Binary Instrument Calculations With Covariate Adjustment

Why Adjust the Instrument Comparisons?

A binary instrument divides observations into two groups, such as those offered a program and those not offered it. The difference in treatment rates between those groups is the first stage; the difference in mean outcomes is the reduced form. With no covariates, their ratio is the just-identified IV estimate. When the groups differ on measured covariates, however, the relevant calculation uses instrument coefficients adjusted for the same covariates in both comparisons.

The adjustment is a regression calculation, not a guarantee that the instrument is valid. It changes the comparisons to account for the specified covariates. Whether the resulting ratio has a causal interpretation still depends on the instrumental-variable assumptions.

Definition: Let \(Z\) be a binary instrument, \(D\) a treatment, \(Y\) an outcome, and \(X\) a vector of specified exogenous covariates. The adjusted first-stage coefficient, \(\hat{\pi}\), is the coefficient on \(Z\) in the regression of \(D\) on an intercept, \(Z\), and \(X\). The adjusted reduced-form coefficient, \(\hat{\rho}\), is the coefficient on \(Z\) in the regression of \(Y\) on an intercept, \(Z\), and the same \(X\). With one excluded instrument and a nonzero first stage, the just-identified 2SLS estimate is \(\hat{\tau}_{IV}=\hat{\rho}/\hat{\pi}\).

The Regression Setup and Its Conditions

The first-stage and reduced-form regressions are descriptive linear projections used to calculate the IV estimate. In sample notation, they are

D_i=a_D+\pi Z_i+X_i^\mathsf{T}\gamma_D+e_{Di}, \qquad Y_i=a_Y+\rho Z_i+X_i^\mathsf{T}\gamma_Y+e_{Yi}.

The covariates in the two equations must be the same, and the regressions must use the same analysis sample and compatible weighting. The coefficient \(\hat{\pi}\) measures the adjusted change in treatment associated with changing \(Z\) from 0 to 1; \(\hat{\rho}\) measures the corresponding adjusted change in outcome. If \(D\) is binary, \(\hat{\pi}\) is in probability points. The units of \(\hat{\rho}\) are the outcome units, so the ratio is in outcome units per treatment unit.

For a causal interpretation, the instrument must be relevant after adjustment, satisfy the appropriate conditional independence and exclusion restrictions given \(X\), and be used consistently with the causal model being interpreted. Relevance requires \(\pi\ne 0\) in the population; a sample estimate near zero makes the ratio unstable. Under a correctly specified additive structural model with a constant treatment effect and an instrument orthogonal to the structural error conditional on \(X\), the ratio identifies that treatment effect. Without such a model, the ratio remains a calculation, and its causal estimand requires additional assumptions and interpretation.

Computing the Adjustment From Covariate Strata

When \(X\) consists of stratum indicators, the adjusted coefficient on \(Z\) in an additive regression can be computed from the within-stratum instrument contrasts. For stratum \(s\), let \(n_s\) be its sample size, \(p_s\) the proportion with \(Z=1\), and \(\Delta_{Qs}\) the difference in mean \(Q\) between \(Z=1\) and \(Z=0\), where \(Q\) is either \(D\) or \(Y\). The regression coefficient is a weighted average of those contrasts:

\hat{\beta}_{Z,Q} = \frac{\sum_s n_s p_s(1-p_s)\Delta_{Qs}} {\sum_s n_s p_s(1-p_s)}.

The weight \(n_s p_s(1-p_s)\) is larger when a stratum has more observations and when the instrument varies more within it. A stratum with no observations in one instrument group has zero weight: it supplies no within-stratum contrast. These weights apply to the additive regression with stratum indicators; they are not generally equal to stratum sample proportions.

Worked Example: Equal Instrument Variation Across Two Strata

Suppose a hypothetical study compares an offer instrument (\(Z=1\)) with no offer (\(Z=0\)). The treatment is binary, and the outcome is a symptom score in points. There are two equally sized covariate strata, with 100 observations in each instrument group in each stratum. The cell means are:

Stratum	Mean treatment, \(Z=0\)	Mean treatment, \(Z=1\)	Mean score, \(Z=0\)	Mean score, \(Z=1\)
A	0.20	0.50	2.0	3.2
B	0.40	0.60	4.0	4.6

Each stratum has \(n_s=200\) and \(p_s=0.5\), so each weight is \(200(0.5)(0.5)=50\). The normalized weights are therefore \(50/(50+50)=0.5\) for each stratum.

The within-stratum treatment contrasts are \(0.50-0.20=0.30\) in A and \(0.60-0.40=0.20\) in B. Thus the adjusted first stage is

\hat{\pi} = \frac{50(0.30)+50(0.20)}{100} =0.25.

The reduced-form contrasts are \(3.2-2.0=1.2\) score points in A and \(4.6-4.0=0.6\) points in B. Therefore,

\hat{\rho} = \frac{50(1.2)+50(0.6)}{100} =0.90 \text{ score points}.

The covariate-adjusted IV ratio is \(0.90/0.25=3.60\) score points per treatment unit. This is the just-identified estimate from the two adjusted coefficients. Calling it a causal treatment effect additionally requires valid-instrument assumptions; the arithmetic alone does not establish them.

Why the Ratio Equals the Just-Identified 2SLS Estimate

The ratio is not an unrelated shortcut: with one excluded instrument, it is the 2SLS coefficient on treatment when the same exogenous covariates are included in both stages. A short algebraic justification makes the role of the shared adjustment explicit. Let \(\tilde Z\), \(\tilde D\), and \(\tilde Y\) denote the sample residuals after removing their linear projections on the intercept and \(X\). The coefficients on \(Z\) in the two adjusted regressions are

\hat{\pi}=\frac{\tilde Z^\mathsf{T}\tilde D}{\tilde Z^\mathsf{T}\tilde Z}, \qquad \hat{\rho}=\frac{\tilde Z^\mathsf{T}\tilde Y}{\tilde Z^\mathsf{T}\tilde Z}.

After accounting for the covariates, the first-stage fitted treatment component is the projection of \(\tilde D\) onto the single instrument \(\tilde Z\):

\widehat{\tilde D} = \tilde Z\frac{\tilde Z^\mathsf{T}\tilde D}{\tilde Z^\mathsf{T}\tilde Z} =\hat{\pi}\tilde Z.

The 2SLS coefficient for the treatment component is the fitted-treatment/outcome covariance divided by the fitted-treatment sum of squares. Substituting the expression above gives

\frac{\widehat{\tilde D}^\mathsf{T}\tilde Y} {\widehat{\tilde D}^\mathsf{T}\widehat{\tilde D}} = \frac{\hat{\pi}\tilde Z^\mathsf{T}\tilde Y} {\hat{\pi}^{\,2}\tilde Z^\mathsf{T}\tilde Z} = \frac{\tilde Z^\mathsf{T}\tilde Y}{\tilde Z^\mathsf{T}\tilde D} = \frac{\hat{\rho}}{\hat{\pi}},

provided the first stage is nonzero. This identity also shows why using different covariates in the two regressions would not produce the stated just-identified 2SLS calculation: the two coefficients would no longer represent comparisons adjusted in the same way.

Worked Calculations With Unequal Stratum Weights

Worked Example: Unequal Instrument Prevalence Across Strata

Consider a hypothetical study with two covariate strata. In stratum A there are 200 people, split evenly between \(Z=0\) and \(Z=1\). In stratum B there are also 200 people, but 50 have \(Z=1\) and 150 have \(Z=0\). The treatment and outcome means are:

Stratum	Mean treatment, \(Z=0\)	Mean treatment, \(Z=1\)	Mean outcome, \(Z=0\)	Mean outcome, \(Z=1\)
A	0.20	0.60	10.0	11.0
B	0.30	0.50	12.0	12.5

For A, \(n_Ap_A(1-p_A)=200(0.5)(0.5)=50\). For B, \(p_B=50/200=0.25\), so its weight is \(200(0.25)(0.75)=37.5\). The normalized weights are \(50/87.5=4/7\) and \(37.5/87.5=3/7\).

The treatment contrasts are \(0.60-0.20=0.40\) in A and \(0.50-0.30=0.20\) in B. The adjusted first stage is

\hat{\pi} = \frac{50(0.40)+37.5(0.20)}{87.5} = \frac{27.5}{87.5} =0.3143.

The outcome contrasts are \(11.0-10.0=1.0\) and \(12.5-12.0=0.5\). The adjusted reduced form is

\hat{\rho} = \frac{50(1.0)+37.5(0.5)}{87.5} = \frac{68.75}{87.5} =0.7857.

Consequently, \(\hat{\tau}_{IV}=0.7857/0.3143=2.50\) outcome units per treatment unit. Using equal weights would give a different calculation because the additive adjusted regression gives more weight to stratum A, where the instrument has greater within-stratum variation.

Worked Example: Forming the Ratio From Adjusted Regression Results

Suppose a hypothetical analysis of 600 eligible patients uses offer status as a binary instrument, treatment receipt as \(D\), and a recovery score measured in points as \(Y\). Both regressions include the same prespecified age and baseline-severity covariates. The reported adjusted coefficient on offer is \(0.18\) in the treatment regression and \(0.72\) points in the outcome regression.

The adjusted first stage is \(\hat{\pi}=0.18\): conditional on the included covariates, offer status is associated with an 18-percentage-point increase in treatment receipt. The reduced form is \(\hat{\rho}=0.72\) recovery-score points. The just-identified ratio is

\hat{\tau}_{IV} = \frac{\hat{\rho}}{\hat{\pi}} = \frac{0.72 \text{ points}}{0.18} = 4.00 \text{ points per treatment unit}.

This example illustrates how to calculate the ratio from regression coefficients; it does not establish that the offer is independent of potential outcomes or has no effect on recovery except through treatment. Those validity conditions must be argued using the study design and substantive knowledge. Also, the nonzero first stage makes the ratio computable, but by itself does not show that the instrument is sufficiently strong for reliable inference.

Common Mistakes and Practical Checks

Dividing crude differences after fitting adjusted models. The ratio must use the adjusted reduced-form and first-stage coefficients from models with the same covariate set, sample, and weighting scheme.
Using an ordinary average of stratum contrasts automatically. In an additive regression with stratum indicators, the weights depend on both stratum size and within-stratum instrument variation: \(n_sp_s(1-p_s)\).
Interpreting relevance as validity. A nonzero adjusted first stage is necessary for the ratio, but it does not establish conditional independence or exclusion.
Ignoring the denominator. If \(\hat{\pi}=0\), the ratio is undefined. If it is close to zero, even moderate uncertainty in the two coefficients can translate into an unstable ratio.
Calling the ratio an effect without stating assumptions. The computed ratio is an IV estimate. Its causal meaning depends on the causal model and instrument assumptions, not just on the fact that the instrument is binary.

Before reporting the estimate, verify that the instrument is coded consistently in both regressions, that the same covariates and observations were used, and that the first-stage and reduced-form coefficients are expressed in compatible units. Report the two coefficients as well as their ratio so readers can see the source of the calculation.

Check Your Understanding

Use the definitions and calculations above to answer the following.

In an adjusted IV calculation, what must be held the same between the first-stage and reduced-form regressions?
A stratum has 120 observations, with 30% assigned \(Z=1\). Calculate its weight \(n_sp_s(1-p_s)\).
Suppose the adjusted first stage is 0.20 and the adjusted reduced form is 1.10 outcome units. Calculate the IV ratio and state its units if treatment is binary.
Why is a stratum with \(p_s=0\) assigned zero weight in the within-stratum contrast formula?
Does a positive adjusted first-stage coefficient establish the exclusion restriction? Explain.