Conditional Strength With Multiple Endogenous Regressors

Why Separate First Stages Can Mislead

With one endogenous regressor, a first-stage F statistic summarizes how strongly the excluded instruments predict that regressor after accounting for included exogenous covariates. With several endogenous regressors, the same calculation can be made separately for each first stage. But these separate statistics do not answer a crucial question: do the instruments predict each endogenous regressor in a way that is distinct from their prediction of the other endogenous regressors?

Suppose two instruments move two endogenous regressors in almost exactly the same proportions. Each regressor may be highly predictable from the instruments considered together, so each separate first-stage F can be large. Yet the instruments may provide little information for distinguishing the two endogenous regressors from one another. That distinction matters because 2SLS needs instrument-induced variation that can separately identify the coefficients on all endogenous regressors.

The diagnostic for this issue is a conditional first-stage strength statistic, commonly reported as a Sanderson–Windmeijer conditional F statistic. It assesses the excluded instruments’ contribution to predicting one endogenous regressor conditional on the other endogenous regressors’ instrument-predicted variation. It complements, rather than replaces, separate first-stage statistics.

Definition: Let \(Y\) be an outcome, \(D=(D_1,\ldots,D_K)\) a vector of \(K\) endogenous regressors, \(X\) included exogenous covariates, and \(Z=(Z_1,\ldots,Z_L)\) excluded instruments. A conditional first-stage strength diagnostic for \(D_j\) assesses how much identifying information the excluded instruments provide for \(D_j\), conditional on the instrument-predicted variation in the other endogenous regressors \(D_{-j}\) and on \(X\). It is usually reported as a conditional F statistic for testing the relevant excluded-instrument contribution.

The first-stage reduced-form equations express each endogenous regressor in terms of the same exogenous variables and instruments:

D_j=X\gamma_j+Z\pi_j+v_j,\qquad j=1,\ldots,K.

Here \(\pi_j\) is the vector of instrument coefficients in the reduced form for \(D_j\), and \(v_j\) is its disturbance. Separate first-stage tests ask whether \(\pi_j=0\) for each \(j\). Conditional diagnostics instead ask whether the instrument-driven variation relevant to one \(D_j\) remains informative after allowing for the instrument-driven variation in the other endogenous regressors. The key issue is therefore not only the size of each \(\pi_j\), but also how their directions relate across equations.

How Conditional Strength Arises

For a geometric explanation, first remove the linear contribution of \(X\) from the instruments and endogenous regressors. Denote the resulting residualized variables by \(\widetilde Z\) and \(\widetilde D_j\). The instrument-predicted components of the endogenous regressors are generated by the same residualized instrument matrix, with coefficient vectors \(\pi_j\). If the coefficient vectors are nearly proportional, the instrument-predicted components will also be nearly proportional: the instruments move the endogenous regressors together.

For two endogenous regressors, suppose \(\pi_2=c\pi_1\) for some constant \(c\). Then the instrument-predicted component of \(D_2\) is \(c\) times that of \(D_1\). Although each component can have substantial variation, the instrument-predicted component of \(D_2\) left after removing the component associated with \(D_1\) is zero. If the coefficient vectors are only approximately proportional, this remaining component is small rather than zero.

This is the logic behind conditioning. Conditional strength focuses on the instrument contribution that is not shared with the other endogenous regressors. A conventional conditional test uses a partial F calculation: compare the reduction in residual sum of squares from adding the excluded instruments to a conditional first-stage relation with the residual variation left in that relation.

F=\frac{(SSE_R-SSE_U)/q}{SSE_U/df_U}.

Here \(SSE_R\) and \(SSE_U\) are the restricted and unrestricted sums of squared errors, \(q\) is the number of excluded-instrument restrictions tested, and \(df_U\) is the unrestricted residual degrees of freedom. For a single endogenous regressor, this reduces to the usual excluded-instrument first-stage F calculation. With multiple endogenous regressors, use the appropriate conditional diagnostic, such as the Sanderson–Windmeijer statistic; do not assume that an ordinary regression F from an ad hoc equation with another endogenous regressor inserted as a control is automatically the correct statistic. Implementations can use the model’s first-stage information to construct the conditional test.

Worked Example: Strong Separate F Statistics, Weak Conditional Strength

Consider hypothetical data with two endogenous regressors, two excluded instruments, one included covariate, and \(n=120\) observations. The separate first-stage output reports \(F=34.0\) for \(D_1\) and \(F=31.0\) for \(D_2\). Both regressors are strongly predicted when each is considered on its own. Suppose the conditional diagnostic reports a conditional F of \(2.40\) for \(D_1\) given \(D_2\), and \(1.73\) for \(D_2\) given \(D_1\).

For illustration, the conditional output for \(D_1\) corresponds to an excluded-instrument sum-of-squares contribution of \(0.48\), an unrestricted residual sum of squares of \(11.52\), \(q=2\) restrictions, and \(df_U=115\). Its statistic is

F_{D_1\mid D_2} =\frac{0.48/2}{11.52/115} =\frac{0.2400}{0.1002} =2.396.

For \(D_2\) given \(D_1\), suppose the corresponding values are \(0.36\), \(12.00\), \(q=2\), and \(df_U=115\). Then

F_{D_2\mid D_1} =\frac{0.36/2}{12.00/115} =\frac{0.1800}{0.1043} =1.725.

The numbers are internally consistent: in each calculation, the numerator is the added instrument contribution per restriction and the denominator is the unrestricted residual mean square. The conditional statistics are much smaller than the separate first-stage F statistics. The appropriate conclusion is not that the instruments fail to predict either endogenous regressor at all; rather, their predictive information is largely shared, leaving limited conditional information for distinguishing the regressors. These hypothetical values flag a serious weakness that calls for further assessment and careful inference.

The displayed sums of squares illustrate the F calculation; in applied work, obtain the conditional statistic from a procedure that implements the conditional first-stage test for the fitted IV model. Do not treat the illustrative arithmetic as a recipe for constructing a valid statistic from any arbitrary regression.

Interpreting the Statistic Alongside Other Diagnostics

A conditional F statistic is an instrument-strength diagnostic, not a test of whether an instrument is valid. A small value signals that the instruments contribute little distinct information for a particular endogenous regressor, conditional on the others. A large value is reassuring about that conditional relevance in the model and under the statistic’s assumptions, but it does not establish the exclusion restriction, rule out other specification problems, or guarantee reliable conventional 2SLS inference.

The ordinary first-stage F statistics and partial \(R^2\) remain useful. They show how well the instruments predict each endogenous regressor separately after accounting for included covariates. Conditional diagnostics add a different perspective by exposing whether that prediction is redundant across endogenous regressors. Neither measure should be used alone, and a universal cutoff should not be treated as a definitive pass-or-fail rule. The rule of ten has important limitations, as covered in the earlier material on the first-stage F statistic and its limitations.

Worked Example: One Endogenous Regressor

Suppose a hypothetical sample has \(n=200\), one endogenous regressor, one included covariate, and two excluded instruments. There are no other endogenous regressors to condition on, so the conditional first-stage test coincides with the ordinary partial F test. Suppose the excluded instruments have partial \(R^2=0.120\) and the residual degrees of freedom are \(196\).

For a joint test of \(q=2\) excluded-instrument coefficients, the partial-\(R^2\) form of the F statistic is

F=\frac{R^2_{\mathrm{partial}}/q}{(1-R^2_{\mathrm{partial}})/df} =\frac{0.120/2}{0.880/196} =\frac{0.0600}{0.004490} =13.36.

In this special case, there is no second endogenous regressor whose predicted variation might overlap with the first, so the conditional and ordinary first-stage questions are the same. With multiple endogenous regressors, a value like this from a separate first stage cannot stand in for the conditional statistic.

For conventional conditional F calculations, the usual linear-model reference distribution relies on the assumptions appropriate to the reported test, including its homoskedastic-error specification. If errors are heteroskedastic or observations are clustered, a conventional statistic may not provide valid inference about the relevant first-stage restrictions. Use a diagnostic suited to the error structure and estimator; heteroskedasticity-robust and cluster-robust strength procedures require their own methods and interpretation.

A Practical Diagnostic Sequence

A disciplined assessment keeps the separate and conditional questions distinct. Confirm the model’s endogenous regressors, included exogenous covariates, and excluded instruments first. Then inspect the first-stage results for each endogenous regressor and the conditional statistics for each endogenous regressor given the others. Large separate statistics combined with small conditional statistics are a warning that the instrument-predicted components overlap substantially.

Specify the IV system.
List \(D\), \(X\), and \(Z\), and verify that the number and arrangement of instruments permit the endogenous coefficients to be identified. Strength diagnostics do not repair underidentification.

Inspect separate first stages.
Record each excluded-instrument F statistic and partial \(R^2\). These summarize the instruments’ marginal predictive contribution for each endogenous regressor.

Inspect conditional statistics.
For each \(D_j\), use a procedure that reports its conditional first-stage strength given the other endogenous regressors. Note which conditional statistic is smallest.

Match the diagnostic to the assumptions.
Check whether the reported statistic is conventional, heteroskedasticity-robust, or cluster-robust, and whether its assumptions fit the sampling and error structure.

Report the pattern, not just a verdict.
Present separate and conditional results with the model specification and relevant degrees of freedom. Explain any mismatch rather than labelling the instruments simply “strong” or “weak.”

Worked Example: Locating the Weak Conditional Direction

A hypothetical model contains three endogenous regressors and three excluded instruments. The reported separate first-stage F statistics are \(28.0\), \(22.0\), and \(19.0\). The corresponding conditional F statistics are \(25.0\), \(4.00\), and \(18.0\), each computed for one endogenous regressor conditional on the other two.

The smallest conditional statistic is \(4.00\), for \(D_2\). The comparison is direct: \(D_2\)'s separate F is \(22.0\), but its conditional F is \(4.00\). Thus the instruments predict \(D_2\) in a separate first stage, while much of that predictive information is shared with the other endogenous regressors. The conditional statistic is \(18.0\) points lower, since \(22.0-4.00=18.0\).

This pattern directs attention to \(D_2\)'s conditional identification; it does not prove that the coefficient on \(D_2\) is unidentified or determine the quality of the entire instrument set by itself. The appropriate interpretation also depends on the diagnostic’s assumptions and the full IV design. A joint multivariate strength assessment can add information, but it answers a related system-level question rather than making these conditional results unnecessary.

Common Mistakes and Limitations

Using separate F statistics as a complete assessment. Large marginal statistics do not show that the instruments distinguish the endogenous regressors from one another. Report conditional diagnostics when there is more than one endogenous regressor.
Calling the conditional F a validity test. It measures conditional relevance, not whether \(Z\) is independent of the structural error or excluded from the outcome equation.
Applying a single cutoff mechanically. A threshold is not a universal guarantee of adequate identification or valid inference. Interpret the statistic in context and alongside appropriate diagnostics.
Ignoring the error structure. A conventional statistic may be inappropriate under heteroskedasticity or clustering. Use a test constructed for the design rather than assuming the conventional reference distribution remains valid.
Confusing strength with identification. Conditional strength concerns the amount of distinct predictive information. The rank and instrument-count requirements for identification are separate conditions; a strength statistic cannot make an unidentified model identified.

Conditional diagnostics are most informative when treated as a map of where instrument-induced variation is shared, not as a single score for the entire analysis. They also have limits: they depend on the specified linear first-stage system, the instruments and covariates included, and the assumptions behind the reported statistic. They cannot establish causal validity or substitute for inference methods designed to remain reliable under weak instruments.

Key takeaway: With multiple endogenous regressors, a large first-stage F for each regressor can coexist with weak conditional strength. Examine whether the instruments provide distinct predictive variation for every endogenous regressor, and match the diagnostic to the model and error structure.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In one sentence, distinguish what a separate first-stage F statistic measures from what a conditional first-stage F statistic measures.
Two endogenous regressors each have a separate first-stage F of \(30\), but their conditional F statistics are \(3\) and \(2\). What does this pattern suggest?
A conditional test has \(SSE_R=9.6\), \(SSE_U=8.0\), \(q=2\), and \(df_U=100\). Calculate its F statistic.
Does a large conditional F statistic establish that the excluded instruments satisfy the exclusion restriction? Explain.
Why should a conventional conditional F statistic be reconsidered when observations are clustered?

Solutions. (1) A separate F measures the instruments’ predictive contribution for one endogenous regressor on its own; a conditional F assesses the distinct contribution after accounting for the other endogenous regressors’ instrument-predicted variation. (2) The instruments predict each regressor separately but provide limited distinct information for distinguishing them; both conditional statistics are a warning, especially the value of \(2\). (3) The added sum of squares is \(9.6-8.0=1.6\), so \(F=(1.6/2)/(8.0/100)=0.800/0.0800=10.0\). (4) No. The statistic concerns conditional relevance, not instrument validity or the exclusion restriction. (5) Clustering changes the dependence structure and can invalidate the conventional reference distribution; a cluster-robust method suited to the design is needed.