Anderson-Rubin Tests With Multiple Instruments

Why Multiple Instruments Require a Joint Test

With one excluded instrument, the Anderson–Rubin (AR) test asks whether that instrument predicts an outcome transformed using a candidate treatment effect. With several excluded instruments, the same logic applies, but the null is a joint restriction: every excluded-instrument coefficient in the transformed-outcome regression must be zero. Testing the coefficients one at a time does not, in general, give the same test or control the same overall error rate.

The joint form is especially useful when the first stage may be weak. Under a candidate value of the treatment effect, the test does not divide an IV estimate by its standard error. Instead, it tests whether the transformed outcome is related to the excluded instruments after controlling for included covariates. The validity of this inference still depends on the instrument and covariate assumptions, and the choice of covariance estimator must reflect the error structure.

Model and Null Restrictions

Consider one endogenous treatment \(D_i\), an outcome \(Y_i\), a vector of included covariates \(X_i\), and a vector of \(q\) excluded instruments \(Z_i\). Write the structural equation as

where \(\beta\) is the treatment effect of interest. For a candidate value \(\beta_0\), define the transformed outcome \(Y_i(\beta_0)=Y_i-\beta_0D_i\). Under \(H_0:\beta=\beta_0\), the structural equation implies \(Y_i(\beta_0)=X_i'\gamma+u_i\). The AR test examines whether the excluded instruments have zero coefficients in a regression of this transformed outcome on \(X_i\) and \(Z_i\).

The restriction follows from the null only under suitable assumptions. One sufficient condition is \(E[u_i\mid X_i,Z_i]=0\) under \(H_0\). This condition includes exogeneity of the included covariates as well as instrument orthogonality. A weaker formulation for the population linear regression is that, after linearly projecting \(Z_i\) on \(X_i\), the residualized instruments are orthogonal to \(u_i\), and that the residualized instruments have a nonsingular second-moment matrix. In particular, \(E[Z_i u_i]=0\) alone is not enough if included covariates may be correlated with \(u_i\).

The treatment \(D_i\) may be correlated with \(u_i\); that is the endogeneity the IV design is intended to address. Under the null, subtracting \(\beta_0D_i\) removes the treatment component from the outcome. The transformed regression can therefore test the candidate value without relying on the conventional IV Wald statistic. For the \(q\) restrictions to be separately estimable, the instruments must have full column rank after controlling for \(X_i\). If they do not, the effective number of testable restrictions is the rank of the residualized instrument matrix, not necessarily \(q\).

Classical Joint F Test

Let \(SSR_R\) be the residual sum of squares from the restricted regression of \(Y_i(\beta_0)\) on \(X_i\) alone, and \(SSR_U\) the residual sum of squares from the unrestricted regression on \(X_i\) and \(Z_i\). If there are \(p\) linearly independent columns in \(X\), including an intercept if used, and \(q\) full-rank excluded instruments, the classical statistic is

Under the null, this has an exact \(F_{q,n-p-q}\) distribution conditional on the regressors when the transformed-regression errors are independent, normally distributed, and homoskedastic with common variance. The numerator measures the reduction in residual variation from adding the \(q\) instruments, per restriction; the denominator estimates the error variance using the unrestricted residual degrees of freedom. Without the normal, homoskedastic error assumptions, this particular finite-sample reference distribution need not be valid.

The justification is the standard nested-regression decomposition. Under the null, the unrestricted model adds \(q\) coefficients whose true values are zero. With fixed regressors and normal errors of variance \(\sigma^2\), \((SSR_R-SSR_U)/\sigma^2\) has a chi-square distribution with \(q\) degrees of freedom, while \(SSR_U/\sigma^2\) has a chi-square distribution with \(n-p-q\) degrees of freedom; these two quantities are independent. Dividing each by its degrees of freedom cancels \(\sigma^2\), giving the stated F statistic. The null distribution does not require a strong first stage. Weak relevance can, however, make the test have low power against some alternatives.

Heteroskedasticity-Robust Joint Wald Test

If the transformed-regression errors may be heteroskedastic, use a heteroskedasticity-robust covariance estimate for the unrestricted regression coefficients. Let \(\hat{\delta}\) be the estimated vector of the \(q\) instrument coefficients, and let \(\widehat V_{\delta}\) be its robust estimated covariance matrix. The joint Wald statistic is

When the restrictions are full rank and the usual large-sample conditions hold, \(W_{\mathrm{AR}}\) is compared with a chi-square distribution with \(q\) degrees of freedom. Equivalently, \(W_{\mathrm{AR}}/q\) is sometimes reported as a Wald statistic with an asymptotic chi-square reference after accounting for the scaling. The chi-square calibration is asymptotic, not an exact finite-sample F result. A suitable sandwich covariance estimate accounts for heteroskedasticity; if observations are clustered, the covariance estimate and inferential calibration must instead reflect clustering, including the number of independent clusters.

How the Joint Restrictions Work in a Small Example

A simple orthogonal design illustrates why the test combines all instrument coefficients and how the restricted and unrestricted regressions relate. Suppose eight hypothetical transformed-outcome observations satisfy \(Y_i(\beta_0)=2+0.5Z_{1i}-0.25Z_{2i}+e_i\), where the three vectors \(Z_1,Z_2,e\) are centered, mutually orthogonal, and each has sum of squares 8. Assume the classical regression conditions and that the intercept is the only included regressor.

Diagnostics, Alternatives, and Limitations

Before interpreting a multiple-instrument AR result, check the actual reduced-form regression and its assumptions. Confirm that the transformed response is \(Y-\beta_0D\), that all intended included covariates are present in both restricted and unrestricted regressions, and that the excluded-instrument columns have full rank after controlling for those covariates. Collinearity can make the inverse covariance or the requested \(q\)-restriction test undefined. If the design is rank deficient, identify the estimable restrictions rather than reporting a statistic for a singular matrix.

• Use a joint test. Separate t tests do not constitute the same test as the \(q\)-restriction AR test. Their results can be misleading about the joint null, particularly when coefficient estimates are correlated.
• Match covariance estimation to the data. Heteroskedasticity or clustering can invalidate the classical F calibration. A robust covariance estimate addresses the specified variance structure, but it does not repair invalid instruments or dependence handled at the wrong level.
• Inspect instrument variation and leverage. Near-collinearity, influential observations, or many instruments relative to the sample can make estimates and approximations unstable. A nominally robust statistic is not a substitute for checking the design and sample size.
• Do not confuse this with a first-stage test. The AR test concerns the transformed outcome under a candidate \(\beta_0\); a first-stage diagnostic concerns prediction of the treatment by the instruments. Neither test establishes the exclusion restriction or other substantive validity assumptions.

A conventional IV Wald test tests a treatment-effect estimate using its estimated standard error and can have unreliable size when instruments are weak. The AR test instead uses a joint reduced-form restriction and remains valid under weak relevance when its error, exogeneity, and covariance assumptions are satisfied. Its limitation is that it can be conservative or have limited power, and a rejected candidate alone does not identify the correct effect. How a collection of accepted and rejected candidates forms a confidence set is a separate inference step.

Common mistake: “Each instrument coefficient has a p-value above 0.05, so the AR test cannot reject.” This reasoning replaces a joint test with separate tests and ignores covariance among estimates. The correction is to form the vector \(\hat{\delta}\), use its full covariance matrix, and test \(H_0:\delta=0\) jointly with the appropriate F or Wald statistic.