15.4 Testing overidentifying restrictions
Theorem 15.4.1 (Hansen’s \(J\) test). Let \(\hat \theta \) be the efficient two-step GMM estimator, with \(\hat S \xrightarrow {p} S\). If all \(\ell \) moment conditions hold, \[ J = n\,\bar g_n(\hat \theta )'\hat S^{-1}\bar g_n(\hat \theta ) \xrightarrow {d} \chi ^2_{\ell - k} . \]
Sketch of proof. Write \(S^{-1} = C'C\). Expanding \(\bar g_n(\hat \theta )\) about \(\theta _0\) and using the expression for \(\hat \theta - \theta _0\) in the proof of Theorem 15.2.2 with \(W = S^{-1}\), \[ \sqrt n\,C\bar g_n(\hat \theta ) = \bigl [I - CG(G'C'CG)^{-1}G'C'\bigr ]\sqrt n\,C\bar g_n(\theta _0) + o_p(1) . \] The vector \(\sqrt n\,C\bar g_n(\theta _0)\) tends to \(N(0, I_\ell )\), since \(CSC' = I\). The matrix in brackets is the projection orthogonal to the \(k\) columns of \(CG\), of rank \(\ell - k\). So \(J = \|\sqrt n\,C\bar g_n(\hat \theta )\|^2\) is asymptotically a quadratic form of a standard normal vector in a projection of rank \(\ell - k\), which is \(\chi ^2_{\ell - k}\) by Theorem 5.4.2. □
Of the \(\ell \) moment conditions, \(k\) are used up in estimating \(\theta \), and the \(J\) statistic tests whether the remaining \(\ell - k\) are consistent with them. In the linear IV model under homoscedasticity it reduces to the Sargan statistic of Theorem 13.4.3, and it has the same limitation: it detects disagreement among the moments, not a failure common to all of them. A rejection may also indicate a misspecified model rather than invalid instruments, since the moment conditions come from the model.
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