Stock-Yogo Critical Values and Their Scope

Why Stock–Yogo Critical Values Need a Scope

A first-stage F statistic can be useful evidence about instrument strength, but a rule such as “F must exceed 10” does not specify what degree of weak-instrument concern is being controlled. Stock–Yogo critical values make that choice more explicit: they calibrate a test statistic to a particular weakness criterion. Their interpretation, however, depends on the statistical setting used to derive them. A threshold from the classical tables is not a universal standard that can be applied to any robust, clustered, or otherwise modified statistic.

This distinction matters because weak instruments can affect both the behavior of 2SLS estimates and the size of tests based on those estimates. The same instruments can satisfy one stated criterion and fail another. A careful report therefore identifies the statistic, the criterion, the relevant instrument dimensions, and whether the assumptions supporting the table are reasonable.

Two Different Weakness Criteria

Stock–Yogo tables are often described using two kinds of criteria. They address different consequences of weak identification, so their critical values are not interchangeable.

A maximal relative-bias criterion limits the maximum relative bias of 2SLS compared with the bias of OLS in the framework used to construct the critical values. For example, the “10% maximal relative-bias” criterion asks whether the statistic is large enough to rule out relative bias above the specified 10% bound in that framework. It does not promise that the realized bias in every application is at most 10%, nor does it bound bias under arbitrary violations of the model.

A maximal test-size criterion concerns the rejection probability of a nominal test under the null hypothesis. For example, a 5% test subject to a 15% maximal-size criterion is assessed against a bound on its actual rejection probability under the null. The “15%” label is a maximum size, not an allowance for an additional 15 percentage points on top of the nominal 5%. This criterion does not directly state a bound on 2SLS relative bias.

For one endogenous regressor and one excluded instrument, commonly used Stock–Yogo values include 9.08 for a 10% maximal relative-bias criterion, 16.38 for a 10% maximal-size criterion, and 8.96 for a 15% maximal-size criterion. The test-size values refer to the stated maximum size for the nominal test considered in the table. These numbers illustrate why the criterion must be named: 9.08 and 8.96 are close numerically, but refer to different questions.

The Classical Setting Behind the Thresholds

The conventional Stock–Yogo critical values are derived for the classical linear IV setting. Their use presumes independent observations and homoskedastic errors in the structural and first-stage equations, together with the distributional conditions of the classical derivation, which use normally distributed errors. In this setting, the Cragg–Donald statistic has the reference behavior used to calibrate the cutoffs. The instrument and structural-model conditions required for an IV causal interpretation remain separate: a strong first stage does not establish that an instrument is relevant in the population, independent of unmeasured causes of the outcome, or excluded from the structural equation.

With one endogenous regressor, the classical first-stage partial F statistic is the relevant statistic for comparison. With multiple endogenous regressors, the Cragg–Donald statistic is based on the weakest direction of the first-stage relationship, not on choosing the largest individual first-stage F. The table’s dimensions must match the number of endogenous regressors and excluded instruments. A cutoff for one endogenous regressor and one instrument cannot simply be reused for a different configuration.

The scope restriction is especially important when using heteroskedasticity-robust or cluster-robust diagnostics. A robust statistic may be appropriate for the data’s variance or dependence structure, but it does not thereby acquire the classical Cragg–Donald reference distribution. In particular, a heteroskedasticity-robust or cluster-robust first-stage statistic should not automatically be compared with a conventional Stock–Yogo cutoff. Robust diagnostics and effective F statistics address this problem using different methods; their results should be interpreted according to the method’s own calibration.

Calculating and Comparing a Classical First-Stage F

For one endogenous regressor \(D\), the first stage regresses \(D\) on the included exogenous controls and the excluded instrument or instruments. The partial F statistic tests whether the excluded instruments’ coefficients are jointly zero, conditional on the controls. For one excluded instrument, this is a one-restriction test.

Let \(SSR_R\) be the residual sum of squares from the restricted regression that includes the controls but omits the excluded instrument, and let \(SSR_U\) be the residual sum of squares from the full first stage. If the full regression has \(df_U\) residual degrees of freedom and there is one excluded instrument, the classical partial F statistic is:

The numerator measures the reduction in residual sum of squares from adding the excluded instrument, per restriction. The denominator is the full model’s residual mean square. With one restriction, this F statistic equals the square of the corresponding classical t statistic. This calculation gives a strength diagnostic; its comparison with a Stock–Yogo value is warranted only under the classical assumptions already stated.

Reading the Criterion, Not Just the Number

A larger first-stage statistic generally provides stronger evidence against weak first-stage relevance, but the decision remains criterion-specific. The following example uses the same one-regressor, one-instrument configuration and classical assumptions to show how a different statistic can yield a different conclusion across criteria.

Why a Robust Statistic Cannot Borrow a Classical Cutoff

The numerical size of a diagnostic is not enough to determine which reference value applies. A statistic whose standard error or reference distribution has been changed to address heteroskedasticity or clustering is not automatically the same statistic on which conventional Stock–Yogo values were calibrated. The appropriate comparison is determined by the diagnostic’s derivation, not by whether its reported number resembles an F statistic.

A Practical Interpretation Sequence

A reliable Stock–Yogo comparison can be organized into a short sequence. Each item matters because a correct numerical calculation can still support an incorrect claim if the wrong criterion or reference setting is used.

Common Mistake: Treating a Cutoff as a Universal Rule

A common mistake is to write, “The F statistic exceeds 10, so weak instruments are not a concern.” This reasoning fails for several reasons. Ten is not a universal Stock–Yogo cutoff; the appropriate critical value depends on the criterion and the number of endogenous regressors and instruments. The first-stage statistic may also have a robust or clustered calibration that does not match the classical table. Finally, even a statistic that exceeds a properly selected cutoff does not test the exclusion restriction or independence of the instrument from unmeasured causes of the outcome.

A better statement is specific: “Under the classical independent, homoskedastic-error setting, the one-regressor, one-instrument first-stage statistic exceeds the 9.08 cutoff for the 10% maximal relative-bias criterion.” If those assumptions are not credible, say that the classical cutoff is not directly applicable and interpret an appropriate robust diagnostic using its own framework.