Endogeneity Tests With Multiple Regressors And Instruments

Why Test Several Endogenous Regressors Together?

A structural equation may contain several regressors whose relationship with the outcome disturbance is uncertain. For example, both training participation and training intensity might be related to unobserved motivation. A test that examines only one regressor at a time can miss dependence that is apparent when the residual components are considered jointly. Conversely, a joint rejection does not tell us which component is responsible.

Residual-inclusion tests address this question by adding first-stage residuals to an outcome equation. With multiple potentially endogenous regressors, each has its own first-stage residual, and the coefficients on those residuals can be tested individually or jointly. The interpretation depends on the maintained structural and instrument assumptions: these tests assess restrictions about exogeneity, not whether the instruments themselves are valid.

Model and Restrictions

Let \(y_i\) be an outcome, \(d_i\) a \(q\)-vector of potentially endogenous regressors, \(w_i\) a vector of included exogenous controls, and \(z_i\) a vector of excluded instruments. Write \(r_i=(w_i',z_i')'\). A linear first-stage system is

where \(v_i\) is the \(q\)-vector of first-stage disturbances. Estimate each first stage using the same included exogenous variables and the specified excluded instruments, then collect the fitted residuals in \(\hat v_i\). The structural outcome equation is

The residual-inclusion specification adds the estimated residual vector:

Here \(\lambda=(\lambda_1,\ldots,\lambda_q)'\) contains the residual-inclusion coefficients. The control-function interpretation requires a correctly specified conditional relationship in which the first-stage residual vector captures the relevant dependence between \(d_i\) and the structural disturbance. It also relies on appropriate instrument conditions: excluded instruments must be relevant for identification and satisfy the required exclusion and exogeneity assumptions. The linear form of the control function must be justified for the application.

The phrase “under the maintained assumptions” matters. A residual coefficient can indicate that the specified conditional mean still depends on a first-stage residual. But if the instruments are invalid, the control function is misspecified, or relevant variables are omitted, the test result need not have the intended exogeneity interpretation.

Individual and Joint Wald Tests

Estimate the augmented equation and obtain \(\hat\lambda\) and its estimated covariance matrix \(\widehat V\). Inference must use a covariance procedure appropriate to the estimated residuals and the assumed error structure. For a vector of linear restrictions \(H_0:R\lambda=r\), the Wald statistic is

Under the null and standard regularity conditions, \(W\) has an asymptotic chi-square distribution with degrees of freedom equal to the rank of \(R\). For one coefficient, the signed statistic \(\hat\lambda_j/\operatorname{SE}(\hat\lambda_j)\) is a signed \(z\)-statistic; the one-restriction Wald chi-square statistic is its square. For a joint test, the off-diagonal covariances in \(\widehat V\) matter, so the individual statistics cannot simply be added unless the relevant estimates are uncorrelated.

A joint test instead evaluates whether all residual coefficients are zero. Let \(R\) be the identity matrix and \(r=0\); then the Wald statistic uses the full covariance matrix of \(\hat\lambda\). This is a single test of \(q\) restrictions, not a collection of separate tests whose conclusions are combined informally.

Where the Test Comes From

The Wald calculation follows from the estimated restriction and its sampling covariance. Under \(H_0:R\lambda=r\), the estimator of the restriction is centered at zero. Under standard large-sample conditions, its standardized form is approximately multivariate normal:

If \(A=R\widehat V R'\) is nonsingular, premultiplying by a matrix that standardizes \(A\) gives a vector with approximately identity covariance. The sum of the squares of its components is therefore approximately chi-square with \(\operatorname{rank}(R)\) degrees of freedom. That sum of squares is exactly the Wald quadratic form. This justification also explains why a joint test uses the covariance among restrictions rather than treating each estimated coefficient as isolated.

This is an asymptotic result, not a claim that every finite-sample test has an exact chi-square distribution. The covariance estimate must match the inferential setting, including heteroskedasticity where relevant and the fact that the residuals were estimated in a first stage. The earlier material on inference with estimated control functions is relevant to that variance calculation; one should not assume that any convenient second-stage standard error is automatically valid.

What the Test Does—and Does Not—Say

The joint null is useful when the substantive question is whether the potentially endogenous regressors, considered as a set, can be treated as exogenous under the specified model. Individual tests answer narrower questions: whether one residual coefficient is zero conditional on the other residuals and regressors. A joint rejection can occur even when neither individual test is decisive, because the joint statistic accounts for the covariance pattern and assesses a different set of restrictions.

Neither outcome certifies the instruments. Instrument validity concerns assumptions about the instruments’ relationship to the structural disturbance and their exclusion from the outcome equation. A residual-inclusion test is constructed to test restrictions on \(\lambda\), conditional on the model and instruments being appropriate. Failure to reject the exogeneity null can also reflect limited precision or weak identifying information; it is not proof of exogeneity.