## Thursday, July 7, 2011

### Alexander Aitken Can you imagine what it would be like  trying to learn and teach econometrics without the use of matrix algebra? O.K., I know that some of you are probably thinking, "that would be great!" But give it some serious thought. We'd be extremely limited in what we could do. It would be a nightmare to go beyond the absolute basics.

Only in the 1960's, with the classic texts by Johnston (1963) and Goldberger (1964), did the use of matrix algebra become standard practice in the teaching of econometrics. We used that first edition of Johnston's text in the first undergraduate econometrics course I took. Thank goodness!

Every student of econometrics is indebted to Alexander Craig Aitken (1895 - 1967) for his development of what is now the standard vector/matrix notation for the linear  regression model (and its extensions). Econometricians also use the Generalised Least Squares ("Aitken") estimator when this model has a non-standard error covariance matrix.

The seminal Generalised Least Squares contribution, together with the first matrix formulation of the linear regression model appeared in Aitken's paper, "On Least Squares and Linear Combinations of Observations", Proceedings of the Royal Society of Edinburgh, 1935, vol. 55, pp. 42-48. In this paper we find the well-known extension of the Gauss-Markhov Theorem to the case where the regression error vector has a non-scalar covariance matrix - the Aitken estimator is shown to be "Best Linear Unbiased".

Aitken's most influential statistical paper was co-authored with (another New Zealander) Harold  Silverstone - "On the Estimation of Statistical Parameters", Proceedings of the Royal Society of Edinburgh, 1942, vol. 61, pp. 186-194. This paper extends earlier ideas by Sir Ronald Fisher to derive (only for the unbiased case) the result that we now usually refer to as the "Cramér-Rao Inequality" for the lower bound on the variance of an estimator. Interestingly, this contribution pre-dates the 1945 work by Rao and Cramér's 1946 paper.

So who was Alexander Craig Aitken?