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406 Chapter 10 Regression with Panel Data

implies that Yit - Yi = b1(Xit - Xi) + (uit - ui). Let Y∼it = Yit - Yi, X∼it = Xit - Xi
and ∼uit = uit - ui; accordingly,

∼Yit = b1X∼it + ∼u it. (10.14)

Thus b∼Y1itcaonn be estimated by the OLS regression of the “entity-demeaned” vari-
ables X∼it. In fact, this estimator is identical to the OLS estimator of b1

obtained by estimation of the fixed effects model in Equation (10.11) using n - 1

binary variables (Exercise 18.6).

The “before and after” (differences) regression versus the binary variables
specification. Although Equation (10.11) with its binary variables looks quite dif-
ferent from the “before and after” regression model in Equation (10.7), in the special
case that T = 2 the OLS estimator of b1 from the binary variable specification and
that from the “before and after” specification are identical if the intercept is excluded
from the “before and after” specifications. Thus, when T = 2, there are three ways
to estimate b1 by OLS: the “before and after” specification in Equation (10.7) (with-
out an intercept), the binary variable specification in Equation (10.11), and the
“entity-demeaned” specification in Equation (10.14). These three methods are
equivalent; that is, they produce identical OLS estimates of b1 (Exercise 10.11).

The sampling distribution, standard errors, and statistical inference. In multiple
regression with cross-sectional data, if the four least squares assumptions in Key
Concept 6.4 hold, then the sampling distribution of the OLS estimator is normal
in large samples. The variance of this sampling distribution can be estimated from
the data, and the square root of this estimator of the variance—that is, the stan-
dard error—can be used to test hypotheses using a t-statistic and to construct
confidence intervals.

Similarly, in multiple regression with panel data, if a set of assumptions—
called the fixed effects regression assumptions—hold, then the sampling distribu-
tion of the fixed effects OLS estimator is normal in large samples, the variance of
that distribution can be estimated from the data, the square root of that estimator
is the standard error, and the standard error can be used to construct t-statistics
and confidence intervals. Given the standard error, statistical inference—testing
hypotheses (including joint hypotheses using F-statistics) and constructing confi-
dence intervals—proceeds in exactly the same way as in multiple regression with
cross-sectional data.

The fixed effects regression assumptions and standard errors for fixed effects
regression are discussed further in Section 10.5.