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Fixed and Random Effects Models
A. Introduction
- consider a model of the form
for i = 1, N and t = 1, T. Let E(" (^) i ) = E(g it ) = 0, Var(" (^) i ) = F" , Var(g it (^) ) = Fg , and E(" (^) i g it ) = 0 2 2
- the presence of " i (^) leads to serial correlation in the uit , E( u u it is ) = F" for t Ö s ; thus, failure to account for " i 2 leads, at a minimum, to incorrect standard errors and inefficient estimation
- if " i (^) is correlated with xit , failure to account for " i leads to heterogeneity (omitted variables) bias in the estimate of $; to see this consider the following illustration
Heterogeneity Bias
B. Fixed Effects Model
- least squares dummy variable model a. note that in the model above, we could rewrite the " i terms as coefficients on a set of dummy variables indicating membership in cross-sectional unit i and estimate the model simply by including the appropriate dummy variables b. this approach is straightforward; however, for large N , it may be impractical to specify so many dummy variables
- mean-differenced model a. let b. similarly, define as the vector of unit i specific means for the explanatory variables c. then the mean-differenced estimator is
where and
d. the mean square error in this model is
where and K is the number of columns in x it
b. in addition, assume that the " i are unobserved random variables which follow a probability distribution known up to some finite set of parameters c. also assume
E(" (^) i ) = E(g it (^) ) = 0 Var(" i (^) ) = F" Var(g it (^) ) = Fg 2 2 E(" (^) i g it (^) ) = 0 E(g it (^) g js (^) ) = 0 E(" i (^) " (^) j ) = 0
d. can write the covariance matrix as
e. a generalized least squares procedure is possible if we can transform the dependent and independent
variables by where
i.e. run a regression with and as the dependent and independent variables f. sometimes described as the “quasi-differenced estimator”
- feasible generalized least squares (FGLS) estimation
a. run fixed effects regression to obtain
b. use slope coefficient from any consistent regression (e.g. OLS) to form and
c. then ; note that it is possible for this
estimator to be negative
- Breusch and Pagan (1980) specification test a. Lagrange multiplier test based on OLS residuals b. test statistic is
D. Fixed or Random Effects
- key consideration is the orthogonality of " i a. if " i (^) is uncorrelated with the variables in xit , then random effects is the appropriate estimator b. if " i (^) is correlated with the variables in xit , then the fixed effects model is appropriate
- can be examined using a Hausman-Wu test a. run both FE and RE models b. test statistic is
- one-way random effects model – the model specification is
xtreg dependent_variable list of independent variables , re i( index_var )
- specification tests: a. xttest0 command used after random effects specification conducts the Breusch-Pagan specification test b. hausman command used after random effects specification conducts the Hausman specification test - after the fixed effects regression, include the command est store fixed - after the random effects regression, include the command hausman fixed.
References
Greene, William. Econometric Analysis , 3 rdEdition, Upper Saddle River, N.J.: Prentice-Hall, 1997, chapter 14.
