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Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes


  • Dufour, Jean-Marie
  • Torres, Olivier


In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.
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  • Dufour, Jean-Marie & Torres, Olivier, 2000. "Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes," Journal of Econometrics, Elsevier, vol. 99(2), pages 255-289, December.
  • Handle: RePEc:eee:econom:v:99:y:2000:i:2:p:255-289

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    References listed on IDEAS

    1. Dufour, Jean-Marie & Kiviet, Jan F., 1996. "Exact tests for structural change in first-order dynamic models," Journal of Econometrics, Elsevier, vol. 70(1), pages 39-68, January.
    2. Kiviet, Jan F. & Dufour, Jean-Marie, 1997. "Exact tests in single equation autoregressive distributed lag models," Journal of Econometrics, Elsevier, vol. 80(2), pages 325-353, October.
    3. Savin, N.E. & Wurtz, A., 1996. "The Effect of Nuisance Parameters on the Power of LM Tests in Logit and Probit Models," Working Papers 96-05, University of Iowa, Department of Economics.
    4. Craig Burnside & Martin Eichenbaum, 1994. "Small sample properties of generalized method of moments based Wald tests," Working Paper Series, Macroeconomic Issues 94-12, Federal Reserve Bank of Chicago.
    5. Dufour, Jean-Marie, 1989. "Nonlinear Hypotheses, Inequality Restrictions, and Non-nested Hypotheses: Exact Simultaneous Tests in Linear Regressions," Econometrica, Econometric Society, vol. 57(2), pages 335-355, March.
    6. Burnside, Craig & Eichenbaum, Martin S, 1996. "Small-Sample Properties of GMM-Based Wald Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(3), pages 294-308, July.
    7. Miyazaki, Shigetaka & Griffiths, William E., 1984. "The properties of some covariance matrix estimators in linear models with AR(1) errors," Economics Letters, Elsevier, vol. 14(4), pages 351-356.
    8. Campbell, Bryan & Dufour, Jean-Marie, 1997. "Exact Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(1), pages 151-173, February.
    9. Nankervis, John C & Savin, N E, 1996. "The Level and Power of the Bootstrap t Test in the AR(1) Model with Trend," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 161-168, April.
    10. Jean-Marie Dufour, 1997. "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, Econometric Society, vol. 65(6), pages 1365-1388, November.
    11. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
    12. DeJong, David N. & Nankervis, John C. & Savin, N. E. & Whiteman, Charles H., 1992. "The power problems of unit root test in time series with autoregressive errors," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 323-343.
    13. Savin, N.E., 1984. "Multiple hypothesis testing," Handbook of Econometrics,in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 14, pages 827-879 Elsevier.
    14. Marc Hallin & Jean-Marie Dufour & Ivan Mizera, 1998. "Generalized run tests for heteroscedastic time series," ULB Institutional Repository 2013/2077, ULB -- Universite Libre de Bruxelles.
    15. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
    16. Park, Rolla Edward & Mitchell, Bridger M., 1980. "Estimating the autocorrelated error model with trended data," Journal of Econometrics, Elsevier, vol. 13(2), pages 185-201, June.
    17. Maasoumi, Esfandiar, 1992. "Fellow's opinion : Rules of thumb and pseudo-science," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 1-4.
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    Cited by:

    1. Dufour, Jean-Marie & Khalaf, Lynda & Bernard, Jean-Thomas & Genest, Ian, 2004. "Simulation-based finite-sample tests for heteroskedasticity and ARCH effects," Journal of Econometrics, Elsevier, vol. 122(2), pages 317-347, October.
    2. Luger, Richard, 2003. "Exact non-parametric tests for a random walk with unknown drift under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 115(2), pages 259-276, August.
    3. Dufour, Jean-Marie & Khalaf, Lynda, 2002. "Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions," Journal of Econometrics, Elsevier, vol. 106(1), pages 143-170, January.
    4. Jean-Marie Dufour & Abdeljelil Farhat, 2001. "Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions," CIRANO Working Papers 2001s-56, CIRANO.
    5. Gossner, Olivier & Schlag, Karl H., 2013. "Finite-sample exact tests for linear regressions with bounded dependent variables," Journal of Econometrics, Elsevier, vol. 177(1), pages 75-84.

    More about this item

    JEL classification:

    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C42 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Survey Methods


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