Why Test Overidentifying Moments?
A structural model can imply more population moment conditions than it has unknown parameters. Estimation uses those moments to choose parameter values, but the fitted model generally cannot match every sample moment exactly. The remaining discrepancy provides a test of whether the restrictions implied by the model are compatible with the data.
This test answers a different question from parameter identification. Identification asks whether the model’s population implications uniquely determine its parameters. An overidentification test asks whether the collection of population implications fits the observed moments. A model can fail one check while passing the other, so the distinction matters for both interpretation and inference.
The model’s restrictions are often implicit: one parameter vector must account for all \(q\) targets at once. For instance, a model with one parameter and three implied moments restricts the three population targets to lie on a one-dimensional curve. The test evaluates how far the sample targets lie from the model’s set of possible predictions, allowing for their sampling variability.
The Minimum-Distance Goodness-of-Fit Statistic
Suppose the sample moments obey a central limit theorem under the null: \(\sqrt{n}\{\hat m-h(\theta_0)\}\) converges in distribution to a mean-zero normal vector with covariance matrix \(S\). Assume \(S\) is positive definite and can be consistently estimated. The optimally weighted minimum-distance estimator chooses \(\hat\theta\) to minimize the squared discrepancy using \(S^{-1}\) as the weight matrix.
Here \(J\) is the minimized, sample-size-scaled discrepancy. If \(S\) is unknown, its consistent estimate is used in the calculation; the displayed expression uses \(S\) to show the underlying population weighting. Under the null and standard regularity conditions, \(J\) has an asymptotic chi-square distribution with \(q-p\) degrees of freedom. The difference \(q-p\) counts the number of independent directions of moment discrepancy left over after fitting \(p\) locally identified parameters.
For a test at significance level \(\alpha\), reject the null when \(J\) exceeds the \(1-\alpha\) quantile of the chi-square distribution with \(q-p\) degrees of freedom. Equivalently, calculate the upper-tail probability under that reference distribution and compare it with \(\alpha\). The p-value describes how unusual a discrepancy at least this large would be if the null restrictions and the test’s assumptions held; it is not the probability that the null is true.
Worked Example: A Small Discrepancy Across Three Moments
Suppose a hypothetical application has \(n=100\), sample moment vector \(\hat m=(2.1,4.2,6.6)'\), and structural implications \(h(\theta)=(\theta,2\theta,3\theta)'\). For this illustration, take the asymptotic covariance matrix \(S\) to be the identity matrix, so optimal weighting gives equal weight to the three squared discrepancies. There are \(q=3\) moments and \(p=1\) parameter.
The minimum-distance estimate is the least-squares fit of the three targets to \((1,2,3)'\):
The residual vector is approximately \((-0.06429,-0.1286,0.1071)'\). Thus,
The reference distribution has \(q-p=2\) degrees of freedom. Its upper-tail probability is approximately \(P(\chi^2_2\geq 3.214)=0.2005\). At the 5% level, this is not evidence against the joint moment restrictions. This conclusion does not establish that the model is true; it says the observed discrepancy is not unusually large under the stated null and assumptions.
Why the Degrees of Freedom Are \(q-p\)
The degrees of freedom follow from the geometry of the local fit. Let \(D\) be the Jacobian of \(h\) at \(\theta_0\), a \(q\)-by-\(p\) matrix. Locally, changing the parameter moves the model’s moment vector in directions spanned by the columns of \(D\). The estimator uses those \(p\) directions to absorb part of the sampling error; the test measures what remains in the other directions.
To see this, write the standardized sample error as \(z=S^{-1/2}\sqrt n\{\hat m-h(\theta_0)\}\), which converges to a standard \(q\)-dimensional normal vector. Define \(A=S^{-1/2}D\). A local parameter change of size \(u/\sqrt n\) changes the standardized fitted moments, to first order, by \(Au\). Consequently, the local fitting problem is to choose \(u\) to minimize \(\lVert z-Au\rVert^2\).
Full column rank of \(D\) implies full column rank of \(A\), so the inverse in this expression exists. The matrix \(P_A\) is the orthogonal projection onto the \(p\)-dimensional column space of \(A\). Therefore \(I-P_A\) projects onto its orthogonal complement, which has dimension \(q-p\). The asymptotic version of \(J\) is the squared length of this projected standard normal vector. In an orthonormal basis of the complement, it is the sum of squares of \(q-p\) independent standard normal variables.
This argument also explains why optimal weighting matters. Whitening by \(S^{-1/2}\) makes the sampling error standard normal, and the weighted criterion then measures ordinary squared distance in those standardized coordinates. With a different weight matrix, the minimized criterion does not in general have this same chi-square reference distribution.
Worked Example: A Large Discrepancy in Two Moments
Consider a second hypothetical application with \(n=1185\), \(\hat m=(2.1,4.5)'\), and \(h(\theta)=(\theta,2\theta)'\). Assume again that \(S\) is the identity matrix. There are two moments and one parameter, so the regular overidentification test has one degree of freedom.
The unscaled criterion is \(Q(\theta)=(2.1-\theta)^2+(4.5-2\theta)^2\). Expanding gives \(Q(\theta)=5\theta^2-22.2\theta+24.66\). Its derivative is \(10\theta-22.2\), so the minimizer is \(\hat\theta=2.22\). Its second derivative is \(10>0\), confirming that the quadratic is strictly convex and this stationary point is its minimum.
At the minimum, the residuals are \(2.1-2.22=-0.12\) and \(4.5-2(2.22)=0.06\). Hence,
The one-degree-of-freedom upper-tail probability is \(P(\chi^2_1\geq21.33)\approx 3.9\times10^{-6}\). At conventional significance levels, this is strong evidence against the joint moment restrictions, assuming the covariance and regularity conditions are appropriate. The test does not identify which moment restriction is responsible for the discrepancy.
Fit and Identification Are Different Questions
The rank condition in the chi-square result is not a cosmetic technicality. It ensures that the \(p\) parameters move the implied moments in \(p\) distinct local directions. If it fails, the parameter vector is not locally identified by these moments in the usual regular sense, and the standard \(q-p\) reference distribution is not justified. Conversely, a well-identified parameter vector does not guarantee that the model’s restrictions fit the data.
Worked Example: A Perfect Fit Without Identified Parameters
Suppose three hypothetical moments are \(\hat m=(1,2,3)'\), with \(n=100\) and \(S=I\). The structural implications are \(h(\theta_1,\theta_2)=(\theta_1+\theta_2,2(\theta_1+\theta_2),3(\theta_1+\theta_2))'\). Although there are \(q=3\) moments and \(p=2\) parameters, the model depends on the parameters only through their sum \(\gamma=\theta_1+\theta_2\).
The criterion can be written entirely in terms of that sum:
The minimum is zero when \(\gamma=1\), but every pair satisfying \(\theta_1+\theta_2=1\) attains it. For example, \((\theta_1,\theta_2)=(0,1)\) and \((2,-1)\) imply exactly the same moments. The Jacobian has two identical columns, each equal to \((1,2,3)'\), so its rank is 1 rather than the required \(p=2\).
Thus a zero discrepancy here does not show that both structural parameters have been identified. Nor is the nominal \(q-p=1\) chi-square result justified by the standard theorem. The example separates two questions: the model can match the moments while leaving parameters undetermined, and a goodness-of-fit statistic cannot by itself diagnose which parameters are identified.
Interpreting the Test Carefully
A rejection says that the observed moment vector is too far from the model’s fitted moment vector to be readily explained by sampling variation under the maintained assumptions. It does not establish that a particular structural assumption is false: misspecification, a poor covariance estimate, data problems, or failure of the asymptotic approximation can also affect the result. A non-rejection is likewise not proof of correctness; it may reflect limited information or a test with little power against the departures of interest.
When a test rejects, it is useful to inspect the moment residuals and the substantive restrictions behind them, rather than treating \(J\) as an explanation. Such inspection is diagnostic, however: selecting a problematic moment after seeing the data does not automatically provide a new test with its original error rate. In all cases, report the statistic, degrees of freedom, p-value, significance level, and the assumptions supporting the reference distribution.
- Common mistake: “There are three moments and two parameters, so the test has one degree of freedom,” even when the Jacobian has rank less than two.
- Correction: The usual \(q-p\) result requires full column rank \(p\), as well as the other regularity conditions. Count locally independent fitted directions, not only parameters written in the model.
- Common mistake: “The test did not reject, so the model is identified and correct.”
- Correction: Non-rejection is not proof of correct specification, and identification is a separate property of the mapping from parameters to population implications.
Check Your Understanding
Answer each question using the assumptions and distinctions developed above.
- A model has five moment conditions and three locally identified parameters. What is the usual asymptotic degrees of freedom for its overidentification test?
- What role does the covariance matrix \(S\) play in the optimally weighted goodness-of-fit statistic?
- Why does the local projection argument leave \(q-p\) dimensions of discrepancy under the full-rank condition?
- A test has a large p-value. What can you conclude, and what can you not conclude, about the structural model?
- How can a model fit all sample moments exactly while failing to identify every parameter?