Bayesian Inference in the Seemingly Unrelated Regressions Model

25
4
∑β i = 1
i= 1
4
∑
j= 1
β = 0
ij
β
ij
= β
ji
3. Inequality restrictions are required for the functions to satisfy concavity
and monotonicity. These restrictions are
• Monotonicity 0 < S i < 1
• Concavity B− S + ss′ is negative semidefinite where
⎡β
⎢
⎢β
B =
⎢β
⎢
⎢⎣
β
11
21
31
41
β
β
β
β
12
22
32
42
β
β
β
β
13
23
33
43
β
β
β
β
14
24
34
44
⎤
⎥
⎥
⎥
⎥
⎥⎦
⎡S
⎢
S =
⎢
⎢
⎢
⎢⎣
1
S
2
S
3
⎤
⎥
⎥
⎥
⎥
S ⎥
4 ⎦
⎡S
⎢
⎢S
s =
⎢S
⎢
⎢⎣
S
1
2
3
4
⎤
⎥
⎥
⎥
⎥
⎥⎦
Note that B− S + ss′ is negative semidefinite if and only if its largest
eigenvalue is nonpositive.
Since
S i depends on the input prices, a decision concerning the input
prices at which S i is evaluated, and the inequality restrictions imposed,
needs to be made. The inequality restrictions were imposed at average
input prices for each of the 23 years.
4. Given the severe inequality restrictions that were imposed, the Metropolis-
Hastings algorithm was used.
5. The quantities of interest are nonlinear functions of the parameters. They
are the elasticities of substitution