one percent of lectures increases test score by 0.07 points.23
The RE estimates for all other regressors are quite similar to the ones
obtained with OLS. All the ability indicators have positive and significant
coefficients. Hours of study per week have a positive and marginally sig-
nificant effect on performance. Both subject and teacher evaluation have
positive coefficients, although only the latter is statistically significant. Year
of registration, living away from home and foreign language all have large
negative and significant coefficients, although the latter is only marginally
statistically significant in the full specification.
The fixed effect model (column 1) produces an estimated coefficient for
attendance that is positive and statistically significant at the 5 per cent
level. The point estimate (0.039) is about half the size of the OLS and RE
estimates, suggesting that there is indeed positive correlation between unob-
served effects and time varying regressors, even after controlling for ability,
effort, motivation and other individual characteristics. This is confirmed by
the Hausman test statistic (29.28) that strongly rejects the null hypothesis
of unobservable characteristics uncorrelated with attendance (p-value=0.01).
This result is quite important, as it indicates that the inclusion of proxy vari-
ables is not sufficient to capture all the correlation between the regressor of
interest and unobservable ability, effort and motivation.
Besides statistical significance, is the estimated effect of attendance on
performance quantitatively relevant? Given that each two-hour lecture is
equivalent to 12.5 per cent of total attendance, the 0.04 estimate in the
fixed effect model implies that missing one lecture is associated to about a
half percentage point drop in test score. This also implies that an average
student who has not missed any lectures obtains a test score 1.2 percentage
points higher than a student who has the average attendance level (70.8 per
cent). It is interesting to observe that the return to each hour of self-study
is substantially lower than that to each hour spent attending lectures.24
Summing up, the results for the panel estimators provide three main
indications: first, proxy variables are not sufficient to control for omitted
variable bias; second, even after eliminating the omitted variable bias, using
a fixed effect estimator, attendance has a positive and significant impact
23Note also that the Breusch-Pagan LM test statistic strongly rejects OLS against RE.
24Given that each two-hour lecture is equivalent to 12.5 per cent of total attendance,
using the 0.04 estimate for lecture attendance in the fixed effect model one obtains an
estimated effect of each hour of lecture attendance of 0.25 (0.04*6.25), as opposed to 0.17
for self study.
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