When resampling the player stats, are you taking into account the dependence structure of box score stats? E.g., if the sampled 3P% is higher than usual, then the sampled PPG should tend to be higher as well.
Also, is the errors-in-variables bias being accounted for at all? Since each player has errors modeled into their counting stats, the estimates of the coefficients are naturally going to get shrunk towards zero. So not only does this affect the width of the intervals, but it also biases the point estimates.
Thanks for the reply and really great post. I wonder how much the intervals would benefit from a dependence structure for the sampling. Under the Gaussian assumption it's easy to model the dependence since you only need an estimate of the correlation matrix. But if you assume different marginal distributions you can still flexibly model the dependence structure with something called "copulas". Or, even just something nonparametric like bootstrapping the outcomes from the games themself, since they will implicitly carry that dependence.
For the estimates getting shrunk towards zero, it's just a consequence of your covariates having noise themselves. This probably isn't easy to parse over comment, but the idea is this:
if you are doing a simple linear regression of y onto x, then you have
y = b0 + b1x + e, (1)
where e is your error term. Then your estimate for the coefficient of x is
b1 = Cov(x, y) / Var(x). (2)
Assuming you meet OLS assumptions, b1 is unbiased. But if you assume that the covariates themselves have error such that
x* = x + n, (3)
where x* is now a "noisy" version of x, and n is now some kind of error, then you have
y = a0 + a1x* + e, (4)
and now your estimate for the coefficient is
a1 = Cov(x*, y) / Var(x*). (5)
But because of (3), you have that
Var(x*) = Var(x) + Var(n), (6)
and so your estimate is (if you assume n is independent of e):
There are ways to adjust for this, depending on the scenario. It probably doesn't affect much, but it could be another small improvement down the line.
Again, thanks for such a great free post. Really like to read your stuff when I get the chance.
When resampling the player stats, are you taking into account the dependence structure of box score stats? E.g., if the sampled 3P% is higher than usual, then the sampled PPG should tend to be higher as well.
Also, is the errors-in-variables bias being accounted for at all? Since each player has errors modeled into their counting stats, the estimates of the coefficients are naturally going to get shrunk towards zero. So not only does this affect the width of the intervals, but it also biases the point estimates.
To the first question, I seperate out all the stats so none of them are dependent. For example, I don't include PPG, but just 2s/3s/FTs
As for coefficient estimates getting shrunk towards zero - that's not my experience
Thanks for the reply and really great post. I wonder how much the intervals would benefit from a dependence structure for the sampling. Under the Gaussian assumption it's easy to model the dependence since you only need an estimate of the correlation matrix. But if you assume different marginal distributions you can still flexibly model the dependence structure with something called "copulas". Or, even just something nonparametric like bootstrapping the outcomes from the games themself, since they will implicitly carry that dependence.
For the estimates getting shrunk towards zero, it's just a consequence of your covariates having noise themselves. This probably isn't easy to parse over comment, but the idea is this:
if you are doing a simple linear regression of y onto x, then you have
y = b0 + b1x + e, (1)
where e is your error term. Then your estimate for the coefficient of x is
b1 = Cov(x, y) / Var(x). (2)
Assuming you meet OLS assumptions, b1 is unbiased. But if you assume that the covariates themselves have error such that
x* = x + n, (3)
where x* is now a "noisy" version of x, and n is now some kind of error, then you have
y = a0 + a1x* + e, (4)
and now your estimate for the coefficient is
a1 = Cov(x*, y) / Var(x*). (5)
But because of (3), you have that
Var(x*) = Var(x) + Var(n), (6)
and so your estimate is (if you assume n is independent of e):
a1 = Cov(x*, y) / Var(x*) = Cov(x, y) / [ Var(x) + Var(n) ], (7)
which is necessarily closer to zero than b1.
There are ways to adjust for this, depending on the scenario. It probably doesn't affect much, but it could be another small improvement down the line.
Again, thanks for such a great free post. Really like to read your stuff when I get the chance.