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10.3 Fixed Effects Regression 405

The Fixed Effects Regression Model Key Concept

The fixed effects regression model is 10.2

Yit = b1X1,it + g + bkXk,it + ai + uit, (10.12)

where i = 1, c, n; t = 1, c, T; X1,it is the value of the first regressor for entity
i in time period t, X2,it is the value of the second regressor, and so forth; and
a1, c, an are entity-specific intercepts.

Equivalently, the fixed effects regression model can be written in terms of a
common intercept, the X’s, and n - 1 binary variables representing all but one entity:

Yit = b0 + b1X1,it + g + bkXk,it + g2D2i
+ g3D3i + g + gnDni + uit,(10.13)
where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth.

Estimation and Inference

In principle the binary variable specification of the fixed effects regression model
[Equation (10.13)] can be estimated by OLS. This regression, however, has k + n
regressors (the k X’s, the n - 1 binary variables, and the intercept), so in practice
this OLS regression is tedious or, in some software packages, impossible to imple-
ment if the number of entities is large. Econometric software therefore has special
routines for OLS estimation of fixed effects regression models. These special rou-
tines are equivalent to using OLS on the full binary variable regression, but are
faster because they employ some mathematical simplifications that arise in the
algebra of fixed effects regression.

The “entity-demeaned” OLS algorithm. Regression software typically computes

the OLS fixed effects estimator in two steps. In the first step, the entity-specific

average is subtracted from each variable. In the second step, the regression is esti-

mated using “entity-demeaned” variables. Specifically, consider the case of a single

regressor in the version of the fixed effects model in Equation (10.10) and take the

average of both sides of Equation (10.10); then Yi = b1Xi + ai + ui, where
T
Yi = (1 > T) g t= 1Yit, and Xi and ui are defined similarly. Thus Equation (10.10)