A Bayesian approach to analyze regional elasticities

1. Introduction

Bayes analysis is straightforward, but it is still not often used in applied
econometrics in regional science. I present a brief introduction to this approach¹. Let θ a
vector of parameters in which we are interested and y a vector of observations from a
density f (y / θ) that is identical to the likelihood function l(θ / y) that contains all the
sample information about the parameters. A priori analyst’s knowledge about
parameters is summarized by a subjective probability distribution f (θ). Therefore the
joint distribution over parameters and observations is:

h (θ, У ) = f (θ / У ).f (У ) = f (У / θ) f (θ)

that yields the Bayes theorem:

f (θ / y )=          θ = θ θ - l(θ / y )f (θ)

J ∖f∕ J                   J ∖f∕ J

which states that the posterior density function for the parameters after the sample is
proportional to the likelihood times the prior information. This way we can update our
prior information that is modified by the sample information and attainment of the
posterior density can be viewed as the end point of any scientific investigation.

Anyone who is familiar with the Bayesian approach is familiar with the
everlasting arguments concerning the proper philosophical and probability foundations,
pros and cons of this approach with respect to classical econometrics and difficulty to
elicit prior distributions in most empirical analyses. These discussions are intellectually
stimulating, but of little interest to practitioners. Therefore I prefer a more pragmatic
approach and present an example where bayesian methods are the only meaningful
solution to derive inferences about quantity of interest such as regional elasticity of
substitutions. We know from previous research that interval of confidence can be
obtained only with a bayesian approach (Gallant and Monham 1985) while all the
neoclassical properties can be imposed only with bayesian a priori restrictions (Barnett,
Geweke and Wolf 1991a, 1991b, Chalfant and Wallace 1991, Chalfant, Gray and White
1991). Then a bayesian transformation from prior to posterior knowledge must be
adopted in our regional analysis and the very point is to adopt a reasonable prior. In

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