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I'm performing a regression where the dependent variable is a ratio, and the explanatory variable is the denominator of that ratio. In other words, I'm normalizing the dependent variable by $X$ and including $X$ as an explanatory variable:

Dependent variable: $Y/X$

Explanatory variable: $X$

For example, I might be interested in whether countries with more people tend to have a higher proportion of women, in which case $X$ is total population and $Y$ is the number of women. Or I might be interested in how the number of dogs per cat varies with the size of the cat population in a country, and so on.

$X$ appears in both the left hand side and right hand side of the regression equation, and I want to avoid any issues that could arise from this (e.g. mechanical correlations).

Should I be worried about mechanical correlations (either generally or in certain situations)? If so, what kinds of situations, and how can I remedy them? And what other issues might arise?

wwl
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1 Answers1

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So you want to estimate $$y_i = \beta_0 + \beta_1 \cdot x_i + u_i$$ where $y_i = y_i^*/x_i$ and $y_i^*$ is the "original" value (like number of women), not the relative value you are interested in. Is this correct?

I do not see any general problems. Your $u_i$ may be heteroscedastic, it may not be normal, $u$ may be correlated with $x$... All well known issues.

If you multiply both sides by $x_i$ you get $$y_i^* = \beta_0 \cdot x_i + \beta_1 \cdot x_i^2 + x_i \cdot u_i$$ This hints that if your relative $y_i$ is defined as above, your "original" model should not have an intercept, and it should have a quadratic term. The first is not particularly common model but here it makes sense (if population is 0 the number of women is 0 too...). If you assume $u \sim i.i.d$, your original model should have linear heteroscedasticity in x.

I would be suspicious though if you have small $x$ values with potentially large measurement errors. These will be blown up here.

Happy to be corrected if anyone can spot something I cannot see...

Ott Toomet
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