Paul Johnson, Steven Durlauf and Jonathan Temple 1135change an element ofSiby one unit. This assumes that the ultimate goal of the
exercise is to estimate a parameter, rather than to identify the “best” growth model.
Bayesian model averaging is still relatively new, and many practical questions
arise. The implementation of the approach in economics has often closely fol-
lowed the early work by Raftery (1995) and Rafteryet al.(1997). One issue concerns
the specification of priors on parameters within a model. In line with Raftery’s
approach, Brock and Durlauf (2001a), Brocket al.(2003), and Sala-i-Martinet al.
(2004) assume a diffuse prior on the model specific coefficients. This has the advan-
tage that, when the errors are normal with known variance, the posterior expected
value ofψ, conditional on the dataDand modelm, is the ordinary least squares
(OLS) estimatorψˆm. One disadvantage of this approach is that, since the diffuse
prior on the regression parameters is improper, one has to be careful that the pos-
terior model probabilities associated with the prior are interpretable. But as long as
the posterior model probabilities include an appropriate penalty for model com-
plexity, we do not see any conceptual problem in interpreting this approach as
strictly Bayesian. Brock and Durlauf (2001a), Brocket al.(2003), and Sala-i-Martin
et al.(2004) all compute posterior model probabilities using Bayesian information
criterion (BIC)-adjusted likelihoods. Fernandezet al.(2001a) and Masanjala and
Papageorgiou (2005) employ proper priors and therefore avoid any such concerns.^9
So far there is only limited evidence that use of improper versus proper priors has
important consequences in practice. Masanjala and Papageorgiou compare results
using proper priors with the improper priors we have described, and find that
the choice is unimportant in their application. Arguably the most important evi-
dence that the conventional approach is problematic can be found in Ciccone and
Jarocinski (2007), and we will discuss their study in detail at the end of this section.
Other work has examined the effects of different proper priors. LettingZidenote
(
Si,Ri,m
)
andηm=(
ψm
πm)
, Fernandez et al.(2001b) propose the use of theZellner (1986)g-prior:
μ(
ηm∣∣
∣m,σε,m)
∝N(
0,σε^2 ,m(
gZm′Zm)− 1 )
, (24.21)withg= 1 /max
{
n,k^2 m}
,ndenoting the number of observations andkmdenotingthe number of regressors in modelm. Ley and Steel (2008) extend this analysis by
considering the performance of this within-model prior for different model space
priors; the model space priors all assume that each variable appears in the true
model with equal probability and that variable inclusions exhibit conditional inde-
pendence as described above. They conclude that, for growth contexts, models of
interest typically involve a sufficiently large number of growth determinants such
thatk^2 m>n, so thatg=k−m^2. They additionally argue that the frequentist/Bayesian
hybrid employed by Sala-i-Martinet al.(2004) is well approximated by theg=n−^1
case and therefore may be criticized on the grounds that this value ofgis generally
inappropriate. A different approach to within-model parameter priors is proposed
by Eicher, Papageorgiou and Raftery (2008). They suggest the use of what Kass and