Bayesian VAR Modeling

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The VARMAX Procedure

Bayesian VAR Modeling

Consider the VAR(p) model

$y_{t}={\delta} + \Phi_1{y}_{t-1} + ... + \Phi_p{y}_{t-p} + {\epsilon}_t$

$y=(X \otimes I_k){\beta} + e.$

When the parameter vector ${\beta}$ has a prior multivariate normal distribution with known mean ${\beta}^{*}$ and covariance matrix $V_{\beta}$ , the prior density is written as

$f({\beta})=(\frac{1}{2\pi})^{k^2p/2}| V_{\beta}|^{-1/2} \exp[-\frac{1}2({\beta}-{\beta}^{*}) V_{\beta}^{-1}({\beta}-{\beta}^{*})].$

The likelihood function for the Gaussian process becomes

$\ell({\beta}|{y}) &=& (\frac{1}{2\pi})^{kT/2}| I_T\otimes\Sigma|^{-1/2}x \ &... ...}2(y-(X\otimes I_k){\beta})' (I_T\otimes\Sigma^{-1})(y-(X\otimes I_k){\beta})].$

Therefore, the posterior density is derived as

$f({\beta}|{y}) \propto \exp[-\frac{1}2({\beta}-\bar{{\beta}})' \bar{\Sigma}_{\beta}^{-1}({\beta}-\bar{{\beta}})]$

where the posterior mean is

$\bar{{\beta}}=[V_{\beta}^{-1}+(X'X\otimes \Sigma^{-1} )]^{-1} [V_{\beta}^{-1}{\beta}^{*}+( X'\otimes \Sigma^{-1})y]$

and the posterior covariance matrix is

$\bar{\Sigma}_{\beta}=[V_{\beta}^{-1} +(X'X\otimes \Sigma^{-1})]^{-1}.$

In practice, the prior mean ${\beta}^{*}$ and the prior variance $V_{\beta}$ need to be specified. If all the parameters are considered to shrink toward zero, the null prior mean should be specified. According to Litterman (1986), the prior variance can be given by

$v_{ij}(l)=\{ ({\lambda}/l)^2 & {if i=j} \ ({\lambda\theta\sigma_{ii}}/{l\sigma_{jj}})^2 & {if i\neq j} .$

where v_ij(l) is the prior variance of the (i,j)th element of $\Phi_l$ , $\lambda$ is the prior standard deviation of the diagonal elements of $\Phi_l$ , $\theta$ is a constant in the interval (0,1), and $\sigma^2_{ii}$ is the ith diagonal element of $\Sigma$ . The deterministic terms have diffused prior variance. In practice, you replace the $\sigma^2_{ii}$ by the diagonal element of the ML estimator of $\Sigma$ in the nonconstrained model.

For example, for a bivariate BVAR(2) model,

$y_{1t} &=& 0 + \phi_{1,11}y_{1,t-1} + \phi_{1,12}y_{2,t-1} + \phi_{2,11}y_{1,t... ...{1,22}y_{2,t-1} + \phi_{2,21}y_{1,t-2} + \phi_{2,22}y_{2,t-2} + \epsilon_{1t}$

with the prior covariance matrix

$V_{\beta}={\rm Diag} &(& \infty, \lambda^2, (\lambda \theta\sigma_1/\sigma_2... ...\sigma_1)^2, \lambda^2, (\lambda \theta\sigma_2/2\sigma_1)^2, (\lambda/2)^2 ).$

For the Bayesian Estimation of integrated systems, the prior mean is set to the first lag of each variable equal to one in its own equation and all other coefficients at zero. For example, for a bivariate BVAR(2) model,

$y_{1t} &=& 0 + 1 y_{1,t-1} + 0 y_{2,t-1} + 0 y_{1,t-2} + 0 y_{2,t-2} + \epsilo... ...&=& 0 + 0 y_{1,t-1} + 1 y_{2,t-1} + 0 y_{1,t-2} + 0 y_{2,t-2} + \epsilon_{2t}$

Forecasting of BVAR Modeling

The bootstrap procedure is used to estimate standard errors of the forecast (Litterman 1986). NREP=B simulations are performed. In each simulation the following steps are taken:

The procedure generates the available number of observations, T, and uniform random integers I_t, where t = 1, ... T.
A new observation, ${\tilde y}_{t}$ , is obtained as a sum of the forecast based on the estimates coefficients plus the vector of residuals from the I_t; that is,
${\tilde y}_t=\sum_{j=1}^p{\hat \Phi_j y}_{t-j} + \hat {{\epsilon}}_{I_t}.$
A new BVAR model is estimated by using the most recent observations, and a prediction value is made of the most recent observations.

The MSE measure of the l-step-ahead forecast is

$MSE(l)={\frac 1{B}} \sum_{i=1}^B ( {\tilde y_{t+l| t}}^i - {\bar {\tilde y}}_t )^2$

where ${\bar {\tilde y}}_t=(1/B)\sum_{i=1}^B\bar {\tilde y}_t^i$ .

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