Economic Models - Convex Optimization

Time Varying Responses 47
Combining Eqs. (14) and (14a) we have
BX ′ =−ƔU ′ + R ′ ⇒ X ′ =−B −1 ƔU ′ + B −1 R ′
⇒ X ′ = U ′ + W ′ (16)
where W ′ = B −1 R ′
Denoting the ith column of matrix X ′ by x i and the ith column of matrix
W ′ by w i , we can write
⎡
⎤
u 1i 0 ··· 0 u 2i 0 ··· 0 u gi 0 ··· 0
0 u 1i ··· 0 0 u 2i ··· 0 0 u gi ··· 0
x i =
· · · · · · · · ·
⎢ · · · · · · · · ·
⎥
⎣ · · · · · · · · · ⎦
0 0 ··· u 1i 0 0 ··· u 2i ··· 0 ... u gi
π + w i
(17)
where u ij is the element of the jth column and ith row of matrix U. The
vector π ∈ E ng , as mentioned earlier, consists of the g column of matrix .
Equation (17) can be written in a compact form, as
x i = H i π + w i , i = 1, 2,...,N (17a)
where x i ∈ E n ,w i ∈ E n and the observation matrix H i is defined on
E n × E ng .
In a time-invariant econometric model, the coefficients vector π is
assumed random with constant expectation overtime, so that
π i+1 = π i , for all i. (18)
In a time-varying and stochastic model we can have
π i+1 = π i + ε i
(18a)
where ε i ∈ E ng is the noise.
Based on these, we can re-write Eq. (17a) as
x i+1 = H i+1 π i+1 + w i+1 i = 0, 1,...,N − 1. (19)
We make the following assumptions.
(a) The vector x i+1 and matrix H i+1 can he measured exactly for all i.
(b) The noises ε i and w i+1 are independent discrete white noises with
known statistics, i.e.,
E(ε i ) = 0; E(w i+1 ) = 0
E(ε i w ′ i+1 ) = 0