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Computing Numerical Distribution Functions in Econometrics


  • James G. MacKinnon

    () (Queen's University)


Many test statistics in econometrics have asymptotic distributions that cannot be evaluated analytically. In order to conduct asymptotic inference, it is therefore necessary to resort to simulation. Techniques that have commonly been used yield only a small number of critical values, which can be seriously inaccurate. In contrast, the techniques discussed in this paper yield enough information to plot the distributions of the test statistics or to calculate P values, and they can yield highly accurate results. These techniques are used to obtain asymptotic critical values for a test recently proposed by Kiefer, Vogelsang, and Bunzel (2000) for testing linear restrictions in linear regression models. A program to compute P values for this test is available from the author's web site.

Suggested Citation

  • James G. MacKinnon, 2001. "Computing Numerical Distribution Functions in Econometrics," Working Papers 1037, Queen's University, Department of Economics.
  • Handle: RePEc:qed:wpaper:1037

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    References listed on IDEAS

    1. MacKinnon, James G & Haug, Alfred A & Michelis, Leo, 1999. "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 14(5), pages 563-577, Sept.-Oct.
    2. Neil R. Ericsson & James G. MacKinnon, 2002. "Distributions of error correction tests for cointegration," Econometrics Journal, Royal Economic Society, vol. 5(2), pages 285-318, June.
    3. Nicholas M. Kiefer & Timothy J. Vogelsang & Helle Bunzel, 2000. "Simple Robust Testing of Regression Hypotheses," Econometrica, Econometric Society, vol. 68(3), pages 695-714, May.
    4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    5. Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 39(3), pages 106-135.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    7. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    8. MacKinnon, James G, 1996. "Numerical Distribution Functions for Unit Root and Cointegration Tests," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(6), pages 601-618, Nov.-Dec..
    9. MacKinnon, James G, 1994. "Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 167-176, April.
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    Cited by:

    1. Fan, Yanqin & Gençay, Ramazan, 2010. "Unit Root Tests With Wavelets," Econometric Theory, Cambridge University Press, vol. 26(05), pages 1305-1331, October.
    2. Peter Sephton, 2008. "Critical values of the augmented fractional Dickey–Fuller test," Empirical Economics, Springer, vol. 35(3), pages 437-450, November.

    More about this item


    unit root test; cointegration test; simulation; critical values; distribution functions; numerical distribution;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • 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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