place of w-1, and (NL - N)-1 he used H-2 and X-1. This is mentioned here because of its possible bearing on certain anomalies in the CLI estimates of equations 1.0 and 4.0 of group I. The matter is discussed in the text in Section 11, part (f).
This appendix is a note on the restricted least-squares method of estimating reduced form parameters, referred to in Section 4. We first describe the method assuming that a one-element subset of structural equations is chosen to provide the restrictions.
Suppose there is a model consisting of G equations in G jointly dependent variables y and K predetermined variables z. Suppose that one of its
(1) β1y1 + ⋯ + βHγH + 0 + ⋯ + 0 + γ1Z1 + ⋯
+ γK*ZK* + 0 + ⋯ + 0 = u
where H < G and K* < K. Consider H equations of the reduced form,
where K** is the number of predetermined variables assumed to be known to be in the model but not in 1. Then K** ≦ K - K*. The parameters πik can be estimated by least-squares. The least-squares estimates can be made more efficient by altering them to take account of the restrictions implied by the zeros in 1, as follows. It must be possible to get equation I from a linear combination of equations 2, in fact, from that combination obtained by taking βi times the ith equation of 2, i = 1, ⋯, H, and summing the results. This means that there are K** equations, one for each zk excluded from 1, thus:
Now if K** > H - 1, i.e., if 1 is overidentified, 3 is overdetermined. Hence if 3 is to hold, and it must, a restriction is implied on the matrix of the πik, i = 1, ⋯, H, k = K* + 1, ⋯, K* + K**, keeping its rank down to H - 1. This restriction may be applied to the matrix of least-squares estimates of the πik, to make them conform to the restrictions implied by the zeros in 1. The computation is not difficult, once the limited information estimates for 1 are obtained.
Similarly, if there are other structural equations besides I which also contain some one of the jointly dependent variables y1, ⋯, yH, say y1, the estimates of the parameters of the reduced-form equation for y1 can be