The Annals of Statistics

Semiparametric estimation of fractional cointegrating subspaces

Willa W. Chen and Clifford M. Hurvich

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We consider a common-components model for multivariate fractional cointegration, in which the s≥1 components have different memory parameters. The cointegrating rank may exceed 1. We decompose the true cointegrating vectors into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters. We estimate each cointegrating subspace separately, using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations, based on the first m Fourier frequencies, with m fixed. The angle between the true and estimated cointegrating subspaces is op(1). We use the cointegrating residuals corresponding to an estimated cointegrating vector to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to ∞ more slowly than n. We use these estimates to test for fractional cointegration and to consistently identify the cointegrating subspaces.

Article information

Ann. Statist., Volume 34, Number 6 (2006), 2939-2979.

First available in Project Euclid: 23 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15.

Fractional cointegration long memory tapering periodogram


Chen, Willa W.; Hurvich, Clifford M. Semiparametric estimation of fractional cointegrating subspaces. Ann. Statist. 34 (2006), no. 6, 2939--2979. doi:10.1214/009053606000000894.

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  • Anderson, T. W. and Das Gupta, S. (1963). Some inequalities on characteristic roots of matrices. Biometrika 50 522--524.
  • Barlow, J. L. and Slapničar, I. (2000). Optimal perturbation bounds for the Hermitian eigenvalue problem. Linear Algebra Appl. 309 19--43.
  • Chen, W. W. and Hurvich, C. M. (2003a). Estimating fractional cointegration in the presence of polynomial trends. J. Econometrics 117 95--121.
  • Chen, W. W. and Hurvich, C. M. (2003b). Semiparametric estimation of multivariate fractional cointegration. J. Amer. Statist. Assoc. 463 629--642.
  • Davis, C. and Kahan, W. M. (1970). The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7 1--46.
  • Hausman, J. (1978). Specification tests in econometrics. Econometrica 46 1251--1271.
  • Huadle, J. and Robinson, P. (2002). Root-$n$-consistent estimation of weak fractional cointegration. Working Paper WP08/02, School of Economics and Business Administration, Univ. Navarra.
  • Hurvich, C. M. and Chen, W. W. (2000). An efficient taper for potentially overdifferenced long-memory time series. J. Time Ser. Anal. 21 155--180.
  • Hurvich, C. M., Moulines, E. and Soulier, P. (2002). The FEXP estimator for potentially non-stationary linear time series. Stochastic Process. Appl. 97 307--340.
  • Künsch, H. R. (1987). Statistical aspects of self-similar processes. In Proc. 1st World Congress of the Bernoulli Society (Tashkent, 1986) 1 67--74. VNU Sci. Press, Utrecht.
  • Lobato, I. N. (1999). A semiparametric two-step estimator in a multivariate long memory model. J. Econometrics 90 129--153.
  • Magnus, J. R. and Neudecker, H. (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester.
  • Marinucci, D. and Robinson, P. M. (2001). Semiparametric fractional cointegration analysis. J. Econometrics 105 225--247.
  • Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763--765.
  • Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.
  • Robinson, P. M. (1994). Semiparametric analysis of long-memory time series. Ann. Statist. 22 515--539.
  • Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630--1661.
  • Robinson, P. M. and Marinucci, D. (2001). Narrow-band analysis of nonstationary processes. Ann. Statist. 29 947--986.
  • Robinson, P. M. and Marinucci, D. (2003). Semiparametric frequency-domain analysis of fractional cointegration. In Time Series with Long Memory (P. M. Robinson, ed.) 334--373. Oxford Univ. Press.
  • Robinson, P. M. and Yajima, Y. (2002). Determination of cointegrating rank in fractional systems. J. Econometrics 106 217--241.
  • Siskind, V. (1972). Second moments of inverse Wishart-matrix elements. Biometrika 59 690--691.
  • Stewart, G. W. and Sun, J. (1990). Matrix Perturbation Theory. Academic Press, San Diego.
  • Velasco, C. (2003). Gaussian semi-parametric estimation of fractional cointegration. J. Time Ser. Anal. 24 345--378.
  • Wedin, P. A. (1983). On angles between subspaces of a finite dimensional inner product space. In Matrix Pencils. Lecture Notes in Math. 973 263--285. Springer, New York.