Economic Models - Convex Optimization

Time Varying Responses 61
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Appendix A
Probability Density Function and Recursive Process
for Re-Estimation
Under the assumptions stated in Section 3 and according to Bayes’ rule, the
conditional probability density function of π i given x i+1 is Gaussian and is
given by:
∫
p(π i+1 |x i+1 ) = p(π i |x i )p(π i+1 ,x i+1 |π i ,x i )dπ i
where
(
p(π i+1 |x i ) = constant exp − 1 )
2 ||π i+1 − πi ∗ ||2 Si
−1
πi ∗ = E(π i |x i )
S i = cov(π i |x i )
(S i assumed to be invertible)
[
P(π i+1 ,x i+1 |π i ,x i ) = constant exp − 1 ∥ π i+1 − π i ∥∥∥
2
2 ∥ x i+1 − H i+1 π i+1
C =
[ ] [
Q1 0
, C
0 Q −1 Q
−1
=
2
1
0
0 Q −1
2
]
.
C −1 ]