The Time Varying Parameter Panel VAR Model

The time varying parameter panel vector autoregressive (TVP-PVAR) model has a state-space representation in which the observation equation is:

Yit =cit+ p

X

j=1

Ait,jYt−j +uit, (4.2)

for datest= 1, . . . , T, whereYit,citanduitrespectively denoteG×1vectors containing variables of interest, intercepts and stochastic disturbances for the i−th country, i = 1, . . . N, Yt= (Y 0 1t, . . .Y 0 N t) 0 _{is an}

N G×1vector which stacks the variables of interest for each country, andAit,jis aG×N Gmatrix of time varying autoregressive coefficients corresponding to lagj = 1, . . . , p. Note that wheni= 1the TVP-PVAR model collapses into a regular time varying parameter vector autoregression (TVP-VAR) of the form in Cogley and Sargent (2001).

For estimation purposes Eq. (4.2) can be written in the form of a seemingly unre- lated regression (SUR) model:

Yt =Xtβt+ut, (4.3) whereXt=IN G⊗ h Yt−10 . . . Y0t−p 1 i ,βt=vec h At ct i0 ,ct= c0_1,t . . . c0_N,t 0 , At = h A0_1t . . . A0_{N t} i0 ,Ait= h Ait,1 . . . Ait,p i0 and ut = u0_1,t . . . u0_N,t 0 .21

21_{Note that⊗denotes the Kronecker Product andvec(·)is a vectorization operation that takes the}

N G×1intercept matrix: ct,j, and theN G×N Gmatrix of VAR coefficients: At, and the and stacks

The cost of the added flexibility in allowing for time varying coefficients is that we now require the estimation of an additional (T −1)N Gk states. Following Canova

et al. (2007, 2012), this computational burden can be reduced by implementing a cross- sectional parameter shrinkage procedure, which exploits the panel dimension of the model. More precisely, βt is factorized as:

βt= Ξ1θ1,t+ Ξ2θ2,t+ Ξ3θ3,t+vt, (4.4) where vt is a N Gk×1 vector of stochastic disturbances and θi,t, i= 1,2,3 are mutu- ally orthogonal vectors with associated (deterministic) loading matrices Ξi which are respectively of dimension N Gk ×N1 < N, N Gk×N and N Gk×G. Note that this factorization has the practical advantage of reducing the computational burden from

N Gk coefficients to just N1 +N +G factors which drive the coefficients. Moreover, writing (4.4)in matrix notation and substituting the result into (4.3) gives:

Yt=Ztθt+wt, (4.5) where Zt = XtΞ, Ξ = h Ξ1 Ξ2 Ξ3 i , θt = θ1,t θ2,t θ3,t 0 and wt = ut+Xtvt. Economically, the expression in (4.5) allows us to measure the relative importance of world, country and variable specific factors in driving the observed data. In particular,

θ1,t is an N Gk ×N1 vector that captures movements in the time varying coefficient vector that are common across both countries and variables. If N1 = 1 then θ1,t is a scalar and the first term: XtΞ1θ1,t, can be interpreted as a common or “world leading indicator”. Similarly, if we split the countries into small and large economies (as in Section 4.4) then N1 = 2, and θ1,t contains elements that represent common indicators across small and large economies. Next,θ2,t is anN×1vector that captures movements in the time varying coefficient vector which are common between countries, implying that the second term: XtΞ2θ2,t, can be interpreted as “country specific leading indicators”. Finally, θ3,t is a G× 1 vector that captures any time varying variable specific movements, implying that the final term: XtΞ3θ3,t can be interpreted as a vector of “variable specific leading indicators”. Since Xt enters into the construction of each indicator they are correlated by construction, however this correlation goes to zero as the number of countries increases (Canova et al., 2007). Also note that if the factorization in Eq. (4.4) is not exact (i.e. the variance of vt is non-zero), then the error term in (4.5) contains heteroscedasticity of known form. To reduce the model complexity we follow the common practice of imposing an exact factorization (Canova et al., 2007; Canova and Ciccarelli, 2012, 2009; Canova et al., 2012; Carriero et al., 2016; Poon, 2017a). To illustrate the structure of these indicators, Appendix 4.6.1 presents

a simple two country, two variable model with a single lag.

To complete the state space specification of the TVP-PVAR model, we need to describe the law of motion for the latent factors (i.e. θi, i = 1,2,3) along with their prior distributions. To this end, we follow Canova et al. (2007, 2012) and specify the state equations for datest = 2, . . . , T, as:

θt = θt−1+ηt, ηt∼N(0,Ω), (4.6) where Ω = diag(ω2

1, . . . , ωm2) and all elements of Ω are assumed to follow independent inverse-gamma distributions. The states are initialized with θ1 ∼ N(θ0,Vθ) where

θ0 and Vθ are known hyperparameters. As discussed by Primiceri (2005) the random walk specification is useful because it allows for the possibility of permanent shifts in the relationships between macroeconomic variables. Computationally, it has the advantage of parsimony. To enhance the readability of the paper, we defer the estimation details to Appendix 2.5.1.