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Proposition 3.7 (testing a subset of orthogonality conditions13): Suppose
Assumptions 3.1-3.5 hold. Let xil be a subvector of xi, and strengthen Assumption
3.4 by requiring that the rank condition for identification is satisfied for xil (SO
E(xil zi) is of full column rank). Then, for any consistent estimators g of S and gl I
of s11,C=J-J1依分布收敛于自由度为（k-k1）的卡方分布。
where K = #xi (dimension of xi), K1 = #xil (dimension of xil), and J and J1
are defined in (3.6.8) and (3.6.10).
Clearly, the choice of g and gII does not matter asymptotically as long as they are
consistent. But in finite samples, the test statistic C can be negative. This problem
can be avoided and C can be made nonnegative in finite samples if the same g is
used throughout, that is, i f g l l in (3.6.9) and (3.6.10) is the submatrix ofgin (3.6.7)
and (3.6.8). This is accomplished by taking the following steps:
(1) do the efficient two-step GMM with full instruments xi to obtain g from the
first step, 8 and J from the second step;
(2) extract the submatrix g1 from g obtained from (I), calculate 8 by (3.6.9) using
this g11 and J1 by (3.6.10). Then take the difference in J.
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