What the Reduced-Form Coefficient Measures
In an instrumental-variables analysis, the instrument \(Z\) is used to study a treatment \(D\) and an outcome \(Y\). Before asking how the instrument helps estimate a treatment effect, it is useful to measure its relationship with the outcome itself. The coefficient of \(Z\) in a regression of \(Y\) on \(Z\) and covariates is called the reduced-form coefficient, or the instrument effect on the outcome when a causal interpretation is justified.
This coefficient answers a focused question: how much does the outcome differ, or change, with the instrument, after accounting for the included covariates? Its units are outcome units per instrument unit. It does not, by itself, describe the effect of treatment on the outcome. Nor does the act of including covariates guarantee that the coefficient is causal.
For observations \(i=1,\ldots,n\), the fitted regression can be written as
Here, \(X_i\) is a vector of included covariates, \(\beta_0\) is an intercept, and \(\tau\) is the reduced-form regression coefficient of interest. The ordinary least squares coefficient is defined by the fitted linear projection; it can be calculated without assuming the equation is a correct causal model. A causal interpretation requires additional assumptions, discussed below.
Binary Instruments: Adjusting Across Covariate Strata
For a binary instrument, the unadjusted outcome difference is the mean outcome among observations with \(Z=1\) minus the mean among observations with \(Z=0\). With covariates in the regression, the coefficient need not equal that raw difference: it compares outcomes after accounting for the covariates according to the regression specification.
A clear special case is a binary covariate \(X\), with a regression that includes an intercept and \(X\), but no interaction between \(Z\) and \(X\). This model imposes one common instrument coefficient across the two covariate strata. When both instrument values occur in each stratum, the fitted coefficient can be expressed as a weighted average of the two within-stratum outcome differences. The weights depend on how much variation in \(Z\) there is within each stratum.
Worked Example: A Binary Instrument With Balanced Strata
Suppose a hypothetical randomized encouragement study records a binary encouragement \(Z\), a binary baseline covariate \(X\), and a one-year outcome \(Y\) measured in symptom-score points. There are two observations in each \(Z,X\) cell, with the following cell means.
| Covariate stratum | Mean outcome when \(Z=0\) | Mean outcome when \(Z=1\) | Within-stratum difference |
|---|---|---|---|
| \(X=0\) | 10 points | 12 points | 2 points |
| \(X=1\) | 14 points | 17 points | 3 points |
Fit the additive regression of \(Y\) on an intercept, \(Z\), and \(X\), without a \(Z \times X\) interaction. Each stratum has equal numbers at \(Z=0\) and \(Z=1\), and both strata have the same size. Thus, after adjusting for \(X\), the common coefficient is the average of the two within-stratum differences:
The adjusted regression associates encouragement with an outcome that is, on average under this common-slope model, 2.5 symptom-score points higher when \(Z=1\) than when \(Z=0\), at the same covariate value. The stratum-specific differences are not identical, so the common coefficient is a summary rather than a claim that the difference is exactly 2.5 points in each stratum. A model containing a \(Z \times X\) interaction would instead allow separate instrument coefficients for the two strata.
With unequal instrument variation across strata, the common-slope regression does not generally give each stratum equal weight. In the binary-covariate case just described, each stratum’s weight is proportional to its sample size times the within-stratum variance of \(Z\). A stratum in which nearly everyone has the same instrument value contributes less information about the adjusted instrument slope.
Worked Example: Unequal Instrument Variation Across Strata
Consider a second hypothetical dataset with 200 people. In stratum \(X=0\), 50 people have \(Z=0\) with mean outcome 10 points, and 50 have \(Z=1\) with mean outcome 12 points. In stratum \(X=1\), 90 people have \(Z=0\) with mean outcome 20 points, and 10 have \(Z=1\) with mean outcome 25 points. Fit an additive regression with an intercept, \(Z\), and \(X\).
The two within-stratum outcome differences are \(12-10=2\) points and \(25-20=5\) points. For binary \(Z\), the within-stratum contribution to the slope weight is proportional to \(n_x p_x(1-p_x)\), where \(n_x\) is the stratum size and \(p_x\) is the fraction with \(Z=1\). The weights are therefore
The common adjusted coefficient is the weighted average:
The \(X=0\) stratum receives more weight because it contains more within-stratum variation in the instrument. The coefficient is not the unweighted average \((2+5)/2=3.5\) points. This calculation describes the fitted adjusted association; it does not establish that changing the instrument would cause the outcome to change by 2.794 points.
Continuous Instruments and Residualization
For a continuous instrument, \(\tau\) is the adjusted change in the fitted mean of \(Y\) for a one-unit increase in \(Z\), with the included covariates held fixed in the linear model. A one-unit increase must be interpreted on the instrument’s actual scale. If \(Z\) is measured in tens of kilometres, the coefficient is per ten kilometres, not per kilometre.
The role of covariate adjustment can be stated precisely using residualization. Regress \(Z\) on an intercept and \(X\), and call the residuals \(\tilde Z_i\). Regress \(Y\) on the same intercept and covariates, and call those residuals \(\tilde Y_i\). The coefficient on \(Z\) in the regression of \(Y\) on \(Z\) and \(X\) is the slope from regressing \(\tilde Y\) on \(\tilde Z\).
This formula says that the adjusted slope is determined by the co-movement of the parts of \(Z\) and \(Y\) left after their linear relationships with the included covariates have been removed. If \(\sum_i\tilde Z_i^2=0\), the instrument is perfectly predicted by the included regressors in this sample, so its separate coefficient cannot be estimated.
Why the Residual Formula Holds
The least-squares fit chooses coefficients to minimize the sum of squared residuals. Its normal equations require the fitted regression residuals to have zero sample inner product with each included regressor, including \(Z\) and every column of \(X\). The residualized variables \(\tilde Y\) and \(\tilde Z\) are each orthogonal to the intercept and columns of \(X\), by the least-squares equations for their separate regressions on those variables.
Write the fitted outcome regression as \(Y_i=\hat\beta_0+\hat\tau Z_i+\hat\gamma^\top X_i+\hat e_i\). Removing from both sides the linear projections on the intercept and \(X\) leaves
The outcome-regression residual \(\hat e_i\) is orthogonal to \(Z\) and to the included covariates. Since \(\tilde Z\) is \(Z\) minus a linear combination of those covariates and the intercept, it is also orthogonal to \(\hat e\). Multiplying the last equation by \(\tilde Z_i\), summing over observations, and using \(\sum_i \tilde Z_i\hat e_i=0\) gives
Dividing by \(\sum_i\tilde Z_i^2\) yields the stated formula. Thus the adjusted coefficient uses only the instrument variation not linearly explained by the included covariates.
Worked Example: A Continuous Instrument With a Covariate
Suppose a hypothetical dataset records an outcome \(Y\) in recovery-score points, a continuous instrument \(Z\) in travel-time units, and a covariate \(X\). The four observations are shown below. To make the arithmetic transparent, the covariate is coded as \(-1\) or \(1\).
| Observation | \(X_i\) | \(Z_i\) | \(Y_i\) |
|---|---|---|---|
| 1 | \(-1\) | 1 | 6.6 |
| 2 | 1 | 3 | 10.2 |
| 3 | 1 | 5 | 11.8 |
| 4 | \(-1\) | 7 | 11.4 |
The mean of \(Z\) is 4. Its centered values are \((-3,-1,1,3)\), and their products with \(X\) sum to
Therefore, after regressing \(Z\) on an intercept and \(X\), its residuals are \(\tilde Z=(-3,-1,1,3)\). The mean outcome among observations with \(X=-1\) is \((6.6+11.4)/2=9\), and the mean for \(X=1\) is \((10.2+11.8)/2=11\). Subtracting the relevant covariate-stratum mean gives \(\tilde Y=(-2.4,-0.8,0.8,2.4)\). Hence
The fitted adjusted association is 0.8 recovery-score points per one-unit increase in travel time. It is a slope over the instrument values represented in these data; it is not automatically a causal effect of travel time.
When Is the Instrument Coefficient Causal?
A regression coefficient is a statistical summary. To interpret it as the causal effect of instrument assignment on the outcome, the assignment mechanism must support that interpretation. For example, let \(Y(z)\) denote the outcome that would be observed under instrument value \(z\). In a study with covariates \(X\), a standard condition is conditional exchangeability, \(Y(z)\perp Z\mid X\): within levels of \(X\), instrument assignment is independent of the potential outcome. Consistency requires the observed outcome to equal the potential outcome under the instrument value actually received. Positivity requires that instrument values being compared have support within the covariate levels of interest.
These conditions allow an adjusted comparison to represent an effect of \(Z\) on \(Y\), provided the regression specification also summarizes that comparison appropriately. A single coefficient from an additive linear regression imposes a common linear slope. If the causal effect of \(Z\) varies across covariate values, the coefficient is a regression-weighted summary, not necessarily the effect for every person or stratum. Exogeneity of covariates, or simply including them in a regression, does not itself establish conditional exchangeability of the instrument.
The reduced-form causal effect concerns the instrument’s effect on the outcome. It is distinct from the effect of the treatment \(D\) on \(Y\). The exclusion restriction is not needed merely to define or estimate the causal effect of \(Z\) on \(Y\); it is relevant when attributing the instrument’s effect on the outcome solely to its effect through treatment. Relevance, too, is a separate question: an instrument can affect the outcome without the regression coefficient alone showing how strongly it changes treatment.
Common Mistakes and Practical Checks
- Calling the reduced-form coefficient the treatment effect. It is the coefficient of the instrument in the outcome regression. The treatment effect is a different estimand and requires further assumptions and analysis.
- Interpreting an adjusted coefficient as a raw group difference. Covariates change which comparisons contribute to the regression coefficient. With uneven instrument variation across strata, the adjusted coefficient can weight stratum-specific differences unequally.
- Assuming adjustment guarantees causality. Regression controls for the specified covariates in the specified form. It cannot, by itself, establish that the instrument is independent of potential outcomes conditional on those covariates.
- Ignoring the instrument’s scale. A continuous-instrument coefficient is per one unit of \(Z\). State what that unit is and report the corresponding outcome units.
- Overlooking weak variation after adjustment. If the covariates nearly predict \(Z\), little residualized instrument variation remains. The coefficient may be imprecisely estimated even though the regression can technically be fit.
Check Your Understanding
Use the definitions and calculations above to answer the following questions.
- A binary instrument is coded 0 and 1, and the outcome is measured in days. What are the units of its reduced-form coefficient?
- In the balanced-strata example, why is the common adjusted coefficient 2.5 points even though the stratum-specific differences are 2 and 3 points?
- In the unequal-variation example, which stratum receives greater weight, and what feature of the data determines that weight?
- For a continuous instrument measured in tens of kilometres, how should a coefficient of \(-0.4\) outcome units be interpreted?
- Does including covariates in the outcome regression establish a causal effect of the instrument on the outcome? State one additional condition needed for that interpretation.
- Why should the reduced-form coefficient not be described as the treatment effect?