9.2.5 Factor-Augmented VARs

9.2.5 Factor-Augmented VARs

If the number of predictor variables is very large, unrestricted VARs may not be well suited for constructing prediction models as they would involve the estimation of a very large number of coefficients. One approach that can be used to address this is to keep the key variables of interest in the VAR and then augment the VAR with a small set of common factors, $F_t.$ , that summarize the information from a potentially very large set of conditioning variables. Letting ${\tilde{y}}_t=(y_t^{\prime },F_t^{\prime }{)}^{\prime }$ , the factor-augmented VAR (FAVAR) model proposed by Bernanke, Boivin, and Eliasz (2005) takes the form

$${\tilde{y}}_t=c+\sum _{i=1}^P{A}_i{\tilde{y}}_{t-i}+{\tilde{u}}_t.$$

The key to the FAVAR model is that the dimension of $F_t$ is far smaller than that of an original set of conditioning variables. In practice $F_t$ will be extracted from a set of factors. Pesaran, Pick, and Timmermann (2011) suggest treating ${\gamma }_t$ and $F_t$ asymmetrically so that lagged values of y are used to predict $F_{;}$ , but not the other way round. This means that fewer parameters have to be estimated, although this constraint can be relaxed. Forecasting with factors is covered more extensively in chapter 10.

9.3 BAYESIAN VARS

The Bayesian approach complicates the problem of estimating a VAR in two ways: first, a prior for the model must be specified; second, estimation is no longer undertaken by OLS, apart from in very simple cases. Despite these complications, Bayesian VAR (BVAR) methods are popular for a number of reasons. First, even from a classical perspective, BVARs often provide better forecasts than those from an unrestricted OLS estimation of VARs. Second, BVARs can facilitate a closer relationship between the underlying economics of the forecast problem and the estimation, for instance by imposing economic theory through the prior. Such theory-based information can sometimes be used to improve estimation. Early Bayesian methods made assumptions not only to follow economic theory but also to address computational difficulties. Subsequent methods, developed when computational limitations were less binding, focus less on assumptions that ease computation.

Bayesian forecasts can often be thought of as shrinkage methods that reduce the effect of estimation errors. Bayesian methods have the additional advantage that a by-product is the construction of a posterior density for the outcome. Hence, unlike some of the classical methods, Bayesian methods are not tied to MSE loss in particular.

Under the assumption that the innovations are normally distributed, the likelihood of the VAR is fully specified and has a well-known form. Combined with a set of priors, one can construct posteriors and the desired forecasts. How complicated this process gets depends both on distributional assumptions and on the form of the model.

We provide here a general overview of the literature. A number of sources give details of the precise methods for construction of forecasts from Bayesian VARs. Koop and Korobilis (2010) provide an overview from an estimation perspective. Karlsson (2013) is an excellent and detailed reference on the assumptions of different models and also provides algorithms that can be used to estimate and generate forecasts from VARs.