Economic Models - Convex Optimization

48 Dipak R. Basu and Alexis Lazaridis
E(c i ε ′ i ) = Q 1δ ij ,
E(w i w ′ i ) = Q 2δ ij .
where δ ij is the Kronecker delta, and
The above co-variance matrices, assumed to be positive definite.
(c) The state vector is normally distributed with a finite co-variance matrix.
(d) Regarding Eqs. (18a) and (19), the Jacobians of the transformation of
ε i into π i+1 and of w i+1 into x i+1 are unities. Hence, the corresponding
conditional probability densities are:
p(π i+1 |π i ) = p(ε i )
p(x i+1 |π i+1 ) = p(w i+1 ).
Under the above assumptions and given Eqs. (18a) and (19), the problem
set is to evaluate
and
E(π i+1 |x i+1 ) = π ∗ i+1
cov (π i+1 |x i+1 ) = S i+1
(the error co-variance matrix)
where x i+1 = x 1 ,x 2 ,x 3 ,...,x i+1 .
The solution to this problem (Basu and Lazaridis, 1986; Lazaridis,
1980) is given by the following set of recursive equations, as it is briefly
shown in Appendix A.
π ∗ i+1 = π∗ i + K i+1(x i+1 − H i+1 π ∗ i ) (20)
K i+1 = S i+1 H i+1 ′ Q−1 2
(21)
Si+1 −1 i+1 + H i+1 ′ Q−1 2 H i+1 (22)
Pi+1 −1 1 + S i ) −1 . (23)
The recursive process is initiated by regarding K 0 and H 0 as null matrices
and computing π ∗ 0 and S 0 from
π0 ∗ =ˆπ i.e., the reduced form coefficients (columns of matrix ˆ)
S 0 = P 0 = ˜.
The reduced form coefficients, along with their co-variance matrices,
can be updated by this recursive process and at each stage the set of Riccati