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10.5 The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression 411

10.5 The Fixed Effects Regression Assumptions

and Standard Errors for Fixed Effects
Regression

In panel data, the regression error can be correlated over time within an entity.
Like heteroskedasticity, this correlation does not introduce bias into the fixed
effects estimator, but it affects the variance of the fixed effects estimator and
therefore it affects how one computes standard errors. The standard errors for
fixed effects regressions reported in this chapter are so-called clustered standard
errors, which are robust both to heteroskedasticity and to correlation over time
within an entity. When there are many entities (when n is large), hypothesis tests
and confidence intervals can be computed using the usual large-sample normal
and F critical values.

This section describes clustered standard errors. We begin with the fixed
effects regression assumptions, which extend the least squares regression assump-
tions to panel data; under these assumptions, the fixed effects estimator is asymp-
totically normally distributed when n is large. To keep the notation as simple
as possible, this section focuses on the entity fixed effects regression model of
Section 10.3, in which there are no time effects.

The Fixed Effects Regression Assumptions

The four fixed effects regression assumptions are summarized in Key Concept 10.3.
These assumptions extend the four least squares assumptions, stated for cross-
sectional data in Key Concept 6.4, to panel data.

The first assumption is that the error term has conditional mean zero, given
all T values of X for that entity. This assumption plays the same role as the first
least squares assumption for cross-sectional data in Key Concept 6.4 and implies
that there is no omitted variable bias. The requirement that the conditional mean
of uit not depend on any of the values of X for that entity—past, present, or
future—adds an important subtlety beyond the first least squares assumption for
cross-sectional data. This assumption is violated if current uit is correlated with
past, present, or future values of X.

The second assumption is that the variables for one entity are distributed iden-
tically to, but independently of, the variables for another entity; that is, the variables
are i.i.d. across entities for i = 1, c, n. Like the second least squares assumption
in Key Concept 6.4, the second assumption for fixed effects regression holds if enti-
ties are selected by simple random sampling from the population.