Heteroskedasticity-Robust Weak-Instrument Tests

Why robust weak-instrument tests matter

Instrumental-variable Wald inference can be unreliable when the first stage is weak. Anderson–Rubin (AR) and score-based tests address a different question: for a proposed value of the structural coefficient, do the instruments satisfy the implications of that value? Their null distributions do not require the first stage to be strong. In applications with nonconstant error variance, however, the covariance estimate used to calibrate the test must also be appropriate for heteroskedasticity.

This tutorial treats one endogenous regressor and a scalar coefficient, with \(q\) excluded instruments. The focus is the test of one proposed coefficient value, not joint restrictions on several structural coefficients. Robustness to weak instruments does not remove the need for valid instruments, correct moment conditions, or regularity conditions for the large-sample approximation.

Model, null, and moment conditions

For independent observations \(i=1,\ldots,n\), write the structural equation as

y_i=x_i\beta+c_i'\gamma+u_i,

where \(y_i\) is the outcome, \(x_i\) is the endogenous regressor, \(c_i\) is a vector of included exogenous controls (including an intercept if needed), and \(z_i\) is a \(q\)-vector of excluded instruments. The parameter of interest is \(\beta\). To test \(H_0:\beta=b\), the null implies \(y_i-x_i b=c_i'\gamma+u_i\).

Definition: Let \(\tilde z_i\) denote the instruments residualized with respect to the included controls. The null moment condition is \(E[\tilde z_i u_i]=0\). Its covariance matrix is \(\Omega=E[\tilde z_i\tilde z_i'u_i^2]\). Heteroskedasticity is allowed: the variance of \(u_i\) may vary with instruments, controls, or other observed characteristics, provided the stated moment and regularity conditions hold.

The moment condition expresses instrument exogeneity relative to the controls; it is not a claim that \(x_i\) is uncorrelated with \(u_i\). Indeed, \(x_i\) may be endogenous. Under \(H_0\), the moment is zero because the null-imposed structural error is \(u_i\). No lower bound on first-stage strength is needed for this null implication.

Robust Anderson–Rubin test

For a candidate \(b\), form the outcome \(y_i-x_i b\) and regress it on both the controls and the excluded instruments. Write the auxiliary regression as

y_i-x_i b=c_i'\gamma_b+z_i'\pi_b+e_i.

Under \(H_0:\beta=b\) and instrument validity, the population coefficient on the excluded instruments is \(\pi_b=0\). The AR test is a Wald test of this restriction in the auxiliary regression. Use a heteroskedasticity-consistent covariance estimate, such as HC0; do not use the homoskedastic regression covariance unless constant conditional variance is justified.

AR_{\mathrm{rob}}(b) =\hat\pi_b'\widehat{\mathrm{Var}}_{\mathrm{HC0}}(\hat\pi_b)^{-1}\hat\pi_b.

For clarity, if \(A\) is the auxiliary regression design matrix containing controls and instruments, and \(\hat e_i\) are its unrestricted OLS residuals, the HC0 covariance for all auxiliary coefficients is

(A'A)^{-1}\left(\sum_{i=1}^n a_i a_i'\hat e_i^2\right)(A'A)^{-1},

where \(a_i\) is the design row written as a column vector. The covariance for \(\hat\pi_b\) is the instrument-coefficient block of this matrix. Under the null and regularity conditions below, the robust statistic is asymptotically \(\chi^2_q\). This is generally an asymptotic result, not an exact finite-sample result.

Worked Example: Two-Instrument Robust AR Test

Suppose a hypothetical sample has \(n=100\). For the proposed coefficient \(b\), the auxiliary regression gives \(\hat\pi_b=(0.12,-0.08)'\). Suppose the instrument-coefficient block of its HC0 covariance estimate is

\widehat{\mathrm{Var}}_{\mathrm{HC0}}(\hat\pi_b) =\frac{1}{100} \begin{pmatrix}4&1\\1&2\end{pmatrix}.

The inverse covariance is \(\frac{100}{7}\begin{pmatrix}2&-1\\-1&4\end{pmatrix}\). Thus

AR_{\mathrm{rob}}(b) =100 \begin{pmatrix}0.12&-0.08\end{pmatrix} \frac{1}{7} \begin{pmatrix}2&-1\\-1&4\end{pmatrix} \begin{pmatrix}0.12\\-0.08\end{pmatrix} =100\left(\frac{0.0736}{7}\right) \approx 1.051.

For two restrictions, the reference distribution is \(\chi^2_2\). Its upper-tail probability at 1.051 is \(\exp(-1.051/2)\approx0.5913\). At \(\alpha=0.05\), the test does not reject \(H_0\). This means the auxiliary-regression evidence is not strong enough to reject this proposed coefficient; it does not establish that the coefficient is correct.

Robust score test from the null moment

A score-based construction evaluates the instrument moment at the proposed value, without estimating the unrestricted instrument coefficients for the numerator. Let \(r_i^0\) be the residual from the restricted regression of \(y_i-x_i b\) on the controls alone. Let \(\tilde z_i\) be the sample residual from projecting each instrument on those same controls. The sample null moment and its HC covariance estimate are

\bar g=\frac{1}{n}\sum_{i=1}^n\tilde z_i r_i^0, \qquad \hat\Omega_0=\frac{1}{n}\sum_{i=1}^n\tilde z_i\tilde z_i'(r_i^0)^2.

The robust score statistic is

S_{\mathrm{rob}}(b)=n\bar g'\hat\Omega_0^{-1}\bar g.

The numerator is the null-imposed moment. It must not be formed from residuals of the unrestricted auxiliary regression: those OLS residuals are orthogonal to the included residualized instruments, making their sample instrument moments zero by construction. The restricted residuals \(r_i^0\), in contrast, retain the moment discrepancy being tested. The robust Wald and score statistics use different finite-sample covariance calculations, but under the null and standard regularity conditions they have the same limiting \(\chi^2_q\) distribution.

Worked Example: Computing a Robust Score Statistic

In a hypothetical sample with \(n=50\) and one residualized instrument, suppose the null-restricted residuals and instrument values give \(\bar g=0.040\), and the null-residual HC estimate is \(\hat\Omega_0=0.160\). Then

S_{\mathrm{rob}}(b) =50\frac{(0.040)^2}{0.160} =50\frac{0.0016}{0.160} =0.500.

With \(q=1\), compare this statistic to \(\chi^2_1\); the upper-tail p-value is approximately 0.4795. The test does not reject at 0.05. Here the statistic is based on the restricted moment, not on the unrestricted auxiliary-regression residuals. The calculation presumes the moment conditions and regularity conditions hold; merely calculating a robust covariance does not establish instrument validity.

Why the chi-square approximation applies

The justification is a central limit argument for the null moment. Let \(g_i=\tilde z_i u_i\). Under the null, \(E[g_i]=0\); if observations are independent and identically distributed, \(E[g_i g_i']\) is finite, and standard regularity conditions ensure consistent estimation of the controls and residualized instruments, then

\sqrt n\,\bar g\ \longrightarrow_d\ N(0,\Omega), \qquad \hat\Omega_0\ \longrightarrow_p\ \Omega.

If \(\Omega\) is positive definite, premultiplication by its inverse square root gives a standard normal \(q\)-vector. Its squared Euclidean length has a \(\chi^2_q\) distribution. Replacing \(\Omega\) by a consistent estimate therefore yields the score limit:

n\bar g'\hat\Omega_0^{-1}\bar g\ \longrightarrow_d\ \chi^2_q.

The same null restriction in the auxiliary regression yields the robust AR Wald limit. The first-stage coefficient does not enter the null moment restriction \(\pi_b=0\), which explains why the test can remain valid when the first stage is weak or zero. Weak identification can make the resulting confidence set wide or otherwise uninformative, but it does not by itself invalidate this null-based calibration.

In applied work, the argument requires more than the zero-mean restriction. Typical sufficient conditions include a fixed number \(q\) of instruments, independent observations (or a suitable dependence-robust alternative), finite moments adequate for the central limit theorem and covariance estimation, and no single observation dominating the sample moments. The covariance matrix \(\Omega\) must be nonsingular. If observations are clustered or serially dependent, the displayed HC0 covariance is not generally valid; the covariance estimator must reflect that dependence. Exact finite-sample validity also does not follow from these asymptotic conditions.

Worked Example: Why the Robust Covariance Changes the Decision

Suppose a hypothetical one-instrument auxiliary regression gives \(\hat\pi_b=0.25\). The conventional homoskedastic standard error is 0.10, while the HC0 standard error is 0.20. The one-restriction Wald statistics are

W_{\mathrm{homo}}=(0.25/0.10)^2=6.25, \qquad W_{\mathrm{rob}}=(0.25/0.20)^2=1.5625.

Using the asymptotic \(\chi^2_1\) reference distribution, the p-values are approximately 0.0124 and 0.211, respectively. The homoskedastic calculation would reject at 0.05, whereas the robust calculation would not. If the errors are heteroskedastic, only the robust covariance calculation is justified by the conditions stated here. These numbers illustrate the effect of a covariance choice; they do not show that heteroskedasticity is present in any real study.

Interpretation and common pitfalls

A robust AR or score test assesses a proposed coefficient through instrument moments. A small p-value is evidence against that value under the maintained model and moment conditions. A large p-value is not the probability that the null is true, and non-rejection is not proof of identification or validity. When the instruments are weak, a set of coefficient values may all be difficult to reject; reporting that uncertainty is more informative than substituting conventional Wald inference.

Using unrestricted residuals for the score numerator. The reasoning “use the residuals from the auxiliary regression in every moment” fails because OLS forces their sample moments with included instruments to zero. Use \(r_i^0\), the null-restricted residual, in \(\bar g\).
Assuming robust means exact. HC covariance estimates protect the asymptotic variance calculation against heteroskedasticity; they do not generally produce an exact finite-sample reference distribution.
Confusing weak-instrument robustness with instrument validity. These tests do not rescue a design in which the excluded instruments violate the required exogeneity moments.
Using the wrong covariance structure. HC0 addresses independent observations with potentially unequal variances. Clustering or serial dependence requires an estimator that accounts for that dependence.

Key takeaway: For robust weak-instrument inference, test the null-implied instrument restriction and estimate its covariance robustly. In the score statistic, form the numerator from null-restricted residuals; in the AR Wald statistic, use the heteroskedasticity-consistent covariance of the unrestricted auxiliary-regression coefficients.

Check Your Understanding

Work through these questions using the definitions and conditions above.

Why can the AR test of \(H_0:\beta=b\) be written as a test that the excluded-instrument coefficients are zero in an auxiliary regression?
For the robust score statistic, which residuals enter \(\bar g\), and why would unrestricted auxiliary-regression residuals be unsuitable?
With one instrument, \(n=80\), \(\bar g=0.05\), and \(\hat\Omega_0=0.20\), calculate the robust score statistic.
Name two regularity conditions needed for the chi-square approximation beyond the zero-mean moment restriction.
What does a failure to reject a proposed coefficient mean, and what does it not establish?