The Yule-Walker Equations and Algorithms for Their Solution

"Every cell phone call solves the Yule-Walker equations every ten microseconds." Thierry Dutoit

The Yule-Walker equations are pervasive in science and technology, and, even though we are most concerned with their role in financial time series, we would be like ostriches with our heads in the sand if we chose to ignore the huge investment that the scientific community has made to master the understanding of these equations.

Here you will find some of the resources that I found on the web that provide additional background on the Yule-Walker equations, their solutions, and the competing methods for the fitting of AR(p) models. As usual, I give a brief introduction to the link and suggest how the material fits into our course, but, if you have the time, it is always best to let each article speak for itself.

Links suffice to get you to the good stuff, but for citation purposes you may eventually need complete references. I am sure that you can get these when the need arises, so, for the moment, I take the liberty of being brief. Eventually, I (or a work-study student!) will flush out the incomplete references.

Introductory or Culturally Oriented Pieces

Gidon Eshel provides a useful teaching note on the Yule-Walker equations which discusses their relation to between the least squares approach for fitting an AR(p) model. This note is written at the level of our class. For the full story one should note the links in the final paragraph below.

Rick Katz of NCAR provides us with gentle presentation "Sir Gilbert Walker and a Connection Between El Nino and Statistics." It is in outline form, but it is still fully informative. With some of your own exploration, it will not be difficult to fill in the blanks, but you should still check out Katz's full article in Statistical Science, 17 (2002), 97-117. This is a beautifully written article which represents Statistical Science at its best. The article has many pictures, including one of Walker who looks the model of a classic Edwardian. Incidentally, here is a picture of the world as seen from Darwin, Australia .

Incidentally, Katz does climate and economics, and he has a web site of case studies. Many of these case studies have a good statistical story.

William Meeker has a useful class note which introduces AR(p) models and discusses the Wolfer sunspot data that was analyzed by Yule (1927) in the world's first use of an autoregressive model. As Katz's notes in his lecture, Yule considered only the case AR(2), but in this note Meeker also discusses the models for p=3, 4, etc.

The National Geophysical Data Center has more on Johann Rudolph Wolf (1816-1893) and his method for estimating solar activity. The center also has the sunspot numbers for 1700--1995. In this series the maximum occurs in 1957, and one sees that there is much more to be found than just the well-know approximate period of 11.1 years.

Yule and Walker Originals

G. Udny Yule "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers" Philosophical Transactions of the Royal Society of London, Ser. A, Vol. 226, (1927) 267--298.

Gilbert Walker "On Periodicity in Series of Related Terms," Proceedings of the Royal Society of London, Ser. A, Vol. 131, (1931) 518--532.

Incidentally, these links are to the electronic archives of the Bibliothèque National de France, so it helps to recall that "Allez Page" means "go to page." Sadly, this site is not particularly user-friendly. Despite instructions that suggest otherwise, it seems that you can print just one page at a time. Still, for the moment, this is perhaps the best places to check for older originals.


Criticisms, Advanced Algorithms, and Research Oriented Pieces

The Yule-Walker approach to the estimation of the AR(p) coefficients is interesting historically and attractive mathematically. Nevertheless, the estimates provided by the Yule-Walker are prone to bias certain biases, and in some cases the confidence intervals reported by widely used software can be badly wrong.

I will append a page of references that speak to these pathologies and that offer some suggestions for improvements. I will also include a few pointers to the HUGE literature on algorithms for solving the YW equations and related Toeplitz systems.