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Appendix D CHOICE OF PREDETERMINED VARIABLES FOR ESTIMATION BY THE ABBREVIATED VARIANT OF THE LIMITED INFORMATION METHOD

The limited information estimates of any structural equation depend upon observations of a subset of the predetermined variables that are not in the equation being estimated but are in the system. The elements of this subset are called z**'s and there must be at least as many as H - 1 of them if H is the number of jointly dependent variables in the equation being estimated (see text, Sec. 1). Of course, there may be more than H - 1; if so, the estimates will be better. In our model the largest value of H - 1 for any equation is 4, for equation 3.0; if this is excepted, the largest value is 2, for each of several equations. Therefore the number of z**'s required for any equation is 2 except in the case of equation 3.0, which requires 4.

Now there are 25 predetermined variables in the complete model, and no equation contains more than 4. Thus, for each equation there are at least 21 variables available for use as z**'s, and so there is an arbitrary choice of z**'s to be made for each equation. If there were no costs in money and in degrees of freedom, one would always use all the available variables as z**'s. Because of these costs, a proper subset of the available variables has been used in each case, i.e., the abbreviated variant of the limited information method has been used.

The stochastic equations have been divided into four groups in such a way as to minimize the intersection of the set of jointly dependent variables in any group with the corresponding set for any other group; in fact every such intersection is empty. Then for any equation the set of z**'s is the set of all predetermined variables in the group to which the equation belongs, minus the set of predetermined variables appearing in the equation (see the accompanying table).

VARIABLES
GROUP EQUATIONS Jointly Dependent Predetermined
I (1.0), (2.0),
(3.4), (4.0),
(4.2), (5.1)
(X - ΔH), X, N,t, (pX - ε) -1, w-1,
W1, (pX - ε), w/p, (NL - N)-1, p-1
w, Δp, (NL - N)
II (6.2), (7.0),C, Y, D1, r/q1, (M/p) -1, t ΔF,
(9.0) (Y + Y-1 + Y-2), Δrv-1, 1/r-1
III (10.0)D2, ir-1, (q1) -1, (q1) -2
ΔF-1, i-1, t
IV (11.0) Δii-1, εR, t

Klein's grouping of equations was quite similar. In particular for group I he used exactly the same predetermined variables as I did, except that in

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