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.
Best Answer
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 $E\left[\log y\right].$
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?.