# Bayesian vector autoregression

In statistics and econometrics, Bayesian vector autoregression (BVAR) uses Bayesian methods to estimate a vector autoregression (VAR) model. BVAR differs with standard VAR models in that the model parameters are treated as random variables, with prior probabilities, rather than fixed values.

Vector autoregressions are flexible statistical models that typically include many free parameters. Given the limited length of standard macroeconomic datasets relative to the vast number of parameters available, Bayesian methods have become an increasingly popular way of dealing with the problem of over-parameterization. As the ratio of variables to observations increases, the role of prior probabilities becomes increasingly important.[1]

The general idea is to use informative priors to shrink the unrestricted model towards a parsimonious naïve benchmark, thereby reducing parameter uncertainty and improving forecast accuracy.[2]

A typical example is the shrinkage prior, proposed by Robert Litterman (1979)[3][4] and subsequently developed by other researchers at University of Minnesota,[5][6] (i.e. Sims C, 1989), which is known in the BVAR literature as the "Minnesota prior". The informativeness of the prior can be set by treating it as an additional parameter based on a hierarchical interpretation of the model.[7]

In particular, the Litterman/Minnesota prior considers a normal prior on a set of parameters with fixed and known covariance matrix, which will be estimated with one of three techniques: Univariate AR, Diagonal VAR, or Full VAR.

This type model can be estimated with Eviews, Stata, Python[8] or R[9] Statistical Packages.

Recent research has shown that Bayesian vector autoregression is an appropriate tool for modelling large data sets.[10]

## References

1. ^ Koop, G.; Korobilis, D. (2010). "Bayesian multivariate time series methods for empirical macroeconomics" (PDF). Foundations and Trends in Econometrics. 3 (4): 267–358. CiteSeerX 10.1.1.164.7962. doi:10.1561/0800000013. SSRN 1514412.
2. ^ Karlsson, Sune (2012). Forecasting with Bayesian Vector Autoregression. Handbook of Economic Forecasting. 2 B. pp. 791–897. doi:10.1016/B978-0-444-62731-5.00015-4. ISBN 9780444627315.
3. ^ Litterman, R. (1979). "Techniques of forecasting using vector autoregressions". Federal Reserve Bank of Minneapolis Working Paper. no. 115: pdf.
4. ^ Litterman, R. (1984). "Specifying VAR's for macroeconomic forecasting". Federal Reserve Bank of Minneapolis Staff Report. no. 92.
5. ^ Doan, T.; Litterman, R.; Sims, C. (1984). "Forecasting and conditional projection using realistic prior distributions" (PDF). Econometric Reviews. 3: 1–100. doi:10.1080/07474938408800053.
6. ^ Sims, C. (1989). "A nine variable probabilistic macroeconomic forecasting model". Federal Reserve Bank of Minneapolis Discussion Paper. no. 14: pdf.
7. ^ Giannone, Domenico; Lenza, Michele; Primiceri, Giorgio (2014). "Prior Selection for Vector Autoregressions". Review of Economics and Statistics. 97 (2): 436–451. CiteSeerX 10.1.1.375.7244. doi:10.1162/rest_a_00483.
8. ^ joergrieger/pybvar 2019: 'pybvar' is a package for bayesian vector autoregression in Python. This package is similar to bvars.
9. ^ Kuschnig N; Vashold L. BVAR: Bayesian Vector Autoregressions with Hierarchical Prior Selection in R
10. ^ Banbura, T.; Giannone, R.; Reichlin, L. (2010). "Large Bayesian vector auto regressions". Journal of Applied Econometrics. 25 (1): 71–92. doi:10.1002/jae.1137.