Linear Instrumental Variables Model Research Articles

LASSO-type instrumental variable selection methods with an application to Mendelian randomization.

Valid instrumental variables (IVs) must not directly impact the outcome variable and must also be uncorrelated with nonmeasured variables. However, in practice, IVs are likely to be invalid. The existing methods can lead to large bias relative to standard errors in situations with many weak and invalid instruments. In this paper, we derive a LASSO procedure for the k-class IV estimation methods in the linear IV model. In addition, we propose the jackknife IV method by using LASSO to address the problem of many weak invalid instruments in the case of heteroscedastic data. The proposed methods are robust for estimating causal effects in the presence of many invalid and valid instruments, with theoretical assurances of their execution. In addition, two-step numerical algorithms are developed for the estimation of causal effects. The performance of the proposed estimators is demonstrated via Monte Carlo simulations as well as an empirical application. We use Mendelian randomization as an application, wherein we estimate the causal effect of body mass index on the health-related quality of life index using single nucleotide polymorphisms as instruments for body mass index.

A POWERFUL SUBVECTOR ANDERSON–RUBIN TEST IN LINEAR INSTRUMENTAL VARIABLES REGRESSION WITH CONDITIONAL HETEROSKEDASTICITY

We introduce a new test for a two-sided hypothesis involving a subset of the structural parameter vector in the linear instrumental variables (IVs) model. Guggenberger, Kleibergen, and Mavroeidis (2019, Quantitative Economics, 10, 487–526; hereafter GKM19) introduce a subvector Anderson–Rubin (AR) test with data-dependent critical values that has asymptotic size equal to nominal size for a parameter space that allows for arbitrary strength or weakness of the IVs and has uniformly nonsmaller power than the projected AR test studied in Guggenberger et al. (2012, Econometrica, 80(6), 2649–2666). However, GKM19 imposes the restrictive assumption of conditional homoskedasticity (CHOM). The main contribution here is to robustify the procedure in GKM19 to arbitrary forms of conditional heteroskedasticity. We first adapt the method in GKM19 to a setup where a certain covariance matrix has an approximate Kronecker product (AKP) structure which nests CHOM. The new test equals this adaptation when the data are consistent with AKP structure as decided by a model selection procedure. Otherwise, the test equals the AR/AR test in Andrews (2017, Identification-Robust Subvector Inference, Cowles Foundation Discussion Papers 3005, Yale University) that is fully robust to conditional heteroskedasticity but less powerful than the adapted method. We show theoretically that the new test has asymptotic size bounded by the nominal size and document improved power relative to the AR/AR test in a wide array of Monte Carlo simulations when the covariance matrix is not too far from AKP.

Constrained Conditional Moment Restriction Models

Shape restrictions have played a central role in economics as both testable implications of theory and sufficient conditions for obtaining informative counterfactual predictions. In this paper, we provide a general procedure for inference under shape restrictions in identified and partially identified models defined by conditional moment restrictions. Our test statistics and proposed inference methods are based on the minimum of the generalized method of moments (GMM) objective function with and without shape restrictions. Uniformly valid critical values are obtained through a bootstrap procedure that approximates a subset of the true local parameter space. In an empirical analysis of the effect of childbearing on female labor supply, we show that employing shape restrictions in linear instrumental variables (IV) models can lead to shorter confidence regions for both local and average treatment effects. Other applications we discuss include inference for the variability of quantile IV treatment effects and for bounds on average equivalent variation in a demand model with general heterogeneity.

Discussion on "Instrumented difference-in-differences" by Ting Ye, Ashkan Ertefaie, James Flory, Sean Hennessy & Dylan S. Small.

We reinterpret the instrumented difference-in-differences (iDID) under a linear instrumental variables (IV) model. Under the linear IV model, we show why iDID is a clear improvement over two existing methods, difference-in-differences (DID) and a cross-sectional, IV analysis. We also re-express some of the assumptions of iDID using familiar, regression-based identification assumptions. We conclude with a method inspired by the linear IV model that can potentially remedy the weak identification problem iniDID.

Ivcrc: An instrumental-variables estimator for the correlated random-coefficients model

We discuss the ivcrc command, which implements an instrumental-variables (IV) estimator for the linear correlated random-coefficients model. The correlated random-coefficients model is a natural generalization of the standard linear IV model that allows for endogenous, multivalued treatments and unobserved heterogeneity in treatment effects. The estimator implemented by ivcrc uses recent semiparametric identification results that allow for flexible functional forms and permit instruments that may be binary, discrete, or continuous. The ivcrc command also allows for the estimation of varying-coefficient regressions, which are closely related in structure to the proposed IV estimator. We illustrate the use of ivcrc by estimating the returns to education in the National Longitudinal Survey of Young Men.

Ivcrc: An Instrumental Variables Estimator for the Correlated Random Coefficients Model

<ns3:p>We discuss the ivcrc module, which implements an instrumental variables (IV) estimator for the linear correlated random coefficients (CRC) model. The CRC model is a natural generalization of the standard linear IV model that allows for endogenous, multivalued treatments and unobserved heterogeneity in treatment effects. The estimator implemented by ivcrc uses recent semiparametric identification results that allow for flexible functional forms and permit instruments that may be binary, discrete, or continuous. The ivcrc module also allows for the estimation of varying coefficients regressions, which are closely related in structure to the proposed IV estimator. We illustrate use of ivcrc by estimating the returns to education in the National Longitudinal Survey of Young Men.</ns3:p>

Inference for the Linear IV Model Ridge Estimator Using Training and Test Samples

The asymptotic distribution is presented for the linear instrumental variables model estimated with a ridge penalty and a prior where the tuning parameter is selected with a holdout sample. The structural parameters and the tuning parameter are estimated jointly by method of moments. A chi-squared statistic permits confidence regions for the structural parameters. The form of the asymptotic distribution provides insights on the optimal way to perform the split between the training and test sample. Results for the linear regression estimated by ridge regression are presented as a special case.

Robust and Optimal Estimation for Partially Linear Instrumental Variables Models with Partial Identification

This paper studies robust and optimal estimation of the slope coefficients in a partially linear instrumental variables model with nonparametric partial identification. We establish the root-n asymptotic normality of a penalized sieve minimum distance estimator of the slope coefficients. We show that the asymptotic normality holds regardless of whether the nonparametric function is point identified or only partially identified. However, in the presence of nonparametric partial identification, the model is not regular in the sense of Bickel et al. (1993) and the asymptotic variance matrix may depend on the penalty, so classical efficiency analysis does not apply. We then develop an optimally penalized estimator which minimizes the asymptotic variance of a linear functional of the slope coefficients estimator through employing an optimal penalty, and propose a feasible two-step procedure. To conduct inference, a consistent variance matrix estimator is provided. Monte Carlos simulations examine finite sample performance of our penalized estimators.

Robust and optimal estimation for partially linear instrumental variables models with partial identification

This paper studies robust and optimal estimation of the slope coefficients in a partially linear instrumental variables model with nonparametric partial identification. We establish the root-n asymptotic normality of a penalized sieve minimum distance estimator of the slope coefficients. We show that the asymptotic normality holds regardless of whether the nonparametric function is point identified or only partially identified. However, in the presence of nonparametric partial identification, the slope coefficients may not be continuous in the underlying distribution and the asymptotic variance matrix may depend on the penalty, so classical efficiency analysis does not apply. We instead develop an optimally penalized estimator that minimizes the asymptotic variance of a linear functional of the slope coefficients estimator by employing an optimal penalty for a given weight, and propose a feasible two-step procedure. We also propose an iterated procedure to address how to choose both penalty and weight optimally and further improve efficiency. To conduct inference, we provide a consistent variance matrix estimator. Monte Carlo simulations examine the finite sample performance of our estimators.

Ivcrc: An Instrumental Variables Estimator for the Correlated Random Coefficients Model

We present the ivcrc command, which implements an instrumental variables (IV) estimator for the linear correlated random coefficients (CRC) model. This model is a natural generalization of the standard linear IV model that allows for endogenous, multivalued treatments and unobserved heterogeneity in treatment effects. The proposed estimator uses recent semiparametric identification results that allow for flexible functional forms and permit instruments that may be binary, discrete, or continuous. The command also allows for the estimation of varying coefficients regressions, which are closely related in structure to the proposed IV estimator. We illustrate this IV estimator and the ivcrc command by estimating the returns to education in the National Longitudinal Survey of Young Men.

Generic results for establishing the asymptotic size of confidence sets and tests

This paper provides a set of results that can be used to establish the asymptotic size and/or similarity in a uniform sense of confidence sets and tests. The results are generic in that they can be applied to a broad range of problems. They are most useful in scenarios where the pointwise asymptotic distribution of a test statistic is a discontinuous function of a parameter.The results are illustrated in several examples. These are: (i) the conditional likelihood ratio test of Moreira (2003) for linear instrumental variables models with instruments that may be weak, extended to the case of heteroskedastic errors; (ii) the grid bootstrap confidence interval of Hansen (1999) for the sum of the AR coefficients in a kth order autoregressive model with unknown innovation distribution, and (iii) the standard quasi-likelihood ratio test in a nonlinear regression model where identification is lost when the coefficient on the nonlinear regressor is zero. In addition, as a simple running example, we consider a two-sided equal-tailed CI for the AR coefficient in an AR(1) model, which is a simplified version of the CI in Andrews and Guggenberger (2014).

Robust estimation with many instruments

Linear instrumental variables models are widely used in empirical work, but often associated with low estimator precision. This paper proposes an estimator that is robust to outliers and shows that the estimator is minimax optimal in a class of estimators that includes the limited maximum likelihood estimator (LIML). Intuitively, this optimal robust estimator combines LIML with Winsorization of the structural residuals and the Winsorization leads to improved precision under thick-tailed error distributions. Consistency and asymptotic normality of the estimator are established under many instruments asymptotics and a consistent variance estimator which allows for asymptotically valid inference is provided.

Chinese imports’ impacts on Brazil’s inter-industry wage premium

PurposeThe purpose of this paper is to investigate the effects of Chinese and non-Chinese import penetration on the inter-industry wage premium, and how such effects vary according to the unskilled-labor intensity of the industry and to the implementation of the Nova Matriz Economica policy in 2008.Design/methodology/approachThe paper empirically examines the effects of the Chinese and non-Chinese import penetration on the wage premium using a linear instrumental variables model and data from Brazilian household surveys and censuses.FindingsThe estimates show the Chinese import penetration positively affecting the wage premium in unskilled-labor intensive. And the implementation of the new macroeconomic policy strengthened this effect.Originality/valueTo the best of the author’s knowledge, this is the first paper to study the effects of Chinese and non-Chinese import penetration on the inter-industry wage premium.

A more powerful subvector Anderson Rubin test in linear instrumental variables regression

We study subvector inference in the linear instrumental variables model assuming homoskedasticity but allowing for weak instruments. The subvector Anderson and Rubin (1949) test that uses chi square critical values with degrees of freedom reduced by the number of parameters not under test, proposed by Guggenberger, Kleibergen, Mavroeidis, and Chen (2012), controls size but is generally conservative. We propose a conditional subvector Anderson and Rubin test that uses data‐dependent critical values that adapt to the strength of identification of the parameters not under test. This test has correct size and strictly higher power than the subvector Anderson and Rubin test by Guggenberger et al. (2012). We provide tables with conditional critical values so that the new test is quick and easy to use. Application of our method to a model of risk preferences in development economics shows that it can strengthen empirical conclusions in practice.

On the structure of IV estimands

When the overidentifying restrictions of the constant-effect linear instrumental variables model fail, common IV estimators converge to different probability limits. I characterize the estimands of two stage least squares, two step GMM, and limited information maximum likelihood as functions of the single-instrument estimands from the just-identified IV regressions which consider each instrument separately. The limited information maximum likelihood estimand is found to be discontinuous on a set of dimension equal to the number of instruments minus one, and to equal the full parameter space on a set of dimension equal to the number of instruments minus two.

Implementing Valid Two-Step Identification-Robust Confidence Sets for Linear Instrumental-Variables Models

In this article, we consider inference in the linear instrumental-variables models with one or more endogenous variables and potentially weak instruments. I developed a command, twostepweakiv, to implement the two-step identification-robust confidence sets proposed by Andrews (2018, Review of Economics and Statistics 100: 337–348) based on Wald tests and linear combination tests (Andrews, 2016, Econometrica 84: 2155–2182). Unlike popular procedures based on first-stage F statistics (Stock and Yogo, 2005, Testing for weak instruments in linear IV regression, in Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg), the two-step identification-robust confidence sets control coverage distortion without assuming the data are homoskedastic. I demonstrate the use of twostepweakiv with an example of analyzing the effect of wages on married female labor supply. For inference on subsets of parameters, twostepweakiv also implements the refined projection method (Chaudhuri and Zivot, 2011, Journal of Econometrics 164: 239–251). I illustrate that this method is more powerful than the conventional projection method using Monte Carlo simulations.

On the Use of the Lasso for Instrumental Variables Estimation with Some Invalid Instruments

ABSTRACTWe investigate the behavior of the Lasso for selecting invalid instruments in linear instrumental variables models for estimating causal effects of exposures on outcomes, as proposed recently by Kang et al. Invalid instruments are such that they fail the exclusion restriction and enter the model as explanatory variables. We show that for this setup, the Lasso may not consistently select the invalid instruments if these are relatively strong. We propose a median estimator that is consistent when less than 50% of the instruments are invalid, and its consistency does not depend on the relative strength of the instruments, or their correlation structure. We show that this estimator can be used for adaptive Lasso estimation, with the resulting estimator having oracle properties. The methods are applied to a Mendelian randomization study to estimate the causal effect of body mass index (BMI) on diastolic blood pressure, using data on individuals from the UK Biobank, with 96 single nucleotide polymorphisms as potential instruments for BMI. Supplementary materials for this article are available online.

On Bootstrap inconsistency and Bonferroni-based size-correction for the subset Anderson–Rubin test under conditional homoskedasticity

We focus on the linear instrumental variable model with two endogenous regressors under conditional homoskedasticity, and study the subset Anderson and Rubin (1949, AR) test when the nuisance structural parameter, the unrestricted slope coefficient of endogenous regressor, may be weakly identified. Weak identification leads to nonstandard null limiting distributions, and alternative to the usual chi-squared critical value is needed. We first investigate the bootstrap validity for the subset AR test based on various plug-in estimators, and show that the bootstrap provides asymptotic refinement when the nuisance structural parameter is strongly identified, but is inconsistent when it is weakly identified. This is in contrast to the result of bootstrap validity in Moreira et al. (2009). Then, we propose a Bonferroni-based size-correction method that yields correct asymptotic size for all the test statistics considered. The power performance of size-corrected tests can be further improved by applying the mapping between structural and endogenous parameters in the model. Monte Carlo experiments confirm the bootstrap inconsistency and demonstrate that all the subset tests based on our correction technique control the size.

Minimum distance approach to inference with many instruments

I analyze a linear instrumental variables model with a single endogenous regressor and many instruments. I use invariance arguments to construct a new minimum distance objective function. With respect to a particular weight matrix, the minimum distance estimator is equivalent to the random effects estimator of Chamberlain and Imbens (2004), and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator. This weight matrix is inefficient unless the errors are normal, and I construct a new, more efficient estimator based on the optimal weight matrix. Finally, I show that when the minimum distance objective function does not impose a proportionality restriction on the reduced-form coefficients, the resulting estimator corresponds to a version of the bias-corrected two-stage least squares estimator. I use the objective function to construct confidence intervals that remain valid when the proportionality restriction is violated.

Double/debiased machine learning for treatment and structural parameters

We revisit the classic semiparametric problem of inference on a low dimensional parameter θ_0 in the presence of high-dimensional nuisance parameters η_0. We depart from the classical setting by allowing for η_0 to be so high-dimensional that the traditional assumptions, such as Donsker properties, that limit complexity of the parameter space for this object break down. To estimate η_0, we consider the use of statistical or machine learning (ML) methods which are particularly well-suited to estimation in modern, very high-dimensional cases. ML methods perform well by employing regularization to reduce variance and trading off regularization bias with overfitting in practice. However, both regularization bias and overfitting in estimating η_0 cause a heavy bias in estimators of θ_0 that are obtained by naively plugging ML estimators of η_0 into estimating equations for θ_0. This bias results in the naive estimator failing to be N^(-1/2) consistent, where N is the sample size. We show that the impact of regularization bias and overfitting on estimation of the parameter of interest θ_0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman-orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ_0, and (2) making use of cross-fitting which provides an efficient form of data-splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in a N^(-1/2)-neighborhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid confidence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of DML applied to learn the main regression parameter in a partially linear regression model, DML applied to learn the coefficient on an endogenous variable in a partially linear instrumental variables model, DML applied to learn the average treatment effect and the average treatment effect on the treated under unconfoundedness, and DML applied to learn the local average treatment effect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.