# Interpret regression coefficients when independent variable is a ratio

I am running an OLS regression of the form

$$\log\left(Y\right)=x_0 + \log\left(x_1\right)\beta_1+x_2\beta_2 + \epsilon$$

where the dependent variable Y and some independent variables are log transformed. Their interpretation in terms of %changes is straightforward.

However, I have one covariate $x_2$ which is a fraction $\in [0,1]$. Infact, it's a ratio of $x_3$ and $x_1$ i.e. $x_2 =\frac{x_3}{x_1}$. Note that $x_1$ by itself is in the model but $x_3$ is not.

I was wondering how would I interpret its coefficient, since one unit change in it would not make much sense in terms of interpretation? For instance, what if $\beta_2$=0.3. Any help is greatly appreciated.

## 1 Answer

For a more useful answer you should tell us more about your real application. As the question only seems to be about the role of a ratio variable $x \in [0,1]$, so I simplify the question by removing the other parts of the model. It then becomes: $$\log Y = \beta_0 + \beta x + E$$ which in multiplicative form becomes $$Y = C e^{\beta x} E$$ where $C=e^{\beta_0}, E = e^\epsilon$ . The derivative of $Y$ with respect to $x$ is the $\frac{\partial Y}{\partial x}= \beta Y$ so that $\beta = \frac{\partial Y}{\partial x} / Y$. The fact that $x$ is a ratio plays no part, the interpretation is the same. You seem to be preoccupied with the fact that increasing a ratio with 1 does'nt make sense, then increase it by a smaller amount, say 0.01, the the relative increase in $Y$ (that is, increase as a proportion of $Y$ is $0.01 \beta$.

There might be other issues unrelated to this problem with interpretation, that is, if your proportion is based on few cases, that is, of the form $z/n$ where $z$ is a count and $n$ is small, it will be measuered with error, which would need some elaboration to take into account.

• Thanks! I just edited the question a little. The other consideration I have is that the fraction covariate is a ratio of another covariate and something unrelated, as I explain in the question. I agree that I can interpret it as 0.01 change and so on, but 0.01 change in ratio could mean that either I could increase the numerator or decrease the denominator. However, in my case decreasing the denominator is not an option. So, can I just interpret the 0.01 change as an increase in numerator, or do I need to transform the variables in some other way than a ratio? – Blade Runner May 12 '16 at 13:44
• No, you can use it as a ratio. The only thing the model sees is the ratio, it dosn't know about numerators or denominators! If you want a more direct interpretation, multiply the ratio by 100 to convert it into a %, then the parameter is directly the effect og a 1% change. – kjetil b halvorsen May 12 '16 at 14:21