<p><a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3605606" rel="nofollow noreferrer">Bartlett and Partnoy (BP) (2020)</a> show that OLS with natural log dependent variables that are ratios must include the ln(denominator) on the RHS or they must assume that the numerator and denominator have a perfectly linear relationship (see pages 24-28). This paper, by law professors, is generalizing the ratio problem from a specific ratio problem: <a href="https://scholarship.law.vanderbilt.edu/vlr/vol73/iss2/1/" rel="nofollow noreferrer">The misuse of Tobin's Q</a>.</p>
<p>If we want unbiased <span class="math-container">$\beta_1$</span> in <span class="math-container">$(\text{E}1)$</span>, we have to assume a perfectly linear relationship between the numerator and denominator:</p>
<p><span class="math-container">$$
\ln\left(\frac{GDP_{it}}{population_{it}}\right) = \beta_0 + \beta_1 treatment_{it} + \epsilon_{it} 
\tag{E1}
$$</span>
Because:
<span class="math-container">$$
\ln\left(\frac{GDP_{it}}{population_{it}}\right) = \ln(GDP_{it}) - \ln(population_{it})
\tag{E2}
$$</span></p>
<p>Thus, there is an implicit assumption that the coefficient for <span class="math-container">$\ln(population_{it}) = 1$</span> in:
<span class="math-container">$$
\ln(GDP_{it}) = \beta_0 + \beta_1 treatment_{it} + 1 \ln(population_{it}) + \epsilon_{it} 
\tag{E3}
$$</span></p>
<p>To the degree that <span class="math-container">$\beta_2 \neq 1$</span> in <span class="math-container">$(\text{E}4)$</span>, OLS will generate biased <span class="math-container">$\beta_1$</span> in <span class="math-container">$(\text{E}1)$</span>, and they show it can be severe. The solution is to correct the not perfectly linear relationship by putting the <span class="math-container">$\ln(denominator_{it})$</span> on the RHS:</p>
<p><span class="math-container">$$
\ln\left(\frac{GDP_{it}}{population_{it}}\right) = \beta_0 + \beta_1 treatment_{it} + \beta_2 \ln(population_{it}) + \epsilon_{it} 
\tag{E4}
$$</span></p>
<p>Their paper is discussing a different ratio as DV, but &quot;the ratio problem&quot; should apply to any ratio that does not have a perfectly linear relationship between numerator and denominator (there are additional issues when the DV is not logged, actually). I'm currently observing this in a paper I'm writing that started with <span class="math-container">$(\text{E}1)$</span> with two-way fixed-effects, which seemed like the intuitive choice; however, I get substantially different results with <span class="math-container">$(\text{E}4)$</span>, and indeed <span class="math-container">$\beta_2 \neq 1$</span> in my model. I was unable to find any economics methods literature that discusses this issue, but I found (obviously) dozens of paper that use equations like <span class="math-container">$(\text{E}1)$</span> along with other controls.</p>
<p>Is it true that <span class="math-container">$(\text{E}1)$</span> should exhibit biased <span class="math-container">$\beta_1$</span> when <span class="math-container">$\beta_2 \neq 1$</span>? If so, doesn't this potentially blow up results from all the research using <span class="math-container">$(\text{E}1)$</span>? Does anyone know of econ methods literature that deals with this issue?</p>
<p>Is there any reason to think that <span class="math-container">$(\text{E}4)$</span> would not be necessary over <span class="math-container">$(\text{E}1)$</span> in a two-way fixed effects panel model?</p>
<p><strong>Edit:</strong> I wrote a <a href="https://stats.stackexchange.com/questions/633748/natural-logged-dependent-variable-and-the-ratio-problem-with-two-way-fixed-effec">stats version of this question</a> that has more detail (but less direct discussion of economics).</p>

CC BY-SA 4.0
2023-12-15T11:42:38.153Z
Interesting! My gut feeling is, it depends on how you treat your variables from the subject-matter point of view. If you want to model GDP per capita (instead of GDP and population), you treat it a a fundamental unit of analysis and you do not care how it could be defined by using other variables (namely, GDP and population). Then this is a non-issue. If, on the other hand, you want to model GDP and population and their ratio is not a fundamental unit of analysis, using log(GDP per capita) as the dependent variable might cause some trouble. But this is just my gut feeling.
Richard Hardy
CC BY-SA 4.0
2023-12-15T18:08:52.547Z
That's a helpful way of looking at it. Certainly, log(GDP per capita) is "a thing" or concept of its own that is useful in comparing economies. I think the issue might arise if one wants unbiased $\beta_1$ for each of two samples, and these samples would each generate a distinct $\beta_2$. It's almost like the respective $\beta_1$ coefficients without $\beta_2$ are predicting slightly different concepts of log(GDP per capita). Gut feeling: seems like you would you get more accurate predictions of log(GDP per capita) with $\beta_1$ across distinct samples by including $\beta_2$ in each sample.
dcoy
CC BY-SA 4.0
2023-12-15T18:20:19.373Z
What two samples do you have in mind?
Richard Hardy
CC BY-SA 4.0
2023-12-15T20:43:05.783Z
In my case, I have small border municipalities that experienced a sudden, exogenous mass migration, plus a matched sample. I suspect Ln(GDP) and Ln(pop) are more non-linear in my sample, because each city gets a minimum federal transfer to local GDP to govern that is enough to move the dial in small cities. As cities get larger, this transfer makes up an increasingly small proportion of local GDP. Thus, my small city sample has $\beta_2 = 0.88$, but I'd expect a hypothetical sample of larger cities to be closer to $1$. Just an idea.
dcoy
CC BY-SA 4.0
2023-12-18T15:06:34.060Z
Seems like the specification (E4) will cause an endogeneity problem as every residual correlated with $\ln(population)$ (like measurement error) will affect both left hand and right hand sides.  (A bit the same issue as when estimating Engel curves by regressing demand shares on income)
tdm
CC BY-SA 4.0
2023-12-19T16:55:12.170Z
@tdm Thanks, this is at the heart of the question. Is it causing measurement error or eliminating it? (E4) gives the same $\beta_1$ as (E5) linked in new edit: $Ln(GDP_{it}) = \beta_0 + \beta_1treat_{it} + \beta_2Ln(pop_{it}) + \epsilon_{it}$, which does not have $Ln(pop)$ on the LHS. But one gets a different $\beta_1$ for a model without $Ln(pop)$ on either side in any circumstance that adding it to the RHS would have $\beta_2\neq1$. This makes it seem like its accounting for measurement error that comes from non-exact-linearity in $Ln(GDP)$ and $Ln(pop)$. Feel free to write an answer.
dcoy
<macroeconomics><gdp><economic-growth><regression><paneldata>
Controlling for Log population with Log GDP per Capita outcome in OLS
dcoy
# Controlling for Log population with Log GDP per Capita outcome in OLS

(Asked by: username_0 on 2023-12-15)

[Bartlett and Partnoy (BP) (2020)](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3605606) show that OLS with natural log dependent variables that are ratios must include the ln(denominator) on the RHS or they must assume that the numerator and denominator have a perfectly linear relationship (see pages 24-28). This paper, by law professors, is generalizing the ratio problem from a specific ratio problem: [The misuse of Tobin's Q](https://scholarship.law.vanderbilt.edu/vlr/vol73/iss2/1/).

If we want unbiased $\beta\_1$ in $(\text{E}1)$, we have to assume a perfectly linear relationship between the numerator and denominator:

$$
\ln\left(\frac{GDP\_{it}}{population\_{it}}\right) = \beta\_0 + \beta\_1 treatment\_{it} + \epsilon\_{it}
\tag{E1}
$$
Because:
$$
\ln\left(\frac{GDP\_{it}}{population\_{it}}\right) = \ln(GDP\_{it}) - \ln(population\_{it})
\tag{E2}
$$

Thus, there is an implicit assumption that the coefficient for $\ln(population\_{it}) = 1$ in:
$$
\ln(GDP\_{it}) = \beta\_0 + \beta\_1 treatment\_{it} + 1 \ln(population\_{it}) + \epsilon\_{it}
\tag{E3}
$$

To the degree that $\beta\_2 \neq 1$ in $(\text{E}4)$, OLS will generate biased $\beta\_1$ in $(\text{E}1)$, and they show it can be severe. The solution is to correct the not perfectly linear relationship by putting the $\ln(denominator\_{it})$ on the RHS:

$$
\ln\left(\frac{GDP\_{it}}{population\_{it}}\right) = \beta\_0 + \beta\_1 treatment\_{it} + \beta\_2 \ln(population\_{it}) + \epsilon\_{it}
\tag{E4}
$$

Their paper is discussing a different ratio as DV, but "the ratio problem" should apply to any ratio that does not have a perfectly linear relationship between numerator and denominator (there are additional issues when the DV is not logged, actually). I'm currently observing this in a paper I'm writing that started with $(\text{E}1)$ with two-way fixed-effects, which seemed like the intuitive choice; however, I get substantially different results with $(\text{E}4)$, and indeed $\beta\_2 \neq 1$ in my model. I was unable to find any economics methods literature that discusses this issue, but I found (obviously) dozens of paper that use equations like $(\text{E}1)$ along with other controls.

Is it true that $(\text{E}1)$ should exhibit biased $\beta\_1$ when $\beta\_2 \neq 1$? If so, doesn't this potentially blow up results from all the research using $(\text{E}1)$? Does anyone know of econ methods literature that deals with this issue?

Is there any reason to think that $(\text{E}4)$ would not be necessary over $(\text{E}1)$ in a two-way fixed effects panel model?

**Edit:** I wrote a [stats version of this question](https://stats.stackexchange.com/questions/633748/natural-logged-dependent-variable-and-the-ratio-problem-with-two-way-fixed-effec) that has more detail (but less direct discussion of economics).

**Tags:** macroeconomics, gdp, economic-growth, regression, paneldata

> **Comments:**
> **username_1**: Interesting! My gut feeling is, it depends on how you treat your variables from the subject-matter point of view. If you want to model GDP per capita (instead of GDP and population), you treat it a a fundamental unit of analysis and you do not care how it could be defined by using other variables (namely, GDP and population). Then this is a non-issue. If, on the other hand, you want to model GDP and population and their ratio is not a fundamental unit of analysis, using log(GDP per capita) as the dependent variable might cause some trouble. But this is just my gut feeling.  
> **username_0**: That's a helpful way of looking at it. Certainly, log(GDP per capita) is "a thing" or concept of its own that is useful in comparing economies. I think the issue might arise if one wants unbiased $\beta_1$ for each of two samples, and these samples would each generate a distinct $\beta_2$. It's almost like the respective $\beta_1$ coefficients without $\beta_2$ are predicting slightly different concepts of log(GDP per capita). Gut feeling: seems like you would you get more accurate predictions of log(GDP per capita) with $\beta_1$ across distinct samples by including $\beta_2$ in each sample.  
> **username_1**: What two samples do you have in mind?  
> **username_0**: In my case, I have small border municipalities that experienced a sudden, exogenous mass migration, plus a matched sample. I suspect Ln(GDP) and Ln(pop) are more non-linear in my sample, because each city gets a minimum federal transfer to local GDP to govern that is enough to move the dial in small cities. As cities get larger, this transfer makes up an increasingly small proportion of local GDP. Thus, my small city sample has $\beta_2 = 0.88$, but I'd expect a hypothetical sample of larger cities to be closer to $1$. Just an idea.  
> **username_2**: Seems like the specification (E4) will cause an endogeneity problem as every residual correlated with $\ln(population)$ (like measurement error) will affect both left hand and right hand sides.  (A bit the same issue as when estimating Engel curves by regressing demand shares on income)  
> **username_0**: @username_2 Thanks, this is at the heart of the question. Is it causing measurement error or eliminating it? (E4) gives the same $\beta_1$ as (E5) linked in new edit: $Ln(GDP_{it}) = \beta_0 + \beta_1treat_{it} + \beta_2Ln(pop_{it}) + \epsilon_{it}$, which does not have $Ln(pop)$ on the LHS. But one gets a different $\beta_1$ for a model without $Ln(pop)$ on either side in any circumstance that adding it to the RHS would have $\beta_2\neq1$. This makes it seem like its accounting for measurement error that comes from non-exact-linearity in $Ln(GDP)$ and $Ln(pop)$. Feel free to write an answer.  

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## 0 answers

