8.4.1  TWO-STAGE LEAST SQUARESIf Z

8.4.1  TWO-STAGE LEAST SQUARESIf Z contains more variables than X, then Z X will be L * K with rank K 6 L and will thus not have an inverse—(8-6) is not useable. The crucial result for estimation is plim (Z E/n) = 0. That is, every column of Z is asymptotically uncorrelated with E. That also means that every linear combination of the columns of Z is also uncorrelated with E, which suggests that one approach would be to choose K linear combinations of the columns of Z. Which to choose? One obvious possibility is simply to choose K variables among the L in Z. Discarding the information contained in the extra L - K columns will turn out to be inefficient. A better choice that uses all of the instruments is the projection of the columns of X in the column space of Z,