Fixed Effects Model (FEM)
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Fixed Effects Model (FEM)
Presented byNeals M. FrageApril 26, 2006
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Outline An illustrative example of the Fixed
Effects Model (FEM). Cautions on the use of FEM. Application of FEM.
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An illustrative example Consider the following pooling
estimation model:
20,...,2,1
4,3,2,133221
t
i
XXY itititit
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Assumptions of a pooling estimation The intercept value is the same
across units or entities (in this case, companies).
The slope coefficient is constant across units or entities (companies).
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Limitations of a Pooled Regression Assumptions of constant intercept
and slope coefficients are highly restricted and far-fetched.
May distort the “true” relationship between the dependent and independent variables across entities (companies).
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Account for the “individuality”of each entity. FIXED EFFECTS APPROACHA. Slope coefficients constant but the
intercept varies across entities.B. Slope coefficients are constant but
the intercept varies over individuals and time.
C. Slope coefficients and the intercept vary across entities.
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A. Slope coefficients constant but the intercept varies across entities.
To see this, consider the following model (16.3.2):
The subscript i on the intercept suggests that the intercepts of the four firms may be different.
itititiit XXY 33221
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Fixed Effects Model (FEM) Model (16.3.2) is an example of
the fixed effects (regression) model.
The term fixed effects is due to the fact that, although the intercept may differ across individual (in this case firms), each individual’s intercept does not vary over time.
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How to account for the “individual” intercept? Differential intercept dummies.
The differential intercepts tell us by how much the intercepts of GM, US, and West differ from the intercept of GE.
itititiiiit XXDDDY 33224433221
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The Time Effect Just as we used the dummy
variables to account for individual (firm) effect, we can allow for time effect (the function shifts over time) by introducing time dummies.
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The Time Effect (continued)
Where Dum35 takes a value of 1 for observation in year 1935 or 0 otherwise, etc. (1954 is the base year).
itititit XXDumDumDumY 332219210 53...3635
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B. Slope coefficients are constant but the intercept varies over individuals and time.
Combine models 16.3.3 and 16.3.6 to find individual (firm) effect as well as time effect.
ititit
WESTUSGMit
XXDum
DumDDDYiii
332219
104321
53...
35
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C. Slope coefficients and the intercept vary across entities.
To account for differences in the intercepts and slope coefficients, the individual (firm) dummies are introduced in an additive manner (like model 16.3.3) and in an interactive manner (multiply the dummy by each of the X variables).
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Differential intercepts and slope coefficients. Model (16.3.8)
ititiitiitiiti
itiitiititiiiit
XDXDXDXD
XDXDXXDDDY
)()()()(
)()(
346245334233
32222133224433221
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Cautions on the use of FEM Lost of degrees of freedom. Multicollinearity. Unable to identify the impact of
certain time-invariant variables (e.g. sex, ethnicity, and color).
The error term follows the classical assumptions (which may have to be modified).
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Application of FEM Many studies have been done on
whether FDI and exports are substitutes or complements. There has been no consensus (results have varied). (Archaic).
“New trade theory” brings an industrial organization approach to international trade.
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Motivation for Research I want to investigate the
relationship between FDI (foreign direct investment) and exports given the degree of concentration in an industry.
Research Question: How does market structure impact the link between FDI and exports?
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Data One home country (the UK) and
one host country (the US). 1997 US manufacturing industries. Using NAICS five-digit level data.
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Empirical Specification FDI = f (Exports, Ownership-Location-
Internalization factors). FDI = FDI of the UK in the US
manufacturing industries. Exports = Exports from the UK in the US
manufacturing industries. OLI = Factors in the US manufacturing
industries that impact and/or determine FDI.
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OLI factors Ownership (O) = CR4 Location (L) = Sales Internalization (I) = Advertising
cost
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Using FEM To find out the differences in the
relationship between FDI and exports across market structures, I introduce dummy and interactive dummy variables.
The dummy variables will capture the differential intercepts and the interactive dummies, the differential slope coefficients.
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Using FEM (continued) I am interested in the three main
types of market structure: Perfect competition, Monopoly, and Oligopoly. Hence, I create two dummy variables.
MD = Monopoly Dummy OD = Oligopoly Dummy Base Category = Perfect competition
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Using FEM (continued) The creation of the dummy variables
is based on the HHI (Herfindahl-Herschman Index) level. The HHI ranges from 0 to 10,000.
HHI < 1000 implies competition 1000 <=HHI<1800 implies Oligopoly HHI >=1800 implies Monopoly
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Using FEM (continued) OD=1 if 1000<=HHI<1800 0 Otherwise MD=1 if HHI >=1800 0 Otherwise
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Using FEM (continued) Since the goal is to capture the
differences in the link between FDI and exports across market structures, the dummy variables (MD and OD) are interacted with the independent variable Exports. The interactive terms are ExportsMD and ExportsOD.
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Simultaneity? Bi-directional link between Exports
and FDI (e.g. Graham, 1997). Simultaneity problems can result
from the bi-directional link . To deal with the simultaneity
problems, I change the specification of the model. Instead of estimating a linear model, I estimate a log-linear model.
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Estimating FEM The estimated model is the
following:
iiiii
iiii
iii
MDExportsLogODExportsLog
MDODADVLogSalesLog
CRLogExportsLogFDILog
)(*)(*
)(*)(*)(*)(*
)4(*)(*)(
87
6543
210
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Empirical Results The coefficients of CR4 and Sales
are as expected and significant. The coefficient of advertising cost is negative (as expected) but insignificant.
The relationship between FDI and exports vary across market structures, but never significant.
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Empirical Results (continued) Consider the differential intercepts
and slope coefficients for the three types of market structure.
Competition: Log(FDI) = -3.95 – 0.02 Log(Exports)
Oligopoly: Log(FDI) = -6.74 + 0.14 Log(Exports)
Monopoly: Log(FDI) = -7.43 + 0.12 Log(Exports)
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Heteroscedasticity? Heteroscedasticy is likely when dealing
with a cross-sectional dataset and when having qualitative variables.
Both the graphical test and the Breusch-Pagan-Godfrey (BPG) test suggest that heteroscedasticity is present in the fixed-effects estimation.
Weighted least squares (WLS) is used to correct for heteroscedasticity.
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WLS Results The WLS results are similar to the previous
results except that the coefficient of advertising cost becomes significant.
Consider the differential intercepts and slope coefficients of the WLS for the three types of market structure.
Competition: Log(FD)I = -3.13 – 0.002Log(Exports) Oligopoly: Log(FDI) = -6.09 + 0.16Log(Exports) Monopoly: Log(FDI) = -6.37 + 0.20Log(Exports)
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Conclusion
FEM can be cautiously used to account for “individuality” or differences across units.
The link between FDI and exports is positive in imperfect markets and negative in a perfect market (although the results are insignificant).