Stock-Yogo Weak-Instrument Thresholds

Why a First-Stage F Cutoff Needs a Criterion

A first-stage statistic describes how strongly the instruments predict the endogenous regressor. But a number such as $F=12$ does not, by itself, say how much bias or inferential distortion is acceptable. Stock–Yogo thresholds make that comparison explicit: they calibrate a cutoff to a stated performance criterion under a specified homoskedastic model. The right question is therefore not simply whether $F$ exceeds a familiar rule of thumb, but which criterion, model, and table entry the comparison uses.

The thresholds are associated with the work of Stock and Yogo on weak-instrument diagnostics. They are not a universal definition of a strong instrument, nor do they guarantee a particular realized bias in a given data set. They are reference cutoffs derived under assumptions that must be considered alongside the result.

The Model and the Two Calibration Criteria

For clarity, begin with one endogenous regressor $D$, $q$ excluded instruments collected in $Z$, and $k$ included controls collected in $X$, including an intercept if the model has one. A standard linear setup is

Y=\beta D+X\gamma+u,\qquad D=X\delta+Z\pi+v.

The usual fixed-design calibration assumes that $X$ and $Z$ are nonrandom, that $X$ has rank $k$, that $M_XZ$ has rank $q$, and that the reduced-form errors are independent across observations and jointly normal with a common covariance matrix:

\begin{pmatrix}u\\v\end{pmatrix} \sim N\!\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix} \sigma_u^2 & \sigma_{uv}\\ \sigma_{uv} & \sigma_v^2 \end{pmatrix} \otimes I_n \right),\qquad \sigma_v^2>0.

Here $M_X=I_n-P_X$, where $P_X$ is the projection matrix onto the column space of $X$. Instrument exogeneity is part of the structural setup: the excluded instruments are uncorrelated with the structural error $u$. Weak-instrument calibrations allow the first-stage relevance, represented by $\pi$, to be small. The covariance $\sigma_{uv}$ may be nonzero, which is why ordinary least squares can be biased for $\beta$.

Definition: The maximal relative-bias criterion sets a tolerance for the largest approximate bias of 2SLS relative to the bias of OLS, over the nuisance endogeneity in the specified model. A maximal-size criterion sets a tolerance for the largest rejection probability of a conventional Wald test under the null, again over the nuisance features allowed by the model. Stock–Yogo tables report critical values calibrated to particular choices of criterion and tolerance; the choice must be stated to interpret a cutoff.

For example, a table entry labelled “10% maximal relative bias” addresses a different standard from one labelled “5% maximal relative bias.” A size-based entry is different again. In common size calibrations, the label gives a bound on the actual rejection probability of a conventional test with a stated nominal level; for instance, a 10% maximum size for a nominal 5% test means a tolerance of actual rejection probability up to 10%. Do not treat “10% bias” and “10% size” as interchangeable descriptions.

First-Stage F and the Concentration Parameter

After controlling for $X$, the first-stage F statistic tests whether the excluded-instrument coefficients in the regression of $D$ on $X$ and $Z$ are jointly zero. Let $P_{M_XZ}$ denote projection onto the column space of $M_XZ$, and let $M_{[X,Z]}$ denote projection onto the orthogonal complement of the joint column space of $X$ and $Z$. The partial F statistic is

F=\frac{D'P_{M_XZ}D/q}{D'M_{[X,Z]}D/(n-k-q)}.

The denominator degrees of freedom are $n-k-q$: the fitted first stage uses $k+q$ linearly independent regressors. The concentration parameter in this fixed-design model is

\lambda=\frac{\pi'Z'M_XZ\pi}{\sigma_v^2}.

It summarizes first-stage signal relative to reduced-form error variance. In the classical Stock–Yogo setting, the relevant first-stage statistic is compared with a cutoff calibrated to how performance changes as this signal varies. The cutoff is not obtained by asking whether the null hypothesis $\pi=0$ is rejected at a conventional significance level.

Result: Under the fixed-design Gaussian first-stage assumptions above, the partial first-stage statistic has a noncentral F distribution: $$ F\sim F_{q,\,n-k-q}(\lambda). $$ The numerator degrees of freedom are $q$, the residual degrees of freedom are $n-k-q$, and the noncentrality parameter is $\lambda$.

Proof. Choose an $n\times(n-k)$ matrix $Q$ whose columns form an orthonormal basis for the space orthogonal to $X$. Then $Q'Q=I_{n-k}$ and $Q'X=0$. Write $d_*=Q'D$ and $W=Q'Z$. In this reduced space the first stage is $d_*=W\pi+Q'v$, with $Q'v\sim N(0,\sigma_v^2I_{n-k})$. The projection onto the columns of $W$ has rank $q$, so the projected sum of squares divided by $\sigma_v^2$ is a noncentral chi-squared variable with $q$ degrees of freedom and noncentrality

\frac{\pi'W'W\pi}{\sigma_v^2} =\frac{\pi'Z'M_XZ\pi}{\sigma_v^2} =\lambda.

The residual projection in this space has rank $(n-k)-q=n-k-q$. Its sum of squares divided by $\sigma_v^2$ is central chi-squared with $n-k-q$ degrees of freedom. The two projections are orthogonal, so these Gaussian quadratic forms are independent. Dividing each by its degrees of freedom gives the stated noncentral F distribution. $\square$

How to Read a Stock–Yogo Threshold

A Stock–Yogo cutoff $c$ is meaningful only with its table heading: number of excluded instruments, number of endogenous regressors when relevant, performance criterion, and tolerance. In the single-endogenous-regressor case, tables commonly give first-stage F cutoffs for particular relative-bias bounds. With several endogenous regressors, the relevant Stock–Yogo statistic is not in general an ordinary F for one first stage; the applicable table and statistic must match that setting.

For a single endogenous regressor and one excluded instrument, a commonly reported Stock–Yogo cutoff for a 10% maximal relative-bias criterion is 8.96. A cutoff of 16.38 is commonly reported for a stricter 5% maximal relative-bias criterion in that same one-instrument setting. These are not two estimates of one universal boundary. They answer different calibration questions. Always verify the exact table heading and model configuration rather than transferring a number to a different criterion or setup.

The comparison is often phrased as follows: if the observed statistic exceeds the selected cutoff, it meets that table’s stated calibration standard under its assumptions; if it does not, it fails to meet that standard. Neither conclusion establishes that the instruments are “valid,” and exceeding the cutoff does not prove that 2SLS bias or test-size distortion in the realized sample is below the stated bound.

Worked Example: Choosing the Bias Tolerance

Suppose a model has one endogenous regressor and one excluded instrument. Its observed first-stage F statistic is $F=12.4$. Compare it with the two relative-bias cutoffs just described.

For the 10% maximal relative-bias criterion, the cutoff is $8.96$, and

$$ 12.4-8.96=3.44>0. $$

Thus $F$ exceeds that cutoff. For the 5% maximal relative-bias criterion, the cutoff is $16.38$, and

$$ 12.4-16.38=-3.98<0. $$

Thus the same statistic does not exceed the stricter cutoff. The appropriate conclusion is conditional: the observed first-stage statistic meets the 10% table criterion but not the 5% table criterion, under the corresponding calibration assumptions. It would be incorrect to report simply that the instruments are “strong” without saying which standard was used.

Worked Example: Computing the Partial F With Controls

There are $n=120$ observations, $k=4$ included controls, and $q=2$ excluded instruments. In the first-stage regression, suppose the restricted residual sum of squares (using the controls only) is 360 and the unrestricted residual sum of squares (using controls and instruments) is 300. Compute the partial F statistic and its degrees of freedom.

The reduction in residual sum of squares due to adding the instruments is $360-300=60$. There are two restrictions, so the numerator mean square is $60/2=30$. The residual degrees of freedom in the unrestricted first stage are

$$ n-k-q=120-4-2=114. $$

The denominator mean square is $300/114=2.6316$ (to four decimal places). Therefore,

F=\frac{30}{300/114} =\frac{30(114)}{300} =11.4.

The statistic has numerator degrees of freedom 2 and denominator degrees of freedom 114 under the central null in the Gaussian model. The Stock–Yogo comparison still requires a table entry that matches two excluded instruments and the chosen criterion; a one-instrument cutoff cannot be substituted.

Worked Example: A Threshold Is Not a Significance Test

Suppose a researcher has a one-instrument first stage with $F=12.4$, and plans to use the 10% maximal relative-bias cutoff of $8.96$. The statistic exceeds the cutoff by $12.4-8.96=3.44$. What does this comparison establish, and what does it not establish?

It establishes that the observed statistic is above the selected calibration threshold. Under the table’s model and criterion, this is evidence that the specified maximal relative-bias tolerance is met according to that calibration. It does not establish that the first-stage null is significant at a particular $p$-value: a significance test uses a null distribution and a chosen significance level, whereas this threshold was chosen to address a bias criterion.

It also does not establish a 10% probability that the estimator’s bias exceeds a bound. The label describes the tolerance used in the calibration, not a posterior probability or a repeated-sampling probability attached directly to this observed F statistic.

Assumptions, Limits, and a Common Mistake

The classical thresholds are calibrated for specified homoskedastic models. Their interpretation relies on the model’s error assumptions, the number of instruments and endogenous regressors, and the chosen bias or size criterion. When these features differ, a familiar cutoff may no longer have its stated calibration. In particular, heteroskedasticity changes the validity of classical first-stage diagnostics; heteroskedasticity-robust diagnostics require a different analysis.

The thresholds also do not test the exclusion restriction or instrument exogeneity. A strong association between $Z$ and $D$ is a relevance fact; it cannot show that $Z$ is uncorrelated with the structural error $u$. Nor does a favorable first-stage comparison make conventional Wald inference automatically reliable in every setting.

Mistake: “The F statistic exceeds 10, so weak instruments are ruled out.” This treats a rule of thumb as a universal theorem and ignores the number of instruments and the calibration criterion.
Correction: Identify the matching Stock–Yogo table entry, state its assumptions and tolerance, and report whether the observed statistic exceeds that particular cutoff.
Mistake: Use $n-q$ as the residual degrees of freedom after partialling out controls.
Correction: With $n$ observations, $k$ linearly independent controls, and $q$ excluded instruments, the residual degrees of freedom are $n-k-q$.

Stock–Yogo thresholds are useful because they translate an abstract concern about weak identification into an explicit performance standard. Their value is precisely that the standard is stated, not that a single number applies to every design. A careful report names the statistic, the table criterion, the tolerance, and the assumptions supporting the comparison.

Check Your Understanding

Answer each question by specifying the relevant model feature or calibration criterion.

In a model with $n=150$, $k=5$ controls, and $q=3$ excluded instruments, what are the partial first-stage F statistic’s denominator degrees of freedom?
Why can the same observed F statistic pass a 10% relative-bias threshold but fail a 5% relative-bias threshold?
What is the difference between a Stock–Yogo cutoff and the critical value for a conventional test of the null that the excluded-instrument coefficients are zero?
Does exceeding a Stock–Yogo threshold establish instrument exogeneity? Explain.
Why should a classical Stock–Yogo cutoff not automatically be applied when the errors are heteroskedastic?