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 ЛГ2-vector such that Kx + AT2 = K. Partition X conformably as X = (X,, X2). Then we can write X’X0 = X’y as

X’1x1A + x;xJ2 = x’1y

(1.2.10)

and

X’M + X’2X202 = X2y.

(1.2.11)

Solving (1.2.11) for 02 and inserting it into (1.2.10), we obtain

0l=(XM2Xl)-lX[M2y>

(1.2.12)

where M2 = I — X2(X2X2)_IX2. Similarly,

02 = (Х2М, Х2)-1Х2МіУ,

(1.2.13)

where M, = I – X, (X’,X, )~lX[.

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.)

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