Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T15:36:27.778Z Has data issue: false hasContentIssue false
  • English
  • Français

COINTEGRATION RANK TESTING UNDERCONDITIONAL HETEROSKEDASTICITY

Published online by Cambridge University Press:  22 March 2010

Giuseppe Cavaliere ,
Anders Rahbek  and
A.M. Robert Taylor
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We analyze the properties of the conventionalGaussian-based cointegrating rank tests of Johansen(1996, Likelihood-Based Inference in CointegratedVector Autoregressive Models) in the case where thevector of series under test is driven by globallystationary, conditionally heteroskedastic(martingale difference) innovations. We firstdemonstrate that the limiting null distributions ofthe rank statistics coincide with those derived byprevious authors who assume either independent andidentically distributed (i.i.d.) or (strict andcovariance) stationary martingale differenceinnovations. We then propose wild bootstrapimplementations of the cointegrating rank tests anddemonstrate that the associated bootstrap rankstatistics replicate the first-order asymptotic nulldistributions of the rank statistics. We show thatthe same is also true of the corresponding ranktests based on the i.i.d. bootstrap of Swensen(2006, Econometrica 74, 1699–1714).The wild bootstrap, however, has the importantproperty that, unlike the i.i.d. bootstrap, itpreserves in the resampled data the pattern ofheteroskedasticity present in the original shocks.Consistent with this, numerical evidence suggeststhat, relative to tests based on the asymptoticcritical values or the i.i.d. bootstrap, the wildbootstrap rank tests perform very well in smallsamples under a variety of conditionallyheteroskedastic innovation processes. An empiricalapplication to the term structure of interest ratesis given.

Information

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 6 , December 2010 , pp. 1719 - 1760
Copyright
Copyright © Cambridge University Press 2010

Footnotes

Parts of this paper were written while Cavaliereand Taylor both visited CREATES, whose hospitalityis gratefully acknowledged. We are grateful tothree anonymous referees and to Peter Phillips,Søren Johansen, Anders Swensen, and CarstenTrenkler for their helpful and constructivecomments on earlier versions of this paper. Wealso thank Steve Leybourne for providing us withthe data used in Section 6. Cavaliere and Rahbekthank the Danish Social Sciences Research Council,project 2114-04-001, for continuing financialsupport. Cavaliere also acknowledges financialsupport from MIUR PRIN 2007 grants. Rahbek is alsoaffiliated with CREATES, funded by the DanishNational Research Foundation.

References

REFERENCES

Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.CrossRefGoogle Scholar
Burridge, P. & Taylor, A.M.R. (2001) On regression-based tests for seasonal unit roots in the presence of periodic heteroscedasticity. Journal of Econometrics 104, 91–117.CrossRefGoogle Scholar
Campbell, J.Y. & Shiller, R.J. (1987) Cointegration and tests of present value models. Journal of Political Economy 95, 1062–1088.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2009) Co-integration Rank Testing under Conditional Heteroskedasticity. Granger Center Discussion Paper 09/02.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, A.M.R. (2010) Testing for cointegration in vector autoregressions with non-stationary volatility. Journal of Econometrics, forthcoming.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with nonstationary volatility. Econometric Theory 24, 43–71.CrossRefGoogle Scholar
Cheng, X. & Phillips, P.C.B. (2008) Cointegrating Rank Selection in Models with Time-Varying Variance. Cowles Foundation Discussion Paper 1688.Google Scholar
Cheng, X. & Phillips, P.C.B. (2009) Semiparametric cointegrating rank selection. Econometrics Journal 12, S83–S104.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Diebold, F.X., Ji, L., & Li, C. (2006) A three-factor yield curve model: Non-affine structure, systematic risk sources, and generalized duration. In Klein, L.R. (ed.), Long-Run Growth and Short-Run Stabilization: Essays in Memory of Albert Ando, pp. 240–274. Edward Elgar.Google Scholar
Elliott, G. (1998) The robustness of cointegration methods when regressors almost have unit roots. Econometrica 66, 149–158.CrossRefGoogle Scholar
Engle, R.F. (1990) Discussion: Stock market volatility and the crash of ’87. Review of Financial Studies 3, 103–106.CrossRefGoogle Scholar
Giese, J. (2006) Level, Slope, Curvature: Characterising the Yield Curve’s Derivatives in a Cointegrated VAR Model. Working paper, Nuffield College.Google Scholar
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.CrossRefGoogle Scholar
Glosten, L.R., Jaganathan, R., & Runkle, D.E. (1993) On the relation between the expected value and the volatility of nominal excess returns on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2004) Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89–120.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2007) Asymptotic and bootstrap inference for AR(∞) processes with conditional heteroskedasticity. Econometric Reviews 26, 609–641.CrossRefGoogle Scholar
Hall, A.D., Anderson, H., & Granger, C.W.J. (1992) A cointegration analysis of treasury bill yields. Review of Economics and Statistics 74, 116–26.CrossRefGoogle Scholar
Hannan, E.J. & Heyde, C.C. (1972) On limit theorems for quadratic functions of discrete time series. Annals of Mathematical Statistics 43, 2058–2066.CrossRefGoogle Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (2000) Testing for structural change in conditional models. Journal of Econometrics 97, 93–115.CrossRefGoogle Scholar
Harris, R.I.D. & Judge, G. (1998) Small sample testing for cointegration using the bootstrap approach. Economics Letters 58, 31–37.CrossRefGoogle Scholar
Inoue, A. & Kilian, L. (2002) Bootstrapping autoregressive processes with possible unit roots. Econometrica 70, 377–391.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2007) On the law of large numbers for (geometrically) ergodic Markov chains. Econometric Theory 23, 761–766.CrossRefGoogle Scholar
Johansen, S. (1996), Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. 2nd ed. Oxford University Press.Google Scholar
Johansen, S. (2002) A small sample correction of the test for cointegrating rank in the vector autoregressive model. Econometrica 70, 1929–1961.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2005) Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.10.1017/S0266466605050474CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2009) Aymptotics of the QMLE for non-linear ARCH models. Journal of Time Series Econometrics. Berkeley Electronic Press; http://www.bepress.com/itse/vol1/iss1/art2.CrossRefGoogle Scholar
Lee, T.-H. & Tse, Y. (1996) Cointegration tests with conditional heteroskedasticity. Journal of Econometrics 73, 401–410.CrossRefGoogle Scholar
Liu, R.Y. (1988) Bootstrap procedure under some non-i.i.d. models. Annals of Statistics 16, 1696–1708.CrossRefGoogle Scholar
MacKinnon, J.G., Haug, A.A., & Michelis, L. (1999) Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics 14, 563–577.3.0.CO;2-R>CrossRefGoogle Scholar
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255–285.CrossRefGoogle Scholar
Mantalos, P. & Shukur, G. (2001) Bootstrapped Johansen tests for cointegration relationships: A graphical analysis. Journal of Statistical Computation and Simulation 68, 351–371.CrossRefGoogle Scholar
Nelson, D.B. (1991), Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–369.CrossRefGoogle Scholar
Nielsen, B. & Rahbek, A. (2000) Similarity issues in cointegration analysis. Oxford Bulletin of Economics and Statistics 62, 5–22.CrossRefGoogle Scholar
Paparoditis, E.P. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71, 813–855.10.1111/1468-0262.00427CrossRefGoogle Scholar
Park, J.Y. (2003) Bootstrap unit root tests. Econometrica 71, 1845–1895.CrossRefGoogle Scholar
Pesavento, E. (2004) An analytical evaluation of the power of tests for the absence of cointegration. Journal of Econometrics 122, 349–384.CrossRefGoogle Scholar
Rahbek, A., Hansen, E., & Dennis, J.G. (2002) ARCH innovations and their impact on cointegration rank testing. Downloadable from http://www.math.ku.dk/~rahbek/.Google Scholar
Swensen, A.R. (2006) Bootstrap algorithms for testing and determining the cointegration rank in VAR models. Econometrica 74, 1699–1714.CrossRefGoogle Scholar
Trenkler, C. (2009) Bootstrapping systems cointegration tests with a prior adjustment for deterministic terms. Econometric Theory 25, 243–269.10.1017/S0266466608090087CrossRefGoogle Scholar
van der Weide, R. (2002) GO-GARCH: A multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics 17, 549–564.CrossRefGoogle Scholar
van Giersbergen, N.P.A. (1996), Bootstrapping the trace statistics in VAR models: Monte Carlo results and applications. Oxford Bulletin of Economics and Statistics 58, 391–408.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap, and other resampling methods. Annals of Statistics 14, 1261–1295.Google Scholar
Xu, K.-L. (2008), Bootstrapping autoregression under non-stationary volatility. Econometrics Journal 11, 1–26.10.1111/j.1368-423X.2008.00235.xCrossRefGoogle Scholar