ABSTRACT
X i), ∆Zit = (Zi –
__ Z i). e ∆εit = (εit – εi).
The individual effects are modelled by the intercept, which varies across all observations.This estimation is consistent and unbiased even if the independent variables are correlated with the individual error. The problem with the within transformation is that it drops out all timeinvariant regressors from the model. But time-invariant variables can be important to explain an economic behaviour. Socio-economic variables are generally time invariant or are not available as time series, and the fixed effect specification reduces the explanatory power of the model. In order to maintain timeinvariant variables in the model, we can adopt a random-effects GLS (generalized least squares) procedure (RE). The model can be written as follows:
DEPVAR X Zit it i i it= + + + +( )α β γ α ε In this specification, individual effects are random variables and individual heterogeneity is explained by a second error term, αi iid N (0, σ 2α ), and εit is the
idiosyncratic error iid N (0, σ 2ε ). The random-effects model is consistent and more efficient than the fixed-effect one if there is no correlation between αi and regressors. A Breusch-Pagan test allows us to compare the validity of the pooled OLS versus the random effect estimator. The null hypothesis is that the variance of εi is zero. If the null is not rejected, then we can presume that the pooled OLS is unbiased and consistent; if the null is rejected, the random-effects estimator should be preferred to the pooled OLS. A Hausman test (1978) can be used to compare fixed and random effects. The Hausman test verifies exogeneity of individual effects: rejection of the null hypothesis of no systematic differences between FE and RE coefficients implies that there is correlation between regressors and unobserved individual heterogeneity. If this is the case, RE, as well as OLS, are not consistent and should be rejected, and the fixed-effects model, or alternative models such as the instrumental variables models, should be used instead. Unfortunately, in some cases the application of the Hausman test is not conclusive: when the matrix of the squares of the differences of the variances of the coefficients is not positive definite, the test is not reliable. Alternatively, Wooldridge (2002: 290-291) proposes a test to compare fixed-and random-effects models. This test consists in estimating a random-effects model where the time-demeaning variables of timevariant variables are inserted. Then a Wald test is used to see whether the timedemeaning variables are jointly not significantly different from zero; if this is the case, the random-effects model can be accepted.
