# Regression with Panel Data

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Regression with Panel Data
Chapter 10 Regression with Panel Data

Regression with Panel Data (SW Chapter 10)

Notation for panel data

Panel data notation, ctd.

Why are panel data useful?

Example of a panel data set: Traffic deaths and alcohol taxes

U.S. traffic death data for 1982:

U.S. traffic death data for 1988

Why might there be higher more traffic deaths in states that have higher alcohol taxes?

These omitted factors could cause omitted variable bias.

Example #2: cultural attitudes towards drinking and driving:

Panel Data with Two Time Periods (SW Section 10.2)

Example: Traffic deaths and beer taxes

FatalityRate v. BeerTax:

Fixed Effects Regression (SW Section 10.3)

The regression lines for each state in a picture

Summary: Two ways to write the fixed effects model “n-1 binary regressor” form

Fixed Effects Regression: Estimation

1. “n-1 binary regressors” OLS regression

2. “Entity-demeaned” OLS regression

Entity-demeaned OLS regression, ctd.

Entity-demeaned OLS regression, ctd.

Example: Traffic deaths and beer taxes in STATA

Example, ctd. For n = 48, T = 7:

By the way… how much do beer taxes vary?

Regression with Time Fixed Effects (SW Section 10.4)

Time fixed effects only

Two formulations for time fixed effects

Time fixed effects: estimation methods

Combined entity and time fixed effects

The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression (SW Section 10.5 and App. 10.2)

A. Extension of LS Assumptions to Panel Data

Assumption #1: E(uit|Xi1,…,XiT,i) = 0

Assumption #2: (Xi1,…,XiT,Yi1,…,YiT), i =1,…,n, are i. i. d
Assumption #2: (Xi1,…,XiT,Yi1,…,YiT), i =1,…,n, are i.i.d. draws from their joint distribution.

Assumption #5: corr(uit,uis|Xit,Xis,i) = 0 for t  s

Assumption #5 in a picture:

What if Assumption #5 fails: so corr(uit,uis|Xit,Xis,i) 0?

B. Standard Errors

Sampling distribution of fixed effects estimator, ctd.

Sampling distribution of fixed effects estimator, ctd.

Case I: when uit, uis are uncorrelated

Case II: uit and uis are correlated – so Assumption 5 fails

Case II: Clustered Standard Errors

Comments on clustered standard errors, ctd.

Comments on clustered standard errors, ctd.

Implementation in STATA

Case II: treat uit and uis as possibly correlated

Fixed Effects Regression Results Dependent variable: Fatality rate

Summary: SEs for Panel Data in a picture:

Application: Drunk Driving Laws and Traffic Deaths (SW Section 10.6)

Drunk driving laws and traffic deaths, ctd.

The drunk driving panel data set

Why might panel data help?

Empirical Analysis: Main Results

Digression: extensions of the “n-1 binary regressor” idea

Summary: Regression with Panel Data (SW Section 10.7)