9.3.5 Large-Dimensional Bayesian VARs

9.3.5 Large-Dimensional Bayesian VARs

VARs have traditionally been used to model and predict low-dimensional vectors. Extending the VAR to include large-dimensional vectors poses problems in so far as it is likely to lead to greater parameter estimation errors—recall that $n(np+1)$ mean parameters have to be estimated for an unconstrained VAR( p) model.

Banbura, Giannone, and Reichlin (2010) apply Bayesian shrinkage methods to large-dimensional unrestricted VARs in such a way that the degree of shrinkage is allowed to depend on the number of variables included in the VAR. Specifically, the tightness of the priors increases as more variables are added to the model. This approach can be viewed as an alternative to the type of factor models covered in chapter 10 and deals directly with the challenge that estimation error can be expected to increase with the dimension of the VAR, n.

Writing the BVAR as a $T\times n$ multivariate regression,

$$\begin{array}{r}Y=Z\beta +U,\end{array}\text{\hspace{1em}(9.29)}$$

where, again, $Y=(y_1,\dots ,y_T{)}^{\prime },Z=(z_1,\dots ,z_T{)}^{\prime }$ , for $z_t=(1,y_{t-1}^{\prime },\dots ,y_{t-p}^{\prime }{)}_{:}^{\prime }$ $U=(u_1,\dots ,u_T)$ ), and $\beta =(c^{\prime },A_1^{\prime },\dots ,A_p^{\prime })$ , Banbura, Giannone, and Reichlin (2010) assume standard normal-inverted Wishart priors,

$$\begin{array}{c}\mathrm{vec}(\beta )|\Sigma \sim \mathrm{N}(\mathrm{vec}({\beta }_0),\Sigma \otimes {\Omega }_0),\\ \Sigma \sim \mathrm{iW}(S_0,{\alpha }_0),\end{array}$$

where $E\left[u_t{u}_t^{\prime }\right]=\Sigma =\mathrm{diag}({\sigma }_1^2,\dots ,{\sigma }_n^2)$ and the prior parameters ${\beta }_0,{\Omega }_0,S_0,{\alpha }_0$ are chosen in accordance with what amounts to a set of modified Minnesota priors. To understand the key difference from (9.24), let $E[(A_k{)}_{ij}]={\delta }_i$ when $j=i$ . When the data are stationary, a prior with ${\delta }_i<1$ is more appropriate than the original Minnesota prior which sets ${\delta }_i=1$ and so assumes a random walk. This prior may seem difficult to implement, but Banbura, Giannone, and Reichlin (2010) suggest a simple strategy based on augmenting the model (9.29) with $T_d$ dummy observations $Y_d$ and $Z_d$ given by

$$\begin{array}{rl}& Y_d=\left(\begin{array}{cc}\dim (g_1(1,{\sigma }_1,\dots ,{\delta }_m,{\sigma }_n)/\lambda )& \\ 0& 0\\ 0& \dots \\ \dots & \dots \\ \mathrm{ding}({\sigma }_1,\dots ,{\sigma }_n)& \\ \dots & \dots \\ 0_{\mathrm{tron}}& \end{array}\right),\\ & Z_d=\left(\begin{array}{ccc}\dim (g_1(1,2,\dots ,p))\otimes \dim (g_1({\sigma }_1,\dots ,{\sigma }_n)/\lambda )& 0_{\mathrm{tron}}& \\ \dots & \dots & \\ 0& 0_{\mathrm{ron}}& \dots \\ \dots & \dots & \dots \\ 0_{\mathrm{tron}}& 0_{\mathrm{trot}}& \dots \\ 0& 0_{\mathrm{trot}}& \ell \end{array}\right),\end{array}$$

where $\epsilon$ is a very small number and the hyperparameter $\lambda$ controls the tightness of the prior variance of $A_k$ and thus can be thought of as a shrinkage parameter.

Consider OLS estimation of the augmented $T_{\ast }\times N_{\ast }$ regression model,

$$Y_{\ast }=Z_{\ast }\beta +U_{\ast },\text{\hspace{1em}(9.30)}$$

where

$$T_{\ast }=T+T_d,$$

$$Y_{\ast }=(Y^{\prime },Y_d^{\prime }{)}^{\prime },Z_{\ast }=(Z^{\prime },Z_d^{\prime }),U_{\ast }=(U^{\prime },U_d^{\prime }{)}^{\prime }.$$

Banbura, Giannone, and Reichlin (2010) show that these OLS estimates have the same posterior mean as the posterior of $\beta$ based on a set of normal-inverted Wishart priors with

$${\beta }_0=(Z_d^{\prime }{Z}_d{)}^{-1}{Z}_d^{\prime }{Y}_d,$$

$${\Omega }_0=(Z_d^{\prime }{Z}_d{)}^{-1},$$

$$S_0=(Y_d-Z_d{\beta }_0{)}^{\prime }(Y_d-Z_d{\beta }_0),$$

$${\alpha }_0=T_d-np-1.$$

Specifically, under an improper prior, $\Sigma \sim |\Sigma {|}^{-(n+3)/2}$ , the posterior becomes

$$\mathrm{vec}(\beta )|\Sigma ,Y\sim N(\mathrm{vec}(\tilde{\beta }),\Sigma \otimes (Z_{\ast }^{\prime }{Z}_{\ast }{)}^{-1}),$$

$$\Sigma |Y\sim \mathrm{iW}(\tilde{\Sigma },T_d+2+T-np-1),$$

where $\tilde{\beta }=(Z_{\ast }^{\prime }{Z}_{\ast }{)}^{-1}{Z}_{\ast }^{\prime }Y$ and $\tilde{\Sigma }=(Y_{\ast }-Z_{\ast }\tilde{\beta }{)}^{\prime }(Y_{\ast }-Z_{\ast }\tilde{\beta })$ are the least squares estimates from the augmented model (9.30). The matrix that needs to be inverted in the OLS estimation of the augmented model has dimension $(np+1)$ and so this is feasible even for large values of n and $\mathbf{P}.$

Banbura, Giannone, and Reichlin (2010) apply this approach to a data set used by Stock and Watson which has 131 monthly macro variables. In practice, they set ${\delta }_i=1$ for variables deemed to be nonstationary while ${\delta }_i=0$ for variables deemed to be stationary. The shrinkage parameter, $\lambda ,$ is clearly important in this setup. Banbura, Giannone, and Reichlin (2010) set λ so that it matches the in-sample one-stepahead MSE of a three-variable VAR with 13 monthly lags. Moreover, they consider three different values for the dimension of the VAR, $n=3,n=20$ , and $n=131$ · Their empirical results look at out-of-sample MSE performance for employment, CPI inflation, and the Federal funds rate. For these variables, the medium- and largescale models perform well compared to both a benchmark random walk with drift model and also compared to the small-scale model that includes only three variables. Interestingly, whereas the gains from going from $n=3$ to $n=20$ are very large, the gains from further extending to include $n=131$ variables are more modest.

Similar empirical evidence by Carriero, Kapetanios, and Marcellino (2009, 2012) suggests that large-scale Bayesian VARs can produce accurate forecasts of exchange rates and government bond yields whose performance is comparable to the best univariate benchmarks $(e.g.$ , random walks). Such benchmarks are often found to be hard to beat for these types of variables.