ARIMA and State Space Models
Our unembarrassed (but brief) source for the Kalman Filter is the Wikipedia article.
- It has that great picture that makes clear the different roles of observed and state variables.
- It also has notation that is as clear as one can hope to get.
- It gives the recursions with unambiguous detail. These are a little tedious, but if you try to code some of this, you'll be happier with detail than with ambiguity.
- It also nudges one further along the line of intellectual descendents, including the now wildly popular particle filter.
Naturally, there are lots of other resources. In Stat 956 Fall 2007 we went over the nice elementary e exposition of Meinhof and Singpurwalla. This gives the simplest Bayesian derivation for the simplest one-dimensional problem. Nevertheless, it still requires knowing a bit about the geometry of the conditional multivariate gaussian.
We'll look at Paul Gilbert's Survey on State Space Representations of ARMA models.
A bit later we'll also look at the exciting new idea of particle filters. The literature is explosive, and the only concern is that it therefore becomes difficulty to sort out what is hot and what is not. Still, if you are looking for an area where change is rapid and applications abound, this is a pretty healthy place to look, especially if you take what is said with a grain of salt.
Generic caveats common to any "more elaborate" method:
- To be honorable, you have to trot out the plain vanilla methods first.
- Moreover, you have to make the race fair. It is oh-so tempting to fine tune your new method and compare it to off-the-shelf originals.
- You also have to search hard for some way to discern if the "new" answer is the "right" answer. In engineering, the proof is the machine that you design. In social science, especially economics, you are likely to just be raising one more possibility in a context where everyones numbers may already be widely off the mark.
- You'll need some story to convince folks that you did not fail to see the forest because of the trees. It's a good idea to line up such a story as soon as you can.
- The sweet spot, where these caveats are minimized, is when the "more elaborate" method is the only method. Great when it happens; not too common.
