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When instruments are weakly correlated with endogenous regressors, conventional methods for instrumental variables (IV) estimation and inference become unreliable. A large literature in econometrics has developed procedures for detecting weak instruments and constructing robust confidence sets, but many of the results in this literature are limited to settings with independent and homoskedastic data, while data encountered in practice frequently violate these assumptions. We review the literature on weak instruments in linear IV regression with an emphasis on results for nonhomoskedastic (heteroskedastic, serially correlated, or clustered) data. To assess the practical importance of weak instruments, we also report tabulations and simulations based on a survey of papers published in the American Economic Review from 2014 to 2018 that use IV. These results suggest that weak instruments remain an important issue for empirical practice, and that there are simple steps that researchers can take to better handle weak instruments in applications.

Instrumental variable (IV) methods are widely used to identify causal effects in models with endogenous explanatory variables. Often the instrument exclusion restriction that underlies the validity of the usual IV inference is suspect; that is, instruments are only plausibly exogenous. We present practical methods for performing inference while relaxing the exclusion restriction. We illustrate the approaches with empirical examples that examine the effect of 401(k) participation on asset accumulation, price elasticity of demand for margarine, and returns to schooling. We find that inference is informative even with a substantial relaxation of the exclusion restriction in two of the three cases.

We consider estimation of and inference about coefficients on endogenous variables in a linear instrumental variables model where the number of instruments and exogenous control variables are each allowed to be larger than the sample size. We work within an approximately sparse framework that maintains that the signal available in the instruments and control variables may be effectively captured by a small number of the available variables. We provide a LASSO-based method for this setting which provides uniformly valid inference about the coefficients on endogenous variables. We illustrate the method through an application to demand estimation.

Instrumental variables have been widely used for estimating the causal effect between exposure and outcome. Conventional estimation methods require complete knowledge about all the instruments' validity; a valid instrument must not have a direct effect on the outcome and not be related to unmeasured confounders. Often, this is impractical as highlighted by Mendelian randomization studies where genetic markers are used as instruments and complete knowledge about instruments' validity is equivalent to complete knowledge about the involved genes' functions. In this article, we propose a method for estimation of causal effects when this complete knowledge is absent. It is shown that causal effects are identified and can be estimated as long as less than 50% of instruments are invalid, without knowing which of the instruments are invalid. We also introduce conditions for identification when the 50% threshold is violated. A fast penalized ℓ 1 estimation method, called sisVIVE, is introduced for estimating the causal effect without knowing which instruments are valid, with theoretical guarantees on its performance. The proposed method is demonstrated on simulated data and a real Mendelian randomization study concerning the effect of body mass index(BMI) on health-related quality of life (HRQL) index. An R package sisVIVE is available on CRAN. Supplementary materials for this article are available online.

In the single-IV model, researchers commonly rely on t-ratio-based inference, even though the literature has quantified its potentially severe large-sample distortions. Building on Stock and Yogo (2005), we introduce the tF critical value function, leading to a standard error adjustment that is a smooth function of the first-stage F-statistic. For one-quarter of specifications in 61 AER papers, corrected standard errors are at least 49 and 136 percent larger than conventional 2SLS standard errors at the 5 percent and 1 percent significance levels, respectively. tF confidence intervals have shorter expected length than those of Anderson and Rubin (1949), whenever both are bounded.

We develop results for the use of Lasso and post-Lasso methods to form first-stage predictions and estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p. Our results apply even when p is much larger than the sample size, n. We show that the IV estimator based on using Lasso or post-Lasso in the first stage is root-n consistent and asymptotically normal when the first stage is approximately sparse, that is, when the conditional expectation of the endogenous variables given the instruments can be well-approximated by a relatively small set of variables whose identities may be unknown. We also show that the estimator is semiparametrically efficient when the structural error is homoscedastic. Notably, our results allow for imperfect model selection, and do not rely upon the unrealistic \"beta-min\" conditions that are widely used to establish validity of inference following model selection (see also Belloni, Chernozhukov, and Hansen (2011b)). In simulation experiments, the Lasso-based IV estimator with a data-driven penalty performs well compared to recently advocated many-instrument robust procedures. In an empirical example dealing with the effect of judicial eminent domain decisions on economic outcomes, the Lasso-based IV estimator outperforms an intuitive benchmark. Optimal instruments are conditional expectations. In developing the IV results, we establish a series of new results for Lasso and post-Lasso estimators of nonparametric conditional expectation functions which are of independent theoretical and practical interest. We construct a modification of Lasso designed to deal with non-Gaussian, heteroscedastic disturbances that uses a data-weighted 𝓁₁-penalty function. By innovatively using moderate deviation theory for self-normalized sums, we provide convergence rates for the resulting Lasso and post-Lasso estimators that are as sharp as the corresponding rates in the homoscedastic Gaussian case under the condition that log p = o(n 1/3 ). We also provide a data-driven method for choosing the penalty level that must be specified in obtaining Lasso and post-Lasso estimates and establish its asymptotic validity under non-Gaussian, heteroscedastic disturbances.

Authors often add covariates to a base model sequentially either to test a particular coefficient’s “robustness” or to account for the “effects” on this coefficient of adding covariates. This is problematic, due to sequence sensitivity when added covariates are intercorrelated. Using the omitted variables bias formula, I construct a conditional decomposition that accounts for various covariates’ role in moving base regressors’ coefficients. I also provide a consistent covariance formula. I illustrate this conditional decomposition with NLSY data in an application that exhibits sequence sensitivity. Related extensions include instrumental variables, the fact that my decomposition nests the Oaxaca-Blinder decomposition, and a Hausman test result.

Many studies use shift-share (or “Bartik”) instruments, which average a set of shocks with exposure share weights. We provide a new econometric framework for shift-share instrumental variable (SSIV) regressions in which identification follows from the quasi-random assignment of shocks, while exposure shares are allowed to be endogenous. The framework is motivated by an equivalence result: the orthogonality between a shift-share instrument and an unobserved residual can be represented as the orthogonality between the underlying shocks and a shock-level unobservable. SSIV regression coefficients can similarly be obtained from an equivalent shock-level regression, motivating shock-level conditions for their consistency. We discuss and illustrate several practical insights of this framework in the setting of Autor et al. (2013), estimating the effect of Chinese import competition on manufacturing employment across U.S. commuting zones.

Several methods have been proposed for partially or point identifying the average treatment effect (ATE) using instrumental variable (IV) type assumptions. The descriptions of these methods are widespread across the statistical, economic, epidemiologic, and computer science literature, and the connections between the methods have not been readily apparent. In the setting of a binary instrument, treatment, and outcome, we review proposed methods for partial and point identification of the ATE under IV assumptions, express the identification results in a common notation and terminology, and propose a taxonomy that is based on sets of identifying assumptions. We further demonstrate and provide software for the application of these methods to estimate bounds. Supplementary materials for this article are available online.

Marketing is a field that is rich in data. Our data is of high quality, often at a highly disaggregate level, and there is considerable variation in the key variables for which estimates of effects on outcomes such as sales and profits are desired. The recognition that, in some general sense, marketing variables are set by firms on the basis of information not always observable by the researcher has led to concerns regarding endogeneity and widespread pressure to implement instrumental variables methods in marketing problems. The instruments used in our empirical literature are rarely valid and the IV methods used can have poor sampling properties, including substantial finite sample bias and large sampling errors. Given the problems with IV methods, a convincing argument must be made that there is a first order endogeneity problem and that we have strong and valid instruments before these methods should be used. If strong and valid instruments are not available, then researchers need to look toward supplementing the information available to them. For example, if there are concerns about unobservable advertising or promotional variables, then the researcher is much better off measuring these variables rather than using instruments (such as lagged marketing variables) that are clearly invalid. Ultimately, only randomized variation in marketing variables (with proper implementation and large samples) can be argued to be a valid instrument without further assumptions.