A Bayesian approach to analyze regional elasticities

y_t = ^αL ⁺ Yll ^ln(PLtt / Pκt )⁺ Yyl ^lnYt ^{+ δ}TL ^tt ^{+ ε}t^,

εt ~ N (о, σ² ) is an homoschedastich

error.

3. Statistical analysis

Following Geweke (1986) we assume standard linear regression³ with a proper
prior that presupposes a well behaved cost function:

f (β,σ)-1(β,y)f„(β / σ)fG_θ(σ)               (3.1)

since I(β,^~y) is an indicator function equal to one when inequality constraints are
satisfied (i.e. when posterior distributions don’t violate concavity and monotonicity).
We stress the difference with Barnett, Geweke and Wolf (1991a), (1991b) or Chalfant,
Gray and White (1991) who consider the indicator function depending on parameters
only, while it’s well known that concavity requires that substitution matrix is negative
semidefinite⁴. This in turn constraint parameter and predictive to lie into a neoclassical
regular space. Therefore the posterior is no longer a normal gamma inverse distribution
since:

(3.2)

where

b=(X’X)^-1X’y, ν s² =y’y-b’(X’X) b,ν=T-k.            (3.3)

~

Actually the predictive density function f(y%/y,X^%), with X :(q × k) , and the

β∕y are not multi-t Student distribution with ν degrees of freedom as without inequality
constraints (Zellner 1971). Within this framework sampling theory methods cannot be
adopted due to the absence of any distribution theory and bayesian methods must be
³ More precisely assume z_i= {y_i,x_i’}’ exchangable³. Then, conditional on θ∈Θ , z_i/θ are i.i.d. Let the
parameter vector be decomposed into two separate subvectors θ^, = [φ^,,ψ]', where φ = (β^,,σ)’ and
assume the hypothesis of bayesian cut (see Florens and Mouchart (1985)).

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