Weak-Instrument-Robust Tests of Joint Coefficient Restrictions

Why Joint Restrictions Need Weak-Instrument-Robust Inference

Suppose a structural equation contains several endogenous regressors, with coefficient vector \(\beta\). A question of interest may concern a contrast such as whether two coefficients sum to one, or whether a group of coefficients is jointly zero. When the instruments are weak, a test built from the usual 2SLS estimate and its estimated standard error can have poor size: its rejection probability under the null may differ substantially from the nominal significance level.

A reduced-form test avoids relying on a well-approximated sampling distribution for the 2SLS estimator. For any proposed value of \(\beta\), it tests whether the outcome less the corresponding linear combination of endogenous regressors is related to the excluded instruments. The joint-restriction test then searches over all coefficient vectors satisfying the restriction and uses the smallest reduced-form test statistic. This construction remains valid even when the instruments provide little information about the endogenous regressors, subject to the moment and regularity conditions stated below.

The Reduced-Form Test for a Candidate Coefficient Vector

Fix a candidate vector \(b\) for \(\beta\). Under the candidate, form \(y_i-d_i'b\). If \(b=\beta\), the structural equation says that this quantity is \(x_i'\gamma+u_i\). Under instrument exogeneity, the excluded instruments should therefore have no additional explanatory power after accounting for the included controls.

One implementation regresses \(y_i-d_i'b\) on \(x_i\) and \(z_i\), then uses a heteroskedasticity-robust Wald test of the null that all \(L\) coefficients on \(z_i\) are zero. Equivalently, after accounting for the controls, the moment vector for the candidate is based on \(z_i(y_i-d_i'b)\). Its population expectation is zero at the true coefficient vector. A robust Wald statistic for these \(L\) moment restrictions can be written as

Here \(g_i(b)\) is the \(L\)-vector of instrument moments, with included controls handled by regression adjustment or partialling out, and \(\widehat{\Omega}(b)\) is a heteroskedasticity-robust estimate of the covariance matrix of the moment contributions. In a regression implementation, the equivalent robust Wald statistic uses the appropriate sandwich covariance estimator for the excluded-instrument coefficients. The covariance being estimated is that of the moment contributions, not merely the covariance matrix of the instruments.

For a point null \(b=\beta\), the statistic is asymptotically \(\chi^2_L\) under suitable conditions. For example, sufficient conditions include independent and identically distributed observations, finite \(2+\delta\) moments for some \(\delta>0\) of the relevant regressors and errors, and a nonsingular \(\operatorname{Var}(g_i(\beta))\). The robust covariance estimator must consistently estimate this covariance matrix. Other sampling schemes can also work, but they require appropriate central-limit, covariance-convergence, and covariance-estimation conditions. Independence and finite moments alone, without such conditions, are not enough to guarantee the approximation.

This point-null test is an identification-robust reduced-form test: its validity does not require the first-stage coefficients to be large or the structural coefficient vector to be precisely estimated. The instruments must still satisfy the required exclusion and moment conditions for the structural model. The test does not establish those conditions; it uses them.

Testing a Joint Linear Restriction

For the null \(R\beta=r\), define the restricted statistic by minimizing the point-null statistic over the entire affine set that satisfies the restrictions:

The infimum is used because the restricted set need not have a unique minimizer. In ordinary applications with a continuous statistic and a well-behaved closed restriction set, it is attained; numerical optimization should nevertheless check for convergence and sensitivity to starting values. A rejection means that even the coefficient vector on the null set that best fits the reduced-form moments produces a sufficiently large statistic.

The reference degrees of freedom are \(L\), the number of excluded-instrument moment restrictions in the point-null test—not \(q\), the number of coefficient restrictions. Let \(c_{L,1-\alpha}\) be the \(1-\alpha\) quantile of \(\chi^2_L\). Reject \(H_0:R\beta=r\) when \(T_R(r)>c_{L,1-\alpha}\). The resulting test can be conservative, particularly when minimizing over a large set of coefficient vectors.

The key inequality gives the size justification directly. Because the true coefficient vector belongs to the restricted set, minimizing over that set cannot produce a larger statistic than evaluating the statistic at the truth. Therefore,

This argument does not require the minimizer to estimate \(\beta_0\), or the instruments to identify \(\beta_0\) strongly. It explains both the robustness and the possible conservatism: the restricted statistic is no larger than the statistic at the true value, so its rejection probability cannot exceed the point-test benchmark asymptotically.

A calibrated p-value is the upper-tail probability \(P(\chi^2_L\geq T_R(r))\). It is conservative in the sense that, under the null, its probability of falling below any nominal level is no greater than that level asymptotically. It is not the probability that the null is true.

Confidence Regions by Inverting the Tests

A point-null test can be inverted by retaining every candidate vector whose reduced-form statistic does not exceed the \(\chi^2_L\) critical value. The resulting set is a weak-instrument-robust confidence region for \(\beta\):

Under the same conditions, if \(\beta_0\) is the true coefficient vector, then it belongs to this set whenever the point-null test at \(\beta_0\) does not reject. Thus the region has asymptotic coverage of at least \(1-\alpha\). Inverting the tests for a scalar restriction value \(r\) gives a confidence set for that restriction:

Weak identification can make these sets wide, unbounded, or irregular rather than elliptical. That shape is informative: it indicates that many coefficient values, possibly very distant from one another, are compatible with the reduced-form evidence. A narrow region should not be assumed in advance simply because a conventional 2SLS confidence interval is narrow.

Practical Interpretation and a Common Pitfall

The method matters when researchers need conclusions about several endogenous-regressor coefficients but cannot justify strong identification. It provides tests of specified linear restrictions and confidence regions using reduced-form evidence, rather than treating a potentially unstable 2SLS estimate as if its strong-identification approximation necessarily applied. The trade-off is that valid inference can be conservative and can yield confidence sets that are less visually familiar than Wald ellipses.

• Pitfall: using the number of coefficient restrictions for the critical value. A reader may see \(q\) restrictions in \(R\beta=r\) and compare the minimized statistic with a \(\chi^2_q\) critical value. That is not the calibration justified here. The point reduced-form test has \(L\) instrument-moment restrictions, and the conservative minimized test uses \(\chi^2_L\).
• Correction: Identify the number \(L\) of excluded-instrument moment restrictions in the reduced-form test, use its \(\chi^2_L\) critical value, and minimize the statistic over the full null set before making the decision.
• Pitfall: treating failure to reject as confirmation. Weak instruments may leave the test with limited ability to distinguish among coefficient values. A large p-value does not show that the restriction is true.
• Correction: Report the test decision and, where useful, the inverted confidence region. Its width and shape communicate how much the data constrain the coefficients.

For a reproducible analysis, report the restriction matrix \(R\) and vector \(r\), the number of excluded instruments \(L\), the robust covariance procedure, the optimization method, the minimized statistic, and the chi-square-calibrated p-value or critical value. These details make clear which null was tested and how the weak-instrument-robust calibration was applied.