Instrumental variables (IV) are variables used to identify and estimate relationships when an explanatory variable is correlated with unobserved determinants of the outcome. In econometrics and causal inference, an instrument provides variation in an explanatory variable that can be separated from the sources of endogeneity. The term also denotes the resulting family of estimation methods. An instrument is not simply an additional control variable: its usefulness depends on substantive assumptions about how it affects the explanatory variable and how it relates to the outcome. (aeaweb.org)
The problem instrumental variables address
Consider the linear regression model
[ Y_i=\alpha+\beta X_i+u_i, ]
where (Y_i) is an outcome, (X_i) is an explanatory variable, and (u_i) contains determinants of (Y_i) not explicitly included in the equation. Ordinary least squares generally does not consistently estimate (\beta) when (X_i) and (u_i) are correlated. This can arise from omitted variables, measurement error, or simultaneous determination of (X_i) and (Y_i). (eml.berkeley.edu)
For example, a regression of earnings on years of education may combine the effect of schooling with differences in unobserved characteristics that influence both schooling and earnings. Adding observable controls does not necessarily remove this confounding. IV methods instead seek a source of variation in schooling that is unrelated to the remaining unobserved determinants of earnings. Whether a proposed source actually has this property requires an argument grounded in the setting, not merely a statistical association. (aeaweb.org)
Instrument validity and identification
For the simple linear model, an instrument (Z_i) must satisfy two basic conditions:
- Relevance: (Z_i) is associated with (X_i), so that (\operatorname{Cov}(Z_i,X_i)\ne0).
- Orthogonality: (Z_i) is uncorrelated with the structural error, so that (\operatorname{Cov}(Z_i,u_i)=0).
With multiple regressors and instruments, relevance becomes a rank condition: the instruments must provide sufficiently distinct information to identify all the coefficients. Merely having many instruments is insufficient. (eml.berkeley.edu)
In causal applications, validity is commonly explained through two separate substantive requirements. Exogeneity means that instrument assignment is independent of the relevant unobserved determinants or potential outcomes, possibly conditional on observed covariates. The exclusion restriction means that the instrument affects the outcome only through the treatment or explanatory variable being studied. In the potential-outcomes framework, these are distinct assumptions rather than interchangeable names for the same condition. (nber.org)
An instrument need not be uncorrelated with the outcome. Indeed, a valid instrument can affect the outcome through its effect on the treatment. What is ruled out is an independent outcome pathway or a relationship with unobserved outcome determinants that undermines the identifying assumptions. (aeaweb.org)
The single-instrument estimator
Taking covariances with (Z_i) in the simple model gives
[ \operatorname{Cov}(Z,Y)
\beta\operatorname{Cov}(Z,X) +\operatorname{Cov}(Z,u). ]
Under orthogonality and relevance,
[ \beta= \frac{\operatorname{Cov}(Z,Y)} {\operatorname{Cov}(Z,X)}. ]
The sample IV estimator replaces these population covariances with sample counterparts. Under appropriate sampling and regularity conditions, it is a consistent estimator; consistency does not imply exact unbiasedness in finite samples. (eml.berkeley.edu)
For a binary instrument, the corresponding ratio is often called the Wald estimand:
[ \frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[X\mid Z=1]-E[X\mid Z=0]}. ]
The numerator is the instrument’s reduced-form effect on the outcome; the denominator is its first-stage effect on the explanatory variable. Thus IV divides the outcome change associated with the instrument by the treatment change associated with it. Its causal interpretation depends on additional assumptions about treatment responses. (nber.org)
Two-stage least squares
Two-stage least squares (2SLS or TSLS) is the standard linear IV procedure with multiple instruments or control variables. For
[ Y_i=\alpha+\beta X_i+W_i'\gamma+u_i, ]
where (W_i) contains exogenous controls, it proceeds as follows:
- Regress (X_i) on the excluded instruments and (W_i), obtaining fitted values (\widehat X_i).
- Regress (Y_i) on (\widehat X_i) and (W_i).
With several endogenous regressors, each is projected onto the full instrument set. Exogenous regressors included in the outcome equation also belong in that set. (stata.com)
In matrix notation, let (R) contain all structural regressors and (Z) all instruments. Then
[ \widehat\theta_{\mathrm{2SLS}} =(R'P_ZR)^{-1}R'P_ZY, \qquad P_Z=Z(Z'Z)^{-1}Z'. ]
Here (P_Z) is the projection matrix onto the instruments’ column space, assuming the required inverses exist. The estimator is also a particular generalized method of moments estimator. (users.ssc.wisc.edu)
Ordinary second-stage regression software reproduces the coefficients but generally not the correct IV standard errors. IV uncertainty calculations use the structural residuals and the appropriate covariance estimator. Heteroskedasticity-robust, cluster-robust, or autocorrelation-robust calculations address different error structures; they do not establish instrument validity. (stata.com)
Heterogeneous effects and local average treatment effects
When treatment effects differ across individuals, an IV estimate need not represent the average effect for the entire population. The local average treatment effect (LATE) framework specifies an important case in which it has a precise causal meaning. (nber.org)
Let (D_i(z)) denote whether individual (i) would receive a binary treatment under instrument assignment (z), and let (Y_i(d)) denote the outcome under treatment status (d). These are potential outcomes. Under instrument independence, exclusion, a nonzero first stage, consistency and no interference, and monotonicity,
[ D_i(1)\ge D_i(0), ]
the binary-instrument Wald estimand equals
[ E[Y_i(1)-Y_i(0)\mid D_i(1)>D_i(0)]. ]
This is the average effect for compliers: individuals whose treatment status changes because of the instrument. Monotonicity rules out “defiers” whose response goes in the opposite direction. Always-takers and never-takers do not contribute to the treatment change induced by the instrument. (nber.org)
A randomized offer of a training place, for example, may affect participation without making everyone participate. Under these assumptions, the offer identifies the effect for people induced to participate, not necessarily for all participants or all eligible people. (nber.org)
Different instruments can identify effects for different complier groups. With multiple instruments, interpreting 2SLS as a positively weighted average of local effects requires additional conditions; that interpretation is not automatic. (nber.org)
Applications and historical development
IV methods emerged in the 1920s from attempts to estimate supply and demand relationships. Observed prices and quantities are jointly determined in a market equilibrium, so their association does not by itself trace either curve. A variable shifting supply but excluded from demand can provide identifying variation for a demand equation, provided the required assumptions hold. (aeaweb.org)
Later applications addressed measurement error and omitted-variable problems. The development of natural experiments expanded the use of instruments based on institutional rules, lotteries, and externally generated changes in opportunities. IV also applies to randomized experiments with imperfect treatment compliance, where assignment and actual treatment receipt differ. Random assignment establishes an important source of independence, but does not by itself guarantee exclusion. (aeaweb.org)
Work by Guido Imbens and Joshua Angrist in the 1990s clarified which average treatment effects instruments can identify under heterogeneous responses. Their methodological contributions to the analysis of causal relationships were recognized by the 2021 Nobel Memorial Prize in Economic Sciences. (nobelprize.org)
Weak instruments, diagnostics, and limitations
A weak instrument provides too little predictive information about the endogenous regressor for conventional large-sample approximations to work reliably. Weakness can produce substantial estimator bias and misleading significance tests even in large datasets. Adding many weak instruments can worsen finite-sample behavior. (nber.org)
First-stage partial (R^2) and tests of the excluded instruments’ predictive contribution describe instrument strength. The familiar first-stage (F>10) rule is a heuristic, not a universal guarantee: relevant thresholds depend on the model, number of endogenous regressors, error assumptions, and tolerable bias or test distortion. (nber.org)
Weak-instrument-robust methods, including Anderson–Rubin and conditional likelihood-ratio tests, permit hypothesis testing and construction of confidence sets without relying on the usual strong-instrument approximation. Limited-information maximum likelihood can have better bias properties than 2SLS in some weak-instrument settings, but no estimator creates identifying information that is absent. (nber.org)
When there are more excluded instruments than endogenous regressors, the equation is overidentified, and tests of overidentifying restrictions can assess compatibility of the moment conditions. Such tests do not prove that every instrument is valid; non-rejection is not a substitute for an identification argument. (stata.com)
The central practical limitation is that relevance is observable more directly than exogeneity or exclusion. A strongly predictive instrument can still be invalid, while a small violation of validity can be especially damaging when relevance is weak. Institutional evidence, model assumptions, and the population affected by the instrument therefore remain essential to interpreting an IV estimate. (nber.org)
References
- Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experimentsaeaweb.org
- Econ. 240B: Chapter 4, Instrumental Variableseml.berkeley.edu
- Econometricsusers.ssc.wisc.edu
- Identification and Estimation of Local Average Treatment Effectsnber.org
- Identification of Causal Effects Using Instrumental Variablesnber.org
- Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2021nobelprize.org
- The Causal Interpretation of Two-Stage Least Squares with Multiple Instrumental Variablesnber.org
- Instrumental Variables Regression with Weak Instrumentsnber.org
- The Cure Can Be Worse than the Disease: A Cautionary Tale Regarding Instrumental Variablesnber.org
- Testing for Weak Instruments in Linear IV Regressionnber.org
- Inference with Weak Instrumentsnber.org
- FAQ: Two-stage least-squares regressionstata.com