IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/20125.html
   My bibliography  Save this paper

Bayesian Multivariate Time Series Methods for Empirical Macroeconomics

Author

Listed:
  • Koop, Gary
  • Korobilis, Dimitris

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.

Suggested Citation

  • Koop, Gary & Korobilis, Dimitris, 2009. "Bayesian Multivariate Time Series Methods for Empirical Macroeconomics," MPRA Paper 20125, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:20125
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/20125/1/MPRA_paper_20125.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Chib, Siddhartha & Greenberg, Edward, 1994. "Bayes inference in regression models with ARMA (p, q) errors," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 183-206.
    2. Dimitris Korobilis, 2013. "Var Forecasting Using Bayesian Variable Selection," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(2), pages 204-230, March.
    3. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    4. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility: A Review," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 145-175.
    5. Thomas A. Lubik & Frank Schorfheide, 2004. "Testing for Indeterminacy: An Application to U.S. Monetary Policy," American Economic Review, American Economic Association, vol. 94(1), pages 190-217, March.
    6. Geweke, John & Zhou, Guofu, 1996. "Measuring the Pricing Error of the Arbitrage Pricing Theory," Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 557-587.
    7. Fabio Canova & Matteo Ciccarelli, 2009. "Estimating Multicountry Var Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 50(3), pages 929-959, August.
    8. Stock, James H. & Watson, Mark W., 1999. "Forecasting inflation," Journal of Monetary Economics, Elsevier, vol. 44(2), pages 293-335, October.
    9. Giordani, Paolo & Kohn, Robert, 2008. "Efficient Bayesian Inference for Multiple Change-Point and Mixture Innovation Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 66-77, January.
    10. Canova, Fabio & Gambetti, Luca, 2009. "Structural changes in the US economy: Is there a role for monetary policy?," Journal of Economic Dynamics and Control, Elsevier, vol. 33(2), pages 477-490, February.
    11. Jan J. J. Groen & Richard Paap & Francesco Ravazzolo, 2013. "Real-Time Inflation Forecasting in a Changing World," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 29-44, January.
    12. Canova, Fabio, 1993. "Modelling and forecasting exchange rates with a Bayesian time-varying coefficient model," Journal of Economic Dynamics and Control, Elsevier, vol. 17(1-2), pages 233-261.
    13. Paap, Richard & van Dijk, Herman K, 2003. "Bayes Estimates of Markov Trends in Possibly Cointegrated Series: An Application to U.S. Consumption and Income," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(4), pages 547-563, October.
    14. Geweke, John & Keane, Michael, 2007. "Smoothly mixing regressions," Journal of Econometrics, Elsevier, vol. 138(1), pages 252-290, May.
    15. Christopher A. Sims, 1993. "A Nine-Variable Probabilistic Macroeconomic Forecasting Model," NBER Chapters, in: Business Cycles, Indicators, and Forecasting, pages 179-212, National Bureau of Economic Research, Inc.
    16. Timothy Cogley & Thomas J. Sargent, 2005. "Drift and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 8(2), pages 262-302, April.
    17. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, Oxford University Press, vol. 65(3), pages 361-393.
    18. Dimitris Korobilis, 2013. "Assessing the Transmission of Monetary Policy Using Time-varying Parameter Dynamic Factor Models-super-," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 75(2), pages 157-179, April.
    19. Marco Del Negro & Christopher Otrok, 2008. "Dynamic factor models with time-varying parameters: measuring changes in international business cycles," Staff Reports 326, Federal Reserve Bank of New York.
    20. Sungbae An & Frank Schorfheide, 2007. "Bayesian Analysis of DSGE Models," Econometric Reviews, Taylor & Francis Journals, vol. 26(2-4), pages 113-172.
    21. Mario Forni & Lucrezia Reichlin, 1998. "Let's Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics," Review of Economic Studies, Oxford University Press, vol. 65(3), pages 453-473.
    22. Marco Del Negro & Frank Schorfheide, 2004. "Priors from General Equilibrium Models for VARS," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 45(2), pages 643-673, May.
    23. Marta Banbura & Domenico Giannone & Lucrezia Reichlin, 2010. "Large Bayesian vector auto regressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 25(1), pages 71-92.
    24. Sims, Christopher A & Zha, Tao, 1998. "Bayesian Methods for Dynamic Multivariate Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 949-968, November.
    25. Ingram, Beth F. & Whiteman, Charles H., 1994. "Supplanting the 'Minnesota' prior: Forecasting macroeconomic time series using real business cycle model priors," Journal of Monetary Economics, Elsevier, vol. 34(3), pages 497-510, December.
    26. Belviso Francesco & Milani Fabio, 2006. "Structural Factor-Augmented VARs (SFAVARs) and the Effects of Monetary Policy," The B.E. Journal of Macroeconomics, De Gruyter, vol. 6(3), pages 1-46, December.
    27. Koop,Gary & Poirier,Dale J. & Tobias,Justin L., 2007. "Bayesian Econometric Methods," Cambridge Books, Cambridge University Press, number 9780521671736, June.
    28. Litterman, Robert B, 1986. "Forecasting with Bayesian Vector Autoregressions-Five Years of Experience," Journal of Business & Economic Statistics, American Statistical Association, vol. 4(1), pages 25-38, January.
    29. Otrok, Christopher & Whiteman, Charles H, 1998. "Bayesian Leading Indicators: Measuring and Predicting Economic Conditions in Iowa," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 997-1014, November.
    30. Bernanke, Ben S. & Boivin, Jean, 2003. "Monetary policy in a data-rich environment," Journal of Monetary Economics, Elsevier, vol. 50(3), pages 525-546, April.
    31. Sentana, Enrique & Fiorentini, Gabriele, 2001. "Identification, estimation and testing of conditionally heteroskedastic factor models," Journal of Econometrics, Elsevier, vol. 102(2), pages 143-164, June.
    32. Chib, Siddhartha & Greenberg, Edward, 1995. "Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models," Journal of Econometrics, Elsevier, vol. 68(2), pages 339-360, August.
    33. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility," Microeconomics Working Papers 22058, East Asian Bureau of Economic Research.
    34. Matteo Ciccarelli & Alessandro Rebucci, 2001. "The Transmission Mechanism of European Monetary Policy: Is There Heterogeneity? Is It Changing Over Time?," Working Papers 0115, Banco de España.
    35. Frühwirth-Schnatter, Sylvia & Wagner, Helga, 2008. "Marginal likelihoods for non-Gaussian models using auxiliary mixture sampling," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4608-4624, June.
    36. Korobilis, Dimitris, 2008. "Forecasting in vector autoregressions with many predictors," MPRA Paper 21122, University Library of Munich, Germany.
    37. Stock, James H & Watson, Mark W, 1996. "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 11-30, January.
    38. Giorgio E. Primiceri, 2005. "Time Varying Structural Vector Autoregressions and Monetary Policy," Review of Economic Studies, Oxford University Press, vol. 72(3), pages 821-852.
    39. M. Ayhan Kose & Christopher Otrok & Charles H. Whiteman, 2003. "International Business Cycles: World, Region, and Country-Specific Factors," American Economic Review, American Economic Association, vol. 93(4), pages 1216-1239, September.
    40. Chan,Joshua & Koop,Gary & Poirier,Dale J. & Tobias,Justin L., 2019. "Bayesian Econometric Methods," Cambridge Books, Cambridge University Press, number 9781108423380, May.
    41. Canova, Fabio & Ciccarelli, Matteo, 2004. "Forecasting and turning point predictions in a Bayesian panel VAR model," Journal of Econometrics, Elsevier, vol. 120(2), pages 327-359, June.
    42. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178.
    43. Marta Bańbura, 2008. "Large Bayesian VARs," 2008 Meeting Papers 334, Society for Economic Dynamics.
    44. James H. Stock & Mark W. Watson, 2005. "Implications of Dynamic Factor Models for VAR Analysis," NBER Working Papers 11467, National Bureau of Economic Research, Inc.
    45. Chib, Siddhartha & Nardari, Federico & Shephard, Neil, 2006. "Analysis of high dimensional multivariate stochastic volatility models," Journal of Econometrics, Elsevier, vol. 134(2), pages 341-371, October.
    46. Thomas Doan & Robert B. Litterman & Christopher A. Sims, 1983. "Forecasting and Conditional Projection Using Realistic Prior Distributions," NBER Working Papers 1202, National Bureau of Economic Research, Inc.
    47. Giordani, Paolo & Kohn, Robert & van Dijk, Dick, 2007. "A unified approach to nonlinearity, structural change, and outliers," Journal of Econometrics, Elsevier, vol. 137(1), pages 112-133, March.
    48. Sims, Christopher A, 1980. "Macroeconomics and Reality," Econometrica, Econometric Society, vol. 48(1), pages 1-48, January.
    49. Koop, G, 1992. "Aggregate Shocks and Macroeconomic Fluctuations: A Bayesian Approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 7(4), pages 395-411, Oct.-Dec..
    50. Gary Koop & Simon Potter, 2004. "Forecasting in dynamic factor models using Bayesian model averaging," Econometrics Journal, Royal Economic Society, vol. 7(2), pages 550-565, December.
    51. Ben S. Bernanke & Jean Boivin & Piotr Eliasz, 2005. "Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach," The Quarterly Journal of Economics, Oxford University Press, vol. 120(1), pages 387-422.
    52. Omori, Yasuhiro & Chib, Siddhartha & Shephard, Neil & Nakajima, Jouchi, 2007. "Stochastic volatility with leverage: Fast and efficient likelihood inference," Journal of Econometrics, Elsevier, vol. 140(2), pages 425-449, October.
    53. J. Durbin, 2002. "A simple and efficient simulation smoother for state space time series analysis," Biometrika, Biometrika Trust, vol. 89(3), pages 603-616, August.
    54. Ballabriga, Fernando & Sebastian, Miguel & Valles, Javier, 1999. "European asymmetries," Journal of International Economics, Elsevier, vol. 48(2), pages 233-253, August.
    55. James H. Stock & Mark W. Watson, 1993. "Business Cycles, Indicators, and Forecasting," NBER Books, National Bureau of Economic Research, Inc, number stoc93-1, March.
    56. Esfandiar Maasoumi & Michael McAleer, 2006. "Multivariate Stochastic Volatility: An Overview," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 139-144.
    57. Jean Boivin & Marc P. Giannoni, 2006. "Has Monetary Policy Become More Effective?," The Review of Economics and Statistics, MIT Press, vol. 88(3), pages 445-462, August.
    58. Koop, Gary & Potter, Simon M., 2011. "Time varying VARs with inequality restrictions," Journal of Economic Dynamics and Control, Elsevier, vol. 35(7), pages 1126-1138, July.
    59. Koop, Gary & Leon-Gonzalez, Roberto & Strachan, Rodney W., 2009. "On the evolution of the monetary policy transmission mechanism," Journal of Economic Dynamics and Control, Elsevier, vol. 33(4), pages 997-1017, April.
    60. Stock, James H. & Watson, Mark W. (ed.), 1993. "Business Cycles, Indicators, and Forecasting," National Bureau of Economic Research Books, University of Chicago Press, edition 1, number 9780226774886.
    61. Stock, James H & Watson, Mark W, 2002. "Macroeconomic Forecasting Using Diffusion Indexes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 147-162, April.
    62. George, Edward I. & Sun, Dongchu & Ni, Shawn, 2008. "Bayesian stochastic search for VAR model restrictions," Journal of Econometrics, Elsevier, vol. 142(1), pages 553-580, January.
    63. Kadiyala, K Rao & Karlsson, Sune, 1997. "Numerical Methods for Estimation and Inference in Bayesian VAR-Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 12(2), pages 99-132, March-Apr.
    64. Berg, Andreas & Meyer, Renate & Yu, Jun, 2004. "Deviance Information Criterion for Comparing Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 22(1), pages 107-120, January.
    65. Chib, Siddhartha & Nardari, Federico & Shephard, Neil, 2002. "Markov chain Monte Carlo methods for stochastic volatility models," Journal of Econometrics, Elsevier, vol. 108(2), pages 281-316, June.
    66. Gary Koop & Simon Potter, 2000. "The Vector Floor and Ceiling Model," Discussion Papers in Economics 04/15, Division of Economics, School of Business, University of Leicester.
    67. Mattias Villani, 2009. "Steady-state priors for vector autoregressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(4), pages 630-650.
    68. Litterman, Robert, 1986. "Forecasting with Bayesian vector autoregressions -- Five years of experience : Robert B. Litterman, Journal of Business and Economic Statistics 4 (1986) 25-38," International Journal of Forecasting, Elsevier, vol. 2(4), pages 497-498.
    69. Chang-Jin Kim & Charles R. Nelson, 1999. "State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262112388.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Karlsson, Sune, 2013. "Forecasting with Bayesian Vector Autoregression," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 791-897, Elsevier.
    2. Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2016. "Common Drifting Volatility in Large Bayesian VARs," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 375-390, July.
    3. Dimitris Korobilis, 2013. "Var Forecasting Using Bayesian Variable Selection," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(2), pages 204-230, March.
    4. Carriero, Andrea & Clark, Todd E. & Marcellino, Massimiliano, 2019. "Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors," Journal of Econometrics, Elsevier, vol. 212(1), pages 137-154.
    5. Stock, J.H. & Watson, M.W., 2016. "Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics," Handbook of Macroeconomics, in: J. B. Taylor & Harald Uhlig (ed.), Handbook of Macroeconomics, edition 1, volume 2, chapter 0, pages 415-525, Elsevier.
    6. Fuentes-Albero, Cristina & Melosi, Leonardo, 2013. "Methods for computing marginal data densities from the Gibbs output," Journal of Econometrics, Elsevier, vol. 175(2), pages 132-141.
    7. Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2016. "Large Vector Autoregressions with Stochastic Volatility and Flexible Priors," Working Papers (Old Series) 1617, Federal Reserve Bank of Cleveland.
    8. Andrea Carriero & Francesco Corsello & Massimiliano Marcellino, 2022. "The global component of inflation volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 37(4), pages 700-721, June.
    9. Joshua C.C. Chan & Gary Koop & Roberto Leon-Gonzalez & Rodney W. Strachan, 2012. "Time Varying Dimension Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 30(3), pages 358-367, January.
    10. Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2015. "Large Vector Autoregressions with Asymmetric Priors," Working Papers 759, Queen Mary University of London, School of Economics and Finance.
    11. Bekiros, Stelios D. & Paccagnini, Alessia, 2014. "Bayesian forecasting with small and medium scale factor-augmented vector autoregressive DSGE models," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 298-323.
    12. Nima Nonejad, 2021. "An Overview Of Dynamic Model Averaging Techniques In Time‐Series Econometrics," Journal of Economic Surveys, Wiley Blackwell, vol. 35(2), pages 566-614, April.
    13. Nalan Basturk & Cem Cakmakli & S. Pinar Ceyhan & Herman K. van Dijk, 2014. "On the Rise of Bayesian Econometrics after Cowles Foundation Monographs 10, 14," Tinbergen Institute Discussion Papers 14-085/III, Tinbergen Institute, revised 04 Sep 2014.
    14. Miranda-Agrippino, Silvia & Ricco, Giovanni, 2018. "Bayesian Vector Autoregressions," The Warwick Economics Research Paper Series (TWERPS) 1159, University of Warwick, Department of Economics.
    15. Magnus Reif, 2020. "Macroeconomics, Nonlinearities, and the Business Cycle," ifo Beiträge zur Wirtschaftsforschung, ifo Institute - Leibniz Institute for Economic Research at the University of Munich, number 87.
    16. Dimitris Korobilis, 2013. "Assessing the Transmission of Monetary Policy Using Time-varying Parameter Dynamic Factor Models-super-," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 75(2), pages 157-179, April.
    17. Stelios D. Bekiros & Alessia Paccagnini, 2016. "Policy‐Oriented Macroeconomic Forecasting with Hybrid DGSE and Time‐Varying Parameter VAR Models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 35(7), pages 613-632, November.
    18. Bekiros Stelios & Paccagnini Alessia, 2015. "Estimating point and density forecasts for the US economy with a factor-augmented vector autoregressive DSGE model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 19(2), pages 107-136, April.
    19. Knut Are Aastveit & Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2017. "Have Standard VARS Remained Stable Since the Crisis?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(5), pages 931-951, August.
    20. Zhou, Xiaocong & Nakajima, Jouchi & West, Mike, 2014. "Bayesian forecasting and portfolio decisions using dynamic dependent sparse factor models," International Journal of Forecasting, Elsevier, vol. 30(4), pages 963-980.

    More about this item

    Keywords

    Empirical macroeconometrics; Bayesian estimation; MCMC; vector autoregressions; factor models; time-varying parameters;
    All these keywords.

    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • E58 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit - - - Central Banks and Their Policies
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C87 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Econometric Software
    • E52 - Macroeconomics and Monetary Economics - - Monetary Policy, Central Banking, and the Supply of Money and Credit - - - Monetary Policy
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:20125. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.