Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-27T15:44:33.820Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTRIBUTION-FREE TESTS OF FRACTIONAL COINTEGRATION

Published online by Cambridge University Press:  06 September 2007

Javier Hualde  and
Carlos Velasco
Javier Hualde
Affiliation:
Universidad de Navarra
Carlos Velasco
Affiliation:
Universidad Carlos III de Madrid
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series. Our test statistics have an asymptotic chi-square distribution under the null and rely on generalized least squares–type of corrections that control for the short-run correlation of the weak dependent components of the fractionally integrated processes. We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions that are standard in the literature and, in particular, being able to consider simultaneously (asymptotically) stationary or nonstationary processes. Relatively weak conditions on the corresponding short-run and memory parameter estimates are assumed. The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree. Finite-sample properties of the tests are analyzed by means of a Monte Carlo experiment.We thank Helmut Lütkepohl and two referees for helpful comments and suggestions. We also thank participants at the NSF/NBER Time Series Conference at the University of Heidelberg, Germany, at the Unit Root and Cointegration Testing Conference at the University of Algarve, Faro, Portugal, and seminar participants at the Universidad de Navarra and Ente Luigi Einaudi for helpful comments. Javier Hualde's research is supported by the Spanish Ministerio de Educación y Ciencia through Juan de la Cierva and Ramón y Cajal contracts and ref. SEJ2005-07657/ECON. Carlos Velasco's research is supported by the Spanish Ministerio de Educación y Ciencia, ref. SEJ2004-04583/ECON.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 1 , February 2008 , pp. 216 - 255
Copyright
© 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Beran, J. (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. Journal of the Royal Statistical Society, Series B 57, 659–672.Google Scholar
Breitung, J. & U. Hassler (2002) Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics 110, 167–185.Google Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 59–66.Google Scholar
Chen, W.W. & C.M. Hurvich (2003) Semiparametric estimation of multivariate fractional cointegration. Journal of the American Statistical Association 463, 629–642.Google Scholar
Chen, W.W. & C.M. Hurvich (2006) Semiparametric estimation of fractional cointegrating subspaces. Annals of Statistics 34, 2939–2979.Google Scholar
Dueker, M. & R. Startz (1998) Maximum-likelihood estimation of fractional cointegration with an application to U.S.and Canadian bond rates. Review of Economics and Statistics 80, 420–426.Google Scholar
Dunsmuir, W. (1979) A central limit theorem for parameter estimation in stationary vector time series and its applications to models for a signal observed with noise. Annals of Statistics 7, 490–506.Google Scholar
Dunsmuir, W. & E.J. Hannan (1976) Vector linear time series models. Advances in Applied Probability 8, 339–364.Google Scholar
Engle, R.F. & C.W.J. Granger (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.Google Scholar
Flôres, R.G. & A. Szafarz (1996) An enlarged definition of cointegration. Economics Letters 50, 193–195.Google Scholar
Geweke, J. & S. Porter-Hudak (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4, 221–238.Google Scholar
Gil-Alaña, L.A. (2003) Multivariate tests of fractionally integrated hypothesis. South African Statistical Journal 37, 1–28.Google Scholar
Hassler, U. & J. Breitung (2006) A residual-based LM-type test against fractional cointegration. Econometric Theory 22, 1091–1111.Google Scholar
Hassler, U., F. Marmol, & C. Velasco (2006) Residual log-periodogram inference for long-run relationships. Journal of Econometrics 130, 165–207.Google Scholar
Hualde, J. & P.M. Robinson (2004) Semiparametric Estimation of Fractional Cointegration. Preprint, Universidad de Navarra.
Hualde, J. & P.M. Robinson (2005) Semiparametric Inference in Multivariate Fractionally Cointegrated Systems. Preprint, Universidad de Navarra.
Hualde, J. & P.M. Robinson (2007) Root-n-consistent estimation of weak fractional cointegration. Journal of Econometrics 140, 450–484.Google Scholar
Johansen, S. (1988) Statistical analysis of co-integration vectors. Journal of Economic Dynamics and Control 12, 231–254.Google Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press.
Kim, C.S. & P.C.B. Phillips (2000) Fully Modified Estimation of Fractional Co-integration Models. Preprint, Yale University.
Künsch, H.R. (1987) Statistical aspects of self-similar processes. In Y.A. Prohorov & V.V. Sazonov (eds.), Proceedings of the 1st World Congress of the Bernoulli Society, pp. 67–74. VNU Science Press.
Kurtz, T.G. & P. Protter (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.Google Scholar
Lasak, K. (2005) Likelihood Based Testing for Fractional Cointegration. Preprint, Universitat Autònoma de Barcelona.
Lobato, I.G. (1999) A semiparametric two-step estimation on a multivariate long memory model. Journal of Econometrics 90, 129–153.Google Scholar
Marinucci, D. & P.M. Robinson (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.Google Scholar
Marinucci, D. & P.M. Robinson (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103–120.Google Scholar
Marinucci, D. & P.M. Robinson (2001) Semiparametric fractional cointegration analysis. Journal of Econometrics 105, 225–247.Google Scholar
Marmol, F. & C. Velasco (2004) Consistent testing of cointegrating relationships. Econometrica 72, 1809–1844.Google Scholar
Park, J.Y., S. Ouliaris, & B. Choi (1988) Spurious Regressions and Tests for Cointegration. CAE Working paper 88–07, Cornell University.
Phillips, P.C.B. & S. Ouliaris (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.Google Scholar
Robinson, P.M. (1994) Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437.Google Scholar
Robinson, P.M. (1995a) Gaussian semiparametric estimation of long-range dependence. Annals of Statistics 23, 1630–1661.Google Scholar
Robinson, P.M. (1995b) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.Google Scholar
Robinson, P.M. (2005a) The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283–300.Google Scholar
Robinson, P.M. (2005b) Diagnostic Testing for Cointegration. Preprint, London School of Economics.
Robinson, P.M. & J. Hualde (2003) Cointegration in fractional systems with unknown integration orders. Econometrica 71, 1727–1766.Google Scholar
Robinson, P.M. & F. Iacone (2005) Cointegration in fractional systems with deterministic trends. Journal of Econometrics 129, 263–298.Google Scholar
Robinson, P.M. & D. Marinucci (2001) Narrow-band analysis of nonstationary processes. Annals of Statistics 29, 947–986.Google Scholar
Robinson, P.M. & D. Marinucci (2003) Semiparametric frequency-domain analysis of fractional cointegration. In P.M. Robinson (ed.), Time Series with Long Memory, pp. 334–373. Oxford University Press.
Robinson, P.M. & Y. Yajima (2002) Determination of cointegrating rank in fractional systems. Journal of Econometrics 106, 217–241.Google Scholar
Velasco, C. (2003) Gaussian semiparametric estimation of fractional cointegration. Journal of Time Series Analysis 24, 345–378.Google Scholar
Velasco, C. (2004) The Periodogram of Fractional Processes. Preprint, Universidad Carlos III de Madrid.
Zygmund, A. (1977) Trigonometric Series. Cambridge University Press.