PANEL INDEX VAR MODELS: SPECIFICATION, ESTIMATION - Ivie

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(1995), define yt ∗ = y t − r i=1 ϕ iyt−i, ∗ x ∗ t = x t − r i=1 ϕ ix ∗ t−i with ys ∗ = x ∗ s =0ifs
Convergence of the Gibbs sampler kernel to the true invariant distribution in our model is somewhat standard since the panel <strong>VAR</strong> model (1) is a time-varying SUR model with serially correlated errors (see e.g. Chib and Greenberg (1995)). Convergence in these types of models typically occurs under a set of mild conditions. For example, a simple sufficient is the following. Let A ∗ = {ψ ∈ A, p(ψ) > 0}. If for every ψ ∗ ∈ A and every A 1 ∈ A with π(ψ ∈ A 1 |Y ) > 0itisthecasethatπ(ψ i+1 ∈ A 1 |ψ i = ψ ∗ ,Y) > 0whereπ is the probability measure induced by the Gibbs sampler, then the transition kernel is ergodic and its unique invariant distribution is the posterior density p(ψ|Y ) (see Geweke (2000)). Inference on any continuous function G(ψ), of the parameters of interest can be easily constructed using the output of the Gibbs sampler and the ergodic theorem. For example E(g(ψ)) = G(ψ)p(ψ|Y )dψ can be approximated using 1¯L[ L+¯L =L+1 G(ψ ) −1 ] −1 . Predictive distribution for future y it ’s can be estimated using the recursive nature of the model and the simple conditional structure of (3). In particular, let Y t+τ =(Y t+1 ,...,Y t+τ ). To compute forecasts and turning points we need to construct F (G(Y t+τ ) | Y t )= F (G(Y t+τ ) | Y t , ψ) p (ψ | Y t ) dψ where F (G(Y t+τ ) | Y t , ψ) is the conditional density of the function G of future Y ’s, given ψ. Thenforecastscanbeobtaineddrawingψ () from the posterior distribution and simulating the vector Y ,t+τ from the density f Y t+τ | Y t , ψ () . Y t+τ L+¯L constitutes a sample, from which we can compute moments and function of interest. A point estimate of the forecast is the ergodic average Ŷ t+τ = ¯L −1 [ L+¯L =L+1 =L+1 Y t+τ −1 ] −1 t+τ or the median of the distribution; its numerical variance can be estimated using var Ŷ = ¯L −1 Q o + r s=1 1 − s (Qs + Q r+1 s) −1 where Q s = ¯L L+¯L −1 =s+1+L Y t+τ − Ŷ t+τ Y t+τ − Ŷ t+τ and interdecile ranges can be obtained by ordering the draws for each t + τ. Impulse response profiles can also be computed using these forecasts. We describe their calculation in some details in section 5. 4 Leading Indicators The panel <strong>VAR</strong> (3) with the hierarchical prior (7)- (11) provides a natural framework for the recursive construction of leading indicators. In fact substituting (7) into (3) one obtains the following observable index structure Y t = W t λ t + A t α t + f Z f,t ρ f,t + v t (19) where W t = X t Ξ 1 , A t = X t Ξ 2 and Z f,t = X t Ξ f . In (19) the NG× 1 vector of endogenous variables depends on a vector of common time indices W t , on a vector of unit specific indices A t , and on a set of indices Z f,t indexed 15
Page 1 and 2: PANEL INDEX VAR MODELS: SPECIFICATI
Page 3 and 4: 1 Introduction There has been a gro
Page 5 and 6: on predictive Bayes factors, to exa
Page 7 and 8: e represented, for example, as: δ
Page 9 and 10: units. In (11) the variance of the
Page 11 and 12: B o1 | Y T ẑ 3 , ψ −Bo1 ∼ IG
Page 13: where v t = E t + X t u t has covar
Page 17 and 18: using its out-of-sample prediction
Page 19 and 20: Iterating on (23) we have δ t+τ =
Page 21 and 22: to those of these three countries,
Page 23 and 24: Figure 2: Testing interdependencies
Page 25 and 26: Figure 4: Fixed effects Evolution o
Page 27 and 28: inflation is at the upper edge of t
Page 29 and 30: 7 Conclusions This paper attempted
Page 31 and 32: References [1] Binder, M, Hsiao, C.

PANEL INDEX VAR MODELS: SPECIFICATION, ESTIMATION - Ivie

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