2. THE GMM MODEL AND THE PROPOSED TEST

The GMM model specifies that {Y_t}_t=1^T is an observed time series and θ is a p × 1 parameter vector with a true value θ₀, in the interior of a compact space Θ, which has the property that

where h(.,.) is a G-dimensional function and E_t(.) denotes the expectation of the argument conditional on the information set at time t. Let Z_t be a k × 1 vector of instruments in this information set, with Gk ≥ p. It follows that E(φ(Y_t,θ₀)) = 0, where

, a Gk × 1 vector. The GMM estimator of

where S(θ) = T^−1/2Σ_t=1^Tφ(Y_t,θ)′ W_T(θ)T^−1/2Σ_t=1^Tφ(Y_t,θ) and W_T(θ) is a symmetric positive-definite weighting matrix.

Under standard assumptions, the GMM estimator is consistent and asymptotically normal and, if Gk > p, Hansen's J-test of the overidentifying restrictions is asymptotically χ²(Gk − p) distributed. But these assumptions include the requirement that the Gk × p matrix B = E [dφ(Y_t,θ₀)/dθ] has rank p. This is the local identification assumption and is the focus of this paper. Another standard assumption is that E(φ(Y_t,θ)) has a unique zero at θ = θ₀, which rules out the possibility of local but not global identification (Hsiao, 1983).

However this asymptotic theory often works poorly in practice as has been documented in Monte Carlo studies by Tauchen (1986), Kocherlakota (1990), Hansen, Heaton, and Yaron (1996), and others, and this problem can arise from weak identification. The weak identification problem in GMM may be effectively circumvented by the use of S-sets, as proposed in Stock and Wright (2000). The approach dispenses with point estimation and instead forms a confidence set for the entire parameter vector θ directly from an objective function, using a nonlinear analog of the Anderson–Rubin confidence set (Anderson and Rubin, 1949; Staiger and Stock, 1997). If T^−1/2Σ_t=1^Tφ(Y_t,θ₀) →_d N(0,A(θ₀)) for some nonsingular matrix A(θ₀) and W_T(θ₀) →_p A(θ₀)⁻¹, then S(θ₀) converges to a χ²(Gk) distribution. No identification assumption is required for this to hold. The confidence set for θ is formed as the inverse of the acceptance region of this test; i.e., the confidence set of coverage 1 − α is S_θ*(α) = {θ : S(θ) ≤ F_χ²(α,Gk)} where F_χ²(a,b) is the upper 100a percentile of a χ² distribution on b degrees of freedom. In a completely unidentified model (E(φ(Y_t,θ)) = 0, uniformly in θ) or with local asymptotic underidentification (E(φ(Y_t,θ)) = O(T^−1/2), uniformly in θ), such a confidence set will have infinite expected volume. But under these circumstances, any valid confidence set must have infinite expected volume (Dufour, 1997). An S-set can be empty or nonconvex.

The focus of this paper is on constructing a test for detecting a lack of identification in GMM. If the test accepts the null, the researcher will know not to use conventional asymptotic theory. The S-set works regardless of whether the model is identified or not. However, a means of detecting underidentification is still useful because (a) if the model is identified, consistent point estimates of the parameters are available, (b) if the model is identified, we have confidence intervals for individual parameters that are not asymptotically conservative, unlike the confidence sets for elements of θ that can be projected from an S-set, and (c) an S-set is hard to represent when there are more than two parameters. All these advantages of using conventional asymptotics rather than the S-set are available if and only if the identification is sufficient for the conventional asymptotics to work well. The identification condition that I wish to test is that the matrix B is of rank p. I shall do this by testing the null of underidentification: that ρ(B) = r, where the operator ρ(.) denotes the rank of its argument and r < p. I define

and introduce some assumptions.

Assumption 1.

for positive-definite finite C.

Assumption 2.

is an estimator of the variance of vec[dφ(Y_t,θ)/dθ] , or of 2π-times its zero-frequency spectral density matrix, such that

.

Assumption 3. T^−1/2Σ_t=1^Tφ(Y_t,θ₀) →_d N(0,A(θ₀)) for some nonsingular matrix A(θ₀).

Assumption 4. W_T(θ) is an estimator of 2π-times the zero-frequency spectral density matrix of φ(Y_t,θ) such that W_T(θ₀) →_p A⁻¹(θ₀).

For example, if φ(Y_t,θ₀) is serially uncorrelated then A(θ₀) = E(φ(Y_t,θ₀) φ(Y_t,θ₀)′), and in this case one would set W_T(θ) = (T⁻¹Σ_t=1^Tφ(Y_t,θ) φ(Y_t,θ)′)⁻¹, giving the standard continuous updating estimator. Assumptions 1–4 are high-level convergence assumptions. They all apply only at the true parameter value, θ₀. Let

where Ω(r) is the space of all Gk × p matrices of rank r. The proposed test statistic of the hypothesis that ρ(B) = r is L*(r) = inf_{θ∈Sθ*(α)} L(θ,r).

THEOREM 1. Under Assumptions 1–4, if B is in fact of rank r, then lim_T→∞ P(L*(r) ≤ F_χ²(α,(Gk − r)(p − r))) ≤ 2α.

Proof. From Theorem 1 of Cragg and Donald (1997), if ρ(B) = r then L(θ₀,r) →_d χ²((Gk − r)(p − r)). But lim_T→∞ P(θ₀∈S_θ*(α)) = 1 − α and if θ₀∈S_θ*(α) then L*(r) ≤ L(θ₀,r), so if ρ(B) = r then lim_T→∞ P(L*(r) ≤ F_χ²(α,(Gk − r)(p − r))) ≤ 2α. █

The term L*(r) yields an asymptotically conservative test; the limiting probability of incorrectly rejecting the null is bounded above by 2α. This is the test statistic proposed in this paper; the test rejects if and only if L*(r) exceeds the χ² critical value. It is a disadvantage that the proposed test is conservative, but in the context of the linear IV model there can still be severe identification difficulties even if the underidentification null is marginally rejected (Staiger and Stock, 1997), and so a conservative test may be quite useful. If ρ(B) is in fact lower than the hypothesized rank r, I conjecture (following Cragg and Donald, 1997) that the test will continue to be asymptotically conservative. If ρ(B) is higher than the hypothesized rank r, the test will reject with probability one, asymptotically. To detect identification difficulties, it is perhaps especially appropriate to test the null that ρ(B) = p − 1. In the linear IV model, L(θ,r) does not depend on θ, and there is therefore no need to eliminate θ by a Bonferroni argument as in Theorem 1; to do so would simply halve the asymptotic size of the test. Accordingly, in the linear IV model, the proposed test is no longer asymptotically conservative and instead has size equal to its nominal level, asymptotically. It reduces to the test proposed by Cragg and Donald (1993). In the case where the hypothesized rank is zero, this reduces further to the usual first-stage F-test.¹