# 4.1 — Panel Data and Fixed Effects - Class Notes

## Contents

*Tuesday, November 10, 2020*

## Overview

Today, we begin our brief look at panel data, where we track multiple individuals over time. Panel data contains its own unique challenges, because it contains a time series component for every individual, giving potential sources of bias.

We now need to understand the third assumption about \(u_i\): no autocorrelation. The errors of our observations are likely going to be correlated within each individual and within each time period.

We can correct for these with a **fixed effects** model that isolates and absorbs some of that bias. In general, for a **two-way** fixed effects model:

\[\widehat{\text{Y}}_{it} = \beta_0+\beta_1 \text{X}_{1it} + \beta_2 \text{X}_{2it}+\alpha_{i} + \theta_{t} + \nu_{it}\]

Each observation is an individual \(i\) at time \(t\) (pay attention to the subscripts).

- Let \(Y_{it}\) be our dependent variable, and \(X_{1it}\) be the independent variable of interest. We would like to estimate the causal effect of \(X_{1it} \rightarrow Y_{it}\).
- \(\alpha_i\) is the
**group fixed-effect**. It absorbs all unobservable factors that**vary by group**but**don’t change over time**. - \(\tau_t\) is the
**time fixed-effect**. It absorbs all unobservable factors that**do not vary by group**but**change over time**. - Since the fixed effects
*do not*pick up factors that, we need to include other variables that might cause \(X_{1it}\) to be endogenous: hence, \(X_{2it}\)*both*vary by group*and*change over time - \(\nu_{it}\) is the remaining (random) error term (after we have pulled \(\alpha_i\) and \(\tau_t\) out of \(u_{it}\).

## Slides

## New Packages Mentioned

`plm`

for working with panel data and fixed effects

Note there are several other popular packages (that I have not extensively worked with), such as `fixest`

, `lfe`

.

## Assignments

Homework 5 answers are posted on that page.

## Live Class Session on Zoom

The live class Zoom meeting link can be found on Blackboard (see `LIVE ZOOM MEETINGS`

on the left navigation menu), starting at 11:30 AM.

If you are unable to join today’s live session, or if you want to review, you can find the recording stored on Blackboard via Panopto (see `Class Recordings`

on the left navigation menu).