# How to make sense of “luck” in a multilinear regression?

In baseball statistics, there is a statistic called "luck" which is the difference between a team's win-loss record and their Pythagorean win-loss record. This statistic is supposed to measure how lucky or unlucky a team was to win however many games they did in a season.

Suppose one has a big data set which, for each year n, includes

1. team winning percentage $P(n)$
2. team winning percentage the previous year $P(n-1)$
3. team luck the previous year $L(n-1)$

and wants to create a linear regression model using $P(n-1)$ and $L(n-1)$ to estimate $P(n)$.

There's no apparent relationship between $L(n-1)$ and $P(n)$, but it seems as though we could use $L(n-1)$ in conjunction with $P(n-1)$ to better predict $P(n)$ based on how "flukey" $P(n-1)$ was and in what way.

So, the question is, how could one incorporate a luck-type measure into a linear regression model like I've discussed? I'm not concerned with this particular luck-type measure, but rather any measure which does something similar to what this one is supposed to do.

• Intuitively, if $L$ truly is "luck" it shouldn't be able to predict anything! As such, your hope is that $L(n-1)$ captures deviations (of team record relative to run production) that were not due to luck but perhaps to other factors, and that such factors may persist to time $n$. Therefore your best hope is to see whether $L(n-1)$ is useful for predicting $L(n)$ (rather than $P(n)$). – whuber Mar 2 '12 at 23:02
• @mike - I don't understand what your obstacle/difficulty is. If you think you "could use L(n−1) in conjunction with P(n−1) to better predict P(n)" then I'd have expected you to try that in a straightforward, 2-predictor regression (checking some diagnostics, probably needless to say). If you do, please share the results! – rolando2 Mar 3 '12 at 17:14

It's not clear to me that luck buys you much, but I believe you could regress $P(n)$ on $P(n-1)$, $L(n-1)$, and $P(n-1) \times L(n-1)$. I think you'd want to scale $L$ so that $0 < L < 2$ with a mean of 1.
At least that's my novice opinion. When using diagnostics to see if your coefficients are statistically significant, I believe that the fact that you're using a time series will cause significance to be over-stated (because presumably $P(n)$ is positively correlated with $P(n-1)$), and I'm not sure how you compensate for that.