3.9: Logarithmic Regression - Class Notes

Overview
Slides
R Practice Problems
Problem Set 4 Due TODAY

Thursday, November 21, 2019

Overview

Today, we finish up our view of nonlinear models with logarithmic models, which are more frequently used. We also discuss a few other tests and transformations to wrap up multivariate regression before we turn to panel data: standardizing variables to compare effect sizes, and joint hypothesis tests.

Interpretting logged variables can often be difficult to remember, so here I reproduce the tables that describe the interpretations of the marginal effect of $X \to Y$ , as well as some visual examples from the slides:

Model	Equation	Interpretation
Linear-Log	$Y = β_{0} + β_{1} l n (X)$	1% change in $X \to \frac{\hat{β_{1}}}{100}$ unit change in $Y$
Log-Linear	$l n (Y) = β_{0} + β_{1} X$	1 unit change in $X \to \hat{β_{1}} \times 100$ % change in $Y$
Log-Log	$l n (Y) = β_{0} + β_{1} l n (X)$	1% change in $X \to \hat{β_{1}}$ % change in $Y$

Hint: the variable that gets logged changes in percent terms, the variable not logged changes in unit terms

Linear-Log	Log-Linear	Log-Log

$\hat{Y_{i}} = \hat{β_{0}} + \hat{β_{1}} l n (X_{i})$	$l n (\hat{Y_{i}}) = \hat{β_{0}} + \hat{β_{1}} X_{i}$	$l n (\hat{Y_{i}}) = \hat{β_{0}} + \hat{β_{1}} l n (X_{i})$
$R^{2} = 0.65$	$R^{2} = 0.30$	$R^{2} = 0.61$