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412 Chapter 10 Regression with Panel Data

Key Concept The Fixed Effects Regression Assumptions

10.3 Yit = b1Xit + ai + uit, i = 1, c, n, t = 1, c, T,

where
1. uit has conditional mean zero: E(uit ͉ Xi1, Xi2, c, XiT, ai) = 0.
2. (Xi1, Xi2, c, XiT, ui1, ui2, c, uiT), i = 1, c, n are i.i.d. draws from their

joint distribution.
3. Large outliers are unlikely: (Xit, uit) have nonzero finite fourth moments.
4. There is no perfect multicollinearity.
For multiple regressors, Xit should be replaced by the full list X1,it, X2,it, c, Xk,it.

The third and fourth assumptions for fixed effects regression are analogous to the

third and fourth least squares assumptions for cross-sectional data in Key Concept 6.4.

Under the least squares assumptions for panel data in Key Concept 10.3, the

fixed effects estimator is consistent and is normally distributed when n is large.

The details are discussed in Appendix 10.2.

An important difference between the panel data assumptions in Key Concept

10.3 and the assumptions for cross-sectional data in Key Concept 6.4 is Assump-

tion 2. The cross-sectional counterpart of Assumption 2 holds that each observa-

tion is independent, which arises under simple random sampling. In contrast,

Assumption 2 for panel data holds that the variables are independent across enti-

ties but makes no such restriction within an entity. For example, Assumption 2

allows Xit to be correlated over time within an entity.
If Xit is correlated with Xis for different values of s and t—that is, if Xit is cor-

related over time for a given entity—then Xit is said to be autocorrelated (correlated
with itself, at different dates) or serially correlated. Autocorrelation is a pervasive

feature of time series data: What happens one year tends to be correlated with what

happens the next year. In the traffic fatality example, Xit, the beer tax in state i in
year t, is autocorrelated: Most of the time, the legislature does not change the beer

tax, so if it is high one year relative to its mean value for state i, it will tend to be high

the next year, too. Similarly, it is possible to think of reasons why uit would be auto-
correlated. Recall that uit consists of time-varying factors that are determinants of
Yit but are not included as regressors, and some of these omitted factors might be
autocorrelated. For example, a downturn in the local economy might produce