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?

He was born in Dunedin, New Zealand on 1 April, 1895. Aitken  became one of the leading mathematical scholars to have been produced by that country, and his contributions in mathematical statistics and linear algebra are of fundamental interest to econometricians.

Aitken attended Otago Boys' High School from 1908 to 1912, and had the distinction of gaining first place in the nation-wide University Scholarship Examination of 1912. He then studied at the University of Otago in 1913, 1914, and 1918, achieving First Class Honours in Latin and French and (remarkably, only) Second Class Honours in Mathematics.

His studies were broken by active service during the First World War, including the Gallipoli landing and the Battle of the Somme. After teaching at Otago Boys' High School from 1920 to 1923, Aitken travelled to Scotland in 1923 to study with E.T. Whittaker at the University of Edinburgh. At that time, Whittaker was apparently the only scholar in the United Kingdom giving classes in matrix algebra. Aitken lived in Edinburgh for the rest of his life. 

His initial work on the smoothing of data pre-empted what we'd now call the Hodrick-Prescott filter, and earned him a D.Sc. in 1925, In that same year he was appointed Lecturer in Statistics and Mathematical Economics at the University of Edinburgh. He assumed the Chair of Mathematics in 1946 on Whittaker's retirement, and is reputed to have been a brilliant lecturer. 

During his outstanding career, Aitken received numerous honours. In particular, he was elected a Fellow of the Royal Society of Edinburgh, and a Fellow of the Royal Society of London. He was an Honorary Fellow of the Royal Society of New Zealand, of the Society of Engineers, and of the Faculty of Actuaries. He was awarded honorary degrees by the University of Edinburgh and by the (then) University of New Zealand.

Aitken was undoubtedly one of the greatest mathematicians to be born and educated in New Zealand. His main mathematical interests were in the areas of Actuarial Mathematics, Linear Algebra, Numerical Methods and Statistics.

His phenomenal skill in mental arithmetic made him the greatest "mental calculator" for whom there is any reliable record. His powers of retention were legendary.

Econometricians have benefited especially from his applications of matrix algebra to problems in numerical analysis, as well as his statistical contributions to the theory of linear models, as noted earlier. He published on such topics as symmetric groups, invariants, the solution of linear and polynomial equations, eigenvalue problems, and computational algorithms. His books, Determinants and Matrices, and Canonical Matrices (with Turnbull) are classics, and I'm lucky to have copies of both of these in my bookcase. (Thanks, Jacob, for gifting me the second of these!)

A man of great and disparate talents, Aitken was devoted to music and was regarded as a fine violinist and viola player, as well as being an occasional composer. He was a poet and writer. His memoir, Gallipoli to the Somme: Recollections of a New Zealand Infantryman (Oxford University Press, 1963), is regarded as one of the most moving accounts of the appalling reality of life in the trenches during the Great War.

Alexander Aitken died in Edinburgh, Scotland, on 3 November, 1967, and is remembered as a warm and gentle man.

So, the next time that you write down the formula for the linear regression model, you might remember that your life is actually being made easier, not harder, thanks to one brilliant and very interesting man.

Further Reading:

Phillips, P. C. B. (2010). Two pioneer New Zealand econometricians. New Zealand Economic Papers, 44, 1-26.

Tee, G. J., "Two New Zealand Mathematicians", in J. H. Crossley (ed.), Proceedings of the First Australian Conference on the History of Mathematics, Monash University, 1981, 182-199.

Tee, G. J., "Mathematics in the Pacific Basin", British Journal of the History of Science, 1988, vol. 21, 401-417.

© 2011, David E. Giles


  1. I am not sure but Aitken also seems to be the first to compute restricted least squares estimators without resorting to Lagrange multipliers. The idea is an older incarnation of the Theil/Goldberger mixed estimator.

  2. Andrew - nice! I wasn't aware of that.

  3. It can be found in the last few sections (with numerical demonstration) of Aitken (1945) "On Linear Approximation by Least Squares".

  4. Andrew: Thanks for this reference! Much appreciated!