Kociecki, Andrzej (2012): Orbital Priors for TimeSeries Models.

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Abstract
We propose the unified approach to construct the non–informative prior for time–series econometric models that are invariant under some group of transformations. We show that this invariance property characterizes some of the most popular models hence the applicability of the proposed framework is quite general. The suggested prior enjoys many desirable properties both from the Bayesian and non–Bayesian perspective. We provide detailed derivations of our prior in many standard time–series models including, AutoRegressions (AR), Vector AutoRegressions (VAR), Structural VAR and Error Correction Models (ECM).
Item Type:  MPRA Paper 

Original Title:  Orbital Priors for TimeSeries Models 
Language:  English 
Keywords:  Bayesian; Model invariance; Groups; Free group action; Orbit; Right Haar measure; Orbital decomposition; Maximal invariant; Cross section; Intersubjective prior; Vector AutoRegression (VAR); Structural VAR; Error Correction Model (ECM) 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C10  General C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C11  Bayesian Analysis: General 
Item ID:  42804 
Depositing User:  Andrzej Kociecki 
Date Deposited:  24. Nov 2012 17:49 
Last Modified:  04. Jul 2015 23:18 
References:  Akaike, H. (1980), “Likelihood and the Bayes Procedure”, in: J.M. Bernardo et al., eds., Bayesian Statistics, Valencia University Press. Andersson, S. (1982), “Distributions of Maximal Invariants Using Quotient Measures”, The Annals of Statistics, 10, pp. 955–961. Barndorff–Nielsen, O.E., P. Blaesild and P.S. Eriksen (1989), Decomposition and Invariance of Measures and Statistical Transformation Models, Lecture Notes in Statistics, No 58, Springer–Verlag, Berlin. Basu, D. (1977), “On the Elimination of Nuisance Parameters”, Journal of the American Statistical Association, 72, pp. 355–366. Berger, J.O. (1985), Statistical Decision Theory and Bayesian Analysis, Second edition, Springer–Verlag, New York. Berger, J.O. and D. Sun (2006), “Objective Priors for the Bivariate Normal Model With Multivariate Generalizations”, Duke University ISDS Technical Report 2007–06. Berger, J.O. and D. Sun (2007), “Objective Bayesian Analysis for the Multivariate Normal Model”, in: J.M. Bernardo et al., eds., Bayesian Statistics 8, Oxford University Press, Oxford. Berger, J.O. and D. Sun (2008), “Objective Priors for the Bivariate Normal Model”, The Annals of Statistics, 36, pp. 963–982. Berger J.O. and R.L. Wolpert (1988), The Likelihood Principle (second edition), Institute of Mathematical Statistics, Hayward, California. Bondar, J.V. (1977), “A Conditional Confidence Principle”, The Annals of Statistics, 5, pp. 881–891. Bunke, H. (1975), “Statistical Inference: Fiducial and Structural vs. Likelihood”, Statistics, 6, pp. 667–676. Chamberlain, G. (2007), “Decision Theory Applied to an Instrumental Variables Model”, Econometrica, 75, pp. 609–652. Chamberlain, G. and M.J. Moreira (2009), “Decision Theory Applied to a Linear Panel Data Model”, Econometrica, 77, pp. 107–133. Chang, T. and C. Villegas (1986), “On a Theorem of Stein Relating Bayesian and Classical Inferences in Group Models”, The Canadian Journal of Statistics, 14, pp. 289–296. Chang, T. and D. Eaves (1990), “Reference Priors for the Orbit in a Group Model”, The Annals of Statistics, 18, pp. 1595–1614. Chao, J.C. and P.C.B. Phillips (1998), “Posterior Distributions in Limited Information Analysis of the Simultaneous Equations Model Using the Jeffreys Prior”, Journal of Econometrics, 87, pp. 49–86. Clements, M.P. and D.F. Hendry (1999), Forecasting Non–Stationary Economic Time Series, MIT Press, Cambridge, Massachusetts. Datta, G.S. and J.K. Ghosh (1995), “Noninformative Priors for Maximal Invariant Parameter in Group Models”, Test, 4, pp. 95–114. Dawid, A.P. (1982), “Intersubjective Statistical Models”, in: G. Koch and F. Spizzichino, eds., Exchangeability in Probability and Statistics, North–Holland Pub. Co. Dawid, A.P. (2006), ”Invariant Prior Distributions”, in: S. Kotz et al., eds. Encyclopedia of Statistical Sciences, Second edition, vol. 6, John Wiley & Sons, New Jersey. Dawid, A.P. (2007), “Comment” on “Objective Bayesian Analysis for the Multivariate Normal Model”, by J.O. Berger and D. Sun, in: J.M. Bernardo et al., eds., Bayesian Statistics 8, Oxford University Press, Oxford. Dawid, A.P., M. Stone and J.V. Zidek (1973), “Marginalization Paradoxes in Bayesian and Structural Inference”, Journal of the Royal Statistical Society, series B, 35, pp. 189–213. Eaton, M.L. (1989), Group Invariance Applications in Statistics, Regional Conference Series in Probability and Statistics, vol. 1, Institute of Mathematical Statistics, Hayward, California. Eaton, M.L. (2008), “Dutch Book in Simple Multivariate Normal Prediction: Another Look”, in: D. Nolan and T. Speed, eds., Probability and Statistics: Essays in Honor of David A. Freedman, Institute of Mathematical Statistics, Beachwood, Ohio. Eaton, M.L. and D.A. Freedman (2004), “Dutch Book Against Some ‘Objective’ Priors”, Bernoulli, 10, pp. 861–872. Eaton, M.L. and W.D. Sudderth (1998), “A New Predictive Distribution for Normal Multivariate Linear Models”, Sankhya , series A, 60, pp. 363–382. Eaton, M.L. and W.D. Sudderth (1999), “Consistency and Strong Inconsistency of Group–Invariant Predictive Inferences”, Bernoulli, 5, pp. 833–854. Eaton, M.L. and W.D. Sudderth (2001), “Best Invariant Predictive Distributions”, in: M. Viana and D. Richards, eds., Algebraic Methods in Statistics and Probability, American Mathematical Society, Providence. Eaton, M.L. and W.D. Sudderth (2002), “Group Invariant Inference and Right Haar Measure”, Journal of Statistical Planning and Inference, 103, pp. 87–99. Eaton, M.L. and W.D. Sudderth (2004), “Properties of Right Haar Predictive Inference”, Sankhya , 66, pp. 487–512. Eaton, M.L. and W.D. Sudderth (2010), “Invariance of Posterior Distributions Under Reparametrization”, Sankhya , series A, 72, pp. 101–118. Farrell, R.H. (1985), Multivariate Calculation, Use of the Continuous Groups, Springer–Verlag, New York. Fraser, D.A.S. (1956), “Sufficient Statistics with Nuisance Parameters”, The Annals of Mathematical Statistics, 27, pp. 838–842. Fraser, D.A.S. (1968), The Structure of Inference, John Wiley & Sons, New York. Fraser, D.A.S. (1979), Inference and Linear Models, McGraw–Hill, New York. Geisser, S. (1984), “On Prior Distributions for Binary Trials”, American Statistician, 38, pp. 244–251. Hamilton, J.D., D.F. Waggoner and T. Zha (2007), “Normalization in Econometrics”, Econometric Reviews, 26, pp. 221–253. Heath, D. and W.D. Sudderth (1978), “On Finitely Additive Priors, Coherence and Extended Admissibility”, The Annals of Statistics, 6, pp. 333–345. Helland, I.S. (2010), Steps Towards a Unified Basis for Scientific Models and Methods, World Scientific Pub., Singapore. Hill, B.M. (1988), “Discussion by Professor Bruce M. Hill” in, Berger J.O. and R.L. Wolpert, The Likelihood Principle (second edition), Institute of Mathematical Statistics, Hayward, California. Hora, R.B. and R.J. Buehler (1966), “Fiducial Theory and Invariant Estimation”, The Annals of Mathematical Statistics, 37, pp. 643–656. James, A.T. (1954), “Normal Multivariate Analysis and the Orthogonal Group”, The Annals of Mathematical Statistics, 25, pp. 40–75. Jeffreys, H. (1961), Theory of Probability, 3–rd edition, Oxford University Press. Kleibergen, F. and H.K. van Dijk (1994), “On the Shape of the Likelihood/Posterior in Cointegration Models”, Econometric Theory, 10, pp. 514–551. Kocięcki, A. (2011), “Some Remarks on Consistency and Strong Inconsistency of Bayesian Inference”, MPRA Paper 28731, University Library of Munich. Koop, G. and M.F.J. Steel (1991), “A Comment” on “To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends, by P.C.B. Phillips”, Journal of Applied Econometrics, 6, pp. 365–370. Koop, G. and M.F.J. Steel (1992), “A Comment” on “On the Development of Reference Priors” by J.O. Berger and J.M. Bernardo, in: J.M. Bernardo et al., eds. Bayesian Statistics 4, Clarendon Press, Oxford. Koop, G., R. Strachan, H.K. van Dijk and M. Villani (2006), “Bayesian Approaches to Cointegration”, in T.C. Mills and K. Patterson, eds., The Palgrave Handbook of Econometrics, Vol. 1: Econometric Theory, Palgrave–Macmillan, Basingstoke. Lane, D.A. (1988), “Discussion by Professor David A. Lane” in, Berger J.O. and R.L. Wolpert, The Likelihood Principle (second edition), Institute of Mathematical Statistics, Hayward, California Leamer, E.E. (1991), “Comment” on “To Criticize the Critics”, Journal of Applied Econometrics, 6, pp. 371–373. Lehmann, E.L. (1986), Testing Statistical Hypotheses, Second edition, Springer–Verlag, New York. Lehmann, E.L. and G. Casella (1998), Theory of Point Estimation, 2–nd edition, Springer–Verlag, New York. Lindley, D. (1979), “A Comment” on “Reference Posterior Distributions for Bayesian Inference” by J.M. Bernardo, Journal of the Royal Statistical Society, series B, 41, pp. 133–134. Litterman, R.B. (1986), “Forecasting with Bayesian Vector Autoregressions – Five Years of Experience”, Journal of Business and Economic Statistics, 4, pp. 25–38. Lo, A.Y and J. Cabrera (1987), “Bayes Procedures for Rotationally Symmetric Models on the Sphere”, The Annals of Statistics, 15, pp. 1257–1268. McCullagh, P. (2002), “What Is a Statistical Model?” with discussion, The Annals of Statistics, 30, pp. 1225–1310. Muirhead, R.J. (1982), Aspects of Multivariate Statistical Theory, John Wiley & Sons, New York. Nachbin, L. (1965), The Haar Integral, D. Van Nostrand Company, Princeton, New Jersey. Ni, S. and D. Sun (2003), “Noninformative Priors and Frequentist Risks of Bayesian Estimators of Vector–Autoregressive Models”, Journal of Econometrics, 115, pp. 159–197. Ni, S. and D. Sun (2005), “Bayesian Estimates for Vector Autoregressive Models”, Journal of Business and Economic Statistics, 23, pp. 105–117. Phillips, P.C.B. (1991), “To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends”, Journal of Applied Econometrics, 6, pp. 333–364. Poirier, D.J. (1991), “A Comment” on “To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends”, Journal of Applied Econometrics, 6, pp. 381–386. Poirier, D.J. (1992), “A Comment” on “On the Development of Reference Priors” by J.O. Berger and J.M. Bernardo, in: J.M. Bernardo et al., eds. Bayesian Statistics 4, Clarendon Press, Oxford. Schotman, P.C. and H.K. van Dijk (1991a), “On Bayesian Routes to Unit Roots”, Journal of Applied Econometrics, 6, pp. 387–401. Schotman, P.C. and H.K. van Dijk (1991b), “A Bayesian Analysis of the Unit Root in Real Exchange Rates”, Journal of Econometrics, 49, pp. 195–238. Severini, T.A., R. Mukerjee and M. Ghosh (2002), “On an Exact Probability Matching Property of Right–Invariant Priors”, Biometrika, 89, pp. 952–957. Sims, C.A. (1988), “Bayesian Skepticism on Unit Root Econometrics”, Journal of Economic Dynamics and Control, 12, pp. 463–474. Sims, C.A. (1991), “Comment by Christopher A. Sims on ‘To Criticize the Critics’, by Peter C.B. Phillips”, Journal of Applied Econometrics, 6, pp. 423–434. Sims, C.A. and T. Zha (1998), “Bayesian Methods for Dynamic Multivariate Models”, International Economic Review, 39, pp. 949–968. Sims, C.A. and T. Zha (1999), “Error Bands for Impulse Responses”, Econometrica, 67, pp. 1113–1155. Stein, C. (1965), “Approximation of Improper Prior Measures by Prior Probability Measures”, in: J. Neyman and L.M. Le Cam, eds., Bernoulli, Bayes, Laplace: Anniversary Volume, Springer–Verlag, New York. Stone, M. (1970), “Necessary and Sufficient Condition for Convergence in Probability to Invariant Posterior Distributions”, The Annals of Mathematical Statistics, 41, pp. 1349–1353. Stone, M. (1976), “Strong Inconsistency from Uniform Priors” with discussion, Journal of the American Statistical Association, 71, pp. 114–125. Strachan, R.W. and B. Inder (2004), “Bayesian Analysis of the Error Correction Model”, Journal of Econometrics, 123, pp. 307–325. Uhlig, H. (1994), “On Jeffreys Prior When Using the Exact Likelihood Function”, Econometric Theory, 10, pp. 633–644. Villani, M. (2005), “Bayesian Reference Analysis of Cointegration”, Econometric Theory, 21, pp. 326–357. Villani, M. (2009), “Steady–State Priors for Vector Autoregressions”, Journal of Applied Econometrics, 24, pp. 630–650. Waggoner, D.F. and T. Zha (2003), “Likelihood Preserving Normalization in Multiple Equation Models”, Journal of Econometrics, 114, pp. 329–347. Wijsman, R. A. (1986), “Global Cross Sections as a Tool for Factorization of Measures and Distribution of Maximal Invariants”, Sankhya , series A, 48, pp. 1–42. Wijsman, R. A. (1990), Invariant Measures on Groups and Their Use in Statistics, Institute of Mathematical Statistics Lecture Notes–Monograph Series, vol. 14, Hayward, California. Yang, R. and J.O. Berger (1994), “Estimation of a Covariance Matrix Using the Reference Prior”, The Annals of Statistics, 22, pp. 1195–1211. Zha, T. (1999), “Block Recursion and Structural Vector Autoregressions”, Journal of Econometrics, 90, pp. 291–316. Zidek, J.V. (1969), “A Representation of Bayes Invariant Procedures in Terms of Haar Measure”, Annals of the Institute of Statistical Mathematics, 21, pp. 291–308. Zivot, E. (1994), “A Bayesian Analysis of the Unit Root Hypothesis Within An Unobserved Components Model”, Econometric Theory, 10, pp. 552–578. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/42804 