PANEL INDEX VAR MODELS: SPECIFICATION, ESTIMATION - Ivie

B o1 | Y T ẑ 3
, ψ −Bo1 ∼ IG
2 , ˆQ
3
;
2
Boi
−1 | Y T , ψ −Boi ∼ W
ẑ 4i , ˆQ 4i ;
Ψ −1 | Y T , ψ −Ψ ∼ W
ẑ o , ˆQ
o .
where expressions for ˆδ t , ˆV t , ẑ 1 , ˆQ 1 , ˆθ o , ˆΨ −1 , ˆµ, ˆΣ µ ,ẑ 2 , ˆQ 2 , ẑ 3 , ˆQ 3 , ẑ 4i , ˆQ 4i , ẑ o , ˆQ o are given in the
appendix.
Depending on the application, the conditional posterior of (θ 1 , ..., θ T | Y T , ψ −θt ), can be
obtained recursively either with the Kalman filter or the Kalman smoother, as in Chib and
Greenberg (1995). In the first case, we initialize {θ t } t
for each t and save:
ˆθ t|t = ˆθ
t|t−1 ∗ + K t δ t − Ξˆθ t|t−1
∗ (13)
R t|t = (I − K t Ξ) Rt|t−1
∗
K t = Rt|t−1ΞF ∗ −1
t|t−1
F t|t−1 = ΞRt|t−1Ξ ∗ + B 1
where ˆθ t|t−1 ∗ = ˆθ t−1|t−1 ∗ and R∗ t|t−1 = R∗ t−1|t−1 +ξ tB o ,andˆθ
t−1|t−1 ∗ and R∗ t−1|t−1
are, respectively,
the mean and the variance covariance matrix of the conditional distribution of θt−1|t−1 ∗ .
Draws from for θ t aremadefromN(ˆθ t|t ,R t|t ). In the second case, the conditional posterior
of θ 1 , ..., θ T | Y T , ψ −θt is sampled in reverse time order from
θ T
∼ N
ˆθT |T ,R T |T
θ T −1
∼ N
ˆθT −1 ,R T −1 (14)
.
θ 1 ∼ N
ˆθ1 ,R 1
where ˆθ t = ˆθ t|t + Ξ t
θ t+1 − ˆθ t|t
, R t = R t|t − Ξ t R ∗ t+1|t Ξ t,andΞ t = R t|t R ∗−1
t+1|t .
To make the updating scheme described in (13)-(14) operational, initial values at time
t = 1 must be assigned. For instance, one can choose to initialize B 0 = R 0 to be diagonal with
elements φ i equal to small values. ˆθ0 can be initialized by running a VAR for each country
and taking the constant. In the same way, Q 1 can be taken as the variance covariance matrix
of a pooled VAR, and Ω can be initialized by setting it equal to Q 1 .
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