Two interpretations from modeling coins in people’s pockets? I’m deciding on using either a multiple regression model or a bivariate regression model to help identify the relationship x1 and x2 have with my dependent variable y.
Please consider the below scenario:
“Collect a dataset based on the coins in peoples pockets, the y variable/response is the total value of the coins, the variable x1 is the total number of coins and x2 is the number of coins that are not quarters (or whatever the largest value of the common coins are for the local).
It is easy to see that the regression with either x1 or x2 would give a positive slope, but when incuding both in the model the slope on x2 would go negative since increasing the number of smaller coins without increasing the total number of coins would mean replacing large coins with smaller ones and reducing the overall value (y).”
My question: In reality, as x2 rises so does y. Only when we “hold x1 constant” do we think of x2 being negatively related to y.
If x1 and x2 were financial times series data, and we wanted to simply understand the individual relationships between x1 and y, and x2 and y, then wouldn’t we opt to use the interpretations of the separate regressions?
 A: If you had a third variable $x_3$ as the number of the largest coin (so $x_1=x_2+x_3$) and did a regression of $y$ on $x_2$ and $x_3$ together, getting something like
$$y \approx \gamma_0+\gamma_2x_2+\gamma_3x_3$$
you might reasonably expect the coefficient $\gamma_3$ to be large and positive and the coefficient on $\gamma_2$ to be smaller (and positive if the world is made up of coin hoarders and coin users, but negative if the world is made up of people carrying an almost constant weight of coins as people with many small coins would then tend to have fewer large coins and a smaller total value - it is not immediately obvious to me that your description of reality is correct and it would need to be checked empirically).
If instead we look at your original scenario and you do the regression on $x_1$ and $x_2$ together, getting something like $$y \approx \beta_0+\beta_1x_1+\beta_2x_2$$ then (if you use ordinary least squares) you might expect $\beta_0=\gamma_0$, $\beta_1=\gamma_3$ and $\beta_2=\gamma_2-\gamma_3$ and so $\beta_2$ is likely to be negative, about as large as $\beta_1$ but the opposite sign
I would agree with you that $\beta_2$ is not particularly informative.  I would not say that simply regressing $y$ on $x_2$ is meaningful either.  Instead, $\beta_1=\gamma_3$ is valuable information and $\beta_1+\beta_2=\gamma_2$ is interesting additional information.
