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
tor and fa is a ЛГ2vector such that Kx + AT2 = K. Partition X conformably as X = (X,, X2). Then we can write X’X0 = X’y as

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
AWMJX. 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.)
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