Following Primiceri [2005], the traditional VAR(n,p) model can be extended to a time varying parameter VAR with stochastic volatility (TVP-VAR-SV) model by adopting a state space representation with measurement equation defined by:
yt=bt+B1,tyt−1+· · ·+Bp,tyt−p+ut, ut∼ N(0,Σt), (4.1) whereyt is ann×1 vector of variables of interest,bt is ann×1 vector of time vary- ing intercepts, Bi,t, i = 1, . . . ,p, are n×n matrices of time varying VAR coefficients andΣt is ann×ntime varying error covariance matrix.
For estimation purposes (4.1) can be written in the form of a seemingly unrelated regression (SUR) model:
yt=Xtβt+ut, (4.2) whereXt =In⊗ h 1 y0t−1 . . . y0t−p i andβt =vec h bt B1,t . . . Bp,t i0 5.
The time varying error covariance matrix in(4.1)contains time varying variance and covariance terms. Estimating of the model therefore requires identification of each element within the matrix. To this end, Primiceri [2005] proposes an LDL factoriza- tion in which the error covariance matrix is decomposed into two distinct matrices;
LtandDt, in whichLtis a lower triangular matrix with ones along the main diagonal and time varying contemporaneous interactions amongst the endogenous variables in the lower portion of the matrix and Dt is a diagonal matrix of exogenous time varying disturbances. Formally, we have:
Σt =
L0tD−t 1Lt
−1
. (4.3)
For instance, ifn=4 then we have:
Lt = 1 0 0 0 a21,t 1 0 0 a31,t a32,t 1 0 a41,t a42,t a43,t 1 , Dt = eh1,t 0 0 0 0 eh2,t 0 0 0 0 eh3,t 0 0 0 0 eh4,t .
Set in this manner the time varying coefficients can capture any non-linearities within
5Note that⊗denotes the Kronecker Product andvec(·)is a vectorization operation that takes the
intercept and the VAR coefficients and stacks them into a k×1 vector equation by equation where
54The Relationship between Oil Price Shocks and China’s output: A Time-varying Analysis
the (lagged) relationships between the macroeconomic variables within the system. Similarly, the time varying error covariance matrix can distinguish between volatility within the contemporaneous relationships amongst the endogenous macroeconomic variables and volatility stemming from any exogenous shocks. By allowing for time variation in both the coefficients and the variance covariance matrix, the TVP-VAR- SV model allows for the data to determine whether any time variation exists in both the size and frequency of exogenous shocks as well the contemporaneous responses and lagged propagation of the variables to those shocks.
To complete the state space representation we need to specify the various laws of motion for the time varying states. To this end, let h•,t = (h1,t,h2,t,h3,t, . . . ,hn,t)0 and hi,• = (hi,1, . . .hi,T)0. That is, h•,t is an n×1 vector obtained by stacking hi,t by the first subscript whilst hi,• is the T×1 vector obtained by stacking the second subscript. Next, letatdenote the vector of time varying contemporaneous interaction terms collected row wise fromLt i.e. at =
h
a21,a31,a32, . . . ,an(n−1) i0
so that at is an
m×1 vector of parameters where m = n(n−1)/2. With this notation in mind, the laws of motion for the time varying states are given by:
βt = βt−1+νt, νt∼ N 0,Ωβ
, (4.4)
at = at−1+ψt, ψt ∼ N(0,Ωa), (4.5) h•,t = h•,t−1+ηt, ηt∼ N(0,Ωh), (4.6) for t = 2, . . . ,T, where Ωβ = diag
ω2β1, . . . ,ω2βk
, Ωa = diag ω2a1, . . . ,ωam2 and
Ωh=diag ω2h1, . . . ,ωhn2 , where all elements are assumed to follow independent In-
verse Gamma distributions. Finally, the states are initialized asβ1 ∼N β0,Vβ
, a1∼ N(a0,Va) and h1 ∼ N(h0,Vh) where β0, a0, h0, Vβ,Va and Vh are all assumed to
be known. Estimation is completed using a lag length of four quarters. The priors along with the estimation details are provided in Appendix 4.7.2.
In order to distinguish between the importance of allowing for time variation in the coefficients and the volatility of exogenous shocks, we estimate three alternative models, namely:
1. A traditional VAR with constant coefficients and constant error covariance ma- trix (CVAR) put form by Sims [1980];
2. A VAR with time varying coefficients and constant error covariance matrix (TVP-VAR) put forth by Cogley and Sargent [2001]; and
§4.2 Empirical methodology 55
(VAR-SV) in which the error covariance is modeled as above.
We highlight that all of these models are nested in (4.1) and can thus be estimated using the framework described in Appendix 4.7.2. To be clear, the TVP-VAR is a nested version of the TVP-VAR-SV model with the only difference being that the covariance-variance matrix is constant i.e. (Σ=Σ1= · · ·= ΣT). In this the standard natural conjugate prior is Σ ∼ IW(νΣ,SΣ) where IW(·,·) is the Inverse Wishart
distribution with degree of freedom parameter νΣ ≥ p and positive definite scale
matrixSΣ. Next, the constant VAR is a nested version of the TVP-VAR with the only difference being that the parameters are not time varying i.e. β= β1= · · · = βT. In this case we maintain the Gaussian prior and set β ∼ N β¯0, ¯Vβ
. Finally, the VAR- SV model is a nested version of the TVP-VAR-SV model with the only difference being that the parameters are not time varying. In this case we maintain overall consistency between the various specifications and set the same prior for βas in the
case of the VAR and the same prior forΣtas in the TVP-VAR-SV model. Each model is estimated using a lag length of one quarter.6