The Overidentifying Restrictions Test
The asymptotic theory so far has been predicated on the assumption that the model is correctly specified in the sense that E[f(vt, 00)] = 0. If this assumption is false then the argument behind Theorem 1 breaks down, and it is no longer possible to establish the consistency of the estimator. Since the validity of the population moment condition is central to GMM, it is desirable to develop methods for testing whether the data are consistent with this assumption. If p = q then it is not possible to examine this hypothesis directly because, as we have seen, gT (0T) = 0. In this case, the validity of Assumption 3 can only be assessed indirectly using so called conditional moment tests; see Newey (1985) and Tauchen (1985). However, if q > p then the estimator does not set the sample moment to zero and so this leaves scope for testing E [f(vt, 00)] = 0 directly. In Section 3, it is shown that GMM estimation effects a decomposition on the population moment condition into identifying and overidentifying restrictions. By definition, the estimator satisfies the sample analog to the identifying restrictions. However, the overidentifying restrictions are ignored in estimation and so are available as a basis for testing the validity of the population moment condition. This leads us to the overidentifying restrictions test, which is routinely reported in applications of GMM.
In practice, it is desirable to base inference on the the two-step (or iterated) estimator because it yields the most efficient GMM estimator based on E[f (vt, 00)] = 0. Therefore, we restrict attention to this case and so substitute W = S_1. In Section 3, it is shown that Qt(0t) can be interpreted as a measure of how close the sample is to satisfying the overidentifying restrictions. This motivated Sargan (1958) to propose using the statistic
Jt = TQT (0t) (11.29)
to test whether the overidentifying restrictions are satisfied. His analysis is restricted to linear models estimated by instrumental variables. However, Hansen (1982) extends this approach to nonlinear models. The distribution is given in the following theorem.
Theorem 6. Asymptotic distribution of the overidentifying restrictions test. If Assumptions 1-10 and certain other regularity conditions hold
and W = S-1 then Jt A x2q-p.
Notice that the degrees of freedom equal the number of overidentifying restrictions. The limiting distribution can be derived heuristically from our earlier analysis. From (11.25), we have that
TQt (0t) = T1/2gT(0T)’ST1T1/2gT (0t)
= T1/2gT (0o)’S-1/2′[l, – P(0o)]S-1/2T1/2gT (0o) + op(1) (11.30)
where we have used the fact that Iq – P(0o) is a projection matrix. Equation (11.30) implies Jt is asymptotically equivalent to a quadratic form in a random vector with an N(0, Iq) distribution and a projection matrix with rank q – p. The result then follows directly from Rao (1973, p. 186).
To consider the power properties of the test, it is necessary to briefly consider what it means for the model to be misspecified in our context. Taken together, Assumptions 3 and 4 imply the population moment condition is satisfied at some unique value in 0. Therefore, the model is misspecified if there is no value in 0 at which the population moment condition holds. Such a situation is captured by the following assumption.
Assumption 11. Misspecification. E[f(vt, 0)] = p(0) where p : 0 ^ Wq
and || p(0) || > 0 for all 0 Є 0.
One important consequence of Assumption 11 is that the two covariance matrix estimators in (11.21)-(11.22) are inconsistent estimators of the long-run variance because they are calculated under the assumption that E[f(vt, 00)] = 0.21 This inconsistency is not by itself a cause for concern provided p limTA„S-1 is a nonsingular matrix. Such would be the case for the estimator SMD, and in consequence Jt is Op (T) and a consistent test versus the misspecification characterized
in Assumption ll.22 However, Hall (2000a) shows that p limT^„SHACC is a singular matrix, and that this causes JT to be only Op(T/ b(T)) although still consistent. A more powerful test can be constructed by using a version of the HACC which is consistent under both null and alternative hypothesis. Hall (2000a) shows that such an estimator can be constructed by replacing f; in (11.22) with
Гі = T-1 X [fV, Pt(1)) – gT(Pt(1))] [fV-,■, 0t(1)) – gT(Pt(1))]’.
t = І +1
Once this change is made, the overidentifying restrictions test is consistent but now Op(T).