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 [https://www.commerce.gov/data-and-reports/economic-indicators/release-schedule 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.{{cite journal|last1=Koop|first1=G.|last2=Korobilis|first2=D.|year=2010|title=Bayesian multivariate time series methods for empirical macroeconomics|url=http://personal.strath.ac.uk/gary.koop/kk3.pdf|journal=Foundations and Trends in Econometrics|volume=3|issue=4|pages=267–358|citeseerx=10.1.1.164.7962|doi=10.1561/0800000013|ssrn=1514412}}

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.{{cite book|last1=Karlsson|first1=Sune|title=Forecasting with Bayesian Vector Autoregression|journal=Handbook of Economic Forecasting|date=2012|volume=2 B|doi=10.1016/B978-0-444-62731-5.00015-4|url=https://ideas.repec.org/p/hhs/oruesi/2012_012.html|pages=791–897|isbn=9780444627315}}

A typical example is the shrinkage prior, proposed by Robert Litterman (1979){{cite journal

|first1=R. |last1=Litterman

|title=Techniques of forecasting using vector autoregressions

|journal=Federal Reserve Bank of Minneapolis Working Paper

|volume=115 |pages=[http://www.minneapolisfed.org/research/WP/WP115.pdf pdf] |year=1979

}}{{cite journal

|first1=R. |last1=Litterman

|title=Specifying VAR's for macroeconomic forecasting

|journal=Federal Reserve Bank of Minneapolis Staff Report

|volume=92 |year=1984

}} and subsequently developed by other researchers at University of Minnesota,{{cite journal

|first1=T. |last1=Doan

|first2=R. |last2=Litterman

|first3=C. |last3=Sims

|title=Forecasting and conditional projection using realistic prior distributions

|journal=Econometric Reviews

|volume=3 |pages=1–100 |year=1984 |doi=10.1080/07474938408800053

|url=http://www.nber.org/papers/w1202.pdf}}{{cite journal

|first1=C. |last1=Sims

|title=A nine variable probabilistic macroeconomic forecasting model

|journal=Federal Reserve Bank of Minneapolis Discussion Paper

|volume=14 |pages=[http://minneapolisfed.org/research/DP/DP14.pdf pdf] |year=1989

}} (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.{{cite journal|last1=Giannone|first1=Domenico|last2=Lenza|first2=Michele|last3=Primiceri|first3=Giorgio|title=Prior Selection for Vector Autoregressions|journal=Review of Economics and Statistics|date=2014|url=https://ideas.repec.org/p/nbr/nberwo/18467.html|doi=10.1162/rest_a_00483|volume=97|issue=2|pages=436–451|citeseerx=10.1.1.375.7244}}

In particular, the Minnesota prior assumes that each variable follows a random walk process, possibly with drift, and therefore consists of 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[https://github.com/joergrieger/pybvar joergrieger/pybvar 2019: 'pybvar' is a package for bayesian vector autoregression in Python. This package is similar to bvars.] or R[https://cran.r-project.org/package=BVAR Kuschnig N; Vashold L. BVAR: Bayesian Vector Autoregressions with Hierarchical Prior Selection in R] Statistical Packages.

Recent research has shown that Bayesian vector autoregression is an appropriate tool for modelling large data sets.{{cite journal

|first1 = T.|last1 = Banbura|first2 = R.|last2 = Giannone|first3 = L.|last3 = Reichlin|title = Large Bayesian vector auto regressions|journal = Journal of Applied Econometrics|volume = 25|issue = 1|pages = 71–92|year = 2010|doi = 10.1002/jae.1137}}

Factor-Augmented VAR (FAVAR)

Factor-augmented vector autoregression (FAVAR) extends the BVAR framework by incorporating latent factors that capture additional information from a large set of macroeconomic indicators. This approach, developed by Bernanke, Boivin, and Eliasz (2005), combines the advantages of factor models with VAR analysis, allowing researchers to analyze the impact of monetary policy using richer information sets while maintaining a parsimonious model structure. The Bayesian estimation of FAVAR models helps address the uncertainty in both the latent factors and model parameters, providing more robust inference.{{cite journal|last1=Bernanke|first1=B.|last2=Boivin|first2=J.|last3=Eliasz|first3=P.|year=2005|title=Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach|journal=The Quarterly Journal of Economics|volume=120|issue=1|pages=387-422|doi=10.1162/0033553053327452}}

Time-varying parameter FAVAR (TVP-FAVAR) further extends this framework by allowing the model parameters to evolve over time, capturing potential structural changes in the economy. This approach is particularly useful for analyzing the time-varying nature of monetary policy transmission mechanisms and macroeconomic relationships. The combination of time-varying parameters with factor augmentation provides a flexible framework that can capture both cross-sectional and temporal variations in the data, while Bayesian methods help manage the increased parametric complexity.{{cite journal|last1=Koop|first1=G.|last2=Korobilis|first2=D.|year=2010|title=Bayesian multivariate time series methods for empirical macroeconomics|url=http://personal.strath.ac.uk/gary.koop/kk3.pdf|journal=Foundations and Trends in Econometrics|volume=3|issue=4|pages=267–358|doi=10.1561/0800000013}}

TVP-FAVAR models have been widely applied in empirical macroeconomics and monetary policy analysis. Korobilis (2013) used this approach to examine the evolution of monetary policy transmission mechanisms in the United States, finding significant changes in the effects of monetary policy shocks over time.{{cite journal|last1=Korobilis|first1=D.|year=2013|title=Assessing the Transmission of Monetary Policy Using Time-varying Parameter Dynamic Factor Models|journal=Oxford Bulletin of Economics and Statistics|volume=75|issue=2|pages=157-179|doi=10.1111/j.1468-0084.2011.00687.x}} Liu et al. (2017) employed TVP-FAVAR to investigate the time-varying impact of oil price shocks on macro-financial variables.{{cite journal|last1=Liu|first1=P.|last2=Mumtaz|first2=H.|last3=Theophilopoulou|first3=A.|year=2017|title=The transmission of international shocks to the UK. Estimates based on a time-varying factor augmented VAR|journal=Journal of International Money and Finance|volume=72|pages=44-68|doi=10.1016/j.jimonfin.2016.12.002}} More recently, Chen and Valcarcel (2021) utilized the framework to analyze monetary transmission in money markets, providing new insights into the effectiveness of monetary policy tools.{{Cite journal |last1=Chen |first1=Zhengyang |last2=Valcarcel |first2=Victor J. |date=October 2021 |title=Monetary transmission in money markets: The not-so-elusive missing piece of the puzzle |journal=Journal of Economic Dynamics and Control |language=en |volume=131 |pages=104214 |doi=10.1016/j.jedc.2021.104214 | issn=0165-1889}} Del Negro and Otrok (2008) applied the method to study international business cycles, demonstrating its utility in understanding global economic dynamics.{{cite journal|last1=Del Negro|first1=M.|last2=Otrok|first2=C.|year=2008|title=Dynamic factor models with time-varying parameters: measuring changes in international business cycles|journal=Federal Reserve Bank of New York Staff Reports|volume=326|pages=1-43}}

Bayesian vector autoregression

Factor-Augmented VAR (FAVAR)

See also

References

Further reading