4.1 — Panel Data and Fixed Effects — Class Content

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 both vary by group and change over time, we need to include other variables that might cause \(X_{1it}\) to be endogenous: hence, \(X_{2it}\)
• \(\nu_{it}\) is the remaining (random) error term (after we have pulled \(\alpha_i\) and \(\tau_t\) out of \(u_{it}\).

Readings

• Ch. 8.1—8.4 in Bailey, Real Econometrics

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. Note while in going through the slides, you can type h to see a special list of viewing options, and type o for an outline view of all the slides.

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)!

Download as PDF

Assignments

New Packages Mentioned

• fixest for working with fixed effects models

• for working with panel data and fixed effects

Note there are several other popular packages for working with panel data and fixed effects. I used to use (and teach) plm but now find fixest more efficient. There is also lfe which I have not worked with.