Should I scale my variables when using a log-log regression?

Question

I am running a log-log OLS regression in Stata.

My dependent variable is the log of house price, and my regressors of interest are, the log of distance to employment, and the log of jobs created.

My coefficients are very small but statistically significant (e.g. increasing employment by 1% increases house prices by 0.00013%).

I was told that the log-log specification can come up with weirdly small coefficients, and so I should scale my variables by ten to resolve this issue.

My regressions take up to 24 hours to run because of the number of observations, so before I waste a day, I wanted to see if this scaling logic is correct?

Answer 1

Scaling a logged variable will not change the slope coefficients, but it will change the constant. To see this suppose we scale $x_1$ by a constant $c$, \begin{align} \log(y) &= \beta_0 + \beta_1\log(c x_1) + \varepsilon \\ &= \beta_0 + \beta_1 \log(c) + \beta_1\log(x_1) + \varepsilon \\ &= \alpha_0 + \beta_1\log(x_1) + \varepsilon \end{align} where $\alpha_0 = \beta_0 + \beta_1 \log(c)$ is the new intercept term.