Functional Form, why we take logs and read coefficients as percentages
Why econometricians take logs: a log-log slope is an elasticity (percent for percent), a log-level slope a semi-elasticity, and skewed positive variables like wages and prices fit better.
A 3 minute 17 second animated lesson on functional form, why regressions so often take the natural log of a variable and how that changes what a coefficient means. Built for ECON3006 and FIN301 students at Western Sydney University and for anyone learning to read regression output.
It starts from elasticity, the response of one variable to another measured in percentages rather than units, the percent change in Y per percent change in X. A log-log model makes the slope exactly that elasticity. A log-level model makes the slope a semi-elasticity, so a one-unit rise in X gives about beta times 100 percent change in Y, which is the natural way to say a year of schooling raises wages by about 8 percent rather than by a fixed dollar amount.
It closes on why logs are everywhere in applied work. Variables like wages, prices and firm size are positive and right-skewed, so taking logs pulls in the long tail, steadies the error term, and turns a constant-percentage effect into a straight line in the parameters. Pair the video with the Atlas concept page for the formulas, a worked example, a quiz, and citations.