model and censoring assumptions, forming the basis of a closed-form recursive estimator.

“Assume the structural model […] with the assumption that G is an instrumental variable, conditional on L, and further that the censoring time satisfies condition (C). Then E[{G-E(G|L)}e^B_1(t)X+B_2(t)XR(t){dN^j(t)-dB_j(t)X}]=0, for each t, j=1,2.”

The paper develops a Bayesian machine learning algorithm (BCF-IV) that adapts Bayesian Causal Forests to instrumental variable settings with imperfect compliance.

“In particular, we propose to modify a machine learning technique, namely, Bayesian Causal Forests, developed for causal inference goals […] to fit an IV setting. The proposed method, Bayesian Instrumental Variable Causal Forest (BCF-IV), is an ensemble semi-parametric Bayesian regression model that directly builds on the Bayesian Additive Regression Trees (BART) algorithm […].”◌ not checked against the paper’s text as it now stands

The paper formally defines the use of an instrumental variable to identify causal effects when treatment receipt is confounded but assignment is not.

“In these scenarios, where one allows for non-compliance between the treatment assigned and the treatment received, one can assume that the assignment is unconfounded, wherein the receipt is confounded […]. In such cases, one can rely on an instrumental variable (IV), Z_i, to draw proper causal inference.”◌ not checked against the paper’s text as it now stands

The paper states and relies on the four classical instrumental variable assumptions (monotonicity, existence of compliers, unconfoundedness, exclusion restriction) to interpret estimated effects as causal.

“In the following we assume the classical four IV assumptions […] – monotonicity, existence of compliers, unconfoundedness of the IV, exclusion restriction – to hold.”

The paper relies on instrumental variables to construct control variables that address time-varying endogeneity, rather than assuming strict exogeneity of regressors.

“We adopt a control function approach (CFA) and assume that instruments Z_i = (z_i1',...,z_iT') and potentially time-varying control variables are available such that once control variables are conditioned on, the residual is mean independent of the regressors.”◌ not checked against the paper’s text as it now stands

The paper contrasts its CFA-based approach, which does not restrict the joint distribution of instruments and random coefficients, with existing panel approaches that use fixed-effect instrumental variable estimators under strict exogeneity.

“One may use the fixed-effect instrumental variable estimator to consistently estimate the slope, see […].”◌ not checked against the paper’s text as it now stands

The paper notes that a related CRE-based approach uses instruments in a linear panel random coefficient model to show that a fixed-effect instrumental variables estimator is consistent for the average partial effect under conditional mean independence between random coefficients and detrended instruments.

“For instance, […] study a linear panel random coefficient model and assume that the random coefficients are conditionally mean independent of the detrended instrument to show that the fixed-effect instrumental variables estimator is consistent to the average partial effect.”

The paper defines an instrumental variable as a random variable independent of preferences but correlated with choice, whose realizations shift observed choice frequencies without changing preference characteristics.

“We define an instrument as a random variable that is independent of preferences but correlated with choice; in particular, the fraction of individuals who choose an alternative changes for each realization of the instrument while preference characteristics do not.”◌ not checked against the paper’s text as it now stands

The paper shows that the existence of such an instrumental variable can be used to reject the hypothesis that all consumers have complete preferences, thereby shrinking the identification region.

“We show that the existence of an instrumental variable can be used to reject the hypothesis that the preferences of all consumers are complete.”◌ not checked against the paper’s text as it now stands

The paper argues intuitively that if choice shares vary with the instrument, then not everyone can be fully able to rank all alternatives, so incomplete preferences must exist for some individuals.

“If such an instrumental variable exists there must be some individuals who have incomplete preferences. Intuitively, if the fraction of individuals who choose a certain alternative changes with the realization of the instrument, it cannot be that all individuals are always able to rank all alternatives.”◌ not checked against the paper’s text as it now stands

The paper extends its difference-in-differences framework to an instrumental-variable setting where the parallel-trends assumption is imposed on an instrument rather than on the treatment itself.

“First, we consider an instrumental-variable (IV) setting in which the parallel-trends assumption is made with respect to an instrument rather than the treatment.”◌ not checked against the paper’s text as it now stands

It shows that assuming parallel trends with respect to an instrument imposes restrictions on treatment-effect heterogeneity, a point it claims to be the first to note.

“We also appear to be the first to note that a parallel-trends assumption with respect to an instrument restricts treatment-effect heterogeneity.”◌ not checked against the paper’s text as it now stands

The paper derives a Wald-type IV result showing that the ratio of the reduced-form WAS effect of the instrument on the outcome to the first-stage WAS effect of the instrument on the treatment identifies a weighted average of treatment-outcome slopes, with weights depending on first-stage magnitudes.

“we then show that the reduced-form WAS effect of the instrument on the outcome divided by the first-stage WAS effect of the instrument on the treatment equals a weighted average of outcome slopes with respect to the treatment, with more weight given to switchers with larger first-stage effects.”

The paper introduces R packages RobustIV and controlfunctionIV that implement methods for causal inference with possibly invalid instrumental variables under linear and nonlinear outcome models.

“We present R software packages and for causal inference with possibly invalid instrumental variables.”◌ not checked against the paper’s text as it now stands

The package implements the Two Stage Hard Thresholding (TSHT) method to select valid instrumental variables from a set of candidate instrumental variables and make inferences for the causal effect in low- and high-dimensional settings.

“It implements the two-stage hard thresholding method to select valid instrumental variables from a set of candidate instrumental variables and make inferences for the causal effect in both low- and high-dimensional settings.”◌ not checked against the paper’s text as it now stands

The package also implements a high-dimensional endogeneity test and a searching and sampling method that provides uniformly valid confidence intervals robust to errors in instrumental variable selection.

“Furthermore, implements the high-dimensional endogeneity test and the searching and sampling method, a uniformly valid inference method robust to errors in instrumental variable selection.”◌ not checked against the paper’s text as it now stands

The paper notes that prior approaches (PRTE and MPRTE) achieve identification of counterfactual policy effects through the availability of a continuous instrumental variable used to manipulate the treatment proportion.

“Identification relies on the a separable threshold model for the selection equation, and the availability of a continuous instrumental variable. In this setting, the proportion of treated individuals is changed by manipulating the instrumental variable.”◌ not checked against the paper’s text as it now stands

The paper explicitly states that its own identification approach does not require an instrumental variable, unlike the PRTE/MPRTE literature.

“Our analysis does not make any assumptions on the selection equation. We do not require an instrumental variable either.”◌ not checked against the paper’s text as it now stands

The paper shows that the mean-independence exogeneity condition combined with a nonlinear relevance (no multicollinearity) condition on the included exogenous regressor can substitute for the exclusion restriction typically required of instrumental variables.

“In this paper, we show that even in the absence of excluded instruments, the endogenous linear regression model can still be identified by leveraging the nonlinear relevance between the included exogenous regressor and the endogenous variable.”◌ not checked against the paper’s text as it now stands

The paper contrasts its mean-projection first-stage approach with the traditional linear first-stage projection used in standard IV/2SLS regressions that rely on an excluded instrumental variable.

“the standard IV regression, or the two-stage least square (2SLS) regression, relies on the availability of an additional variable Z_i, exc and applies a linear first-stage projection as follows: X_i=λ_0+λ_1Z_i+λ_2Z_i, exc+U_i, where the instrumental variable Z_i, exc is required to be exogenous with respect to ϵ_i, relevant for X_i, and excluded from the regression model […].”◌ not checked against the paper’s text as it now stands

The paper proposes a discretization-based estimator that can be computed as a standard 2SLS estimator using partition dummies of the included exogenous regressor as instrumental variables, without needing an excluded instrument.

“Furthermore, the estimator can be computed as a standard 2SLS estimator with partition dummies as IVs.”