The fixed effect estimator assumes the same numerical value for all three packages. The numerical
differences present in the random effects estimation are due to the different criteria adopted for the variance
component estimation. Maddala and Mount (1973) offer a comprehensive survey on the estimation
methods for the variance component models.
Our analysis focuses on the one-way model in which there is no time-only component in the
disturbance term.
Given the following model yit = a + Xt β + uit i = 1,..., N ; t = 1,..., Ti where the
,
disturbance term can be decomposed in the following way: ut = μi + vit and σ2 = σ^2 + σv2, and
assuming that the idiosyncratic elements μi are random, the estimation of this model consists of the
application of the Ordinary Least Squares technique to the following transformed model:
yit — θi ■ yi = a ■ (1 — θi ) + (X t — θi ■ Xi )■ β + u it — θi ■ ui
it it i i i ■ X i J it it i i ■ / ~ it i i ■
where the bar over the variable represents its temporal average. Let us assume the following:
σv is the standard deviation of the fixed effects estimation (within the regression);
σ 1 = 7σt2ot — σv2 and σtot is the variance of the pooled regression (OLS) achieved with
TSP, if this happens to be negative TSP resorts to the large sample formula given by
σ 1 = ( SSRols — SSRWithin) / NOBS where SSR stands for residuals sum of squares and
NOBS is the total number of observations;
σ12 is the following standard deviation computed with LIMDEP:
~2
σ 1
>θis ■ X ) / (Nfi
12
--σ„ where:
T
ave
bols is the pooled OLS coefficient vector;
Nfirm is the number of different groups;
kis the number of explanatory variables;
depending on certain numerical conditions bols can be replaced by bbetween ;
σ1 is the following standard deviation computed with STATA;
<*1 = max (0, √σbetween
2 / T ) where T is the harmonic mean of the set of Ti.
,
It is clear that all the numerical differences produced by the random effect estimate are caused by the
difference of the small sample formula for the computation of the between-regression variance. The
parameter θi, which discriminates the behaviour of the three packages, is computed in the following ways:
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