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Is there a formula for the covariance between regression slopes from different models, fitted to the same data? For example, if I have a finite and fixed sample, $S$, and models:

  1. $Y = b_1X$
  2. $Y = b_2X + b_3Z$

...is there a general formula to characterize the covariance between $b_1$ and $b_2$, similar to how we can characterize the covariance between $b_2$ and $b_3$? Can it be generalized to an arbitrary number of regression models, including $X$'s effect on $Y$ and many potential covariates?

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