Economic Models - Convex Optimization

62 Dipak R. Basu and Alexis Lazaridis
Hence
∫
p(π i+1 |x i+1 ) = constant
(
exp − 1 )
J
2 ˜ i dπ i
where
J˜
i = (π i − πi ∗ )′ (Si
−1 + Q −1
1 )(π i − πi ∗ )
+ (π i+1 − πi ∗ )′ (Q −1
1
+ H i+1 ′ Q−1 2 H i+1)(π i+1 − πi ∗ )
− 2(π i − πi ∗ )′ Q −1
1 (π i − πi ∗ )
+ (x i+1 − H i+1 πi ∗ )′ Q −1
2 (x i+1 − H i+1 πi ∗ )
− 2(π i+1 − πi ∗ )′ H i+1 ′ Q−1 2 (x i+1 − H i+1 πi ∗ ). (a)
Now, define G −1
i
= (S −1
i
+ Q −1
1
) and consider the expression
J˜
i = [(π i − πi ∗ )′ − G i Q −1
1 (π i+1 − πi ∗ )]′
× G −1
i
[(π i − πi ∗ )′ − G i Q −1
1 (π i+1 − πi ∗ )]. (b)
Note that G i is symmetric, since it is the sum of two (symmetric) co-variance
matrices. Expanding Eq. (b) and noting that G i G −1
i
= I we obtain
J˜
i = (π i − πi ∗ )′ G −1
i
(π i+1 − πi ∗ ) − 2(π i − πi ∗ )′ Q −1
1 (π i+1 − πi ∗ )
+ (π i+1 − πi ∗ )′ Q −1
1 G iQ −1
1 (π i+1 − πi ∗ ). (c)
In view of Eqs. (b) and (c), Eq. (a) can be written as
J˜
i = [(π i − πi ∗ )′ − G i Q −1
1 (π i+1 − πi ∗ )]′
× G −1
i
[(π i − πi ∗ )′ − G i Q −1
1 (π i+1 − πi ∗ )]
+ (π i+1 − πi ∗ )′ (Q −1
1
+ H i+1 ′ Q−1 2 H i+1 − Q −1
1 G iQ −1
1 )(π i+1 − πi ∗ )
+ (x i+1 − H i+1 πi ∗ )′ Q −1
2 (x i+1 − H i+1 πi ∗ )
− 2(π i+1 − πi ∗ )′ H i+1 ′ Q−1 2 (x i+1 − H i+1 πi ∗ ).
∫ ( )
Integration with respect to π i yields constant exp −
1 ˜ 2
J i dπi =
constant exp ( − 1 ...
3 J i)
where
...
J i = ( π i+1 − πi ∗ )′ (Q −1
1
− Q −1
1 G iQ −1
1
+ H i+1 ′ Q−1 2 H i+1) ( π i+1 − πi
∗ )
+ ( x i+1 − H i+1 πi ∗ )′ Q −1 (
2 xi+1 − H i+1 πi
∗ )
− 2 ( π i+1 − πi ∗ )′ H i+1 ′ ( Q−1 2 xi+1 − H i+1 πi
∗ )
.