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10.5 The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression 413

layoffs and diminish commuting traffic, thus reducing traffic fatalities for 2 or more
years. Similarly, a major road improvement project might reduce traffic accidents
not only in the year of completion but also in future years. Such omitted factors,
which persist over multiple years, produce autocorrelated regression errors. Not
all omitted factors will produce autocorrelation in uit; for example, severe winter
driving conditions plausibly affect fatalities, but if winter weather conditions for a
given state are independently distributed from one year to the next, then this com-
ponent of the error term would be serially uncorrelated. In general, though, as long
as some omitted factors are autocorrelated, then uit will be autocorrelated.

Standard Errors for Fixed Effects Regression

If the regression errors are autocorrelated, then the usual heteroskedasticity-robust
standard error formula for cross-section regression [Equations (5.3) and (5.4)] is not
valid. One way to see this is to draw an analogy to heteroskedasticity. In a regres-
sion with cross-sectional data, if the errors are heteroskedastic, then (as discussed
in Section 5.4) the homoskedasticity-only standard errors are not valid because they
were derived under the false assumption of homoskedasticity. Similarly, if the errors
are autocorrelated, then the usual standard errors will not be valid because they
were derived under the false assumption of no serial correlation.

Standard errors that are valid if uit is potentially heteroskedastic and poten-
tially correlated over time within an entity are referred to as heteroskedasticity-
and autocorrelation-consistent (HAC) standard errors. The standard errors used
in this chapter are one type of HAC standard errors, clustered standard errors.
The term clustered arises because these standard errors allow the regression errors
to have an arbitrary correlation within a cluster, or grouping, but assume that the
regression errors are uncorrelated across clusters. In the context of panel data,
each cluster consists of an entity. Thus clustered standard errors allow for hetero-
skedasticity and for arbitrary autocorrelation within an entity, but treat the errors
as uncorrelated across entities. That is, clustered standard errors allow for hetero-
skedasticity and autocorrelation in a way that is consistent with the second fixed
effects regression assumption in Key Concept 10.3.

Like heteroskedasticity-robust standard errors in regression with cross-sectional
data, clustered standard errors are valid whether or not there is heteroskedasticity,
autocorrelation, or both. If the number of entities n is large, inference using clustered
standard errors can proceed using the usual large-sample normal critical values for
t-statistics and Fq, ∞ critical values for F-statistics testing q restrictions.

In practice, there can be a large difference between clustered standard errors
and standard errors that do not allow for autocorrelation of uit. For example, the
usual (cross-sectional data) heteroskedasticity-robust standard error for the BeerTax