Bayesian VARs

There are various approaches to modelling economic time-series. One very popular univariate method is known as the Box-Jenkins methodology and it's based on the idea that the analyst should "let the data speak for itself". Moving from the single-equation framework to a vector of equations, we have vector autoregressions and these are associated with Christopher Sims which he introduced in his paper Macroeconomics and Reality. Other econometric methodologies are linked to David Hendry and Edward Leamer; both of whom take different approaches.

Although I've listed various methodologies and said that they differ, they do, however, contain some elements which are the same across the board. A paper by Adrian Pagan (Three Econometric Methodologies: A Critical Appraisal) gives an excellent overview of the Sims, Hendry and Leamer methodologies and also outlines their similarities and differences.

Atheoretic models, such as univariate Box-Jenkins models and the Unrestricted VAR, do not rely on economic theory as the basis for specifying the model. They let the data do the talking. On the other hand, big simultaneous equation models, often built by a team of economists, have their foundations based on economic theory. While, in the Bayesian approach, beliefs (also known as priors) are allowed to enter the model.

Richard Todd discusses combining two of the aforementioned approaches, namely, the Bayesian approach and Sim's VAR framework, in his paper Improving Economic Forecasting with Bayesian Vector Autoregression. Mixing these two methods results in an economic forecasting model known as the Bayesian vector autoregression (BVAR).

We can immediately see the difference between the data-driven Box-Jenkins methodology and BVARs (data + beliefs) when Todd says:

The Bayesian approach to statistics, a general method for combining beliefs with data, suggests an objective procedure for blending beliefs and data in economic forecasting models. This procedure provides a framework that forecasters can use to document and discuss their beliefs, which can help make economic forecasting more of a science and less of an art.

So, how does the BVAR improve the basic Unrestricted VAR? Well, for a first, a vector autoregression with $N$ variables and $p$ lags, involves $N^{2}p$ parameters. In other words, adding extra variables and lags to a VAR consumes many degrees of freedom. This is a well known feature of vector autoregressions. Consider an example given by Clive Granger:

If N=10, p=4 there are 400 parameters to be estimated from 2000 pieces of data, say, if 200 terms are available for each series. It is clear why there is a desire for simpler models.

Clive's comment about the desire for simpler models contains an appeal to the principle of parsimony which says that we should build models that use the smallest number of coefficients needed to explain the available data. In other words, we should not include any unnecessary coefficients in our models.

Structural econometric models use exclusion restrictions to reduce the number of coefficients to be estimated in a VAR model.

So, how does one go about BVAR modelling in practice? Well, according to these slides by Patrick Brandt at the University of Texas, RATS, Eviews, Stata, and Matlab, are not very conducive to modelling BVARs. However, we're in luck because there is a package that has been developed for modelling BVARs in R. The package is called BMR. Strangely enough, this package was developed by one of my old UCD classmates! Better yet, there appears to be a second package MSBVAR that should also do the trick.