Two-Stage Least Squares as a Weighted Average of Local Effects

Why the Mix of Instruments Matters

With one valid binary instrument and a binary treatment, the Wald ratio can identify a local average treatment effect for units whose treatment changes with the instrument, under the familiar exclusion, independence, and monotonicity assumptions. With several instruments and heterogeneous treatment effects, two-stage least squares (2SLS) need not target one common local effect. Its population coefficient can instead combine instrument-specific local effects, and the coefficients in that combination need not behave like probabilities.

The key issue is that 2SLS uses the joint covariance structure of the instruments. If instruments are correlated, the contribution assigned to one instrument depends not only on its own first-stage effect but also on how it covaries with the other instruments. This tutorial derives a useful weighted-average representation under explicit conditions and shows how a negative weight can arise even when every instrument shifts treatment in the same direction.

Setup and Conditions for Local Effects

Let \(Z=(Z_1,\ldots,Z_K)'\) be a vector of binary instruments, \(D\) a binary treatment, and \(Y\) an outcome. Include an intercept in the 2SLS regression, and define \(\Sigma=\operatorname{Cov}(Z,Z)\). Assume the instrument vector is jointly independent of the potential treatments \(D(z)\) and potential outcomes \(Y(d)\), has full support over its binary combinations, and affects the outcome only through treatment: \(Y=Y(D(Z))\). For each instrument component, assume monotonicity: changing \(Z_j\) from 0 to 1 while holding \(Z_{-j}\) fixed never lowers anyone’s treatment, so \(D(1,z_{-j})\ge D(0,z_{-j})\).

These assumptions alone do not ensure that every marginal Wald ratio is an instrument-specific local effect. A transparent sufficient condition is that the conditional mean treatment and reduced-form outcome are additive in the instruments:

Here \(\pi_j\) and \(\gamma_j\) are the respective changes in conditional mean treatment and outcome when \(Z_j\) changes from 0 to 1, holding the other components fixed. Additivity means these changes do not depend on the values of the other instruments. If \(\pi_j>0\), the ratio \(\theta_j=\gamma_j/\pi_j\) is a local average treatment effect for the units whose treatment is changed by component \(j\), averaged over the distribution of the other instrument components. To see why, under exclusion the outcome change for a unit whose treatment switches is \(Y(1)-Y(0)\); units whose treatment does not switch contribute zero. Monotonicity rules out treatment decreases, so dividing the average outcome change by the average treatment change averages treatment effects among the switchers.

The additive-mean condition is a sufficient condition for the particular representation developed here, not a universal property of instrumental-variable models. Without appropriate conditions, ratios of reduced-form to first-stage coefficients need not be local average treatment effects.

Deriving the 2SLS Weighted Average

Under the additive conditional-mean model, \(\pi\) and \(\gamma\) are the population linear-projection coefficient vectors of \(D\) and \(Y\) on the centered instrument vector. The first-stage fitted value, apart from its intercept, is \(\widehat D=\pi'(Z-E[Z])\). The population 2SLS coefficient is the regression slope of \(Y\) on this fitted value, with an intercept. Its denominator can be written as the covariance between fitted treatment and actual treatment, and its numerator as the covariance between fitted treatment and the outcome.

For the numerator, \(\operatorname{Cov}(\widehat D,Y)=\pi'\operatorname{Cov}(Z,Y)=\pi'\Sigma\gamma\); the second equality follows from \(\operatorname{Cov}(Z,Y)=\Sigma\gamma\). The same reasoning gives \(\operatorname{Cov}(\widehat D,D)=\pi'\operatorname{Cov}(Z,D)=\pi'\Sigma\pi\). Now write each component of \(\gamma\) as \(\gamma_j=\pi_j\theta_j\). Then:

The weights sum to one because \(\sum_j\pi_j(\Sigma\pi)_j=\pi'\Sigma\pi\). If \(\Sigma\) is positive definite and \(\pi\ne 0\), the denominator is positive. But that does not make every numerator positive: \((\Sigma\pi)_j\) can be negative even when \(\pi_j>0\). Thus the weights sum to one but may include negative values or exceed one. They are not necessarily proportions of the population or shares of a group of compliers.

How Correlation Can Produce a Negative Weight

For two instruments, write their covariance matrix as \(\Sigma=\begin{pmatrix}\sigma_1^2&\sigma_{12}\\\sigma_{12}&\sigma_2^2\end{pmatrix}\). Then \((\Sigma\pi)_1=\sigma_1^2\pi_1+\sigma_{12}\pi_2\). Even if both first-stage coefficients are positive, a sufficiently negative covariance \(\sigma_{12}\) can make this expression negative. The instrument’s own positive first-stage shift is then offset, in the fitted-value covariance, by its negative association with the other instrument.

A Local-Effect Calculation with Heterogeneity Across Strata

The additive-mean condition is not the only way a weighted local-effect interpretation can arise. For example, if binary instruments are independent, the marginal change associated with one component averages its controlled changes over the distribution of the other components. With componentwise monotonicity, the resulting reduced-form-to-first-stage ratio is a local effect, even if the controlled changes differ across strata. When instruments are independent, their covariance matrix is diagonal, so the corresponding 2SLS weights are nonnegative.

Interpretation and a Common Pitfall

The weighted-average formula describes the population 2SLS coefficient under the stated model and identification conditions. It does not show that 2SLS always equals the average treatment effect for the population, nor that every instrument-specific ratio is a local effect under minimal instrumental-variable assumptions. The target depends on the instruments’ first stages, their covariance structure, and the assumptions that make the component ratios causal.

A common mistake is to see weights that sum to one and conclude that they must be nonnegative shares, so that 2SLS must lie between the instrument-specific effects. The sum-to-one identity does not imply nonnegativity. In the correlated-instrument example, the weights \(-1/13\) and \(14/13\) sum to one but yield a coefficient greater than both local effects. The correction is to inspect the actual weight formula, including \(\Sigma\), before interpreting the 2SLS coefficient as an average. A negative weight is an algebraic feature of the projection; it is not a negative proportion of people.

In applications, weak first stages create an additional concern: even when the population representation is valid, estimation and inference can be unstable if the instruments predict treatment poorly. The weighted-average derivation is about the estimand and does not remove the need to assess instrument strength or justify the validity assumptions.