Instrumental Variables: An Econometrician's Perspective

Guido W. Imbens

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Instrumental Variables: An Econometrician's Perspective

Guido W. Imbens

arXiv:1410.0163v1Oct 1, 2014stat.ME

TL;DR

The paper reviews instrumental variables methods by connecting modern statistical applications with earlier econometric work, while examining the assumptions that determine when causal inferences are credible. It discusses economic applications, randomized experiments with noncompliance, identification, and instrument relevance to clarify the methods’ applicability.

Problem
Instrumental variables applications require context about their econometric foundations and assumptions, especially because causal inferences depend on when those assumptions are credible.
Method
The paper reviews older econometric applications, relates them to modern statistical settings, and explains instrumental variables using modern framework and notation.
Results
The review shows that observations on equilibrium prices and quantities alone do not identify supply and demand functions, while instrumental variables require both exclusion from outcomes and correlation with treatment.
Takeaways & Limitations
Connecting econometric history with current statistical applications can improve understanding of instrumental variables’ applicability and help identify potential instruments.
Takeaways & Limitations
Instrumental variables interpretations depend on assumptions such as market equilibrium and monotonicity, and violations can affect the estimand’s interpretation.

Abstract

from arXiv · show

I review recent work in the statistics literature on instrumental variables methods from an econometrics perspective. I discuss some of the older, economic, applications including supply and demand models and relate them to the recent applications in settings of randomized experiments with noncompliance. I discuss the assumptions underlying instrumental variables methods and in what settings these may be plausible. By providing context to the current applications, a better understanding of the applicability of these methods may arise.

1. INTRODUCTION

Instrumental variables methods connect econometric work on causal inference without no unmeasured confounding to recent statistical applications. The paper reviews this history and terminology to clarify applicability, assumptions, and links between the literatures.

Instrumental variables methods address causal inference when treatment cannot credibly be viewed as randomly assigned and no unmeasured confounders does not hold.
Recent statistics work differs from earlier econometric literature by focusing primarily on binary treatments, treatment-effect heterogeneity, and explicit assumptions for instrumental-variables analyses.
Instrumental-variables methods have had limited impact on statistical thinking despite technical contributions appearing in statistics journals and noneconomists rarely applying them.
The paper uses early econometric work to provide background for modern instrumental-variables applications and help identify potential instruments.
It relates econometric terminology to statistical terminology to reduce semantic barriers separating the two literatures.

2. CHOICE VERSUS CHANCE IN TREATMENT ASSIGNMENT

Statistics traditionally emphasizes chance through randomized experiments, whereas econometrics emphasizes choice and the behavior producing treatment exposure. The paper contrasts these perspectives and explains how instrumental variables address treatment selection by changing incentives without changing potential outcomes.

2. CHOICE VERSUS CHANCE IN TREATMENT ASSIGNMENT: Statistical and econometric causal analyses differ mainly in their traditional emphasis on chance versus choice in treatment assignment.
2.1 The Statistics Literature: The Focus on Chance: Randomized experiments assign treatment independently of potential outcomes, enabling exact causal inferences in the Fisher–Neyman framework.
2.1 The Statistics Literature: The Focus on Chance: Observational-study methods extend randomized-experiment results using pretreatment covariates and assumptions such as unconfoundedness or strong ignorability.
2.1 The Statistics Literature: The Focus on Chance: Under weak or strong ignorability, the average treatment effect is identified in large samples and can be estimated using matching, subclassification, or regression.
2.2 The Econometrics Literature: The Focus on Choice: Econometric analysis models individuals as agents whose objectives and constraints influence their treatment choices.
2.3 Some Examples: The Roy model challenges unconfoundedness because different treatment choices may indicate different potential outcomes, though its assumption of known ex ante outcomes is implausible in practice.
2.4 Instrumental Variables: Instrumental variables change incentives to choose treatment without affecting treatment-specific potential outcomes, requiring the incentive-related net benefits to differ from the outcome of interest.
2.4 Instrumental Variables: The absence of a direct incentive effect on potential outcomes is controversial and must be assessed case by case.

3. THE CLASSIC EXAMPLE: SUPPLY AND DEMAND

The supply-and-demand example shows why equilibrium prices and quantities do not identify causal effects by themselves, and how instrumental variables can help identify demand while leaving supply harder to identify.

Motivation: Economists model market prices through simultaneous demand and supply functions because equilibrium creates endogeneity.The paper uses the Fulton fish market to illustrate this problem and contrasts economic choice with statistical random assignment.
Supply and demand functions: Demand is represented by potential quantities at each possible price, extending the potential-outcomes framework beyond binary treatments.The specification assumes a common constant price effect across prices and markets, which the paper characterizes as strong.
The statistical demand curve: The equilibrium regression of log quantity on log price is not generally informative about either structural curve because its coefficient depends on supply-and-demand disturbances.Under zero covariance between disturbances, the coefficient becomes a variance-weighted average of the two structural slopes; under particular conditions it can approximate the supply slope.
The effect of a tax increase: A tax changes sellers’ received prices and reduces traded quantity when buyers’ and sellers’ responses operate through the tax-induced prices.The causal effect is proportional to supply and demand elasticities and, for small tax rates, to the tax itself.
Identification with instrumental variables: The paper emphasizes that observations on equilibrium prices and quantities alone cannot identify demand and supply functions separately.Predicting a new tax therefore requires learning the two structural functions rather than relying on the reduced-form regression.
Identification with instrumental variables: In the fish-market illustration, stormy weather serves as an instrument because stormy days have higher average prices and lower quantities, identifying the demand slope.The estimated demand function passes through the average log price and quantity for stormy and fair-weather days.
Identification with instrumental variables: The supply curve is harder to point identify because the data lack an instrument that shifts demand without directly affecting supply.Without such an instrument, the tax effect on quantity and prices is not point identified, although weaker assumptions can yield bounds.

4. A MODERN EXAMPLE: RANDOMIZED EXPERIMENTS WITH NONCOMPLIANCE AND HETEROGENOUS TREATMENT EFFECTS

The paper uses randomized experiments with noncompliance to show how instrumental variables assumptions determine which causal effects are identified. It contrasts point identification for compliers with weighted interpretations and partial-identification bounds when stronger assumptions fail.

4.1 The McDonald, Hiu and Tierney (1992) Data: The influenza study randomized physicians to assign letters rather than directly assigning vaccinations, creating noncompliance between assignment and treatment received.The analysis observes assignment, actual vaccination receipt, and corresponding potential outcomes for 2,861 individuals; physician-level randomization is treated as unclustered, which underestimates standard errors.
4.2 The Assumptions Underlying Instrumental Variables: Instrumental variables analysis requires assumptions about instrument assignment, exclusion of direct assignment effects, and often monotonicity, alongside no interference.Randomized assignment is often satisfied by design, whereas exclusion is substantive and typically controversial; observational applications may instead require unconfounded assignment given covariates.
4.5 Local Average Treatment Effects: Under monotonicity and exclusion, the local average treatment effect is identified for compliers rather than necessarily for the full population.This result relies on the instrument affecting treatment receipt in a directionally consistent way and on assignment having no direct outcome effect.
4.3 Point Identification versus Bounds: When the average treatment effect is not point-identified, partial identification can retain the original estimand by reporting bounds based on restrictions on unobserved potential outcomes.The approach can preserve information about identifiable subgroup effects, although reporting only bounds may omit relevant evidence available under the maintained assumptions.
4.5 Local Average Treatment Effects: Without monotonicity, the ratio of intention-to-treat effects remains a linear-combination average of treatment effects for compliers and defiers.With heterogeneous treatment effects, the resulting weighted average can contain negative weights; a small defier population relative to compliers limits the interpretive impact.

5. THE SUBSTANTIVE CONTENT OF THE INSTRUMENTAL VARIABLES ASSUMPTIONS

Instrumental variables require random assignment or unconfoundedness, exclusion, and monotonicity; each assumption has distinct substantive content and plausibility depends on the application.

The section examines random assignment, exclusion, and monotonicity, while treating instrument relevance separately.The main practical concern for relevance is inference when the assumption is nearly violated.
Random assignment: Random assignment identifies intention-to-treat effects but alone does not causally identify the instrumental-variables estimand.The IV estimand is the ratio of intention-to-treat effects for outcome and treatment.
Exclusion restriction: The exclusion restriction is the most critical and typically most controversial instrumental-variables assumption.It requires assessing whether the instrument affects outcomes through pathways other than treatment receipt.
Exclusion restriction: Under the three assumptions, the intention-to-treatment effect on the outcome equals the product of the complier average effect and the population proportion of compliers.For binary outcomes, this implies that the absolute intention-to-treatment effect on the outcome cannot exceed the corresponding treatment-receipt effect.
Exclusion restriction: In the flu data, the observed restriction is slightly violated: 30/1389 = 0.0216 versus 31/72 = 0.0211, without statistical significance.The example illustrates that the assumptions imply restrictions with empirical content.
Monotonicity: Monotonicity rules out individuals whose treatment receipt moves opposite to the instrument, although its plausibility varies across settings.One-sided noncompliance guarantees monotonicity, whereas administrator-specific assignment criteria may make it less attractive.

6. THE LINK TO THE TEXTBOOK DISCUSSIONS OF INSTRUMENTAL VARIABLES

The textbook IV framework expresses outcomes through a linear regression with endogenous treatment and exogenous variables, linking this formulation to potential outcomes while imposing stronger structure.

Textbook IV discussions typically begin with a linear regression for the observed outcome, endogenous regressor, and additional regressors.This framework is related to both simultaneous-equations models and randomized experiments with noncompliance.
Textbook formulation: The unobserved component is assumed independent of exogenous regressors and instruments but not of the endogenous treatment regressor.Treatment may reflect individual choice or an equilibrium condition, leaving its relation with the unobserved component incompletely specified.
Link to potential outcomes: The regression framework implicitly contains a unit-level causal response function whose conditional expectation is linear in treatment and exogenous covariates.A stronger assumption makes the difference between the response function and its conditional expectation independent of treatment level.
Link to potential outcomes: The textbook setup is more restrictive than necessary because its constant residual-response difference can be relaxed to allow variation in slope coefficients.More flexible models are discussed in the modern econometrics literature.
Textbook formulation: In the linear formulation, exclusion and conditional random assignment are combined in the textbook assumptions rather than separately emphasized.Exclusion appears through the absence of the instrument from the outcome equation, while conditional random assignment appears in the independence condition.

7. EXTENSIONS AND GENERALIZATIONS

The section surveys extensions addressing flexible covariate modeling, compliance-status heterogeneity, principal stratification, weak instruments, many instruments, and nonstandard instrument settings.

Recent statistical work replaces traditional linear covariate adjustment with more flexible models for covariates and the endogenous regressor.Some approaches model the conditional distribution of treatment given instruments and exogenous variables.
Model-based approaches: Structural-mean and doubly robust approaches identify average treatment effects under parametric modeling assumptions and extend to time-varying covariates and dynamic treatment regimes.Their estimators remain consistent under misspecification in the absence of intention-to-treat effects.
Model-based approaches: Compliance-status models represent never-takers, always-takers, and compliers using covariates and potential-outcome distributions.Misspecifying the compliance-status model can produce bias even when intention-to-treat effects are absent.
Principal stratification: Principal stratification defines causal subpopulations by joint potential outcomes of a post-treatment variable, generalizing the latent compliance-type framework.In randomized experiments with noncompliance, the post-treatment variable is treatment receipt and the target subpopulation is compliers.
Weak instruments: Weak-instrument research develops point and interval estimators with better properties, including confidence intervals based on inverting Anderson–Rubin statistics.These intervals remain valid irrespective of instrument strength.
Many instruments and extensions: Many-instrument asymptotics can yield more accurate sampling approximations than conventional asymptotics, while covariance-ratio methods may retain an average causal-effect interpretation.Bekker studies settings where the number of instruments increases with sample size.
Regression discontinuity designs: Fuzzy regression discontinuity identifies a local average treatment effect for individuals marginally affected by crossing the assignment threshold.It does not generally identify the average treatment effect for all units at the threshold.

8. CONCLUSION

The paper connects recent statistical work on instrumental variables with econometrics’ older literature, arguing that the combined perspective enriches both traditions.

The paper reviews the connection between recent statistics research on instrumental variables and the older econometrics literature.
Recent statistics research combines older econometric insights with a separate causality literature, enriching both in the process.

A.1 Set up

This section sets up traditional instrumental-variables models and terminology, then places multiple-instrument methods in the context of empirical practice and recent statistical advances.

The section introduces traditional econometric estimation and inference to provide context for recent instrumental-variables advances.
The textbook model relates scalar outcome Y_i linearly to scalar covariate X_i, potentially including exogenous covariates V_i.
Instrumental variables Z_i form a vector with dimension K, motivating distinctions between single-instrument and multiple-instrument cases.
The statistics literature has paid little attention to overidentified settings with multiple instruments, with Small (2007) noted as an exception.
Multiple instruments may arise from a single continuous or multivalued valid instrument and its transformations, or from multiple basic instruments such as weather measures.

A.2 The Just-Identified Case with no Additional Covariates

In the just-identified case without additional covariates, instrumental variables estimation can be represented through covariance ratios, indirect least squares, or two-stage least squares.

The traditional instrumental-variables estimator without additional covariates is a ratio of two covariances.
For a binary instrument, the estimator is also called the Wald estimator and uses differences in group means.
Indirect least squares: Indirect least squares estimates separate reduced-form regressions of the outcome and endogenous regressor on the instrument, then takes their coefficient ratio.
Indirect least squares: In randomized experiments with binary treatment and instrument, the reduced-form coefficients correspond to intention-to-treat effects.
Two-stage least squares: Two-stage least squares first predicts the endogenous regressor from the instruments and covariates, then regresses the outcome on that prediction and additional covariates.
The instrumental-variables, indirect-least-squares, and two-stage-least-squares estimators are numerically identical in this just-identified setting.

A.3 The Just-Identified Case with Additional Covariates

With additional exogenous covariates, instrumental-variables estimation uses covariate-adjusted reduced forms and prediction, while indirect least squares and two-stage least squares remain identical.

Researchers use covariates to replace unconditional instrument independence with conditional independence and may gain precision.
With additional covariates, the reduced-form regressions include the instrument and exogenous regressors.
Indirect least squares: The indirect least squares estimator remains the ratio of the reduced-form coefficients for the outcome and endogenous regressor.
Two-stage least squares: Two-stage least squares predicts the endogenous covariate from the instrument and exogenous regressors, then regresses the outcome on the prediction and actual exogenous variables.
The two-stage-least-squares estimator is again identical to the indirect-least-squares estimator.
Inference: Traditional inference assumes homoscedastic residuals and uses an approximately normal large-sample distribution centered at the true coefficient.

A.4 The Over-Identified Case

The overidentified case has more instruments than endogenous regressors and belongs to a large econometric literature with multiple proposed estimators.

In the overidentified case, the instrumental-variable vector has dimension K > 1 while the main equation remains unchanged.
The traditional setup assumes residuals are independent of the instruments with mean zero and variance σ^2_ε.
Because overidentified instrumental-variables estimation has a large literature, the section briefly discusses two proposed estimators and refers readers elsewhere for detail.

A.5 Two-Stage-Least-Squares

TSLS extends to multiple instruments by estimating reduced-form relationships, predicting the endogenous variable, and regressing the outcome on that prediction. The mechanics are unchanged when instruments exceed one dimension, as illustrated with trivalued weather data.

TSLS first estimates the reduced form of the endogenous variable on instruments and exogenous variables by least squares.
It then calculates the predicted endogenous-variable value and regresses the outcome on that prediction.
Multiple instruments do not change the mechanics of the TSLS procedure.
Using stormy, mixed, and fair weather generates two instruments instead of one binary storm indicator in the Fulton Fish Market example.

A.6 Limited-Information-Maximum-Likelihood

LIML is an alternative estimator for overidentified instrumental-variables models, motivated by a joint-normal likelihood but computable through eigenvalue calculations. It shares TSLS's consistency and asymptotic variance under stated conditions, while differing in finite practice with weak or numerous instruments.

LIML is a limited-information-maximum-likelihood estimator for overidentified models, based on joint normality conditional on instruments and exogenous variables.
LIML is computationally simple through eigenvalue calculations, though more complicated than TSLS, which requires only matrix inversion.
Both TSLS and LIML are consistent and asymptotically normally distributed with the same variance.
In the just-identified case, TSLS and LIML are numerically identical.
TSLS and LIML can differ substantially in practice with weak instruments or many instruments, corresponding to high overidentification.

A.7 Testing the Over-Indentifying Restrictions

With multiple instruments, indirect least squares does not impose the proportionality required by the structural model, leaving multiple possible estimators. Differences among estimators may reflect sampling variation or violations of maintained assumptions, including differing complier populations.

Indirect least squares does not work well with multiple instruments because the reduced-form regressions do not impose the model's proportionality restriction.
Individual reduced-form components or ratios can each be used to estimate the endogenous-regressor effect when the relevant independence assumption holds.
Under the componentwise independence assumption, differences among these estimators should arise from sampling variation.
Tests based on estimator differences are sensitive to maintained assumptions such as linearity and constant effects.
In local-average-treatment-effect settings, different instruments may target different complier populations, producing estimator differences without necessarily indicating invalid instruments.