Heteroskedasticity-Robust Anderson-Rubin Tests

Why Robustness Matters for the Anderson–Rubin Test

An Anderson–Rubin (AR) test evaluates a proposed value of an instrumental-variables coefficient without first estimating that coefficient by dividing a reduced-form estimate by a first-stage estimate. This distinction matters when the instrument may be weak: the division can make conventional IV Wald inference unreliable, whereas the AR test asks whether the instrument predicts an outcome transformed using the proposed coefficient.

The version developed here is for one endogenous regressor and one excluded instrument, with included covariates. Its heteroskedasticity-robust covariance estimate allows the structural error variance to differ across observations. It does not make an invalid instrument valid, and it does not guarantee high power. The introductory material on Anderson–Rubin tests establishes the basic test; the focus here is how to construct its robust variance and what that changes.

Model, Hypothesis, and Assumptions

Let \(Y_i\) be the outcome, \(D_i\) the endogenous regressor, \(Z_i\) a scalar excluded instrument, and \(X_i\) a vector of included covariates that contains an intercept. The structural equation is

Here \(\beta\) is the coefficient being tested, \(\gamma\) is a vector of nuisance coefficients, and \(u_i\) is the structural error. For a proposed value \(\beta_0\), the null and alternative are \(H_0:\beta=\beta_0\) and \(H_a:\beta\ne\beta_0\). The instrument is excluded from the structural equation: it has no direct effect on \(Y_i\) once \(D_i\) and \(X_i\) are accounted for.

The test’s central validity condition under the null is that the instrument is orthogonal to the structural error after accounting for the covariates: \(E[Z_i u_i\mid X_i]=0\). The model also requires a correctly specified conditional mean for the transformed outcome under the null, independent observations (or an appropriate sampling setup with a valid central limit theorem), finite moments sufficient for the robust variance estimate, and nonzero residual variation in \(Z_i\) after controlling for \(X_i\). Heteroskedasticity is permitted: the variance of \(u_i\) may depend on \(Z_i\), \(X_i\), or other observed characteristics.

These are not assumptions that robust standard errors can create. In particular, robust covariance estimation addresses variation in error variance, not a violation of instrument exogeneity or exclusion. The instrument must still be substantively defensible.

Constructing the Robust Statistic

For a given \(\beta_0\), define the transformed outcome \(R_i(\beta_0)=Y_i-\beta_0D_i\). Under \(H_0\), the structural equation implies \(R_i(\beta_0)=X_i'\gamma+u_i\). The null and instrument orthogonality condition therefore imply that, conditional on \(X_i\), the instrument has zero coefficient in the regression of \(R_i(\beta_0)\) on \(X_i\) and \(Z_i\).

To state the calculation compactly, residualize \(Z_i\) and \(R_i(\beta_0)\) with respect to \(X_i\). Write the resulting values as \(\widetilde Z_i\) and \(\widetilde R_i\). By the Frisch–Waugh–Lovell result, the coefficient on \(Z_i\) in the full regression is

The residual for observation \(i\) from this regression is \(\widehat e_i=\widetilde R_i-\widehat{\delta}(\beta_0)\widetilde Z_i\). The HC0 heteroskedasticity-robust variance estimate for the instrument coefficient is

This is the sandwich variance for the coefficient on the residualized instrument. It uses each observation’s squared residual, rather than imposing a common error variance. Other common HC versions adjust this estimate for sample size or leverage; the version used must be reported, and the resulting calibration is generally asymptotic rather than an exact small-sample result.

The robust Wald statistic is the squared robust \(t\)-statistic. With one excluded instrument and one tested coefficient, its null reference distribution is asymptotically chi-square with one degree of freedom:

At significance level \(\alpha\), reject \(H_0\) when \(W_{\mathrm{AR,rob}}(\beta_0)\) exceeds the \(1-\alpha\) quantile of \(\chi^2_1\), or equivalently when its asymptotic p-value is below \(\alpha\). This is a two-sided test of the instrument coefficient. The reference distribution does not make the test exact in a small sample.

Why Weak Relevance Does Not Break the Test

The key point is visible directly in the null regression. If \(\beta_0=\beta\), then \(Y_i-\beta_0D_i=X_i'\gamma+u_i\), so the instrument coefficient is zero under the stated orthogonality condition. The test estimates and tests that coefficient; it does not estimate the first-stage coefficient and divide by it.

That gives a short justification for weak-instrument validity. Under the null, the robust Wald statistic is based on an estimate of a zero reduced-form coefficient and its sandwich standard error. Under the regularity conditions above, the estimate, standardized by that standard error, has an asymptotic standard normal distribution; its square has an asymptotic \(\chi^2_1\) distribution. The argument does not require the first-stage coefficient to be large or bounded away from zero. By contrast, an IV Wald statistic depends on a ratio involving the first-stage estimate, which can be unstable when that estimate is near zero.

This is a statement about size: under the null and assumptions, rejection occurs at approximately the chosen significance level, even when the instrument is weak. It is not a statement that the test will be informative. A weak instrument may make the AR test have low power, so it may fail to reject many false proposed values.

Worked Example: Robust and Homoskedastic Conclusions

Suppose a hypothetical analysis of 240 independent observations tests a proposed coefficient using the transformed outcome regression. After residualizing on the included covariates, its computed summaries are \(\sum_i\widetilde Z_i^2=100\), \(\sum_i\widetilde Z_i\widetilde R_i=30\), and \(\sum_i\widetilde Z_i^2\widehat e_i^2=400\). For comparison, suppose the regression’s homoskedastic variance calculation gives an estimated error variance of \(1\).

The estimated instrument coefficient is \(30/100=0.300\). The robust variance and standard error are

Thus \(W_{\mathrm{AR,rob}}=0.300^2/0.0400=2.250\), with an asymptotic \(\chi^2_1\) p-value of about \(0.134\). At the 5% level, the robust AR test does not reject.

Under the homoskedastic formula, the variance would instead be \(1/100=0.0100\), giving \(SE=0.100\) and a Wald statistic of \(0.300^2/0.0100=9.000\). Its \(\chi^2_1\) p-value is about \(0.0027\). These different conclusions illustrate why the variance assumption matters. If the errors are heteroskedastic, the homoskedastic calculation need not estimate the correct sampling variance; the robust result is the appropriate one under the stated independent-observation conditions.

Worked Example: Testing a Specific Coefficient

Suppose a hypothetical researcher tests \(H_0:\beta=1.5\), where \(D_i\) is measured in years of a program and \(Y_i\) in outcome points. The coefficient \(1.5\) represents a proposed change of 1.5 outcome points per additional program year. After constructing \(Y_i-1.5D_i\) and regressing it on the included covariates and instrument, the instrument coefficient is \(0.080\), with HC0 standard error \(0.040\).

The robust statistic is

The asymptotic \(\chi^2_1\) p-value is approximately \(0.0455\). At \(\alpha=0.05\), reject the proposed value \(1.5\), conditional on the model, instrument, and sampling assumptions. This result is evidence against that particular coefficient value; it is not by itself evidence that the instrument is valid, nor does it identify which alternative coefficient is correct.

Diagnostics, Comparisons, and Limits

A useful implementation check is to fit the transformed-outcome regression directly and compare its instrument coefficient and HC standard error with the residualized calculation. They should agree apart from numerical rounding and the chosen HC correction. Also confirm that the instrument has residual variation after controlling for \(X_i\); if \(\sum_i\widetilde Z_i^2=0\), its coefficient cannot be estimated in this regression.

The robust test should be interpreted cautiously when the number of independent observations is small, when a few observations have high leverage, or when the instrument coefficient estimate is heavily influenced by a small number of residualized instrument values. HC0 does not automatically solve finite-sample problems. And if observations are dependent within groups, an independent-observation HC covariance estimate is not justified. The covariance estimator must match the sampling dependence.

The robust AR test is therefore a useful alternative when the first stage may be weak and heteroskedasticity is plausible, but it is not a universal remedy. It can be conservative or uninformative when the instrument is weak, and a failure to reject should not be reported as proof that a proposed coefficient is correct. The heteroskedasticity-robust covariance estimate also does not test instrument exogeneity; that assumption requires substantive justification and, where possible, separate diagnostic reasoning.

Common Mistake

• Wrong reasoning: “I used robust standard errors, so the AR test is valid even if the instrument is correlated with the structural error.”
• Correction: Robust standard errors address heteroskedasticity in the variance calculation. The AR test’s null argument still requires the instrument to be orthogonal to the structural error conditional on the included covariates. Robustness to one variance feature is not robustness to instrument invalidity.