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404 Chapter 10 Regression with Panel Data

The state-specific intercepts in the fixed effects regression model also can be
expressed using binary variables to denote the individual states. Section 8.3 con-
sidered the case in which the observations belong to one of two groups and the
population regression line has the same slope for both groups but different inter-
cepts (see Figure 8.8a). That population regression line was expressed mathemat-
ically using a single binary variable indicating one of the groups (case #1 in Key
Concept 8.4). If we had only two states in our data set, that binary variable regres-
sion model would apply here. Because we have more than two states, however,
we need additional binary variables to capture all the state-specific intercepts in
Equation (10.10).

To develop the fixed effects regression model using binary variables, let D1i
be a binary variable that equals 1 when i = 1 and equals 0 otherwise, let D2i equal
1 when i = 2 and equal 0 otherwise, and so on. We cannot include all n binary
variables plus a common intercept, for if we do the regressors will be perfectly
multicollinear (this is the “dummy variable trap” of Section 6.7), so we arbitrarily
omit the binary variable D1i for the first group. Accordingly, the fixed effects
regression model in Equation (10.10) can be written equivalently as

Yit = b0 + b1Xit + g2D2i + g3D3i + g + gnDni + uit, (10.11)

where b0, b1, g2, c, gn are unknown coefficients to be estimated. To derive the
relationship between the coefficients in Equation (10.11) and the intercepts in
Equation (10.10), compare the population regression lines for each state in the
two equations. In Equation (10.11), the population regression equation for the
first state is b0 + b1Xit, so a1 = b0. For the second and remaining states, it is
b0 + b1Xit + gi, so ai = b0 + gi for i Ú 2.

Thus there are two equivalent ways to write the fixed effects regression model,
Equations (10.10) and (10.11). In Equation (10.10), it is written in terms of n state-
specific intercepts. In Equation (10.11), the fixed effects regression model has a
common intercept and n - 1 binary regressors. In both formulations, the slope
coefficient on X is the same from one state to the next. The state-specific intercepts
in Equation (10.10) and the binary regressors in Equation (10.11) have the same
source: the unobserved variable Zi that varies across states but not over time.

Extension to multiple X’s. If there are other observed determinants of Y that
are correlated with X and that change over time, then these should also be
included in the regression to avoid omitted variable bias. Doing so results in the
fixed effects regression model with multiple regressors, summarized in Key
Concept 10.2.