Causal Weights in 2SLS with Multiple Instruments

Why Multiple Instruments Create a Weighting Question

With one valid binary instrument and a binary treatment, the Wald ratio has a familiar local interpretation under independence, exclusion, relevance, and monotonicity: it identifies an average treatment effect for compliers. With several instruments, 2SLS still reports one coefficient, but it combines several sources of instrument-induced treatment variation. The combination depends not only on the instruments’ individual first stages, but also on how their variation is correlated and represented in the regression.

The key distinction is between an algebraic weighting identity and a causal weighting interpretation. The identity follows from the 2SLS projection formula. To interpret its components as treatment effects for particular groups, additional conditions must make those components valid local comparisons. Without those conditions, a 2SLS coefficient is not automatically a convex average of instrument-specific LATEs.

Model and Population 2SLS Formula

Let $D$ be a binary treatment, $Y$ an outcome, $Z=(Z_1,\ldots,Z_K)'$ a vector of excluded instruments, and $X$ a vector of included exogenous controls (including an intercept). Write $\tilde D$, $\tilde Y$, and $\tilde Z$ for the population linear-projection residuals after removing $X$. Thus, the expressions below describe the 2SLS coefficient after partialling out the included controls.

Definition: The first-stage projection coefficient is $\pi=\Omega^{-1}E[\tilde Z\tilde D]$, and the reduced-form projection coefficient is $\gamma=\Omega^{-1}E[\tilde Z\tilde Y]$, where $\Omega=E[\tilde Z\tilde Z']$. If $\Omega$ is nonsingular and the instruments predict treatment, the population 2SLS coefficient is $$ \beta_{2SLS} = \frac{\pi'\Omega\gamma}{\pi'\Omega\pi}. $$

This formula assumes a well-defined linear 2SLS projection and a nonzero first-stage fitted-value variance, $\pi'\Omega\pi>0$. A causal interpretation additionally requires the relevant instrumental-variable assumptions: consistency and no interference; conditional instrument independence from potential outcomes and treatment responses given $X$; exclusion, so the instruments affect $Y$ only through $D$; and adequate instrument relevance. Monotonicity is not needed for the algebra, but is needed for the usual complier-effect interpretation of binary-instrument contrasts.

Deriving the Weights in an Orthogonal Basis

The projection identity becomes especially transparent after expressing the instrument space in an orthogonal basis. Let $R_1,\ldots,R_K$ be residualized instrument scores that are pairwise uncorrelated and have positive variances. These scores span the same instrument space as $\tilde Z$. Define $a_j=E[R_j\tilde D]$ and $b_j=E[R_j\tilde Y]$. The fitted value of treatment from the instruments is $\sum_j (a_j/E[R_j^2])R_j$.

Taking its covariance with the fitted outcome, and using the zero cross-covariances between the $R_j$'s, gives the 2SLS numerator. The same operation with fitted treatment in both positions gives the denominator:

\beta_{2SLS} = \frac{\displaystyle\sum_{j=1}^K \frac{a_jb_j}{E[R_j^2]}} {\displaystyle\sum_{j=1}^K \frac{a_j^2}{E[R_j^2]}} = \sum_{j:a_j\ne 0} \underbrace{\frac{a_j^2/E[R_j^2]}{\sum_{\ell=1}^K a_\ell^2/E[R_\ell^2]}}_{w_j} \underbrace{\frac{b_j}{a_j}}_{\tau_j^{proj}}.

The weights $w_j$ are nonnegative and sum to one over the relevant components. Each is larger when its instrument score has a stronger covariance with treatment, relative to its variance. The quantity $\tau_j^{proj}=b_j/a_j$ is the reduced-form change in outcome per corresponding change in treatment for that score. It is a projection ratio, not necessarily a causal effect for a well-defined group.

This derivation also explains the role of instrument residualization. If, for example, a binary instrument $Z_j$ is residualized on controls and the strata defined by earlier instruments, its residual score isolates variation in $Z_j$ within those strata. With saturated strata $S$, the score is $R_j=Z_j-P(Z_j=1\mid S)$. If the instrument is valid within each stratum, then

E[R_jD]=\sum_s P(S=s)p_s(1-p_s)\Delta D_s, \qquad E[R_jY]=\sum_s P(S=s)p_s(1-p_s)\Delta Y_s,

where $p_s=P(Z_j=1\mid S=s)$, $\Delta D_s=E[D\mid Z_j=1,S=s]-E[D\mid Z_j=0,S=s]$, and $\Delta Y_s$ is the analogous outcome contrast. If $\Delta D_s>0$ in the relevant strata, the ratio $E[R_jY]/E[R_jD]$ is a weighted average of the within-stratum Wald ratios, with weights proportional to $P(S=s)p_s(1-p_s)\Delta D_s$. Under conditional independence, exclusion, and monotonicity within those strata, these Wald ratios have conditional LATE interpretations. Residualization therefore changes which instrument comparisons enter a component; it is not merely a numerical adjustment.

Worked Example: Two Orthogonal Instruments

Consider a hypothetical model with no controls and two independent binary instruments, each with probability one-half. Their variances are both $0.25$, and their covariance is zero. Suppose the first-stage projection slopes are $\pi_1=0.4$ and $\pi_2=0.2$, while the reduced-form slopes are $\gamma_1=0.8$ and $\gamma_2=1.0$. For this illustration, suppose the corresponding instrument contrasts satisfy the conditions for causal interpretation.

Because the instruments are orthogonal, they already form an orthogonal basis. Their component ratios are $0.8/0.4=2$ and $1.0/0.2=5$. The 2SLS numerator and denominator are

\pi'\Omega\gamma =(0.25)(0.4)(0.8)+(0.25)(0.2)(1.0) =0.08+0.05=0.13,

\pi'\Omega\pi =(0.25)(0.4^2)+(0.25)(0.2^2) =0.04+0.01=0.05.

Thus $\beta_{2SLS}=0.13/0.05=2.6$. The normalized weights are $0.04/0.05=0.8$ and $0.01/0.05=0.2$, so the same result is $0.8(2)+0.2(5)=2.6$. The stronger first-stage component receives more weight because its contribution to fitted-treatment variance is larger.

Residualization Changes the Components

When instruments are correlated, their raw columns are not separate sources of independent variation. One useful way to display the calculation is to retain $Z_1$ and residualize $Z_2$ on it. This is a change of basis: it separates variation in $Z_1$ from the remaining variation in $Z_2$. The total 2SLS coefficient is unchanged if the instrument span is unchanged, but the component ratios and their weights refer to these new contrasts.

Worked Example: Residualizing a Correlated Binary Instrument

Suppose $Z_1$ is binary with probability one-half. A second binary instrument has $P(Z_2=1\mid Z_1=0)=0.2$ and $P(Z_2=1\mid Z_1=1)=0.8$. The residual from projecting $Z_2$ on an intercept and $Z_1$ is $R_2=Z_2-(0.2+0.6Z_1)$. It is uncorrelated with $Z_1$, and

E[Z_1^2]=0.25, \qquad E[R_2^2]=\tfrac12(0.2)(0.8)+\tfrac12(0.8)(0.2)=0.16.

Suppose the treatment projection is $E[D\mid Z_1,Z_2]=0.20+0.30Z_1+0.20Z_2$. Since $Z_2=0.2+0.6Z_1+R_2$, this becomes $0.24+0.42Z_1+0.20R_2$. Suppose the reduced-form projection, expressed in that same basis, is $E[Y\mid Z_1,Z_2]=c+0.42Z_1+0.80R_2$, for a constant $c$. The two component ratios are $0.42/0.42=1$ and $0.80/0.20=4$.

The first component contributes $0.25(0.42^2)=0.0441$ to the denominator; the residualized second contributes $0.16(0.20^2)=0.0064$. Therefore

\beta_{2SLS} = \frac{0.0441(1)+0.0064(4)}{0.0441+0.0064} = \frac{0.0697}{0.0505} \approx 1.380.

The corresponding weights are approximately $0.8733$ and $0.1267$. The second instrument’s contribution is based on its variation left after removing the part predictable from $Z_1$, not on its total variation. The component ratios in this calculation are projection ratios; interpreting them as causal effects requires valid causal contrasts for this residualized basis.

Why Raw-Instrument Weights Can Be Negative

The nonnegative weights in the orthogonal-basis identity do not imply that 2SLS is always a convex average of effects attached to the original instrument coordinates. In a nonorthogonal basis, the coefficient can be written as a weighted sum of coordinate ratios, but those algebraic weights can be negative. The coordinate ratios themselves may also fail to be causal effects. Orthogonalization preserves the fitted instrument space; it changes the components being combined.

Worked Example: A Negative Coordinate Weight

For a hypothetical moment calculation, let two centered, standardized instrument scores have covariance matrix

\Omega= \begin{pmatrix} 1 & -0.8\\ -0.8 & 1 \end{pmatrix}, \qquad \pi= \begin{pmatrix} 1\\ 0.5 \end{pmatrix}, \qquad \gamma= \begin{pmatrix} 2\\ 3 \end{pmatrix}.

The coordinate reduced-form-to-first-stage ratios are $\gamma_1/\pi_1=2$ and $\gamma_2/\pi_2=6$. In the raw coordinates, the denominator contributions are $\pi_j(\Omega\pi)_j$. Since $\Omega\pi=(0.6,-0.3)'$, these are $0.6$ and $-0.15$, summing to $0.45$. The 2SLS numerator is $0.6(2)-0.15(6)=0.30$, so

\beta_{2SLS} = \frac{0.30}{0.45} = \frac{0.6}{0.45}(2)+\frac{-0.15}{0.45}(6) \approx 0.6667.

The normalized raw-coordinate weights are approximately $1.333$ and $-0.3333$, not convex weights. This is an algebraic illustration, not a claim that the two coordinate ratios are causal LATEs. A causal reading requires identifying valid instrument-induced contrasts and checking their assumptions; the raw ratios in this example have not been shown to do so.

Common Mistake and Practical Interpretation

A common mistake is to calculate a Wald ratio for each instrument separately, then assume 2SLS is their simple average—or that it must be a positive weighted average of those ratios. The 2SLS numerator and denominator instead depend on the covariance matrix of the instruments and on their joint first-stage responses. Correlation can make raw-coordinate weights negative, while residualization changes which contrasts and first-stage strengths appear in an orthogonal decomposition.

For causal interpretation, state the instrument basis and the assumptions behind each local comparison. If orthogonalized scores correspond to valid conditional instrument contrasts and monotonicity holds for those contrasts, their component ratios may have local causal interpretations, and the orthogonal-basis weights are nonnegative. Otherwise, the projection identity remains correct, but it does not by itself establish a convex average of treatment effects for compliers.

Key takeaway: Multiple-instrument 2SLS weights instrument-induced treatment variation according to its contribution to fitted-treatment variance. Instrument covariance and residualization determine the components. Projection weights are a mathematical fact; causal weights require additional assumptions and a defensible interpretation of the contrasts.

Check Your Understanding

Use the projection formulas and the distinction between algebraic and causal interpretations to answer these questions.

In an orthogonal instrument basis, what determines the weight on a component ratio $b_j/a_j$?
Why does residualizing $Z_2$ on $Z_1$ change the interpretation of the $Z_2$ component without changing 2SLS when the instrument span is preserved?
Under what additional conditions can a residualized binary-instrument ratio be interpreted through conditional LATEs?
Why can a raw-coordinate decomposition of 2SLS have a negative weight even though the orthogonal-basis decomposition has nonnegative weights?
Does the 2SLS projection identity alone show that the coefficient is a convex average of individual treatment effects? Explain.