PANEL INDEX VAR MODELS: SPECIFICATION, ESTIMATION - Ivie

error therefore produces moving averages terms in the residuals of the VAR. Hence, if measurement
error is deemed important, one has two alternatives: (i) specify a long enough lag
length for the VAR so that at least the dominant elements of the MA representation are
accounted for; (ii) impose a particular MA structure on the error of (3). We discuss this
second strategy in the next section.
3 Posterior Estimation
In this section we take (4) to be part of the prior specification and let θ t =[λ t , α t, ρ 1,t,...,ρ f 1 ,t ,f 1 <
F + 2]. Then (4) can be written as
δ t = Ξθ t + u t u t ∼ N(0, Ω ⊗ V ) (7)
where Ξ =[Ξ 1 , Ξ 2 , Ξ 3 ,...,Ξ f1 ]andV is a k × k matrix. We assume a hierarchical structure
for θ t of the form:
Furthermore we let
θ t = (I − C) θ 0 + Cθ t−1 + η t η t ∼ N (0,B t ) (8)
θ 0 = Pµ + ∼ N(0, Ψ) (9)
V = σ 2 I k (10)
B t = γ 1 ∗ B t−1 + γ 2 ∗ B 0 = ξ t ∗ B 0 (11)
with ξ t = γ1 t (1−γ1 + γ t)
2 where B (1−γ 1 ) 0 = diag(B 01 ,B 02 ,B 03 ,...B 0f1 +2). We assume that u t , η t ,
are mutually independent and that γ 1 , γ 2 , P, C are known. Here C is a full rank matrix, P a
matrix which restricts (part of the) initial values for the θ t ’s via an exchangeable prior. Thus,
for example, if the unit specific components are drawn from a distribution with common mean
and there are, e.g. four units, two variables and three components in (4), then:
⎡
P =
⎢
⎣
1 0 0 0
0 1 0 0
0 1 0 0
0 1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
The prior in (7)-(11) is very generally specified: in (8) the components of the coefficients
evolve over time in a geometric fashion and in (9) their initial conditions are linked across
⎤
.
⎥
⎦
8