Advanced Econometrics Takeshi Amemiya
Least Squares Estimator of a Subset of fi
It is sometimes useful to have an explicit^ formula for a subset of the least squares estimates fi. Suppose we partition fi' = (fl, 02),where /?, is a AT, - vec-
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tor and fa is a ЛГ2-vector such that Kx + AT2 = K. Partition X conformably as X = (X,, X2). Then we can write X'X0 = X'y as
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In Model 1 we assume that X is offull rank, an assumption that implies that the matrices to be inverted in (1.2.12) and (1.2.13) are both nonsingular. Suppose for a moment that X, is of hill rank but that X2 is not. In this case 02 cannot be estimated, but 0X still can be estimated by modifying (1.2.12) as
A-WMJX. r'XfMJy, (1.2.14)
where MJ = I — Xf(XJ' XJ)- ’XJ', where the columns of XJ consist of a maximal number of linearly independent columns of X2, provided that XJMJX, is nonsingular. (For the more general problem of estimating a linear combination of the elements of 0, see Section 2.2.3.)
