Economic Models - Convex Optimization

Time Varying Responses 63
Hence
(
p(π i+1 |x i+1 ) = constant exp − 1 )
...
J i .
2
Since p(π i+1 |x i+1 ) is proportional to the likelihood function, by maximizing
the conditional density function, we are also maximizing the likelihood
in order to determine πi+1 ∗ ...
. Note that minimization of J i, is equivalent to
maximizing p(π i+1 |x i+1 ). To minimize ...
J i, we expand Eq. (c) eliminating
terms not containing π i+1 . Then, we differentiate with respect to π
i+1 ′ and
after equating to zero, we finally obtain (Lazaridis, 1980).
πi+1 ∗ = π∗ i + (Q−1 1
− Q −1
1 G iQ −1
1
+ H i+1 ′ Q−1 2 H i+1) −1
× H i+1 ′ Q−1 2 (x i+1 − H i+1 πi ∗ ). (d)
Now, consider the composite matrix:
Q −1
1
− Q −1
1 G iQ1 −1 where G i = (Si
−1 + Q −1
1 )−1 .
Considering the matrix identity of householder, we can write
Q −1
1
− Q −1
1 (S−1 i
+ Q −1
1 )−1 Q −1
1
= (Q 1 + S i ) −1 Pi+1 −1 .
Hence Eq. (d) takes the form:
πi+1 ∗ = π∗ i + K i+1(x i+1 − H i+1 πi ∗ )
where K i+1 ,Si+1 −1 and P−1
i+1
are defined in Eqs. (21)–(23).
It is recalled that H 0 is a null matrix, since no observations exist beyond
period 1 in the estimation process. The same applies for the vector x 0 .
Appendix B
It is recalled that the model was estimated using FIML method. The FIML
estimates are as follows (R 2 and adjusted R 2 refer to the corresponding 2
SLS estimates. Numbers in brackets are the corresponding t-statistics).
Estimated Model
1. A t = 0.842Y t
(3.48)
+ 55191.5
(4.87)
+ 0.024A t−1 − 1.319IR t + 1.051IR t−t − 284EXR t
(1.74) (1.34) (1.16) (1.29)
R 2 = 0.99, R 2 = 0.92, DW = 1.76, ρ = 0.27