4.4 — Nonlinearities and Variable Transformations — Class Content
Problem Set 5 is due on Monday November 28.
Overview
This week we will cover non-linear models, which we anticipated last week with an interaction effect between two continuous variables: \(X_1 \times X_2\), where the marginal effect of \(X_1 \rightarrow Y\) is dependent on the value of \(X_2\).
We cover two types of nonlinear models. First, we cover polynomial models, where we include higher-order \(X\) variables to explain a non-linear effect.
Next we will cover logarithmic models, which are more frequently used — as well as a few final tests and transformations for multivariate regression before we turn to panel data (Unit IV) next week: 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 \(X \rightarrow Y\), as well as some visual examples from the slides:
| Model | Equation | Interpretation |
|---|---|---|
| Linear-Log | \(Y=\beta_0+\beta_1 \mathbf{ln(X)}\) | 1% change in \(X \rightarrow \frac{\hat{\beta_1}}{100}\) unit change in \(Y\) |
| Log-Linear | \(\mathbf{ln(Y)}=\beta_0+\beta_1X\) | 1 unit change in \(X \rightarrow \hat{\beta_1}\times 100\)% change in \(Y\) |
| Log-Log | \(\mathbf{ln(Y)}=\beta_0+\beta_1\mathbf{ln(X)}\) | 1% change in \(X \rightarrow \hat{\beta_1}\)% 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 |
|---|---|---|
 |
 |
 |
| \(\hat{Y_i}=\hat{\beta_0}+\hat{\beta_1}\mathbf{ln(X_i)}\) | \(\mathbf{ln(\hat{Y_i})}=\hat{\beta_0}+\hat{\beta_1}X_i\) | \(\mathbf{ln(\hat{Y_i})}=\hat{\beta_0}+\hat{\beta_1}\mathbf{ln(X_i)}\) |
| \(R^2=0.65\) | \(R^2=0.30\) | \(R^2=0.61\) |
After we finish, we will do another set of practice problems in R., and you will be given your final graded HW 5 to work on this material.
Readings
- Ch. 7 in Bailey, Real Econometrics
R Practice
This week you will be working on R practice problems on nonlinear data. Answers will be posted later on that page.
Assignments
Problem Set 5 Due Mon Nov 28
Problem Set 5 is due by the end of the day on Wednesday, November 28.
Slides
Below, you can find the slides in two formats. Clicking the image will bring you to the html version of the slides in a new tab. The lower button will allow you to download a PDF version of the slides.
I suggest printing the slides beforehand and using them to take additional notes in class (not everything is in the slides)!
