First-Stage F Statistics and Stock-Yogo Critical Values

Why First-Stage Strength Needs a Diagnostic

Instrumental variables requires more than an instrument that is plausibly exogenous: the instrument must also predict the endogenous regressor. When that predictive relationship is weak, conventional 2SLS estimates and Wald inference can behave poorly. The first-stage F statistic summarizes predictive information, while Stock-Yogo critical values calibrate how large that statistic should be for a specified weak-instrument distortion criterion.

These are related but different questions. The conventional first-stage F tests whether excluded instruments have any predictive contribution conditional on included exogenous regressors. A Stock-Yogo comparison asks whether the observed strength meets a particular threshold associated with a bound on bias or distortion of a nominal Wald test. Neither calculation assesses whether the exclusion restriction is credible.

The First-Stage Model and F Statistic

Consider one endogenous regressor \(D\), a vector of included exogenous regressors \(X\), and \(q\) excluded instruments collected in \(Z\). The first-stage linear regression is

D = X\gamma + Z\pi + v.

Here \(\pi\) is the \(q\)-vector of coefficients on the excluded instruments. The conventional homoskedastic first-stage F statistic tests \(H_0:\pi=0\) against the alternative that at least one component of \(\pi\) is nonzero, conditional on \(X\). Let \(k\) be the number of columns of \(X\), including an intercept if one is included, and suppose the regressors have full column rank. The unrestricted regression has \(n-k-q\) residual degrees of freedom.

Definition: The conventional first-stage F statistic is the homoskedastic partial F test of the joint null that all excluded-instrument coefficients in the first-stage regression are zero, controlling for the included exogenous regressors. It measures evidence of conditional predictive relevance, not instrument validity.

One way to calculate it is from the restricted and unrestricted residual sums of squares. The restricted regression includes \(X\) but omits \(Z\); the unrestricted regression includes both. If \(SSR_R\) and \(SSR_U\) are their respective residual sums of squares, then

F = \frac{(SSR_R-SSR_U)/q}{SSR_U/(n-k-q)}.

Equivalently, let \(R^2_{\mathrm{partial}}\) be the fraction of the variation in \(D\) left after controlling for \(X\) that is explained by adding \(Z\). Then

F = \frac{R^2_{\mathrm{partial}}/q}{(1-R^2_{\mathrm{partial}})/(n-k-q)}.

This is a test of the excluded instruments’ incremental explanatory power. The ordinary overall first-stage \(R^2\), which also counts variation explained by \(X\), is not the quantity to use. Under the homoskedastic linear model, the statistic has an \(F\) reference distribution with \(q\) numerator and \(n-k-q\) denominator degrees of freedom under the joint null.

Worked Example: Calculating a First-Stage F Statistic

Suppose a hypothetical study has \(n=200\) observations, \(k=3\) included exogenous regressors, and \(q=1\) excluded instrument. The partial \(R^2\) from adding the instrument to the first stage is \(0.12\). Calculate the conventional first-stage F statistic.

The residual degrees of freedom are \(n-k-q=200-3-1=196\). Substituting the partial \(R^2\) into the formula gives

F = \frac{0.12/1}{(1-0.12)/196} = \frac{0.12}{0.88/196} = \frac{0.12}{0.004490} \approx 26.73.

Thus the reported first-stage statistic is \(F(1,196)=26.73\). It indicates substantial evidence against the zero-coefficient null in this first stage. Whether it meets a particular Stock-Yogo strength criterion is a separate comparison, which depends on the criterion and the applicable table.

Why the Partial F Formula Works

The formula follows directly from the usual nested-regression F statistic. Adding \(q\) excluded instruments reduces the residual sum of squares by \(SSR_R-SSR_U\); dividing by \(q\) gives the reduction per added regressor. The unrestricted residual variance estimate is \(SSR_U/(n-k-q)\). Their ratio is the F statistic above.

To obtain the partial-\(R^2\) form, define \(R^2_{\mathrm{partial}}=(SSR_R-SSR_U)/SSR_R\). Consequently, \(SSR_U=SSR_R(1-R^2_{\mathrm{partial}})\), while \(SSR_R-SSR_U=SSR_RR^2_{\mathrm{partial}}\). Substitution into the nested-regression formula cancels \(SSR_R\), leaving

F = \frac{SSR_R R^2_{\mathrm{partial}}/q} {SSR_R(1-R^2_{\mathrm{partial}})/(n-k-q)} = \frac{R^2_{\mathrm{partial}}/q} {(1-R^2_{\mathrm{partial}})/(n-k-q)}.

The division by \(q\) matters: for the same partial \(R^2\), explaining it with more excluded instruments does not produce the same F statistic as explaining it with one. Sample size and residual degrees of freedom matter too. Therefore, partial \(R^2\) alone is not a complete measure of first-stage strength.

Stock-Yogo Critical Values

Stock-Yogo critical values translate first-stage strength into a specified weak-instrument benchmark in a canonical homoskedastic IV setting. The standard tables depend on the number of excluded instruments and the criterion being controlled. In the familiar case of one endogenous regressor, the criteria include maximal relative bias of 2SLS compared with OLS, and maximal size of a nominal 5% Wald test. These are distinct criteria, so a critical value should always be reported with its criterion.

Definition: A Stock-Yogo critical value is a threshold calibrated, under the assumptions of its table, to limit a specified weak-instrument distortion. For example, a critical value for 10% maximal relative bias concerns a bound on 2SLS bias relative to OLS bias; a critical value for 10% maximal size concerns keeping the actual rejection probability of a nominal 5% Wald test at or below 10% in the calibrated setting.

For one endogenous regressor and one excluded instrument, commonly reported Stock-Yogo thresholds are 8.96 for 10% maximal relative bias and 16.38 for 10% maximal size of a nominal 5% Wald test. For two excluded instruments, the corresponding thresholds are 11.59 and 19.93. These values illustrate why there is no single universal cutoff: even at the same number of instruments, the threshold changes with the distortion being controlled.

In practical use, identify the table’s number of endogenous regressors and excluded instruments, then select the column for the desired criterion. Compare the statistic with that critical value. A statistic at or above the threshold meets that table’s stated benchmark under its assumptions; it does not prove that the instruments are strong in every sense, nor that the assumptions hold in the application.

Worked Example: Comparing With Two Stock-Yogo Criteria

Suppose a hypothetical sample has one endogenous regressor, two excluded instruments, \(n=300\), and \(k=4\) included exogenous regressors. Its first-stage partial \(R^2\) is \(0.08\). Calculate the F statistic and compare it with the two 10% criteria.

The denominator degrees of freedom are \(300-4-2=294\). Therefore,

F = \frac{0.08/2}{(1-0.08)/294} = \frac{0.04}{0.92/294} = \frac{0.04}{0.003129} \approx 12.78.

For two excluded instruments and one endogenous regressor, the 10% maximal-relative-bias threshold is 11.59. Since \(12.78>11.59\), the statistic meets that particular bias benchmark. The 10% maximal-size threshold for a nominal 5% Wald test is 19.93. Since \(12.78<19.93\), it does not meet that size benchmark. The appropriate conclusion is criterion-specific, not simply that the instruments “pass” or “fail.”

Interpreting the Comparison Carefully

A large first-stage F statistic is evidence that the excluded instruments predict the endogenous regressor conditional on the included controls. A Stock-Yogo comparison adds a calibration: it relates that predictive strength to a stated bound on a particular distortion in a stylized model. The comparison is meaningful only if the model setting and assumptions behind the selected critical value are appropriate.

The standard critical values are derived for a homoskedastic framework and the relevant canonical IV configuration. The conventional first-stage F also relies on the homoskedastic regression variance calculation. In applied work, the first stage should be specified with the same included exogenous regressors used in the structural equation, and the statistic should use the excluded instruments jointly. Under heteroskedasticity, this conventional F and its Stock-Yogo comparison are not automatically valid; heteroskedasticity-robust diagnostics require different methods.

The classic tables are also not a universal solution for every IV design. In particular, the familiar one-endogenous-regressor critical values should not be transferred mechanically to models with multiple endogenous regressors or to settings with many instruments. Such cases require diagnostics tailored to their identification structure. A reported F value is most informative when accompanied by the number of excluded instruments, the controls included, and the precise Stock-Yogo criterion used.

Worked Example: Why “F Greater Than 10” Is Not a Conclusion

Consider a hypothetical first stage with one endogenous regressor, one excluded instrument, \(n=150\), \(k=4\), and partial \(R^2=0.06\). Calculate the statistic and assess the two 10% Stock-Yogo benchmarks.

The residual degrees of freedom are \(150-4-1=145\). Thus,

F = \frac{0.06/1}{(1-0.06)/145} = \frac{0.06}{0.94/145} = \frac{0.06}{0.006483} \approx 9.255.

This statistic exceeds 8.96, the one-instrument threshold for 10% maximal relative bias, but is below 16.38, the threshold for 10% maximal size of a nominal 5% Wald test. Saying only “the F statistic is about 9, so it is above 10” would be both numerically wrong and conceptually incomplete: it is below 10, and the relevant conclusion depends on the selected distortion criterion. The statistic still tests the first-stage zero-coefficient null; the Stock-Yogo comparisons address different calibrated performance goals.

Common Mistakes and Practical Reporting

The most common pitfall is treating a rule of thumb such as \(F>10\) as a universal certificate of strength. That rule does not specify the number of instruments, the allowed bias or test-size distortion, or the conditions under which the diagnostic is being used. A more careful report gives the F statistic and its degrees of freedom, states the Stock-Yogo criterion and critical value, and limits the conclusion to that criterion.

A second mistake is interpreting a high first-stage F as evidence that the instruments satisfy the exclusion restriction or are independent of the structural error. The F statistic uses the relationship between \(Z\) and \(D\); it does not test the required relationship between \(Z\) and the structural error. Relevance and validity are separate assumptions.

A useful reporting sequence is:

Specify the first stage.
Name the endogenous regressor, excluded instruments, and included exogenous regressors. Confirm the statistic is conditional on the intended controls.

Report the conventional statistic.
Give the homoskedastic partial F statistic, numerator and denominator degrees of freedom, and, where useful, the partial \(R^2\).

State the calibration.
Identify the applicable Stock-Yogo table and the specific maximal-bias or Wald-test-size criterion.

Qualify the conclusion.
Say whether the threshold is met under the table’s assumptions; do not describe the result as proof of instrument validity or universal strength.

Key takeaway: The conventional first-stage F statistic measures the excluded instruments’ joint predictive contribution under homoskedasticity. Stock-Yogo critical values make its interpretation criterion-specific: the appropriate threshold depends on the number of instruments and the distortion one aims to limit.

Check Your Understanding

Answer each question by referring to the stated model and criterion.

In a first stage with \(n=250\), \(k=5\), \(q=2\), and partial \(R^2=0.10\), calculate the conventional first-stage F statistic.
What null hypothesis does the conventional first-stage F statistic test?
For one endogenous regressor and two excluded instruments, an F statistic is 15. Which of the two 10% Stock-Yogo benchmarks discussed here does it meet, and which does it not meet?
Why is the ordinary overall first-stage \(R^2\) not a substitute for the partial \(R^2\) in the F formula?
Does meeting a Stock-Yogo threshold establish that an instrument satisfies the exclusion restriction? Explain.