This book concerns itself with the mathematics behind the application of classical statistical procedures to econometric models. I first tried to apply such procedures in 1983 when I wrote a book with Roger Bowden on instrumental variable estimation. I was impressed with the amount of differentiation involved and the difficultly I had in recognizing the end product of this process. I thought there must be an easier way of doing things. Of course at the time, like most econometricians, I was blissfully unaware of matrix calculus and the existence of zero-one matrices. Since then several books have been published in these areas showing us the power of these concepts. See, for example Graham (1981), Magnus (1988), Magnus and Neudecker (1999), and Lutkepohl (1996).

This present book arose when I set myself two tasks: first, to make my self a list of rules of matrix calculus that were most useful in applying classical statistical procedures to econometrics; second, to work out the basic building blocks of such procedures – the score vector, the information matrix, and the Cramer–Rao lower bound – for a sequence of econometric models of increasing statistical complexity. I found that the mathematics involved working with operators that were generalizations of the well-known vec operator, and that a very simple zero-one matrix kept cropping up. I called the matrix a shifting matrix for reasons that are obvious in the book.