A Bayesian approach to analyze regional elasticities

used. Actually maximum likelihood estimator of β is equal to b if either the
constraints are not binding or β lies on the boundary of I(β,^~y). Moreover it can be
shown that the distribution of the maximum likelihood estimator depends on the true
value of β, and is not admissable as an estimator (Judge et al. 1985). Then the
"computational coincidence of sampling and posterior distributions does not extend to
inequality constrained linear regression" (Geweke 1986: p.128). For these reasons a
bayesian approach must be preferred, even if we have to apply not standard
computational procedures as Monte Carlo integration, since I(β,^~y) is an indicator
function and analytic integration is not feasible. Our task is to calculate the following
integral with inequality constraints:

∫ g (^β,~ ) l (φ / y ) p (φ) dφ

----<—.----;—<—;------< +∞

∫ l (φ / y ) p (φ) dφ

where φ = (β^,,σ)'. Analitical solutions are not feasible, but a Monte Carlo composition
procedure is straightforward since:

^fR (У^{, β, σ} / У^, ~⁾ = I (^β,~ / ~^{) f}N (У / У^{, β, σ,} ~^{) f}N ^β / У^{, σ}) ^fG (^σ / У )   (3.5)

where I(β, ^~y/ x^~) is the indicator function evaluated at point. Then Monte Carlo method
approximates any function over parameters and predictive with:

∑ g (^βi^, y_t )^fN (y_i / y^{, β}i ^,σi^,~^x )^fN (^βi / y^{, σ} ) ^fG (^σi / У )I (^βi >yi/x )

E [g (β,y )/y,^y ]=^j^½N-------------------------------------------------------- (3.6)

Σ ^fN (У г / У, ^βi ,^σi,^y )^fN (^βi / У, ^σi ) ^fG (^σi / У ) I (^βi ,yi/^y )

i =1

where (β₁, σ₁,~ 1),...., (βN, σ N ,~N ) are i.i.d. draws from the posterior distribution. By
the Law of Large Numbers:

E$ [g (β,y ) / y,^y ]-^±→ E [g (β,y )y,У ]                  (3.7)

while the estimated Monte Carlo standard error of E$g (^)] is:

⁴ This implies that elasticities must satisfy the following conditions:

nn

σ_LL ≤0, σ_KK ≤0, σ_LL σ_KK ≥2 σ_KL ≥0 (where the last inequality obtained by ∑ε_ij =∑S_j σ_ij=0).

j=1j =1

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