# 4.4 — Nonlinearities and Variable Transformations — Class Content

## 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)!