Stock-Yogo Critical Values

Why Stock-Yogo Critical Values Are Criterion-Specific

A first-stage statistic summarizes how strongly the excluded instruments predict the endogenous regressors. But a statistic such as the Cragg-Donald statistic does not, by itself, say how much weakness is acceptable. A critical value gives the statistic a more specific interpretation: it compares instrument strength with a chosen tolerance for relative bias or for the size of a nominal test. The tolerance must be stated because different tolerances produce different critical values and can produce different assessments from the same statistic.

Stock-Yogo critical values are used with the conventional, homoskedastic Cragg-Donald statistic. They depend on the number of endogenous regressors, the number of excluded instruments, and the selected criterion. A valid comparison therefore begins by identifying all three. Looking up a value for the wrong instrument count or the wrong criterion can reverse the practical conclusion.

The Setup and the Assumptions Behind the Comparison

Suppose a linear instrumental-variables model has \(K_1\) endogenous regressors and \(L\) excluded instruments, alongside any included exogenous controls. The Cragg-Donald statistic is a conventional measure of first-stage strength for this setting; with multiple endogenous regressors it reflects the weakest relevant first-stage direction. Stock-Yogo tables translate this statistic into a benchmark assessment for a specified tolerance.

The standard Stock-Yogo critical values are derived for the conventional homoskedastic linear IV setting and its associated distributional framework. In applying them, the observations and errors must fit the assumptions underlying that framework; in particular, the errors must be homoskedastic for the conventional Cragg-Donald comparison to have its intended calibration. The instruments must also satisfy the model's exclusion and exogeneity requirements for an IV analysis to be causally meaningful. However, the Stock-Yogo comparison assesses strength, not whether those validity requirements hold.

The tables offer two distinct kinds of tolerance:

• Maximum relative bias: a specified upper bound on the relative bias of 2SLS compared with the bias of the corresponding OLS estimator, under the framework used to construct the table. For example, a 10% maximum relative-bias criterion asks whether the statistic reaches the threshold associated with limiting that relative bias to the specified level.
• Maximum size: an upper bound on the actual rejection probability of a nominal 5% Wald test of an IV coefficient under the weak-instrument configurations considered. For example, a 20% maximum-size criterion asks whether the statistic reaches the threshold associated with keeping that actual size at or below 20%.

“Maximum size” here means a maximum actual rejection probability, not a 20% probability that a particular null hypothesis is true. A 20% maximum-size benchmark is also not the same as a 20% relative-bias benchmark. The two criteria address different consequences of weak instruments, so their critical values need not agree.

How to Use the Table

First determine \(K_1\) and \(L\) from the model, counting endogenous regressors and excluded instruments, respectively. Then choose the criterion that matches the concern being evaluated and find the critical value for that exact combination. Compare the Cragg-Donald statistic with the lookup value. The conclusion should name the criterion: the statistic either meets or falls short of that specific benchmark.

This comparison is not a conventional hypothesis test that yields a p-value. The critical value is a threshold linked to a performance criterion. Exceeding it means the statistic passes that selected screen; it does not mean that weak instruments have been disproved in every sense.

Maximum Relative Bias and Maximum Size Can Disagree

Choosing the criterion is a substantive part of the assessment, not a minor table detail. A researcher concerned about 2SLS bias relative to OLS may select a maximum relative-bias benchmark. A researcher concerned about the over-rejection of conventional Wald tests may select a maximum-size benchmark. Neither criterion is a universal definition of “strong enough.” The selected benchmark should be named when reporting the result, and the analysis should not switch criteria simply because one yields a more favorable conclusion.

Why the Comparison Works

The justification is a direct consequence of how the table is constructed. For a fixed \(K_1\), \(L\), and selected criterion, the Stock-Yogo critical value is the threshold at which the corresponding bound is attained in the model used to derive the table. The Cragg-Donald statistic measures strength on the scale used by that derivation. Comparing the observed statistic with the threshold therefore tells us whether it reaches that criterion's benchmark.

The logic can be stated explicitly. Let \(c(K_1,L,\tau)\) denote the critical value for dimensions \(K_1,L\) and criterion \(\tau\), and let \(CD\) denote the observed Cragg-Donald statistic. The decision rule is:

If \(CD<c(K_1,L,\tau)\), it falls short of that benchmark. This implication is deliberately limited: meeting a table benchmark does not establish that the model's assumptions are true, and falling short does not prove that a particular coefficient estimate is biased by a specific amount. The table supplies a criterion-based diagnostic, not a sample-specific estimate of bias or size.

Diagnostics, Alternatives, and Limitations

Before relying on a Stock-Yogo comparison, check the instrument count, the number of endogenous regressors, the selected criterion, and the statistic's construction. A common practical error is to use a conventional critical value with a statistic based on a different covariance structure. In particular, heteroskedasticity can invalidate the conventional homoskedastic calibration. The later material on heteroskedasticity-robust first-stage tests addresses diagnostics designed for that setting; conventional Stock-Yogo values should not be treated as robust substitutes.

Other failure modes include reporting a critical value without naming its criterion, using a threshold for one endogenous regressor when the model has several, and interpreting a pass as proof that conventional 2SLS inference is reliable. Even with a pass, the instruments may be invalid, the model may be misspecified, or another performance criterion may not be met. If a comparison fails, it is evidence that strength does not satisfy the selected benchmark, not a reason to change the criterion after seeing the result.

The main alternative is to report a different weak-instrument diagnostic suited to the data and assumptions. For example, a heteroskedasticity-robust first-stage procedure is needed when the conventional homoskedastic assumptions do not hold. A Stock-Yogo result remains useful when its assumptions fit and its target criterion is relevant, but it should be presented alongside the model and assumptions that justify its use. It does not replace assessment of instrument exogeneity, exclusion, or the broader identification argument.