Interpreting 2SLS Under Constant Treatment Effects

What a 2SLS Coefficient Means Under Constant Effects

A two-stage least squares (2SLS) coefficient is a causal effect only under assumptions that connect the regression calculation to a causal model. Under a constant treatment effect, that interpretation is especially direct: if a treatment changes the outcome by the same amount for every unit, then valid and relevant instruments identify that common amount. The interpretation is not automatic, however. A computed coefficient does not establish that the treatment effect is constant or that the instruments are valid.

The distinction from ordinary least squares (OLS) is important. OLS describes a linear relationship between observed treatment and outcome, conditional on included covariates. If treatment is related to unobserved causes of the outcome, that slope need not equal the treatment effect. 2SLS uses variation in treatment associated with instruments instead, and its causal interpretation depends on the structural model and instrument conditions stated below.

Here \(Y_i(d)\) is unit \(i\)'s potential outcome if its treatment were set to \(d\), \(D_i\) is its observed treatment, and \(X_i\) is a vector of included covariates. The observed outcome satisfies consistency, \(Y_i=Y_i(D_i)\). The model also presumes that the potential outcomes are well-defined and that one unit's treatment does not change another unit's outcome. The unobserved term \(u_i\) may be related to treatment; that is one reason to use instrumental variables.

The Structural Model and Instrument Conditions

Suppose \(Z_i\) is a vector of excluded instruments. An excluded instrument is used to predict treatment but is not included as a direct cause in the structural outcome equation. For the interpretation of 2SLS to be justified, the structural model is paired with moment and relevance conditions. Stating these conditions directly avoids treating independence, by itself, as a guarantee that an error has mean zero.

The adjusted instrument condition is the one that matters after controls are included. For example, a raw conditional independence statement about \(Z_i\) and potential outcomes given \(X_i\) does not, without further conditions, imply \(E[\tilde Z_i u_i]=0\). In this tutorial the needed moment is stated explicitly. In an applied analysis, its credibility depends on the study design and substantive knowledge, not merely on the 2SLS calculation.

For a single excluded instrument, relevance means that its adjusted association with treatment is nonzero. With several instruments, relevance is a rank condition: the instruments must contain enough independent variation to predict the endogenous treatment after accounting for \(X_i\). The derivation below assumes this condition holds and that the 2SLS population denominator is nonzero.

Why the 2SLS Coefficient Equals the Common Effect

The basic argument is that the outcome variation associated with the instruments consists of the common treatment effect times the instrument-associated treatment variation, plus an error component orthogonal to those instruments. For clarity, residualize \(Y_i\), \(D_i\), and \(Z_i\) with respect to the intercept and \(X_i\), writing the residuals as \(\tilde Y_i\), \(\tilde D_i\), and \(\tilde Z_i\). The structural equation then gives \(\tilde Y_i=\tau\tilde D_i+\tilde u_i\). Because the excluded instruments are residualized on \(X_i\), the stated moment condition gives \(E[\tilde Z_i\tilde u_i]=0\).

With one excluded instrument, the population IV coefficient is the ratio of the adjusted instrument-outcome association to the adjusted instrument-treatment association. Substituting the structural equation shows why the ratio equals \(\tau\):

With multiple excluded instruments, population 2SLS combines their predictive variation using the usual projection onto the instrument space. Let \(Q_{ZZ}=E[\tilde Z_i\tilde Z_i']\), \(Q_{ZD}=E[\tilde Z_i\tilde D_i]\), and \(Q_{ZY}=E[\tilde Z_i\tilde Y_i]\). When the relevant matrices are invertible and the treatment is identified, the scalar treatment coefficient is

The structural model and instrument moments imply \(Q_{ZY}=\tau Q_{ZD}\). Substitution therefore yields

This is a population identification result. In a finite sample, the estimated 2SLS coefficient will generally differ from \(\tau\) because of sampling variation. The argument also shows what can go wrong: if the instruments are irrelevant, the denominator is zero or too small for reliable identification; if the instrument moment fails, the error term can contribute to the reduced-form association.

Worked Interpretations

What OLS Does—and Does Not—Interpret

In the same structural model, the adjusted population OLS slope is

Thus OLS equals the common treatment effect if the adjusted treatment-error covariance is zero. That is a separate condition; it is not supplied by constant treatment effects. When treatment is endogenous, the covariance term may be nonzero, so OLS need not estimate \(\tau\), even though every unit has the same causal treatment effect in the model.

Conversely, 2SLS is not automatically preferable in every application. Its causal interpretation depends on instrument relevance and the adjusted instrument-error moments, while OLS depends on treatment-error exogeneity. Neither estimator's numerical value establishes its own assumptions. The structural interpretation also relies on the treatment and outcome being consistently defined and on the common additive effect being a reasonable model for the question being asked.

Common Mistake

A frequent error is to write, “The instrument is independent of potential outcomes given the controls, so the IV error moment is zero,” without checking what that statement implies for the structural error or for residualized instruments. Conditional independence alone does not generally imply a zero conditional mean for the error, nor does a statement about raw \(Z\) automatically establish the adjusted moment needed after controls are included. The correction is to specify and defend the moment conditions used in the identification argument—here, \(E[X_i u_i]=0\) and \(E[\tilde Z_i u_i]=0\)—rather than silently treating them as consequences of independence.

A second related mistake is to label every 2SLS coefficient “the causal effect” without stating the model. Under the constant-effect model, the coefficient has a common-effect interpretation if the relevant assumptions hold. If effects vary across units, this particular common-effect conclusion does not follow; the interpretation requires separate analysis.