The Cragg-Donald Statistic

Why a system-level strength diagnostic is needed

With several endogenous regressors, strength is not just a question of whether each regressor has some association with the instruments. The instruments must provide enough information to identify every relevant linear combination of those regressors. Strong-looking first stages considered one at a time can still conceal a combination that is weakly predicted. The Cragg-Donald statistic addresses that problem by summarizing the weakest direction in the first-stage system.

The statistic is designed for a linear instrumental variables model with homoskedastic first-stage errors. It is a diagnostic of instrument strength under those conditions, not a general-purpose test that remains valid under any error structure. Its minimum-eigenvalue construction is particularly useful when there is more than one endogenous regressor.

Model and assumptions

Let \(y\) be the outcome, \(X\) an \(n \times K\) matrix of endogenous regressors, \(Z\) an \(n \times L\) matrix of excluded instruments, and \(W\) a matrix of included exogenous regressors with \(q\) columns. The structural equation and first-stage system are

y = X\beta + W\gamma + u, \qquad X = W\Gamma + Z\Pi + V.

Here \(\Pi\) is the \(L \times K\) matrix of first-stage coefficients, and each row of \(V\) contains the errors for all \(K\) first stages. The statistic uses the first-stage system; the structural equation supplies the instrumental variables context for why instrument strength matters.

Assumptions for the conventional Cragg-Donald statistic: The first stages are linear; the instruments are excluded from the structural equation and exogenous to its error; the first-stage errors have a common, finite covariance matrix and are homoskedastic across observations; and the relevant design matrices have full rank. In particular, \(n-q-L\) must be positive to estimate the first-stage error covariance as defined below. The usual strength interpretation also requires enough excluded instruments to identify the \(K\) endogenous regressors, so \(L \ge K\).

Homoskedasticity here means that the covariance matrix of the vector of first-stage errors is the same for each observation, conditional on the included regressors and instruments. Correlation among the \(K\) first-stage errors for a given observation is allowed. The conventional statistic is not the appropriate robust replacement when this homoskedasticity condition fails.

Definition and derivation

Remove the linear contribution of \(W\) from \(X\) and \(Z\) using the residual-maker \(M_W = I - W(W'W)^{-1}W'\). The first-stage coefficient estimate is

\hat{\Pi} = (Z'M_WZ)^{-1}Z'M_WX.

Let \(\hat V\) be the residual matrix from regressing \(X\) on both \(W\) and \(Z\), and estimate the covariance of its \(K\) columns by \(\hat{\Sigma} = \hat V'\hat V/(n-q-L)\). Define the covariance-standardized instrument relevance matrix

A = \hat{\Sigma}^{-1/2} \hat{\Pi}'Z'M_WZ\hat{\Pi} \hat{\Sigma}^{-1/2}.

The symmetric inverse square root \(\hat{\Sigma}^{-1/2}\) standardizes combinations of endogenous regressors by the estimated covariance of their first-stage errors. Let \(\lambda_{\min}(A)\) be the smallest eigenvalue of \(A\). With the usual F-style scaling by the number of excluded instruments, the Cragg-Donald statistic is

CD = \frac{\lambda_{\min}(A)}{L}.

This definition also shows the statistic’s connection to the single-endogenous-regressor case. When \(K=1\), \(A\) is a scalar: the instrument-explained sum of squares for the first stage divided by its residual variance estimate. Dividing by \(L\) gives the familiar partial first-stage F scaling. For \(K>1\), the smallest eigenvalue extends that idea to the weakest standardized direction in the system.

To see why, consider any unit-length vector \(a\) describing a linear combination of the standardized endogenous regressors. The Rayleigh quotient \(a'Aa\) measures instrument relevance in that direction. The minimum of this quantity over all unit vectors is the smallest eigenvalue:

\lambda_{\min}(A) = \min_{a'a=1} a'Aa.

Thus, a small minimum eigenvalue means that at least one combination is poorly explained by the instruments relative to its first-stage error variation. A large value in one direction cannot compensate for a weak value in another. This is the key reason the statistic is a system diagnostic rather than a summary of average first-stage strength.

Worked Example: Calculating the Statistic From First-Stage Summaries

Suppose a hypothetical analysis has \(n=100\), \(q=1\), \(L=3\), and \(K=2\). Its first-stage residual covariance estimate is

\hat{\Sigma} = \begin{pmatrix} 4 & 0\\ 0 & 9 \end{pmatrix}, \qquad \hat{\Pi}'Z'M_WZ\hat{\Pi} = \begin{pmatrix} 120 & 0\\ 0 & 90 \end{pmatrix}.

The residual degrees of freedom for \(\hat{\Sigma}\) are \(100-1-3=96\), consistent with the stated covariance estimate’s definition. Since \(\hat{\Sigma}^{-1/2} = \operatorname{diag}(1/2,1/3)\), standardizing gives

A = \begin{pmatrix} 1/2 & 0\\ 0 & 1/3 \end{pmatrix} \begin{pmatrix} 120 & 0\\ 0 & 90 \end{pmatrix} \begin{pmatrix} 1/2 & 0\\ 0 & 1/3 \end{pmatrix} = \begin{pmatrix} 30 & 0\\ 0 & 10 \end{pmatrix}.

Its eigenvalues are 30 and 10, so \(\lambda_{\min}(A)=10\). Therefore

CD = \frac{10}{3} = 3.333.

In this example, the second standardized direction is the weaker one. The statistic is 3.333 on the F-style scale; interpreting whether that indicates an acceptable degree of strength requires appropriate critical values and the relevant design dimensions. The value alone is not a universal pass-or-fail threshold.

Worked Example: Strong Marginal Directions, Weak Combination

Consider a second hypothetical system with \(L=2\), \(\hat{\Sigma}=I\), and standardized relevance matrix

A = \begin{pmatrix} 16 & 12\\ 12 & 10 \end{pmatrix}.

The diagonal entries suggest substantial relevance for each coordinate when considered separately. But the joint strength depends on the eigenvalues, not just the diagonal entries. The trace is 26 and the determinant is \(16(10)-12(12)=16\), so the two eigenvalues are

\lambda = \frac{26 \pm \sqrt{26^2-4(16)}}{2} = \frac{26 \pm \sqrt{612}}{2}, \qquad \lambda_{\min} \approx 0.6307.

Consequently, \(CD=0.6307/2=0.3154\). A unit eigenvector for the smaller eigenvalue is approximately \((0.615,-0.788)'\). This means that the combination \(0.615X_1-0.788X_2\), after the stated standardization, is much less well predicted than either coordinate summary suggests. Separate first-stage assessments can miss precisely this kind of weakness.

How to interpret and check the result

The statistic is built from a minimum eigenvalue, so its interpretation is directional: it reports the least instrument-supported standardized combination among the \(K\) endogenous regressors. If the minimum eigenvalue is zero, the estimated relevance matrix is rank deficient; at least one direction has no measured instrument-explained variation in this system. A small positive value indicates a direction with limited measured strength, but deciding how consequential it is requires a reference appropriate to the numbers of instruments and endogenous regressors.

Before interpreting a reported value, check the construction and the conditions behind it. Confirm that included exogenous regressors were accounted for, that the covariance matrix refers to the residuals from the complete first-stage system, and that the scaling uses \(L\), the number of excluded instruments. Check that \(\hat{\Sigma}\) is positive definite and that the design matrices have the required rank. Also examine whether the error structure plausibly supports homoskedastic inference. If the errors are heteroskedastic or clustered, the conventional Cragg-Donald statistic does not provide the corresponding robust diagnostic.

Worked Example: Checking Invariance to a Change of Units

In the first example, suppose the second endogenous regressor is recorded in units twice as large as before. Write \(X^*=XD\), where \(D=\operatorname{diag}(1,2)\). The unstandardized relevance and error covariance matrices transform to \(D'(\hat{\Pi}'Z'M_WZ\hat{\Pi})D\) and \(D'\hat{\Sigma}D\). Their diagonal entries become \((120,360)\) and \((4,36)\), respectively. The standardized relevance values are therefore

\frac{120}{4}=30, \qquad \frac{360}{36}=10.

The eigenvalues remain 30 and 10, and \(CD=10/3=3.333\) as before. A change in the units of an endogenous regressor does not, by itself, change the system’s strength assessment. More generally, the eigenvalues are unchanged by nonsingular linear reparameterizations of the endogenous regressors or excluded instruments, provided the model and matrices are transformed consistently.

Comparison and limitations

A first-stage F statistic for one endogenous regressor summarizes the joint relevance of excluded instruments for that regressor. With multiple endogenous regressors, looking at one F statistic per regressor is a useful initial check but does not assess every linear combination jointly. The Cragg-Donald minimum eigenvalue statistic fills that system-level role under homoskedasticity. Partial \(R^2\) measures instrument contribution to a particular first stage; it does not, on its own, replace the minimum-eigenvalue assessment of the weakest joint direction.

The statistic has important limits. It is not an estimate of the causal effect, a test of instrument exogeneity, or a test that the structural model is correctly specified. Nor does a large value establish that every identification concern has been resolved. Most importantly, the conventional Cragg-Donald statistic relies on homoskedasticity. When that assumption is not credible, use a diagnostic designed for the error structure rather than treating this statistic as robust. Critical values for interpreting the statistic depend on the design; a single rule of thumb should not be substituted for them.

Common mistake: “Each endogenous regressor has a strong first-stage F statistic, so the instruments must be strong for the whole system.” This does not follow: the regressors can be strongly predicted individually while a particular linear combination is weakly predicted. The correction is to assess joint strength with a system diagnostic such as the Cragg-Donald minimum eigenvalue statistic when its homoskedastic assumptions are appropriate.

Check Your Understanding

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

If \(L=4\) and the eigenvalues of \(A\) are 28 and 6, what is the Cragg-Donald statistic?
Why can a small minimum eigenvalue matter even when the diagonal entries of \(A\) are large?
What is the role of \(\hat{\Sigma}^{-1/2}\) in the construction?
Does a large Cragg-Donald statistic establish that the instruments are exogenous? Explain.
Name one feature of the data or model that makes the conventional statistic inappropriate as the strength diagnostic.

Solutions. (1) The minimum eigenvalue is 6, so \(CD=6/4=1.500\). (2) The weakest direction may be a linear combination of regressors rather than one coordinate; the diagonal entries do not reveal its full strength. (3) It standardizes instrument-explained variation by the first-stage error covariance, allowing the system’s weakest direction to be assessed on a common scale. (4) No. Strength and instrument exogeneity are different requirements; the statistic addresses strength under its assumptions, not whether the instruments are independent of the structural error. (5) Heteroskedastic first-stage errors, or clustered errors not handled by the conventional homoskedastic calculation, undermine its use as the corresponding robust diagnostic.

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