# How do I incorporate time-variability in a pair trading framework?

Recently I have been looking at pair trading strategies from a cointegration perspective, as described in chapter 5 of Carol Alexander's Market Risk Analysis volume 2. As most quantitative finance texts the science is well explained, but the description of applications is a bit on the light side.

Theoretically it's pretty straightforward and easy to run the tests for a given time period to see whether a certain pair is likely to be conintegrated or not. To apply the theory to an actual pairs trade however, I would like to add the time dimension to my parameters.

If I simply model the spread as $residuals = y - \alpha - \beta x$, the first thing I'd like to know is the mean and variance of the residuals. In order to setup my bid/ask limits I must have a mean and some measure of the variance. In an ideal case the two would be stable, but how can I quantify this and incorporate the information into the model? Also, how can I make this analysis structural, rather than having to depend on subjective eye-balling of data?

Could anyone point me in the right direction here? My guess is that I should take a look at regime switching models. But since that topic is unknown to me, I'd appreciate very much if someone could give some pointers so I can avoid the worst pitfalls.

EDIT: Perhaps I wasn't clear enough, but my question is not how to do the tests, it's how can do can I quantify the stability of the parameters, i.e., the mean and variance of the spread?

-

## 2 Answers

If you're using R and would like a few examples, here's Paul Teetor's website:

http://quanttrader.info/public/

And, Ernie Chan's website:

http://epchan.blogspot.com/

Energy pairs:
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9506276845932007, "perplexity_flag": "head"}