Anderson-Rubin Tests With Included Covariates

Why Include Covariates in an Anderson–Rubin Test?

An Anderson–Rubin (AR) test evaluates a proposed value of an instrumental-variables coefficient without relying on a strong first stage. In applications, the structural equation often includes exogenous covariates as well as an endogenous regressor. Those covariates belong in the test regression: leaving them out changes the model being tested and can alter the test statistic.

The key operation is to test whether the excluded instrument or instruments explain the outcome after the included covariates have been accounted for. With several excluded instruments, the relevant coefficients are tested jointly. The covariate coefficients are nuisance parameters: they are included in both regressions but are not part of the AR restriction.

The notation distinguishes included covariates from excluded instruments. The included covariates appear directly in the structural equation and in the test regression. The excluded instruments predict \(d\) but do not appear directly in the structural equation. Under the candidate null, their coefficients in the transformed-outcome regression should be zero if the structural equation and instrument restrictions hold.

Model and Conditions for the F Test

For the stated finite-sample F test, assume the structural equation is correctly specified, the included covariates are exogenous, and the excluded instruments satisfy the relevant exclusion and exogeneity conditions. In particular, the structural error has conditional mean zero given the included covariates and instruments under the null. The usual exact F reference distribution requires the stronger conditional error assumption \(u\mid W,Z \sim N(0,\sigma^2 I)\): errors are conditionally normal, have common variance, and are conditionally uncorrelated.

The design must also have full column rank: \(W\) has rank \(k\), and the columns of \(Z\) add \(q\) independent directions beyond \(W\). Require \(n>k+q\), so the unrestricted regression has positive residual degrees of freedom. Full rank alone is not enough; if \(n=k+q\), the residual degrees of freedom are zero and the residual variance estimate and stated F statistic are undefined.

These conditions concern the test's validity and the exact reference distribution. In particular, the test does not require a large first-stage coefficient. Weak instruments can reduce power, but do not by themselves invalidate the AR test under its instrument and error assumptions. Conversely, an AR test does not establish that an instrument is exogenous or excluded; those are substantive design assumptions.

Partialling Out the Covariates

For a candidate \(\beta_0\), write \(r=y-\beta_0d\). The unrestricted regression is \(r=W\gamma+Z\pi+u\); the restricted regression imposes \(\pi=0\) and includes only \(W\). Thus the null of interest is a restriction on the instrument block, not on the covariates.

Let \(P_W=W(W'W)^{-1}W'\) and \(M_W=I-P_W\), the projection onto the covariate space and the residual-maker for that space. Residualizing \(r\) and \(Z\) against \(W\) gives \(\tilde r=M_Wr\) and \(\tilde Z=M_WZ\). The coefficient test in the full regression is equivalent to testing whether the residualized instruments explain the residualized transformed outcome. This is the Frisch–Waugh–Lovell result applied to the AR regression.

Here \(SSR_R=r'M_Wr\) is the residual sum of squares from the restricted regression, and \(SSR_U\) is the residual sum of squares from the unrestricted regression. The numerator measures the reduction in residual sum of squares from adding the \(q\) excluded instruments, divided by the number of restrictions. The denominator is the unrestricted residual variance estimate, based on \(n-k-q\) residual degrees of freedom. Under \(H_0\) and the stated normal, homoskedastic conditions, \(F_{\mathrm{AR}}(\beta_0)\sim F_{q,n-k-q}\).

The equality in the formula follows because the fitted component from adding \(Z\), after accounting for \(W\), is the projection of \(\tilde r\) onto the column space of \(\tilde Z\). Its sum of squares is \(\tilde r'P_{\tilde Z}\tilde r\), which equals \(SSR_R-SSR_U\). The F test therefore compares the improvement from imposing the candidate null with the residual variation left in the unrestricted regression.

Worked Calculations

Diagnostics, Comparisons, and Limitations

Before interpreting a result, verify that the restricted and unrestricted regressions use identical observations and the same \(W\); only the addition of \(Z\) should distinguish them. Check that the covariate matrix and instrument block have the required ranks, that \(n>k+q\), and that \(SSR_R\) is at least \(SSR_U\), as it must be for nested least-squares regressions. A negative reduction in sum of squares usually signals that the models, samples, or reported quantities do not match.

The joint restriction is \(H_0:\pi_1=\cdots=\pi_q=0\), where the \(\pi_j\) are the coefficients on excluded instruments. It is not the restriction that the coefficients on \(W\) are zero, nor is it a first-stage test of whether instruments predict \(d\). The transformed outcome is constructed using the candidate structural coefficient; the AR test evaluates whether the instruments explain that transformed outcome after covariate adjustment.

A conventional IV Wald test estimates the IV coefficient and scales its distance from \(\beta_0\) by a standard error. Its usual approximation can be unreliable when instruments are weak. The AR test instead tests the null directly through the transformed-outcome regression, so its null reference distribution does not depend on a large first-stage coefficient. A first-stage strength statistic addresses instrument relevance, not whether a proposed structural coefficient is compatible with the instrument restrictions.

The exact F reference distribution above is not automatically appropriate under heteroskedasticity, clustering, dependence, or nonnormal finite-sample errors. In those settings, the conventional residual-variance F calculation may have the wrong size. The test also can have low power when instruments explain little variation after covariates are partialled out. Finally, failure to reject a candidate does not prove it is correct; it means the data do not provide sufficient evidence against it under the specified assumptions.

Common Pitfalls

• Testing the covariates instead of the instruments. The AR restriction is on the excluded-instrument coefficients in the null-imposed regression. Covariates remain included and unrestricted.
• Using different covariate adjustments. Both restricted and unrestricted regressions must contain the same \(W\), fitted on the same observations.
• Assuming full rank guarantees an F statistic. Full rank does not ensure positive residual degrees of freedom. Require \(n>k+q\) as well.
• Interpreting a non-significant result as proof. A large p-value is not the probability that the null is true and does not establish instrument validity or strong identification.
• Using the homoskedastic F formula automatically. The stated exact reference needs the conditional normal, common-variance, uncorrelated-error assumptions; it is not a general solution for heteroskedastic or clustered errors.