Paul Johnson, Steven Durlauf and Jonathan Temple 1137Clearly some regressors are similar, such as alternative measures of trade openness,
whereas other regressors are quite disparate, such as measures of geography versus
institutional quality.
To address this, Brocket al.(2003) propose a tree structure to organize model
uncertainty for linear growth models. First, they argue that growth models suffer
from theory uncertainty. Hence, one can identify alternative classes of models
based on what growth theories are included. Second, for each specification of a
body of theories to be embedded, they argue there is specification uncertainty. A
given set of theories requires determining whether the theories interact, whether
they are subject to threshold effects or other types of nonlinearity, and so on. Third,
for each theory and model specification, there is measurement uncertainty; to say
that climate affects growth does not specify the relevant empirical proxies, such as
the number of sunny days or average temperature. Finally, each choice of theory,
specification and measurement is argued to suffer from heterogeneity uncertainty,
which means that it is unclear which sub-sets of countries obey a common linear
model. Brocket al.(2003) argue that one should assign priors that account for the
interdependence implied by this structure in assigning model probabilities.
We now briefly summarize some of the main findings of model averaging studies.
Sala-i-Martinet al.(2004) find that four variables have posterior model inclusion
probabilities above 0.9, namely initial income, the fraction of GDP from mining,
the number of years the economy has been open,^10 and the fraction of the popula-
tion following Confucianism. Fernandezet al.(2001a) also find that four variables
have posterior model inclusion probabilities above 0.9, substantially overlapping
with Sala-i-Martinet al.(2004) despite working with a different model space: initial
income, the fraction of the population following Confucianism, life expectancy,
and the share of equipment investment in GDP.^11 These findings appear to be
somewhat dependent on details of the way in which priors are assigned within
and across the model space.^12 Eicheret al.(2008) work with the same dataset
as Fernandezet al. (2001a) and find that the combination of a unit information
within-model prior and a uniform model space prior generates 16 different growth
variables whose posterior inclusion probabilities are 0.9 or greater. Interestingly,
if one reduces the variable inclusion probability so that the expected number of
variables in a model is seven (which is the prior used by Sala-i-Martinet al., 2004),
only four variables have posterior inclusion probabilities above 0.9; these are the
same as those identified by Fernandezet al.(2001a).
Durlaufet al.(2008) have applied model averaging to an unbalanced panel based
on three time periods spanning 1965–94, with a focus on evaluating the robustness
of various fundamental growth determinants. They confirm the importance of
initial income and investment, and also find a role for population growth and two
macroeconomic variables, the share of government consumption (net of defense
and education spending) in GDP and the rate of inflation. Their approach differs
from those above in that the conventional BIC approximation is applied in the
context of two-stage least squares (2SLS), rather than OLS. The development of a
rigorous combination of model averaging and instrumental variable methods is an
interesting area for further work.