Bayesian Multivariate Time Series Methods for Empirical Macroeconomics

Koop, Gary and Korobilis, Dimitris (2009): Bayesian Multivariate Time Series Methods for Empirical Macroeconomics.

Abstract

Macroeconomic practitioners frequently work with multivariate time series models such as VARs, factor augmented VARs as well as time-varying parameter versions of these models (including variants with multivariate stochastic volatility). These models have a large number of parameters and, thus, over-parameterization problems may arise. Bayesian methods have become increasingly popular as a way of overcoming these problems. In this monograph, we discuss VARs, factor augmented VARs and time-varying parameter extensions and show how Bayesian inference proceeds. Apart from the simplest of VARs, Bayesian inference requires the use of Markov chain Monte Carlo methods developed for state space models and we describe these algorithms. The focus is on the empirical macroeconomist and we offer advice on how to use these models and methods in practice and include empirical illustrations. A website provides Matlab code for carrying out Bayesian inference in these models.

• Item Type: MPRA Paper
• Language: English
• Keywords: Empirical macroeconometrics, Bayesian estimation, MCMC, vector autoregressions, factor models, time-varying parameters
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C50 - General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E58 - Central Banks and Their Policies; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs > C87 - Econometric Software; E - Macroeconomics and Monetary Economics > E5 - Monetary Policy, Central Banking, and the Supply of Money and Credit > E52 - Monetary Policy; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General

References

An, S. and Schorfheide, F. (2007). "Bayesian analysis of DSGE models," Econometric Reviews, 26, 113-172.

Asai, M., McAleer, M. and Yu, J. (2006). "Multivariate stochastic volatility: A review," Econometric Reviews, 25, 145-175.

Ballabriga, F., Sebastian, M. and Valles, J. (1999). "European asymmetries," Journal of International Economics, 48, 233-253.

Banbura, M., Giannone, D. and Reichlin, L. (2008). "Large Bayesian VARs," Journal of Applied Econometrics, forthcoming.

Belviso, F. and Milani, F. (2006). "Structural factor augmented VARs (SFAVARs) and the effects of monetary policy," Topics in Macroeconomics, 6, 2.

Berg, A., Meyer, R. and Yu, J. (2004). "Deviance information criterion for comparing stochastic volatility models," Journal of Business and Economic Statistics, 22, 107-120.

Bernanke, B. and Boivin, J. (2003). "Monetary policy in a data-rich environment," Journal of Monetary Economics, 50, 525-546.

Bernanke, B, Boivin, J. and Eliasz, P. (2005). "Measuring monetary policy: A Factor augmented vector autoregressive (FAVAR) approach," Quarterly Journal of Economics, 120, 387-422.

Boivin, J. and Giannoni, M. (2006) Has monetary policy become more effective? Review of Economics and Statistics, 88, 445-462.

Canova, F. (2007). Methods for Applied Macroeconomic Research published by Princeton University Press.

Canova, F. (1993). "Modelling and forecasting exchange rates using a Bayesian time varying coefficient model," Journal of Economic Dynamics and Control, 17, 233-262.

Canova, F. and Ciccarelli, M. (2009). "Estimating multi-country VAR models," International Economic Review, 50, 929-959.

Canova, F. and Ciccarelli, M. (2004). "Forecasting and turning point predictions in a Bayesian panel VAR model," Journal of Econometrics, 120, 327-359.

Canova, F. and Gambetti, L. (2009). "Structural changes in the US economy: Is there a role for monetary policy?" Journal of Economic Dynamics and Control, 33, 477-490.

Carter, C. and Kohn, R. (1994). "On Gibbs sampling for state space models," Biometrika, 81, 541--553.

Chib, S. and Greenberg, E. (1994). "Bayes inference in regression models with ARMA(p,q) errors," Journal of Econometrics, 64, 183-206.

Chib, S. and Greenberg, E. (1995). "Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models," Journal of Econometrics, 68, 339-360.

Chib, S., Nardari, F. and Shephard, N. (2002). "Markov chain Monte Carlo methods for stochastic volatility models," Journal of Econometrics, 108, 281-316.

Chib, S., Nardari, F. and Shephard, N. (2006). "Analysis of high dimensional multivariate stochastic volatility models," Journal of Econometrics, 134, 341-371.

Chib, S., Omori, Y. and Asai, M. (2009). "Multivariate stochastic volatility," pages 365-400 in Handbook of Financial Time Series, edited by T. Andersen, R. David, J. Kreiss and T. Mikosch. Berlin: Springer-Verlag.

Ciccarelli, M. and Rebucci, A., (2002). "The transmission mechanism of European monetary policy: Is there heterogeneity? Is it changing over time?," International Monetary Fund working paper, WP 02/54.

Cogley, T. and Sargent, T. (2001). "Evolving post-World War II inflation dynamics," NBER Macroeconomic Annual, 16, 331-373.

Cogley, T. and Sargent, T. (2005). "Drifts and volatilities: Monetary policies and outcomes in the post WWII U.S," Review of Economic Dynamics, 8, 262-302.

DeJong, P. and Shephard, N. (1995). "The simulation smoother for time series models," Biometrika, 82, 339-350.

Del Negro, M. and Otrok, C. (2008). "Dynamic factor models with time varying parameters: Measuring changes in international business cycles," Federal Reserve Bank of New York Staff Report no. 326.

Del Negro, M. and Schorfheide, F. (2004). "Priors from general equilibrium models for VARs," International Economic Review, 45, 643-673.

Del Negro, M. and Schorfheide, F. (2009). "Bayesian macroeconometrics," to appear in the Handbook of Bayesian Econometrics, edited by J. Geweke, G. Koop and H. van Dijk. Oxford: Oxford University Press.

Doan, T., Litterman, R. and Sims, C. (1984). "Forecasting and conditional projection using realistic prior distributions," Econometric Reviews, 3, 1-144.

Durbin, J. and Koopman, S. (2001). Time Series Analysis by State Space Methods. Oxford: Oxford University Press.

Durbin, J. and Koopman, S. (2002). "A simple and efficient simulation smoother for state space time series analysis," Biometrika, 89, 603-616.

Fernandes-Villaverde, J. (2009). "The econometrics of DSGE models," Penn Institute for Economic Research working paper 09-008.

Forni, M. and Reichlin, L. (1998). "Let's get real: A factor analytic approach to disaggregated business cycle dynamics," Review of Economic Studies, 65, 453-473.

Fruhwirth-Schnatter, S. (1994). "Data augmentation and dynamic linear models," Journal of Time Series Analysis, 15, 183--202.

Fruhwirth-Schnatter, S. and Wagner, H. (2008). "Marginal likelihoods for non-Gaussian models using auxiliary mixture sampling," Computational Statistics and Data Analysis, 52, 4608-4624.

Gelfand, A. and Dey, D. (1994). "Bayesian model choice: Asymptotics and exact calculations," Journal of the Royal Statistical Society Series B, 56, 501-514.

George, E., Sun, D. and Ni, S. (2008). "Bayesian stochastic search for VAR model restrictions," Journal of Econometrics, 142, 553-580.

Geweke, J. (1977). "The dynamic factor analysis of economic time series," in Latent Variables in Socio-economic Models, edited by D. Aigner and A. Goldberger, Amsterdam: North Holland.

Geweke, J. and Keane, M. (2007). "Smoothly mixing regressions," Journal of Econometrics, 138, 252-291.

Geweke, J. and Zhou, G. (1996). "Measuring the pricing error of the arbitrage pricing theory," Review of Financial Studies, 9, 557-587.

Giordani, P. and Kohn, R. (2008) "Efficient Bayesian inference for multiple change-point and mixture innovation models," Journal of Business and Economic Statistics, 26, 66-77.

Giordani, P., Kohn, R. and Pitt, M. (2009). "Time series state space models," to appear in the Handbook of Bayesian Econometrics, edited by J. Geweke, G. Koop and H. van Dijk. Oxford: Oxford University Press.

Giordani, P., Kohn, R. and van Dijk, D. (2007). "A unified approach to nonlinearity, structural change and outliers," Journal of Econometrics, 137, 112-133.

Groen, J., Paap, R. and Ravazzolo, F. (2008). "Real-time inflation forecasting in a changing world," Erasmus University manuscript, 2008.

Harvey, A. (1989). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press: Cambridge.

Ingram, B. and Whiteman, C. (1994). "Supplanting the Minnesota prior - Forecasting macroeconomic time series using real business cycle model priors," Journal of Monetary Economics, 49, 1131-1159.

Jacquier, E., Polson, N. and Rossi, P. (1994). "Bayesian analysis of stochastic volatility," Journal of Business and Economic Statistics, 12, 371-417.

Johannes, M. and Polson, N. (2009). "Particle filtering," pages 1015-1030 in Handbook of Financial Time Series, edited by T. Andersen, R. David, J. Kreiss and T. Mikosch. Berlin: Springer-Verlag.

Kadiyala, K. and Karlsson, S. (1997). "Numerical methods for estimation and inference in Bayesian VAR models," Journal of Applied Econometrics, 12, 99-132.

Kim, C. and Nelson, C. (1999). State Space Models with Regime Switching. Cambridge: MIT Press.

Kim, S., Shephard, N. and Chib, S. (1998). Stochastic volatility: likelihood inference and comparison with ARCH models, Review of Economic Studies, 65, 361-93.

Koop, G. (1992). "Aggregate shocks and macroeconomic fluctuations: A Bayesian approach," Journal of Applied Econometrics, 7, 395-411.

Koop, G., Leon-Gonzalez, R. and Strachan, R. (2009). "On the evolution of the monetary policy transmission mechanism," Journal of Economic Dynamics and Control, 33, 997--1017.

Koop, G., Poirier, D. and Tobias, J. (2007). Bayesian Econometric Methods. Cambridge: Cambridge University Press.

Koop, G. and Potter, S. (2004). "Forecasting in dynamic factor models using Bayesian model averaging," The Econometrics Journal, 7, 550-565.

Koop, G. and Potter, S. (2006). "The Vector floor and ceiling model," Chapter 4 in Nonlinear Time Series Analysis of the Business Cycle, edited by C. Milas, P. Rothman and D. van Dijk in Elsevier's Contributions to Economic Analysis series.

Koop, G. and Potter, S. (2009). "Time varying VARs with inequality restrictions," manuscript available at http://personal.strath.ac.uk/gary.koop/koop_potter14.pdf.

Korobilis, D. (2008). "Forecasting in vector autoregressions with many predictors," Advances in Econometrics, 23, 403 - 431.

Korobilis, D. (2009a). "Assessing the transmission of monetary policy shocks using dynamic factor models," University of Strathclyde, Discussion Papers in Economics, No. 09-14.

Korobilis, D. (2009b). "VAR forecasting using Bayesian variable selection," manuscript.

Kose, A., Otrok, C. and Whiteman, C. (2003). "International business cycles: World, region and country-specific factors," American Economic Review, 93, 1216-1239.

Kuo, L. and B. Mallick. (1997). "Variable selection for regression models," Shankya: The Indian Journal of Statistics (Series B), 60, 65-81.

Litterman, R. (1986). "Forecasting with Bayesian vector autoregressions -- Five years of experience," Journal of Business and Economic Statistics, 4, 25-38.

Lopes, H. and West, M. (2004). "Bayesian model assessment in factor analysis," Statistica Sinica, 14, 41-67.

Lubik, T. and Schorfheide, F. (2004). "Testing for indeterminacy: An application to U.S. monetary policy," American Economic Review, 94, 190-217.

Omori, Y., Chib, S., Shephard, N. and Nakajima, J. (2007). "Stochastic volatility with leverage: Fast and efficient likelihood inference," Journal of Econometrics, 140, 425-449.

Otrok, C. and Whiteman, C. (1998). "Bayesian leading indicators: Measuring and predicting economic conditions in Iowa," International Economic Review, 39, 997-1014.

Paap, R. and van Dijk, H. (2003). "Bayes estimates of Markov trends in possibly cointegrated series: An application to US consumption and income," Journal of Business and Economic Statistics, 21, 547-563.

Pitt, M. and Shephard, N. (1999). "Time varying covariances: a factor stochastic volatility approach," pages 547--570 in Bayesian Statistics, Volume 6, edited by J.M Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, Oxford: Oxford University Press.

Primiceri. G., (2005). "Time varying structural vector autoregressions and monetary policy," Review of Economic Studies, 72, 821-852.

Sentana, E. and Fiorentini, G. (2001). "Identification, estimation and testing of conditionally heteroskedastic factor models," Journal of Econometrics, 102, 143-164.

Sims, C. (1993). "A nine variable probabilistic macroeconomic forecasting model," pp. 179-204 in J. Stock and M. Watson, eds., Business Cycles Indicators and Forecasting. University of Chicago Press for the NBER).

Sims, C. (1980). "Macroeconomics and reality," Econometrica, 48, 1--48.

Sims, C. and Zha, T. (1998). "Bayesian methods for dynamic multivariate models," International Economic Review, 39, 949-968.

Sims, C. and Zha, T. (2006). "Were there regime switches in macroeconomic policy?" American Economic Review, 96, 54-81.

Stock, J. and Watson, M., (1996). "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business and Economic Statistics, 14, 11-30.

Stock, J. and Watson, M., (1999). "Forecasting inflation," Journal of Monetary Economics, 44, 293-335.

Stock, J. and Watson, M., (2002). "Macroeconomic forecasting using diffusion indexes," Journal of Business and Economic Statistics, 20, 147-162.

Stock, J. and Watson, M., (2005). "Implications of dynamic factor models for VAR analysis," National Bureau of Economic Research working paper 11467.

Stock, J. and Watson, M., (2006). "Forecasting using many predictors," pp. 515-554 in Handbook of Economic Forecasting, Volume 1, edited by G. Elliott, C. Granger and A. Timmerman, Amsterdam: North Holland.

Villani, M. (2009). "Steady-state prior for vector autoregressions," Journal of Applied Econometrics, 24, 630-650.

West, M. (2003). "Bayesian factor regression models in the `large p, small n' paradigm," pages 723-732 in Bayesian Statistics, Volume 7, edited by J.M. Bernardo, M. Bayarri, J.O. Berger, A.P. Dawid, D. Heckerman, A.F.M. Smith, and M. West, Oxford: Oxford University Press.

West, M. and Harrison, P. (1997). Bayesian Forecasting and Dynamic Models. Second edition. Berlin: Springer.