Questions: Generalized Method of Moments (GMM)

Question 1

A researcher estimates a model with 3 parameters using 3 moment conditions (just-identified). She then discovers 2 additional valid instruments, giving her 5 moment conditions. What is the correct next step to improve efficiency?

• A. Discard the extra instruments — additional moment conditions do not improve GMM estimates
• B. Use the additional moments with an initial weighting matrix, then re-estimate using the inverse variance of the moments as the optimal weighting matrix
• C. Switch to OLS, which is always more efficient than GMM with more than k moment conditions
• D. Use the identity matrix as the permanent weighting matrix, since it treats all moments equally and avoids estimation error in W

Answer: (B)

Additional valid moment conditions contain information that can improve efficiency. The two-step procedure implements this: (1) estimate with an initial (e.g., identity) weighting matrix to get a consistent θ̂; (2) compute the sample variance of the moment conditions at θ̂; (3) invert it to form the optimal W; (4) re-estimate. The optimal W downweights noisy moments and upweights precisely estimated ones. Discarding valid instruments throws away information; the identity matrix is suboptimal unless moments happen to have equal variance.

Question 2

A researcher estimates a dynamic panel model using two-step GMM with 12 instruments and 4 parameters. The Hansen J-statistic is 21.3 with 8 degrees of freedom (p = 0.006). What is the most appropriate conclusion?

• A. The model is well-specified — a large J-statistic signals that the moment conditions are highly informative
• B. At least one moment condition appears to be misspecified — some instruments may be invalid or the functional form may be incorrect
• C. The sample size is too small; GMM requires n > 1000 for valid inference
• D. The estimator is inefficient; a different initial weighting matrix would reduce the J-statistic to an acceptable level

Answer: (B)

The J-statistic tests whether all overidentifying restrictions hold simultaneously at the GMM estimates. Under correct specification, J ~ χ²(q) where q = number of moment conditions minus number of parameters = 12 - 4 = 8. A value of 21.3 with p = 0.006 strongly rejects the null that all moments are correctly specified. This is a signal that one or more instruments are invalid (correlated with the error) or that the functional form is misspecified. Passing the J-test is necessary but not sufficient; failing it is a clear warning of misspecification.

Question 3

In a just-identified GMM problem (same number of moment conditions as parameters), the researcher should use a two-step iterative procedure to compute the optimal weighting matrix and obtain estimates.

• T. True
• F. False

Answer: False

When just-identified, there are exactly as many equations (sample moments = 0) as unknowns (parameters), so you can solve the system directly — setting g(θ) = 0 exactly. No weighting matrix is needed or meaningful because there is only one solution regardless of W. The two-step procedure is necessary only when overidentified, because then you cannot satisfy all moments simultaneously and must choose how to weight the residual.

Question 4

OLS can be interpreted as a special case of GMM in which the moment conditions are the sample orthogonality conditions between regressors and residuals.

• T. True
• F. False

Answer: True

The OLS first-order conditions require (1/n)Σ Xᵢ(Yᵢ - Xᵢ'β) = 0 — the sample analogs of E[Xᵢεᵢ] = 0. These are exactly GMM moment conditions with f(Yᵢ, β) = Xᵢ(Yᵢ - Xᵢ'β). When the number of regressors equals the number of moment conditions (just-identified), GMM reduces exactly to OLS. IV is also a GMM special case with instrument orthogonality conditions. This unification is the core appeal of the GMM framework.

Question 5

Why does having more moment conditions than parameters in GMM create both an opportunity and a testable restriction, and how does the Hansen J-test exploit the latter?

Model answer: Opportunity: each additional valid moment condition contains information about θ that can improve estimation precision. By choosing the weighting matrix to exploit the most informative moments, the researcher achieves greater efficiency than any just-identified estimator. Testable restriction: if the model is correctly specified, all moment conditions should hold simultaneously — but with more conditions than parameters, there is no θ that sets all sample moments to exactly zero. At the GMM optimum, the weighted residual g(θ̂)'Ŵg(θ̂) is small but nonzero. The Hansen J-statistic scales this residual by the sample size: nJ ~ χ²(q) under the null of correct specification, where q = (number of moment conditions) - (number of parameters). A large J indicates the moment conditions are in systematic conflict with each other, signaling that at least one is misspecified.

The J-test is the overidentification test: it asks whether the 'extra' moment conditions, beyond what is needed for identification, are consistent with the same θ. Passing provides some evidence of instrument validity; failing is diagnostic of misspecification. Importantly, the J-test cannot detect if all instruments are invalid in the same direction — it only detects internal inconsistency among the moment conditions.