Why Covariates Appear in Two Places
In instrumental variables analysis, a covariate can have two statistical roles at once. If it is an exogenous predictor of the outcome, it belongs in the outcome regression as a control. Because it is exogenous, it also belongs in the instrument set: it instruments itself. This is not the same as treating the covariate as an excluded instrument. An excluded instrument predicts the endogenous treatment but is not included directly in the structural outcome equation.
The distinction matters for the fitted values used by two-stage least squares (2SLS). The first stage predicts the endogenous treatment using both the excluded instruments and the included exogenous covariates. The second stage estimates the outcome equation using the predicted treatment and those same included covariates. Leaving a covariate out of either stage changes the specified estimator.
Here exogeneity means that the structural error has zero conditional mean given the instruments and included covariates, \(E[u_i\mid Z_i,W_i]=0\), under the model being used. In particular, it implies the moment conditions \(E[W_i u_i]=0\) and \(E[Z_i u_i]=0\). Exogeneity is a substantive assumption; putting a variable into the instrument matrix does not make it exogenous. Instrument relevance requires that the instruments provide enough independent predictive variation in \(D_i\), beyond the included covariates, to identify \(\beta_D\). The regressors and instruments must also have no exact linear dependencies that prevent estimation.
Specification of Covariate-Adjusted 2SLS
For a sample of \(n\) observations, let \(X\) be the structural regressor matrix with columns for the intercept, \(D\), and the included covariates. Let \(Q\) be the instrument matrix with columns for the intercept, the included covariates, and the excluded instruments. In the first stage, regress \(D\) on every column of \(Q\). In the second stage, use the fitted treatment together with the included covariates to estimate the structural equation.
The corresponding 2SLS point estimator is
The first-stage fitted values are projections onto the span of the full instrument matrix. The formula for 2SLS therefore uses the same included covariates in the structural regressor matrix and the instrument matrix. The second-stage regression on \(\widehat D\) and \(W\) gives this 2SLS point estimate; ordinary least-squares standard errors from that second-stage regression are not, in general, the appropriate 2SLS standard errors.
Worked Example: Specifying the Two Stages
Suppose a hypothetical study estimates the effect of a program \(D\) on a health outcome \(Y\). Distance to an enrollment office, \(Z\), is proposed as an excluded instrument. Age and baseline severity, \(W_1\) and \(W_2\), are included as exogenous covariates. For clarity, suppose age is measured in decades and severity in scale points.
The structural equation is \(Y_i=\beta_0+\beta_DD_i+\gamma_1W_{1i}+\gamma_2W_{2i}+u_i\). The instrument vector is \(Q_i=(1,W_{1i},W_{2i},Z_i)'\), so both covariates enter the first-stage regression as well as the outcome equation.
For a hypothetical participant aged 50 years, with severity 3 and distance 2, the covariate values are \(W_1=5\), \(W_2=3\), and \(Z=2\). If the fitted first-stage equation in a hypothetical sample is \(\widehat D=0.20+0.03W_1+0.04W_2-0.02Z\), the predicted treatment is $$ 0.20+0.03(5)+0.04(3)-0.02(2)=0.20+0.15+0.12-0.04=0.43. $$
The second stage uses the predicted treatment \(0.43\), age, and severity to estimate the outcome equation. This is a linear first-stage prediction, not necessarily a probability; if \(D\) is binary, a linear projection can in principle produce fitted values outside the interval from 0 to 1. The proposed assumptions still need scrutiny: the analysis must justify that distance is relevant and that, conditional on the included covariates, it is uncorrelated with the structural error.
Why Included Covariates Instrument Themselves
The reason is visible in the moment conditions used by IV estimation. The structural equation contains coefficients on both \(D\) and \(W\), while the instrument vector supplies exogenous variation with which to estimate those coefficients. If \(W\) is exogenous, its own variation satisfies the required orthogonality condition with the error. Including \(W\) as an instrument ensures the instrument set spans the included exogenous regressors as well as the excluded instruments.
A short justification follows from the model. Write \(Y=X\theta+u\). The 2SLS estimator uses the projected regressors \(P_QX\). Under the required moment condition \(E[Q_i u_i]=0\), the population cross-moment between the instruments and the structural error is zero. Since the columns of \(W\) are among the columns of \(Q\), their exogenous variation is available in this projection, just as the variation in \(Z\) is. Omitting \(W\) from \(Q\) would fail to include all the specified exogenous regressors among the instruments, and would not be the standard 2SLS construction for this model.
Worked Example: Covariate-Adjusted 2SLS by Hand
Consider four hypothetical observations. The outcome is generated by \(Y_i=3+2D_i+W_i+u_i\). The proposed excluded instrument is \(Z_i\), and \(W_i\) is an included exogenous covariate.
| Observation | \(W\) | \(Z\) | \(D\) | \(u\) | \(Y\) |
|---|---|---|---|---|---|
| 1 | -1 | -1 | 1 | 2 | 6 |
| 2 | -1 | 1 | 2 | -2 | 4 |
| 3 | 1 | -1 | 1 | -2 | 4 |
| 4 | 1 | 1 | 4 | 2 | 14 |
The outcome values agree with the stated equation: for observation 1, for example, \(3+2(1)+(-1)+2=6\). The full instrument columns are \(1,W,Z\). Their sample cross-products are \(1'W=0\), \(1'Z=0\), and \(W'Z=0\); each column has sum of squares 4. Thus the instrument matrix has full column rank. The residual vector \(u=(2,-2,-2,2)'\) also satisfies \(1'u=0\), \(W'u=0\), and \(Z'u=0\). These sample moments illustrate exogeneity in this constructed example; they would not establish exogeneity in an empirical study.
Regressing \(D\) on \(1,W,Z\) gives the first-stage intercept \(\bar D=(1+2+1+4)/4=2\). Orthogonality of the columns gives the slope on \(W\) as \(W'D/(W'W)=(-1-2+1+4)/4=0.5\), and the slope on \(Z\) as \(Z'D/(Z'Z)=(-1+2-1+4)/4=1\). Therefore, $$ \widehat D=2+0.5W+Z, $$ with fitted values \(0.5,\ 2.5,\ 1.5,\ 3.5\), respectively.
For the 2SLS second-stage point estimate, regress \(Y\) on the intercept, \(\widehat D\), and \(W\). The fitted outcome projection on the instrument columns has intercept \(\bar Y=(6+4+4+14)/4=7\), coefficient \(W'Y/(W'W)=(-6-4+4+14)/4=2\), and coefficient \(Z'Y/(Z'Z)=(-6+4-4+14)/4=2\). Thus the projected outcome is \(7+2W+2Z\). Since \(2\widehat D+W=2(2+0.5W+Z)+W=4+2W+2Z\), this projected outcome can be written as \(3+2\widehat D+W\). The second-stage coefficients are therefore an intercept of 3, a treatment coefficient of 2, and a covariate coefficient of 1. In this constructed data set, the 2SLS estimate is 2 outcome units per one-unit increase in treatment, conditional on the included covariate.
Several Covariates and Instruments
The same rule applies when there is more than one included covariate or excluded instrument. If \(W=(W_1,W_2)'\) and \(Z=(Z_1,Z_2)'\), then \(Q=(1,W_1,W_2,Z_1,Z_2)'\). Each included covariate appears in the structural equation and in the instrument matrix; each excluded instrument appears in the instrument matrix but not directly in the structural equation. Identification requires enough independent excluded-instrument variation to identify the endogenous treatment coefficient after accounting for the included regressors.
Worked Example: Checking a Multi-Instrument First Stage
In another hypothetical design, let \(a,b,c\) each take values \(-1\) and \(1\) in all eight possible combinations. Define two exogenous covariates \(W_1=a\), \(W_2=b\), two excluded instruments \(Z_1=c\), \(Z_2=ab\), and a residual pattern \(R=ac\). Let treatment be generated by $$ D=10+2W_1-W_2+Z_1+0.5Z_2+2R. $$
For example, at \(a=-1,b=1,c=1\), the row has \(W_1=-1\), \(W_2=1\), \(Z_1=1\), \(Z_2=-1\), and \(R=-1\). Its treatment is \(10+2(-1)-1+1+0.5(-1)+2(-1)=5.5\).
Across the eight rows, the columns \(1,W_1,W_2,Z_1,Z_2,R\) are mutually orthogonal: each nonconstant column sums to zero, and the product of any two distinct columns sums to zero. Each column has sum of squares 8. Consequently, the first-stage regression of \(D\) on the full instrument set \(1,W_1,W_2,Z_1,Z_2\) has coefficients $$ \widehat\delta_{W_1}=\frac{W_1'D}{8}=2,\quad \widehat\delta_{W_2}=\frac{W_2'D}{8}=-1,\quad \widehat\pi_{Z_1}=\frac{Z_1'D}{8}=1,\quad \widehat\pi_{Z_2}=\frac{Z_2'D}{8}=0.5. $$ The intercept is \(1'D/8=10\). The term \(2R\) contributes zero to each coefficient because \(R\) is orthogonal to every instrument column. Thus \(\widehat D=10+2W_1-W_2+Z_1+0.5Z_2\). This construction also makes the instrument columns full rank: their cross-product matrix is \(8I\), whose determinant is \(8^5\), not zero. In an actual application, full rank alone is not enough; the excluded instruments must predict treatment beyond the included covariates.
Common Mistakes and Practical Checks
A common mistake is to put covariates into the outcome equation but leave them out of the instrument matrix, on the reasoning that only the excluded instruments should be called “instruments.” The correction is to distinguish the two roles: \(Z\) is excluded from the outcome equation, while exogenous \(W\) is included in it and also instruments itself. The instrument set is not synonymous with the set of excluded instruments.
Another mistake is to add a covariate to both matrices simply because it predicts the outcome. Prediction does not establish exogeneity. If a covariate is correlated with the structural error, treating it as an instrument violates the required moment condition. Researchers must justify the assumed exogeneity and exclusion restrictions using the design and subject-matter knowledge.
Before estimating, write the structural equation and instrument list separately. Confirm that each included exogenous regressor is in both, that every excluded instrument is in the instrument list but not directly in the structural equation, and that the excluded instruments provide relevant variation in the endogenous regressor. These checks clarify what the estimate targets and which assumptions it relies on.
Check Your Understanding
Use the definitions and specification above to answer the following questions.
- In a model with one endogenous treatment, two included exogenous covariates, and one excluded instrument, write the structural regressor list and the complete instrument list.
- Why does an included exogenous covariate appear in the instrument matrix as well as in the outcome equation?
- In the four-observation example, what is the first-stage fitted treatment for observation 4, and which fitted-value equation gives it?
- What assumption would fail if an included covariate were correlated with the structural error but still used as an instrument?
- Why does having a full-rank instrument matrix not, by itself, establish that an excluded instrument is relevant for the endogenous treatment?