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Empirical Limits for Time Series Econometric Models


  • Werner Ploberger

    () (University of Rochester and University of St. Andrews)

  • Peter C. B. Phillips

    () (Yale University, U.S.A., University of Auckland and University of York)


This paper characterizes empirically achievable limits for time series econometric modeling and forecasting. 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 (the DGP) in models that are of econometric interest. The approach utilizes joint probability measures over the combined space of parameters and observables and the results apply for models with stationary, integrated, and cointegrated data. A theorem due to Rissanen is extended 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. For practical purposes, the proximity bound has the asymptotic form ("K"/2)log "n", where "K" is a new dimensionality factor that depends on the nature of the data as well as the number of parameters in the model. When 'good' model selection principles are employed in modeling time series data, we are able to show that our proximity bound quantifies empirical limits even in situations where the models may be incorrectly specified.One of the main implications of the new result is that time trends are more costly than stochastic trends, which are more costly in turn than stationary regressors in achieving proximity to the true density. Thus, in a very real sense and quantifiable manner, the DGP is more elusive when there is nonstationarity in the data. The implications for prediction are explored and a second proximity theorem is given, which provides a bound that measures how close feasible predictors can come to the optimal predictor. Again, the bound has the asymptotic form ("K"/2)log "n", showing that forecasting trends is fundamentally more difficult than forecasting stationary time series, even when the correct form of the model for the trends is known. Copyright The Econometric Society 2003.

Suggested Citation

  • Werner Ploberger & Peter C. B. Phillips, 2003. "Empirical Limits for Time Series Econometric Models," Econometrica, Econometric Society, vol. 71(2), pages 627-673, March.
  • Handle: RePEc:ecm:emetrp:v:71:y:2003:i:2:p:627-673

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    References listed on IDEAS

    1. 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.
    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(01), pages 95-131, April.
    3. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
    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. Phillips, Peter C.B., 2005. "Challenges of trending time series econometrics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(5), pages 401-416.
    4. 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.
    5. 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.
    6. 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.
    7. Durlauf, Steven N., 2001. "Manifesto for a growth econometrics," Journal of Econometrics, Elsevier, vol. 100(1), pages 65-69, January.
    8. 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.
    9. Patrick Marsh, "undated". "A Measure of Distance for the Unit Root Hypothesis," Discussion Papers 05/02, Department of Economics, University of York.
    10. 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, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
    11. 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.
    12. Peter C. B. Phillips, 2003. "Laws and Limits of Econometrics," Economic Journal, Royal Economic Society, vol. 113(486), pages 26-52, March.
    13. Aaron Schiff & Peter Phillips, 2000. "Forecasting New Zealand's real GDP," New Zealand Economic Papers, Taylor & Francis Journals, vol. 34(2), pages 159-181.
    14. 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.
    15. 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.
    16. repec:gam:jecnmx:v:5:y:2017:i:3:p:43-:d:112377 is not listed on IDEAS
    17. 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.
    18. 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.
    19. Offer Lieberman & Peter C.B. Phillips, 2017. "Hybrid Stochastic Local Unit Roots," Cowles Foundation Discussion Papers 2113, Cowles Foundation for Research in Economics, Yale University.

    More about this item

    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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