1136 The Methods of Growth Econometrics
Wasserman (1995) called a unit information prior; the prior is implicitly defined
to ensure that posterior model probabilities are well approximated by the Bayesian
information or Schwarz criterion. This paper concludes that the combination of a
unit information prior for within-model probabilities and a uniform prior across
models produces superior performance against a range of alternatives.
A distinct approach to within-model priors is developed by Magnus, Powell and
Prufer (2008), who employ the distinction between variables that are included in all
models,Si, and variables that are model-specific,Ri,m. For each model they propose
estimating a regression based onSiandei,m, the residual ofRi,mwhen projected
againstSi. A Laplace prior is assigned to theei,mparameters, which corresponds
to the idea that a researcher is ignorant as to whether the absolute value of thet-
statistic (computed at population values) is greater than 1. They argue that this prior
has the advantage that it is not dependent on an arbitrary choice of a parameter,
such as occurs because of the need to setgin (24.21).
There is also no consensus on the appropriate specification of the prior model
probabilitiesμ(m). In the model averaging literature, the usual assumption has
been to assign equal prior probabilities to all models inM. This corresponds to
assuming that the prior probability that a given variable appears in the “true” model
is 0.5 and that the probability that one variable appears in the model is independent
of whether others appear. Sala-i-Martinet al.(2004) consider modifications of this
prior, in which the probability that a given variable appears in the true model is
p<0.5 while preserving the assumption that the inclusion probabilities for each
variable are independent. The Sala-i-Martinet al.probabilities may be written as:
p#(m)(
1 −p)k−#(m)
, (24.22)where #(m)denotes the number of regressors in modelm. This prior can be gen-
eralized by treating (24.21) as the conditional probability of a model givenpand
then assigning a prior top, an idea developed in Brown, Vannucci and Fearn (1998)
and applied to growth regressions by Ley and Steel (2008).
The conditional independence assumption may be unappealing given collinear-
ity between regressors. One reason for this is that the different growth regressors
are sometimes included as proxies for a common growth theory. Durlaufet al.
(2008) address this by using dilution priors, due to George (1999), which down-
weight models that contain potentially “redundant” variables, as when a dataset
contains multiple proxies for the same underlying economic concept. This is
done by multiplying (24.21) by the determinant of the correlation matrix for the
included variables in a given model, which reweights model probabilities so as to
downweight those with redundant variables.
The issue of redundant variables is part of a wider set of conceptual problems
which arise when a researcher uses a prior which treats all models as equally likely.
Brock and Durlauf (2001a), Brocket al.(2003) and Durlaufet al.(2008) criticize
the widespread use of prior model probabilities which assume that the inclusion
of one variable should be independent of the inclusion of another. The conceptual
problem is analogous to the “red bus/blue bus” problem in discrete choice theory.