Interpret regression coefficients when independent variable is a ratio

Question

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.

Answer 1

Ordinarily, we interpret coefficients in terms of how the expected value of the response should change when we effect tiny changes in the underlying variables. This is done by differentiating the formula, which is

$$E\left[\log Y\right] = \beta_0 + \beta_1 x_1 + \beta_2\left(\frac{x_3}{x_1}\right).$$

The derivatives are

$$\frac{\partial}{\partial x_1} E\left[\log Y \right] = \beta_1 - \beta_2\left( \frac{x_3}{x_1^2}\right)$$

and

$$\frac{\partial}{\partial x_3} E\left[\log Y \right] = \beta_2 \left(\frac{1}{x_1}\right).$$

Because the results depend on the values of the variables, there is no universal interpretation of the coefficients: their effects depend on the values of the variables.

Often we will examine these rates of change when the variables are set to average values (and, when the model is estimated from data, we use the parameter estimates as surrogates for the parameters themselves). For instance, suppose the mean value of $x_1$ in the dataset is $2$ and the mean value of $x_3$ is $4.$ Then a small change of size $\mathrm{d}x_1$ in $x_1$ is associated with a change of size

$$\left(\frac{\partial}{\partial x_1} E\left[\log Y \right] \right)\mathrm{d}x_1 = (\beta_1 - \beta_2(4/2^2))\mathrm{d}x_1 = (\beta_1 - \beta_2)\mathrm{d}x_1.$$

Similarly, changing $x_3$ to $x_3+\mathrm{d}x_3$ is associated with change of size

$$\left(\frac{\partial}{\partial x_3} E\left[\log Y \right] \right)\mathrm{d}x_3 = \left(\frac{\beta_{2}}{2}\right)\mathrm{d}x_3$$

in $E\left[\log y\right].$

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For more examples of these kinds of calculations and interpretations, and to see how the calculations can (often) be performed without knowing any Calculus, visit How to interpret coefficients of angular terms in a regression model?, How do I interpret the coefficients of a log-linear regression with quadratic terms?, Linear and quadratic term interpretation in regression analysis, and How to interpret log-log regression coefficients for other than 1 or 10 percent change?.

Answer 2

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 then $\frac{\partial Y}{\partial x}= \beta Y$ 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 doesn't make sense, then increase it by a smaller amount, say 0.01, then 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 measured with error, which would need some elaboration to take into account.

Answer 3

I suppose you could interpret the numerator and denominator with ratio.

If your fraction increases by 1 unit, it means your numerator (x3) increased, if you fraction decreases by 1 unit, it means your denominator (x1) decreased and that would be its effect on dependent variable.

Answer 4

Just as in linear regression it is common to view nonlinear factors such as $x_1^2$ or $x_1 \cdot x_2$ as individual covariates, similarly there is no reason why $x_3/x_1$ can't be a legitimate covariate.

As long as your response variable is indeed linear in that ratio, then that is simply how your system behaves. Suppose you had a model that was linear in body mass index (BMI). That is a ratio commonly used in medicine (although it is highly suspect). Or, the HDL ratio for cholesterol. Or pressure in physics (F/A). If that's the way the system behaves, then that is just how it is.

It is up to you as the modeler to know why that ratio is important in your model, not up to the regression table output to tell you (as in "how to interpret" it).