Treatment-Effect Heterogeneity and 2SLS Weights

Why 2SLS Weights Matter

An instrumental-variable estimate is often reported as a single coefficient. When treatment effects differ across people, that coefficient need not represent an average effect for the whole population. With a binary instrument and covariates, its interpretation depends on which covariate strata experience instrument-induced changes in treatment, and on how much variation in the instrument occurs within each stratum.

This tutorial derives that interpretation for a specific setting: one binary instrument, a binary treatment, and a finite set of covariate strata included as saturated controls in 2SLS. Under the relevant instrumental-variable assumptions, the coefficient is a weighted average of stratum-specific local average treatment effects (LATEs). The weights are nonnegative under monotonicity, but they are not generally the population shares of the strata or the shares of all compliers.

Setup and Assumptions

Let \(Z\in\{0,1\}\) be an instrument, \(D\in\{0,1\}\) treatment, and \(Y\) the outcome. Let \(X\) be a covariate that takes values in a finite set of strata \(x\). Write \(\pi_x=P(X=x)\) and \(q_x=P(Z=1\mid X=x)\). For each stratum, define the reduced-form and first-stage differences

The potential treatment under instrument value \(z\) is \(D(z)\), and the potential outcome under treatment value \(d\) is \(Y(d)\). The assumptions used below are conditional independence of the instrument, exclusion, monotonicity, and overlap within the covariate strata.

The four principal strata are defined by \((D(0),D(1))\): never-takers \((0,0)\), always-takers \((1,1)\), compliers \((0,1)\), and defiers \((1,0)\). Monotonicity rules out defiers. In a stratum with a positive first stage, the stratum-specific LATE is

This parameter is defined only if there are compliers in stratum \(x\), equivalently if \(\Delta D_x>0\) under the assumptions. That distinction will matter when some strata have no instrument-induced treatment change.

From Stratum Differences to a 2SLS Coefficient

Consider 2SLS for the regression of \(Y\) on \(D\) and saturated indicators for \(X\), using \(Z\) and those same indicators as instruments. Saturation means that the controls allow a separate intercept for every stratum. By the Frisch–Waugh–Lovell residualization principle, the coefficient is the ratio of the covariance of the residualized instrument with \(Y\) to its covariance with \(D\).

Within stratum \(x\), the residualized instrument is \(Z-q_x\). Its conditional covariance with any variable \(V\) is

To verify this identity, expand the left side as \(E[ZV\mid x]-q_xE[V\mid x]\). Since \(Z\) is binary, \(E[ZV\mid x]=q_xE[V\mid Z=1,x]\), while \(E[V\mid x]=q_xE[V\mid Z=1,x]+(1-q_x)E[V\mid Z=0,x]\). Substitution and collection of terms give the right side. Averaging the conditional covariance over strata therefore yields

provided the denominator is nonzero. Under conditional independence and exclusion, the reduced-form difference in each stratum is the complier share times the complier average treatment effect. Monotonicity gives \(\Delta D_x=P(D(1)>D(0)\mid X=x)\), and the exclusion restriction means that only compliers’ treatment changes can generate a reduced-form effect. Thus, for every stratum with \(\Delta D_x>0\), \(\Delta Y_x=\Delta D_x\tau_x\). If \(\Delta D_x=0\), there are no compliers in that stratum under monotonicity; its reduced form is also zero, but \(\tau_x\) is not defined there.

This formulation handles zero-first-stage strata directly: no undefined local effect is multiplied by a zero weight. Such a stratum contributes neither to the numerator nor the denominator. It does not receive a LATE in this decomposition.

Worked Example: Calculating the 2SLS Weights

What Determines the Weights?

Each unnormalized weight is the product of three factors: the population share \(\pi_x\), the within-stratum instrument variance \(q_x(1-q_x)\), and the stratum’s first-stage change \(\Delta D_x\). Under monotonicity, the first stage is the share of compliers in that stratum. Consequently, the coefficient gives more weight to strata that are more prevalent, have more variation in instrument assignment, or have a larger complier share. It is not generally a simple average across strata.

The instrument variance factor is largest at \(q_x=0.5\) and approaches zero as \(q_x\) approaches zero or one. A stratum can contain many compliers yet contribute little to this 2SLS estimand if the instrument almost never varies there. Conversely, a smaller stratum may receive substantial weight if it has a strong first stage and considerable instrument variation.

Zero First Stages and the Scope of the Average

A stratum with \(\Delta D_x=0\) has zero unnormalized weight. Under monotonicity, this means that the complier share is zero, so the stratum-specific complier effect \(\tau_x\) is undefined: there is no complier population on which to define that conditional average. The zero weight indicates that the stratum does not contribute to this 2SLS estimand, not that its treatment effect is known to be zero.

Conditions That Preserve the Weight Interpretation

The nonnegative-weight result is conditional on the design and specification, not a universal property of every 2SLS regression. The main conditions are:

• One binary instrument. The derivation uses a single \(Z\in\{0,1\}\), whose within-stratum variance is \(q_x(1-q_x)\).
• Conditional independence and exclusion. These connect stratum-specific reduced forms to treatment effects among compliers rather than to direct effects of the instrument or confounding.
• Monotonicity. This rules out defiers and makes each first stage a nonnegative complier share. Without it, first stages can have other interpretations and the weights need not be nonnegative.
• Saturated covariate controls. The formula uses the conditional instrument residual \(Z-q_x\) in each stratum. Merely entering a restricted, such as linear, function of \(X\) need not produce these stratum-specific residuals or this decomposition.
• Overlap and a nonzero overall first stage. Each included stratum must have instrument variation to identify a within-stratum contrast, and the total first-stage denominator must be positive for the 2SLS coefficient to be defined.

If the covariate adjustment does not saturate the strata, a regression may still produce a 2SLS coefficient, but it need not equal the weighted average derived here. Likewise, if the exclusion restriction or instrument independence fails, an algebraic ratio may remain computable while losing its interpretation as an average of complier effects.

Common Mistake: Calling the Coefficient an ATE

A common but incorrect argument is: “2SLS gives one coefficient, so it must be the average treatment effect for everyone.” The coefficient instead averages effects for the instrument-induced treatment changers represented in the relevant strata, with weights determined by population prevalence, instrument variation, and first-stage size. People whose treatment never changes with the instrument do not directly identify their own treatment effects through this IV comparison.

Another mistake is to include a stratum with zero first stage in the LATE sum and write its term as a zero weight multiplied by an undefined \(\tau_x\). The correction is to sum only over strata with \(\Delta D_x>0\). Zero-first-stage strata contribute zero to the ratio, but their local complier effects are not defined by this design.