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Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition

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  • PERRON, Pierre
  • VODOUNOU, Cosme

Abstract

This paper considers various asymptotic approximations in the near-integrated firstorder autoregressive model with a non-zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous time approximation of Perron (1991). We assess how these alternative methods provide or not an adequate approximation to the finite-sample distribution of the least-squares estimator in a first-order autoregressive model. The results show that, when the initial condition is non-zero, Perron's (1991) continuous time approximation performs very well while the others only offer improvements when the initial condition is zero.

Suggested Citation

  • PERRON, Pierre & VODOUNOU, Cosme, 1998. "Asymptotic Approximations in the Near-Integrated Model with a Non-Zero Initial Condition," Cahiers de recherche 9815, Universite de Montreal, Departement de sciences economiques.
    • Handle: RePEc:mtl:montde:9815
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    1. Perron, Pierre, 1996. "The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors," Journal of Econometrics, Elsevier, vol. 70(2), pages 317-350, February.
    2. Phillips, Peter C B, 1977. "Approximations to Some Finite Sample Distributions Associated with a First-Order Stochastic Difference Equation," Econometrica, Econometric Society, vol. 45(2), pages 463-485, March.
    3. Bergstrom, A.R., 1984. "Continuous time stochastic models and issues of aggregation over time," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 20, pages 1145-1212, Elsevier.
    4. Phillips, P. C. B., 1987. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Econometric Theory, Cambridge University Press, vol. 3(1), pages 45-68, February.
    5. Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
    6. Perron, Pierre, 1991. "A Continuous Time Approximation to the Stationary First-Order Autoregressive Model," Econometric Theory, Cambridge University Press, vol. 7(2), pages 236-252, June.
    7. Knight, J.L. & Satchell, S.E., 1993. "Asymptotic Expansions for Random Walks with Normal Errors," Econometric Theory, Cambridge University Press, vol. 9(3), pages 363-376, June.
    8. Satchell, Stephen Ellwood, 1984. "Approximation to the Finite Sample Distribution for Nonstable First Order Stochastic Difference Equations," Econometrica, Econometric Society, vol. 52(5), pages 1271-1289, September.
    9. Perron, Pierre, 1989. "The Calculation of the Limiting Distribution of the Least-Squares Estimator in a Near-Integrated Model," Econometric Theory, Cambridge University Press, vol. 5(2), pages 241-255, August.
    10. Hisamatsu, Hiroyuki & Maekawa, Koichi, 1994. "The distribution of the Durbin-Watson statistic in integrated and near-integrated models," Journal of Econometrics, Elsevier, vol. 61(2), pages 367-382, April.
    11. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-779, May.
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    Cited by:

    1. Ulrich K. Müller, 2002. "Size and Power of Tests for Stationarity in Highly Autocorrelated Time Series," University of St. Gallen Department of Economics working paper series 2002 2002-26, Department of Economics, University of St. Gallen.
    2. Muller, Ulrich K., 2005. "Size and power of tests of stationarity in highly autocorrelated time series," Journal of Econometrics, Elsevier, vol. 128(2), pages 195-213, October.

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    More about this item

    Keywords

    Edgeworth exnsion; continuous-time asymotics; stochastic exnsion; distribution function; autoregressive model;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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