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
- team winning percentage $P(n)$
- team winning percentage the previous year $P(n-1)$
- 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.