Multiple-Instrument 2SLS With Heterogeneous Effects

Why Multiple Instruments Can Target Different Effects

When treatment effects vary across people, there may be no single effect that every valid instrument identifies. One encouragement might change treatment mainly for people who are relatively easy to persuade; another might change it for people whose treatment decisions respond to a different incentive. If those groups have different effects, combining the instruments can combine different treatment-effect margins.

The 2SLS coefficient is calculated from how the instruments jointly predict treatment and outcome. Its interpretation therefore depends not only on whether the instruments are valid, but also on which instruments are included and how much treatment variation each contributes. The formulas below make that dependence explicit.

Definition: Let \(D\) be a scalar treatment, \(Y\) an outcome, and \(Z=(Z_1,\ldots,Z_J)'\) a vector of instruments. After centering the variables, define the first-stage and reduced-form slopes for instrument \(j\) by \(\pi_j=\operatorname{Cov}(Z_j,D)/\operatorname{Var}(Z_j)\) and \(\gamma_j=\operatorname{Cov}(Z_j,Y)/\operatorname{Var}(Z_j)\). When \(\pi_j\ne 0\), define the instrument-specific ratio \(\tau_j=\gamma_j/\pi_j\). This ratio is an algebraic quantity; it is not automatically a causal subgroup effect.

For clarity, first consider a population model without covariates in which the centered instruments are mutually uncorrelated, so their covariance matrix is diagonal. A corresponding sample calculation uses the same structure when the instrument columns have been made orthogonal in the sample. With covariates, the same reasoning applies after the relevant variables have been residualized with respect to those covariates.

Assumptions for the causal discussion: The instruments are relevant for treatment, satisfy instrument exogeneity and exclusion, and the model has sufficient rank for 2SLS. Exogeneity means the instruments are independent of the relevant potential outcomes and treatment responses (or satisfy the corresponding conditional independence after adjustment); exclusion means they affect the outcome only through treatment. These assumptions support an IV interpretation, but do not by themselves make every \(\tau_j\) a subgroup effect. A local average treatment effect (LATE) interpretation for a single binary instrument additionally requires random assignment, a nonzero first stage, and monotonicity: the instrument does not make some individuals less likely to take treatment while making others more likely.

The Orthogonal-Instrument Formula

With mutually uncorrelated instruments, the fitted treatment is the sum of each instrument’s contribution to predicted treatment. The population 2SLS coefficient is the covariance of that fitted treatment with the outcome divided by its covariance with treatment. In terms of the slopes defined above, this gives:

\beta_{\mathrm{2SLS}} = \frac{\sum_{j=1}^{J}\pi_j\gamma_j\operatorname{Var}(Z_j)} {\sum_{j=1}^{J}\pi_j^2\operatorname{Var}(Z_j)} = \sum_{j=1}^{J}w_j\tau_j, \qquad w_j= \frac{\pi_j^2\operatorname{Var}(Z_j)} {\sum_{k=1}^{J}\pi_k^2\operatorname{Var}(Z_k)}.

The second equality follows from \(\gamma_j=\pi_j\tau_j\), which is the definition of \(\tau_j\) when \(\pi_j\ne 0\). Notice that the variance factor is essential: the weight is proportional to \(\pi_j^2\operatorname{Var}(Z_j)\), not just \(\pi_j^2\), and not \(\pi_j^2\operatorname{Var}(Z_j)\) paired with a ratio defined using a different scaling. The weights are nonnegative and sum to one, so this orthogonal-instrument coefficient is a weighted average of the \(\tau_j\) values when those ratios are defined.

Derivation

Write \(Z_j\) as centered and mutually uncorrelated, with variance \(\sigma_j^2=\operatorname{Var}(Z_j)\). The population first-stage projection of \(D\) on the instruments is \(\widehat D=\sum_j\pi_j Z_j\), where the projection slope on \(Z_j\) is \(\pi_j\). By orthogonality,

\operatorname{Cov}(\widehat D,Y) = \sum_j\pi_j\operatorname{Cov}(Z_j,Y) = \sum_j\pi_j\gamma_j\sigma_j^2,

and, by the same calculation,

\operatorname{Cov}(\widehat D,D) = \sum_j\pi_j\operatorname{Cov}(Z_j,D) = \sum_j\pi_j^2\sigma_j^2.

The just-identified regression of \(Y\) on the fitted treatment has slope \(\operatorname{Cov}(\widehat D,Y)/\operatorname{Cov}(\widehat D,D)\). Substituting the two expressions gives the first fraction in the displayed formula. Substituting \(\gamma_j=\pi_j\tau_j\) gives the weighted-average expression. This derivation also shows why omitting instrument variances from the weighting calculation can give a wrong result.

Worked Example: Combining Two Orthogonal Instruments

Consider hypothetical population moments for an outcome measured in points and a binary treatment. Two centered, orthogonal instruments have variances \(1\) and \(4\). Their first-stage slopes are \(\pi_1=0.20\) and \(\pi_2=0.10\), and their reduced-form slopes are \(\gamma_1=0.10\) and \(\gamma_2=0.02\) points per instrument unit. Assume both instruments satisfy the stated IV validity conditions and the first stage has positive rank.

The instrument-specific ratios are \(\tau_1=0.10/0.20=0.50\) points per treatment and \(\tau_2=0.02/0.10=0.20\) points per treatment. The denominator of the 2SLS formula is

$$ (0.20)^2(1)+(0.10)^2(4)=0.04+0.04=0.08. $$

The numerator is

$$ (0.20)(0.10)(1)+(0.10)(0.02)(4)=0.020+0.008=0.028. $$

Therefore \(\beta_{\mathrm{2SLS}}=0.028/0.08=0.35\) points per treatment. The weights are \(0.04/0.08=0.50\) and \(0.04/0.08=0.50\), so the same result is \(0.50(0.50)+0.50(0.20)=0.35\) points per treatment. These moments are hypothetical; the causal meaning of either ratio requires more than this arithmetic.

Why the Instrument Set Can Change the Coefficient

Adding an instrument can change the fitted treatment and thus change the relative contributions of the instrument-specific ratios. In the orthogonal case, adding an instrument adds its nonnegative quantity \(\pi_j^2\operatorname{Var}(Z_j)\) to the weighting calculation. If its ratio differs from the existing coefficient, the combined coefficient changes. This is not necessarily evidence that one calculation is wrong: the two specifications can place different weights on treatment variation associated with different groups.

Worked Example: One Instrument Versus Two

Use the hypothetical moments in the preceding example. With instrument 1 alone, the 2SLS coefficient is its reduced-form slope divided by its first-stage slope:

\beta_{\mathrm{2SLS},\,Z_1} = \frac{\gamma_1}{\pi_1} = \frac{0.10}{0.20} = 0.50 \text{ points per treatment}.

With instrument 2 alone, the calculation is \(0.02/0.10=0.20\) points per treatment. With both instruments, the result is \(0.35\) points per treatment, as calculated above. Thus the two-instrument estimate differs from either single-instrument estimate because the two ratios differ and both receive positive weight.

For a causal illustration, suppose the two encouragements are randomized and, under the design and assumptions, shift treatment for different sets of people. If the first encouragement’s treatment margin has an effect of \(0.50\) points per treatment and the second’s has an effect of \(0.20\), the combined estimate reflects both margins in the proportions specified by the first-stage contributions and instrument variances. Those values are hypothetical causal margins, not a conclusion warranted by the ratios alone. In a real application, validity and the interpretation of each instrument’s induced variation must be justified from the design.

For a single binary instrument, the Wald ratio can identify a LATE for compliers under independence, exclusion, and monotonicity. With multiple instruments, it is tempting to label each \(\tau_j\) as the LATE for a corresponding subgroup. That label is appropriate only when the design provides a defensible single-instrument comparison for that margin, with the required assumptions. In a joint model, the ratios are defined by reduced-form and first-stage slopes; dependence among instruments, other treatment pathways, or a more complicated response pattern can make a simple subgroup interpretation unavailable.

Correlated Instruments Need More Care

The weighted-average formula above relies on orthogonality. With correlated instruments, it is useful to write the population formula using the instrument covariance matrix \(\Sigma_Z=\operatorname{Cov}(Z,Z)\). Let \(\pi\) and \(\gamma\) be the vectors of first-stage and reduced-form projection slopes, so that \(\operatorname{Cov}(Z,D)=\Sigma_Z\pi\) and \(\operatorname{Cov}(Z,Y)=\Sigma_Z\gamma\). If \(\pi_j\ne0\), retain the definition \(\tau_j=\gamma_j/\pi_j\). The population 2SLS coefficient is then:

\beta_{\mathrm{2SLS}} = \frac{\pi'\Sigma_Z\gamma}{\pi'\Sigma_Z\pi}.

When \(\Sigma_Z\) is diagonal, this reduces to the orthogonal-instrument formula. When it is not diagonal, the simple variance weights no longer apply. Even if \(\gamma_j=\pi_j\tau_j\), the coefficient on a particular \(\tau_j\) in the resulting algebraic expression can be negative. Thus a convex-average interpretation should not be asserted merely because the model has several valid instruments.

Worked Example: Correlation Changes the Weighting Calculation

Suppose two centered instruments have covariance matrix \(\Sigma_Z=\begin{pmatrix}1&0.8\\0.8&1\end{pmatrix}\), first-stage slope vector \(\pi=(0.20,0.10)'\), and reduced-form slope vector \(\gamma=(0.10,0.02)'\). The instrument-specific ratios remain \(0.10/0.20=0.50\) and \(0.02/0.10=0.20\) points per treatment.

First calculate \(\Sigma_Z\gamma=(0.116,0.100)'\), so \(\pi'\Sigma_Z\gamma=(0.20)(0.116)+(0.10)(0.100)=0.0332\). Also, \(\Sigma_Z\pi=(0.28,0.26)'\), so \(\pi'\Sigma_Z\pi=(0.20)(0.28)+(0.10)(0.26)=0.082\). Therefore:

\beta_{\mathrm{2SLS}} = \frac{0.0332}{0.082} \approx 0.4049 \text{ points per treatment}.

This differs from the orthogonal-instrument result even though the slope vectors have the same entries: the covariance between the instruments affects how their predictive variation combines. In this particular example the ratio can be written as a convex combination, with coefficients \(0.056/0.082\approx0.6829\) and \(0.026/0.082\approx0.3171\). That feature is specific to these numbers; the general correlated-instrument formula does not guarantee nonnegative weights.

A nonsingular change of basis among a fixed set of instruments does not change the 2SLS fitted-value space or its coefficient. For instance, replacing the columns by nonsingular linear combinations preserves their span. This is different from adding or removing an instrument: doing so generally changes the span of the predictors for treatment, and can change the 2SLS coefficient. Correlation is therefore not a reason to treat alternative encodings of the same instrument set as different causal designs, but instrument selection can change the estimand under heterogeneous effects.

Common Mistakes and Practical Interpretation

Leaving out instrument variance. For orthogonal instruments, the weight is proportional to \(\pi_j^2\operatorname{Var}(Z_j)\). The ratio \(\tau_j=\gamma_j/\pi_j\) uses slopes, so the variance factor must appear in the weights when deriving the combined coefficient.
Calling every ratio a subgroup effect. \(\gamma_j/\pi_j\) is always an algebraic ratio when defined. A causal LATE interpretation requires a design and assumptions that identify the relevant treatment margin.
Assuming all valid instruments identify the same effect. Validity concerns the relationship of instruments to potential outcomes and treatment pathways. It does not imply homogeneous treatment effects or identical complier populations.
Claiming a weighted average without checking dependence. The nonnegative variance weights apply to orthogonal instruments. For correlated instruments, use the covariance-matrix formula and inspect whether any proposed weights are actually nonnegative and sum to one.
Interpreting a changed estimate as proof of invalidity. Under heterogeneity, different valid instrument sets can place different weights on treatment margins. A changed estimate is a reason to examine the design and estimand, not by itself a test of validity.

Key takeaway: With orthogonal instruments, multiple-instrument 2SLS combines the instrument-specific reduced-form-to-first-stage ratios using weights proportional to first-stage strength squared times instrument variance. Under heterogeneous effects, changing the instrument set can therefore change the coefficient’s causal target. The subgroup interpretation of each ratio requires its own design-based justification.

Check Your Understanding

Answer each question using the assumptions stated in the tutorial.

For two orthogonal instruments, explain why the weight on instrument \(j\) involves both \(\pi_j^2\) and \(\operatorname{Var}(Z_j)\).
Suppose two orthogonal instruments have ratios \(0.4\) and \(0.8\) outcome units per treatment. Can their combined coefficient lie outside this range under the stated formula? Explain.
Why is \(\gamma_j/\pi_j\) not automatically a LATE, even when instrument \(j\) is relevant?
In the correlated-instrument example, identify the numerator and denominator used to calculate the 2SLS coefficient.
Explain how adding a valid instrument can change the coefficient when treatment effects are heterogeneous without implying that the added instrument is invalid.