In these fixed effects models, our variable of interest becomes sometimes insignificant so that
identification of effects needs to be discussed. The relative large standard errors in the models
with country fixed effects in columns (3) to (5) may indicate that the within country variations
in student achievement and government size are too small for identification. Notably, the OLS
R2 is as high as 0.94. However, it may equally be that it is the time-specific effects that
complicate identification, e.g. in column (3). The purpose of the scaling of the test scores
described above is to make the scores comparable over time. In consequence, with time-
specific effects, the model in essence draws inference on which other countries that
participated on a particular test and year. Our motivation for including year effects is that, on
average, all the independent variables have positive trends. Indeed, while the p-value of joint
significance of the time-specific effects is 0.02 in the model in column (3) in Table 4, the p-
value is only 0.13 when a simple trend is added. For this reason, we replace in column (4) the
time-specific effects with such a time trend. The coefficient of the trend variable is negative,
indicating a positive trend in the other independent determinants, as expected. The OLS R2
appears only marginally lowered, while the within-R2 is clearly reduced. Interestingly, the
effect of government consumption spending is significantly negative in this specification. We
conclude that it is not country-specific factors that make the effect of government size
insignificant in column (3), but the handling of the variation over time.
What kind of within-country variation in student achievement and government consumption is
driving the results? Is it country-specific trends, or fluctuations around the trends? Figure 2
above suggests that some countries exhibit a trend-like development in student achievement.
To investigate this question, column (5) in Table 4 expands the model with country-specific
trends. Then the effect of government consumption increases to about the same magnitude as
in the model without country-specific factors in column (2), and is highly significant. Thus, it
seems like it is variations around trends that account for the association between government
consumption and student achievement. The result is independent of whether the model
includes time-specific effects, and the difference between the models in columns (3) and (5) is
not related to the fact that the number of countries that contribute to identification is
necessarily smaller in the latter model.
In columns (6) to (8) of Table 4 we estimate the same models restricting our sample to OECD
countries, to ease comparison with Table 5 where an OECD-specific measure of the welfare
state is employed. We define the subsample of OECD countries by membership in the year
2000, but test later the robustness of our results for post-communist period effects. In the
OECD sample, the effect of government consumption is significant in the model without
country fixed effects (column (6)), with its size appearing independent of whether country
fixed effects and country-specific trends are included in the model (columns (7) and (8)). The
effect of -1 is remarkably similar to that in column (4) - the preferred model for the whole
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