Robust Bayesian Estimation in Causal Two-Stage Least Squares Modeling with Instrumental Variables

Political scientists frequently use instrumental variables (IV) estimation to estimate the causal effect of an endogenous treatment variable. However, when the treatment effect is heterogeneous, this estimation strategy only recovers the local average treatment effect (LATE). The LATE is an average treatment effect (ATE) for a subset of the population: units that receive treatment if and only if they are induced by an exogenous IV. However, researchers may instead be interested in the ATE for the entire population of interest. In this article, we develop a simple reweighting method for estimating the ATE, shedding light on the identification challenge posed in moving from the LATE to the ATE. We apply our method to two published experiments in political science in which we demonstrate that the LATE has the potential to substantively differ from the ATE.

Recent theoretical contributions to the problem of job mobility decisions during the transition from a centrally planned economy to a market economy suggest that individuals self-select into specific forms of job mobility behaviour on the basis of their unobservable characteristics. In this case, standard results from both simple treatment-control comparisons of average income and from OLS regressions of reduced-form earnings equations do generally not identify any causal effect of job mobility on income. This paper addresses the endogeneity problem related to job mobility in a quasi-experimental framework and estimates the returns to (early) job mobility in the Eastern German transition process for the period 1990-96. Identification is based on instrumental variables estimation. Two instruments are suggested that account for some of the variation in the job mobility behaviour throughout transition. They allow further for a natural distinction between forced and voluntary movers on the basis of the local average treatment effect (LATE) interpretation of IV estimates. Estimation results contrast strongly with results from both simple treatment-control comparisons and simple OLS estimation. They suggest negative returns to forced job mobility as opposed to positive returns to voluntary job mobility early in transition, thus stressing the existence of heterogeneous returns to job mobility.

This paper provides a review of methodological advancements in the evaluation of heterogeneous treatment effect models based on instrumental variable (IV) methods. We focus on models that achieve identification by assuming monotonicity of the treatment in the IV and analyze local average and quantile treatment effects for the subpopulation of compliers. We start with a comprehensive discussion of the binary treatment and binary IV case as for instance relevant in randomized experiments with imperfect compliance. We then review extensions to identification and estimation with covariates, multi-valued and multiple treatments and instruments, outcome attrition and measurement error, and the identification of direct and indirect treatment effects, among others. We also discuss testable implications and possible relaxations of the IV assumptions, approaches to extrapolate from local to global treatment effects, and the relationship to other IV approaches.

One practical challenge in observational studies and quasi-experimental designs is selection bias. The issue of selection bias becomes more concerning when data are non-normal and contain missing values. Recently, a Bayesian robust two-stage causal modeling with instrumental variables was developed and has the advantages of addressing selection bias and handle non-normal data and missing data simultaneously in one model. The method provides reliable parameter and standard error estimates when missing data and outliers exist. The modeling technique can be widely applied to empirical studies particularly in social, psychological and behavioral areas where any of the three issues (e.g., selection bias, data with outliers and missing data) is commonly seen. To implement this method, we developed an R package named ALMOND (Analysis of LATE (Local Average Treatment Effect) for Missing Or/and Nonnormal Data). Package users have the flexibility to directly apply the Bayesian robust two-stage causal models or write their own Bayesian models from scratch within the package. To facilitate the application of the Bayesian robust two-stage causal modeling technique, we provide a tutorial for the ALMOND package in this article, and illustrate the application with two examples from empirical research.

Nonadherence to assigned treatment is common in randomized controlled trials (RCTs). Recently, there has been increased interest in estimating causal effects of treatment received, for example, the so-called local average treatment effect (LATE). Instrumental variables (IV) methods can be used for identification, with estimation proceeding either via fully parametric mixture models or two-stage least squares (TSLS). TSLS is popular but can be problematic for binary outcomes where the estimand of interest is a causal odds ratio. Mixture models are rarely used in practice, perhaps because of their perceived complexity and need for specialist software. Here, we propose using multiple imputation (MI) to impute the latent compliance class appearing in the mixture models. Since such models include an interaction term between the latent compliance class and randomized treatment, we use "substantive model compatible" MI (SMC MIC), which can additionally handle missing data in outcomes and other variables in the model, before fitting the mixture models via maximum likelihood to the MI data sets and combining results via Rubin's rules. We use simulations to compare the performance of SMC MIC to existing approaches and also illustrate the methods by reanalyzing an RCT in UK primary health. We show that SMC MIC can be more efficient than full Bayesian estimation when auxiliary variables are incorporated, and is superior to two-stage methods, especially for binaryoutcomes.

A nationally representative dataset from the Philippines is used to derive returns to schooling estimates from instrumental variables, utilizing a supply-side intervention in the education market capable of generating significant changes in schooling. These estimates apply to a subgroup of, mainly, liquidity constrained individuals, in the spirit of the Local Average Treatment Effect (LATE) literature. Returns to schooling estimates which apply to a subgroup of individuals affected by policy intervention may be more interesting from a policy perspective than the return to the ‘average’ individual. The findings are consistent with other recent evidence suggesting that the causal effect of education, at least for certain subgroups of individuals, is as big or bigger than what is suggested by OLS estimates.

Researchers commonly use jackknife-based instrumental variables estimators to eliminate the many-instruments bias of two-stage least squares. Where inference must be clustered, however, the jackknife fails to eliminate the bias. We propose a cluster-jackknife approach in which first-stage predicted values for each observation are constructed from a regression that leaves out the observation’s entire cluster. The cluster-jackknife instrumental variables estimator (CJIVE) eliminates many-instruments bias, and consistently estimates causal effects in the traditional linear model and local average treatment effects in the heterogeneous treatment effects framework. We illustrate the method in an application estimating the effects of pre-trial detention in Miami-Dade County.

Assumptions that are sufficient to identify local average treatment effects (LATEs) generate necessary conditions that allow instrument validity to be refuted. The degree to which instrument validity is violated, however, probably varies across subpopulations. In this article, we use causal forests to search and test for such local violations of the LATE assumptions in a data-driven way. Unlike previous instrument validity tests, our procedure is able to detect local violations. We evaluate the performance of our procedure in simulations and apply it in two different settings: parental preferences for mixed-sex composition of children and the Vietnam draft lottery.

We examine the heterogeneous and dynamic impact of China's New Rural Pension Scheme on intergenerational wealth dependence using a nationally representative longitudinal household survey covering the period 2011-13.We adopt an instrumental quantile regressiondiscontinuity design to address the endogeneity of partial compliance of the pension scheme and the observed individual heterogeneity.Overall, we find that the pension scheme smooths wealth dependence between generations in the short term, but strengthens the persistence of assets among the wealthiest households in the medium term.The mechanisms underlying these distributional effects are intergenerational transfers, time reallocation, and filial adjustments of the wealth portfolio.Complementary policy interventions, particularly for the poor across generations, would be needed to neutralize the distributional impact of the pension in terms of intergenerational wealth persistence.

Recent research has demonstrated the importance of flexibly controlling for covariates in instrumental variables estimation. In this article we study the finite sample and asymptotic properties of various weighting estimators of the local average treatment effect (LATE), motivated by Abadie’s kappa theorem and offering the requisite flexibility relative to standard practice. We argue that two of the estimators under consideration, which are weight normalized, are generally preferable. Several other estimators, which are unnormalized, do not satisfy the properties of scale invariance with respect to the natural logarithm and translation invariance, thereby exhibiting sensitivity to the units of measurement when estimating the LATE in logs and the centering of the outcome variable more generally. We also demonstrate that, when noncompliance is one sided, certain weighting estimators have the advantage of being based on a denominator that is strictly greater than zero by construction. This is the case for only one of the two normalized estimators, and we recommend this estimator for wider use. We illustrate our findings with a simulation study and three empirical applications, which clearly document the sensitivity of unnormalized estimators to how the outcome variable is coded. We implement the proposed estimators in the Stata package kappalate.

The interpretation of estimates from Moving to Opportunity (MTO) as neighborhood effects has created significant controversy among social scientists. This paper presents a framework that clarifies the interpretation of results from the MTO housing mobility experiment. The paper defi nes several neighborhood treatments and estimates their Local Average Treatment Effects (LATEs) using assigned treatment in MTO as an instrumental variable. This framework clarifies that while parameters estimated in the literature do not suffer from selection bias, selection into treatment is an inescapable issue if one seeks to learn about neighborhood effects from MTO. The LATE parameters estimated in this paper are neighborhood effects for the subgroup of MTO families who are compliers with respect to the defined treatment. In contrast, the Treatment-on-the-Treated (TOT) parameters reported in the literature are program effects.

Four alternative but related approaches to empirical evaluation of policy interventions are studied: social experiments, natural experiments, matching methods, and instrumental variables. In each case the necessary assumptions and the data requirements are considered for estimation of a number of key parameters of interest. These key parameters include the average treatment effect, the treatment of the treated and the local average treatment effect. Some issues of implementation and interpretation are discussed drawing on the labour market programme evaluation literature.

Thetdistribution is a useful extension of the normal distribution, which can be used for statistical modeling of data sets with heavy tails, and provides robust estimation. In this paper, in view of the advantages of Bayesian analysis, we propose a new robust coefficient estimation and variable selection method based on Bayesian adaptive Lassotregression. A Gibbs sampler is developed based on the Bayesian hierarchical model framework, where we treat thetdistribution as a mixture of normal and gamma distributions and put different penalization parameters for different regression coefficients. We also consider the Bayesiantregression with adaptive group Lasso and obtain the Gibbs sampler from the posterior distributions. Both simulation studies and real data example show that our method performs well compared with other existing methods when the error distribution has heavy tails and/or outliers.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

Analyzing Zero Returns to Education in Germany: Heterogeneous Effects and Skill Formation