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For example:

GDP = B0_constant + B1_(GDP/Pop) + B2_X2 + B3_X3

Given that GDP appears on both left- and right-hand side this must certainly be problematic.

What is this particular situation called?

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  • $\begingroup$ Hi: I've never heard of such a scenario but that doesn't mean its wrong. $\endgroup$ – mlofton Jan 20 at 17:31
  • $\begingroup$ It almost always helps to include the stochastic part of the model explicitly. Usually such an equation means you suppose GDP does not equal the right hand side, but that it differs from the right hand side by a random amount $\epsilon.$ To be consistent, you must use the same expression for GDP on both sides. Ignoring the covariates $X_i$ for simplicity, your first attempt might look like $$GDP=b_0+b_1(b_0+b_1 GDP/Pop)+\epsilon)+\epsilon.$$This reveals how problematic your formulation is and provides clues for fixing it. $\endgroup$ – whuber Jan 20 at 17:55
  • $\begingroup$ So there's no simultaneity problem here? That's more what I was getting at, not the specific example per se. $\endgroup$ – StatsScared Jan 20 at 22:34
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This is called an opportunity for restructuring your regression equation. For example, why not do this:

$$ GDP/Pop = \beta_0 + \beta_2 X_2 + \beta_3 X_3 $$

Then you only have to multiply to the population to get GDP estimates.

OR

$$ log(GDP / Pop) = \beta_0 + \beta_2 X_2 + \beta_3 X_3 $$

which is effectively:

$$ log(GDP) = log(Pop) + \beta_0 + \beta_2 X_2 + \beta_3 X_3 $$

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    $\begingroup$ Unfortunately, all your examples are substantially different models of GDP. Why not propose the simplest algebraically equivalent reformulation, which (to be fully explicit) would be $GDP(1 - b_1/Pop)=b_0+b_2X_2+b_3X_3+\epsilon$? Although unusual, it's meaningful, interpretable, and fairly easy to fit. $\endgroup$ – whuber Jan 20 at 17:59
  • $\begingroup$ The problem with that re-arrangement is that you still have to estimate the b1 coefficient. You can take that over to the right side, but then you are fitting a non-linear model with coefficients that might not be all independently estimable in the non-linear model. $\endgroup$ – R Carnell Jan 20 at 19:09
  • $\begingroup$ Actually, it's worse than that: the best choice of $b_1$ is one that will make $1-b_1/Pop$ small, which is probably not what was intended. Up to some error, this will cause the two GDP terms on either side of the original equation almost to cancel. I doubt this was the intention of the original model. This calls into question any efforts to manipulate the original model until we can learn more from the OP about what they are trying to accomplish. $\endgroup$ – whuber Jan 20 at 19:25

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