Conditional First-Stage Diagnostics with Multiple Endogenous Regressors

Why Multiple Endogenous Regressors Need Conditional Diagnostics

When a model has more than one endogenous regressor, a strong first stage for each regressor considered separately does not guarantee that the instruments distinguish their effects from one another. Two endogenous regressors may have fitted values that move almost in lockstep with the instruments. In that case, the instruments may predict both variables well overall but provide little independent variation for separating their coefficients.

Sanderson-Windmeijer conditional first-stage F statistics address this issue by assessing instrument strength for each endogenous regressor conditional on the others. They are diagnostics of conditional identification, not tests of whether the instruments satisfy the exclusion restriction and not a substitute for weak-instrument-robust inference.

The null is about rank, not simply whether every coefficient in the first-stage equation for \(D_j\) is zero. For \(m=2\), for example, the instruments may predict both regressors, but if their coefficient vectors are proportional, they do not provide distinct instrument directions for the two endogenous effects. The conditional statistic for each regressor measures evidence against that regressor's rank-deficiency null.

Statistic and Calculation

In the usual homoskedastic formulation, the Sanderson-Windmeijer statistic is an F test of the conditional relevance restrictions. Its numerator degrees of freedom are \(q-m+1\), the number of independent instrument directions left after conditioning on the other \(m-1\) endogenous regressors. The denominator degrees of freedom depend on the sample and the first-stage specification. In a robust implementation, the restrictions are tested with a heteroskedasticity-robust covariance estimate; the resulting Wald statistic can be reported in F form by dividing by the number \(r=q-m+1\) of restrictions.

Here \(\widehat{g}_j\) is the estimated vector of conditional relevance restrictions for \(D_j\), and \(\widehat{V}_j\) is its estimated covariance matrix. The restrictions are formed from the reduced-form coefficient matrix and account for the fact that the directions spanned by the other endogenous regressors are estimated too. Thus, a full calculation uses the joint reduced-form estimates and their covariance; it is not generally correct to take an ordinary first-stage F statistic and divide it by an arbitrary number.

With a conventional homoskedastic finite-sample calculation, the statistic is compared with an F distribution using the appropriate numerator and denominator degrees of freedom. With robust covariance estimation, the Wald form is asymptotic; software may report a robust F approximation or a chi-square version. The reference distribution and covariance estimator should therefore be stated when reporting results. The examples below use the robust Wald-F form and an F approximation with the stated denominator degrees of freedom.

What to Conclude From the Statistics

Read a separate statistic for each endogenous regressor. A small conditional F statistic indicates that, after accounting for the other endogenous regressors, the data provide limited evidence that the excluded instruments supply an additional identifying direction for that regressor. A larger value indicates stronger evidence against that conditional relevance null, but it does not by itself prove that conventional 2SLS inference is reliable.

The ordinary first-stage F statistic and the conditional statistic answer different questions. An ordinary first-stage F tests the excluded instruments' joint contribution in one reduced-form equation. The Sanderson-Windmeijer statistic asks whether that contribution distinguishes one endogenous regressor from the others. When the instrument coefficient vectors are close to linearly dependent, the ordinary statistics can appear reassuring while one or more conditional statistics remain low.

There is no universal cutoff that turns every conditional F statistic into a definitive strong-or-weak classification. Interpretation depends on the design, the number of instruments and endogenous regressors, the covariance estimator, and the reference distribution. Stock-Yogo critical values are not automatically applicable to every conditional statistic or robust implementation. Report the statistic for each regressor, its degrees of freedom, the covariance method, and the reference distribution, then use the diagnostic as one part of an identification and inference assessment.

Common Mistake: Treating Separate First Stages as Independent Identification Checks

A common mistake is to see a large ordinary first-stage F statistic for every endogenous regressor and conclude that all endogenous effects are separately well identified. That reasoning checks whether each regressor is predicted, but not whether the instruments generate sufficiently distinct fitted movements across regressors. If \(\pi_1\) and \(\pi_2\) are nearly proportional, both regressors can have substantial first-stage prediction while the instrument variation is poor at distinguishing their effects.

The correction is to examine the conditional statistic for each regressor and interpret it as a measure of independent instrument information conditional on the others. These statistics diagnose relevance and conditional identification; they do not verify instrument exogeneity, and a failure to reject a conditional null is not proof that the null is true.