Piotr Kębłowski

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In the paper, properties of small sample cointegration rank tests are compared. The Monte Carlo experiments conducted for different data generating processes and areas in the parametric space indicate that (i) the likelihood ratio test with Bartlett correction usually have the best properties, (ii) bootstrapping the Bartlett corrected test does not lead to improvement of test properties, (iii) size of asymptotic test and tests with degrees-of-freedom correction is usually heavily distorted, (iv) the Bartlett correction can lead to a small improvement of power, (v) correlation of error terms between stationary and non-stationary component of canonical form lead to a signifi cant increase of power and size of test, but only size of Bartlett corrected test and bootstrapped test converge to nominal size.


small sample inference, cointegration rank, canonical form, local alternatives


[1] Ahlgren N., Antell J., (2008), Bootstrap and Fast Double Bootstrap Tests of Cointegration Rank with Financial Time Series, Computational Statistics & Data Analysis, 52, 4754-4767.

[2] Ahlgren N., Antell J., (2013), The Power of Bootstrap Tests of Cointegration Rank, Computational Statistics, DOI 10.1007/s00180-013-0425-6.

[3] Ahlgren N., Juselius M., (2012), Tests For Cointegration Rank and the Initial Condition, Empirical Economics, 42, 667–691.

[4] Barndorff-Nielsen O. E., Hall P., (1988), On the Level-Error After Bartlett Adjustment of the Likelihood Ratio Statistic, Biometrika, 75, 374–378.

[5] Bartlett M. S., (1937), Properties of Suffi ciency and Statistical Tests, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 160, 268–282.

[6] Bartlett M. S., (1954), A Note on the Multiplying Factors for Various ?2 Approximations, Journal of the Royal Statistical Society. Series B, 16, 296–298.

[7] Beran R., (1988), Prepivoting Test Statistics. A Bootstrap View of Asymptotic Refi nements, Journal of the American Statistical Association, 83, 687–697.

[8] Bravo F., (1999), A Correction Factor for Unit Root Test Statistics, Econometric Theory, 15, 218–227.

[9] Cavaliere G., Rahbek A., Taylor A. M. R., (2010a), Testing for Cointegration in Vector Autoregressions with Non-stationary Volatility, Journal of Econometrics, 158, 7–24.

[10] Cavaliere G., Rahbek A., Taylor, A. M. R., (2010b), Cointegration Rank Testing under Conditional Heteroskedaticity, Econometric Theory, 26, 1719–1760.

[11] Cavaliere G., Rahbek A., Taylor A. M. R., (2012), Bootstrap Determination of the Co-integration Rank in Vector Autoregressive Model, Econometrica, 80, 1721–1740.

[12] Cheung Y.-W., Lai K. S., (1993), Finite-Sample Sizes of Johansen’s Likelihood Ratio Tests for Cointegration, Oxford Bulletin of Economics and Statistics, 55, 313–328.

[13] Cox D. R., Reid N., (1987), Parameter Orthogonality and Approximate Conditional Inference, Journal of the Royal Statistical Society. Series B, 49, 1–39.

[14] Cox D. R., Reid N., (1992), A Note on the Difference Between Profi le and Modifi ed Profile Likelihood, Biometrika, 79, 408–411.

[15] Cribari-Neto F., Cordeiro G. M., (1996), On Bartlett and Bartlett-type Corrections, Econometric Reviews, 15, 339–367.

[16] Davidson R., MacKinnon J. G., (2000), Bootstrap Tests: How Many Bootstraps?, Econometric Reviews, 19, 55–68.

[17] Edgeworth F. Y., (1905), The Law of Error, Transactions of the Cambridge Philosophical Society, 20, 36–65 i 113–141.

[18] Efron B., (1979), Bootstrap Methods: Another Look at the Jackknife, The Annals of Statistics, 7, 1–26.

[19] van Giersbergen N. P. A., (1996), Bootstraping the Trace Statistics in VAR Models. Monte Carlo Results and Applications, Oxford Bulletin of Economics and Statistics, 58, 391–408.

[20] Gonzalo J., Pitarakis J.-Y., (1999), Dimensionality Effect in Cointegration Analysis, w: R. F. Engle, H. White (ed.) Cointegration, Causality, And Forecasting. A Festschrift in Honour of C.W.J. Granger, Oxford University Press, New York, 212–229.

[21] Hansen H., Rahbek A., (2002), Approximate Conditional Unit Root Inference, Journal of Time Series Analysis, 23, 1–28.

[22] Haug A. A., (1996), Tests for Cointegration. A Monte Carlo Comparison, Journal of Econometrics, 71, 89–115.

[23] Harris R. I. D., Judge G., (1998), Small Sample Testing for Cointegration Using the Bootstrap Approach, Economics Letters, 58, 31–37.

[24] Hendry D. F., (1984), Monte Carlo Experimentation in Econometrics, w: Z. Griliches, M. D. Intriligator (ed.) Handbook of Econometrics, t. 2, 937–976.

[25] Jensen J. L., Wood A. T. A., (1997), On the Non-Existence of a Bartlett Correction for Unit Root Tests, Statistics & Probability Letters, 35, 181–187.

[26] Johansen S., (1996), Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford.

[27] Johansen S., (2002), A Small Sample Correction for the Test of Cointegrating Rank in the Vector Autoregressive Model, Econometrica, 70, 1929–1961.

[28] Johansen S., Hansen H., Fachin S., (2005), A Simulation Study of Some Functionals of Random Walks, materiał powielony.

[29] Kębłowski P., (2006a), Moc testu śladu z poprawką Bartletta w krótkiej próbie, w: A. Welfe (red.) Metody Ilościowe w Naukach Ekonomicznych, Ofi cyna Wydawnicza SGH, Warszawa, 47–59.

[30] Kębłowski P., (2006b), Small Sample Power of Bartlett Corrected Likelihood Ratio Test of Cointegration Rank, w: A. Welfe (red.) Proceedings of the Thirtieth Second International Conference Macromodels, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 73–85.

[31] Kębłowski P., (2009), Małopróbkowe wnioskowanie o rzędzie kointegracji, rozprawa doktorska, UŁ.

[32] Lawley D. N., (1956), A General Method for Approximating the Distribution of Likelihood Ratio Criteria, Biometrika, 43, 296–303.

[33] Nielsen B., (1997), Bartlett Correction of the Unit Root Test in Autoregressive Models, Biometrika, 84, 500–504.

[34] Omtzigt P., Fachin S., (2006), The Size and Power of Bootstrap and Bartlett-corrected Tests of Hypotheses on the Cointegration Vectors, Econometric Reviews, 25, 41–60.

[35] Phillips P. C. B., (1988), Regression Theory for Near-Integrated Time Series, Econometrica, 56, 1021–1043.

[36] Reimers H.-E., (1992), Comparisons of Tests for Multivariate Cointegration, Statistical Papers, 33, 335–359.

[37] Reinsel G. C., Ahn S. K., (1989), Likelihood Ratio Test for Unit Roots and Forecasting Properties in the Nonstationary Vector AR Model, materiał powielony.

[38] Reinsel G. C., Ahn S. K., (1992), Vector AR Models with Unit Roots and Reduced Rank Structure: Estimation, Likelihood Ratio Test, and Forecasting, Journal of Time Series Analysis, 13, 353–375.

[39] Swensen A. R., (2006), Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models, Econometrica, 74, 1699–1714.

[40] Swensen, A. R., (2009), Corrigendum to ’Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models, Econometrica, 77, 1703–1704.

[41] Swensen A. R., (2011), A Bootstrap Algorithm for Testing Cointegration Rank in VAR Models in the Presence of Stationary Variables, Journal of Econometrics, 165, 152–162.

[42] Toda H. Y., (1994), Finite Sample Properties of Likelihood Ratio Tests for Cointegrating Ranks when Linear Trends are Present, The Review of Economics and Statistics, 76, 66–79.

[43] Toda H. Y., (1995), Finite Sample Performance of Likelihood Ratio Tests for Cointegrating Ranks in Vector Autoregressions, Econometric Theory, 11, 1015–1032.

[44] Trenkler C., (2009), Bootstrapping Systems Cointegration Tests with a Prior Adjustment for Deterministic Terms, Econometric Theory, 25, 243–269.

[45] Wilks S. S., (1938), The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses, The Annals of Mathematical Statistics, 9, 60–62.

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