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Empirical Limits for Time Series Econometric Models

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Abstract

This paper seeks to characterize empirically achievable limits for time series econometric modeling. The approach involves the concept of minimal information loss in time series regression and the paper shows how to derive bounds that delimit the proximity of empirical measures to the true probability measure in models that are of econometric interest. The approach utilizes generally valid asymptotic expressions for Bayesian data densities and works from joint measures over the sample space and parameter space. A theorem due to Rissanen is modified so that it applies directly to probabilities about the relative likelihood (rather than averages), a new way of proving results of the Rissanen type is demonstrated, and the Rissanen theory is extended to nonstationary time series with unit roots, near unit roots and cointegration of unknown order. The corresponding bound for the minimal information loss in empirical work is shown not to be a constant, in general, but to be proportional to the logarithm of the determinant of the (possibility stochastic) Fisher-information matrix. In fact, the bound that determines proximity to the DGP is generally path dependent, and it depends specifically on the type as well as the number of regressors. Time trends are more costly than stochastic trends, which, in turn, are more costly than stationary regressors in achieving proximity to the true density. The conclusion is that, in a very real sense, the 'true' DGP is more elusive when there is nonstationarity in the data. Some implications of these results for prediction and for the achieving proximity to the optimal predictor are explored.

Suggested Citation

  • Peter C.B. Phillips & Werner Ploberger, 1999. "Empirical Limits for Time Series Econometric Models," Cowles Foundation Discussion Papers 1220, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1220
    Note: CFP 1062.
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    1. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
    2. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(1), pages 95-131, April.
    3. P. C. B. Phillips & S. N. Durlauf, 1986. "Multiple Time Series Regression with Integrated Processes," Review of Economic Studies, Oxford University Press, vol. 53(4), pages 473-495.
    4. Peter C.B. Phillips & Werner Ploberger, 1991. "Time Series Modelling with a Bayesian Frame of Reference: 1. Concepts and Illustrations," Cowles Foundation Discussion Papers 980, Cowles Foundation for Research in Economics, Yale University.
    5. Peter C.B. Phillips & Werner Ploberger, 1992. "Time Series Modeling with a Bayesian Frame of Reference: Concepts, Illustrations and Asymptotics," Cowles Foundation Discussion Papers 1038, Cowles Foundation for Research in Economics, Yale University.
    6. Kim, Jae-Young, 1994. "Bayesian Asymptotic Theory in a Time Series Model with a Possible Nonstationary Process," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 764-773, August.
    7. Keuzenkamp, Hugo A & McAleer, Michael, 1995. "Simplicity, Scientific Interference and Econometric Modelling," Economic Journal, Royal Economic Society, vol. 105(428), pages 1-21, January.
    8. Phillips, Peter C B & Ploberger, Werner, 1996. "An Asymptotic Theory of Bayesian Inference for Time Series," Econometrica, Econometric Society, vol. 64(2), pages 381-412, March.
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    Cited by:

    1. Juan Rubio-Ramirez & Jesus Fernandez-Villaverde & Pablo A. Guerron-Quintana, 2010. "Fortune or Virtue: Time Variant Volatilities versus Parameter Drifting in U.S. Data," 2010 Meeting Papers 270, Society for Economic Dynamics.
    2. Athanasopoulos, George & de Carvalho Guillén, Osmani Teixeira & Issler, João Victor & Vahid, Farshid, 2011. "Model selection, estimation and forecasting in VAR models with short-run and long-run restrictions," Journal of Econometrics, Elsevier, vol. 164(1), pages 116-129, September.
    3. Patrick Marsh, "undated". "A Measure of Distance for the Unit Root Hypothesis," Discussion Papers 05/02, Department of Economics, University of York.
    4. Neri, Marcelo Côrtes & Soares, Wagner Lopes, 2008. "Turismo sustentável e alivio a pobreza: avaliação de impacto," FGV EPGE Economics Working Papers (Ensaios Economicos da EPGE) 689, EPGE Brazilian School of Economics and Finance - FGV EPGE (Brazil).
    5. Phillips, Peter C.B., 2005. "Challenges of trending time series econometrics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(5), pages 401-416.
    6. Peter C. B. Phillips, 2003. "Laws and Limits of Econometrics," Economic Journal, Royal Economic Society, vol. 113(486), pages 26-52, March.
    7. Hall, Alastair R. & Inoue, Atsushi & Nason, James M. & Rossi, Barbara, 2012. "Information criteria for impulse response function matching estimation of DSGE models," Journal of Econometrics, Elsevier, vol. 170(2), pages 499-518.
    8. Aaron Schiff & Peter Phillips, 2000. "Forecasting New Zealand's real GDP," New Zealand Economic Papers, Taylor & Francis Journals, vol. 34(2), pages 159-181.
    9. Phillips, Peter C. B., 2001. "Trending time series and macroeconomic activity: Some present and future challenges," Journal of Econometrics, Elsevier, vol. 100(1), pages 21-27, January.
    10. Lieberman, Offer & Phillips, Peter C.B., 2020. "Hybrid stochastic local unit roots," Journal of Econometrics, Elsevier, vol. 215(1), pages 257-285.
    11. Phillips, Peter C. B., 2002. "New unit root asymptotics in the presence of deterministic trends," Journal of Econometrics, Elsevier, vol. 111(2), pages 323-353, December.
    12. Thomas M. Fullerton, Jr. & Jorge A. Ibarra Salazar & Mario Elizalde, 2015. "Microeconomic Gasoline Consumption Anomalies in Mexico: 1997-2007," Asian Economic and Financial Review, Asian Economic and Social Society, vol. 5(4), pages 709-722, April.
    13. Offer Lieberman & Peter C.B. Phillips, 2017. "Latent Variable Nonparametric Cointegrating Regression," Cowles Foundation Discussion Papers 3013, Cowles Foundation for Research in Economics, Yale University.
    14. Kelvin Balcombe, 2005. "Model Selection Using Information Criteria and Genetic Algorithms," Computational Economics, Springer;Society for Computational Economics, vol. 25(3), pages 207-228, June.
    15. Peter C.B. Phillips & Zhipeng Liao, 2012. "Series Estimation of Stochastic Processes: Recent Developments and Econometric Applications," Cowles Foundation Discussion Papers 1871, Cowles Foundation for Research in Economics, Yale University.
    16. Peter C. B. Phillips & Ji Hyung Lee, 2015. "Limit Theory for VARs with Mixed Roots Near Unity," Econometric Reviews, Taylor & Francis Journals, vol. 34(6-10), pages 1035-1056, December.
    17. Durlauf, Steven N., 2001. "Manifesto for a growth econometrics," Journal of Econometrics, Elsevier, vol. 100(1), pages 65-69, January.
    18. Jesús Fernández-Villaverde & Juan F. Rubio-Ramírez, 2008. "How Structural Are Structural Parameters?," NBER Chapters, in: NBER Macroeconomics Annual 2007, Volume 22, pages 83-137, National Bureau of Economic Research, Inc.
    19. Ji Hyung Lee & Zhentao Shi & Zhan Gao, 2018. "On LASSO for Predictive Regression," Papers 1810.03140, arXiv.org, revised Feb 2021.
    20. Munehisa Kasuya & Tomoki Tanemura, 2000. "Small Scale Bayesian VAR Modeling of the Japanese Macro Economy Using the Posterior Information Criterion and Monte Carlo Experiments," Bank of Japan Working Paper Series Research and Statistics D, Bank of Japan.
    21. Ronald W. Butler & Marc S. Paolella, 2017. "Autoregressive Lag—Order Selection Using Conditional Saddlepoint Approximations," Econometrics, MDPI, Open Access Journal, vol. 5(3), pages 1-33, September.

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    More about this item

    Keywords

    Proximity bounds; data generating process; empirical measures; Fisher information; minimal information loss; Lebesgue measure; optimal predictor; path dependence; trends; unit roots;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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