a working student and having internet at home should be negatively related
to attendance, while not having a direct impact on performance. Note, in
particular, that in the case of work, this assumption can be maintained given
that (in the complete specification) we are controlling for the number of hours
of study.
A third possibility is to exploit the time dimension of the data set, assum-
ing that the omitted variables do not change over time, to eliminate the effect
of unobservable factors using a panel estimator in the following specification:
yit = β1x1it + β2x2i + αi + ηit (3)
where ηit is the idiosyncratic error component, i.i.d.(0, ση2), uncorrelated with
(x1it, x2i, αi), and αi is i.i.d.(0, σα2 ), potentially correlated with x1it and x2i.
The fixed effect (FE) estimator is based on the assumption that αi is cor-
related with the explanatory variables (or that it represents fixed constants),
and is obtained as OLS on the data transformed in deviations from individual
means:
yit- ÿi. = (xiit- χii. )βi + (ηit- ηk.) (4)
This estimator is consistent even in the presence of unobservable effects
correlated with the regressors, provided ηit and x1t are uncorrelated at all
leads and lags. However, the fixed effects estimator wipes out all time in-
variant regressors and is not efficient. We therefore also consider the random
effects estimator (RE), based on quasi-deviations from individual means:
yit - θVi. = (x 1 it - θxi. ) β 1 + (x2i - θx2i. ) β2 + (αi - θαi) + (Vit - θηi. ) (5)
σ2
σ2+Tσα
where θ = 1-
is a measure of the weight of the between component
in the total variability of the error term. This estimator is inconsistent in the
presence of unobservable effects correlated with the regressors. However, it is
efficient (as it is the GLS estimator) and it allows to estimate the parameters
of time-invariant regressors.17
Sargan tests of overidentifying restrictions distributed as χ2 with two degrees of freedom
under the null hypothesis of instrument validity.
17We report Hausman tests of the null hypothesis that the individual-specific component
of the error term (αi) is uncorrelated with the regressors, based on the comparison of the
estimates obtained for the fixed and random effects models. We also report Breusch-Pagan
Lagrange Multiplier tests of the hypothesis of constant variance of the individual-specific
component of the error term (αi), i.e. a test of the pooled (OLS) model against the
alternative of the random effect model.
10