Frequency domain principal components estimation of fractionally cointegrated processes

Abstract

This study introduces a new frequency domain principal components estimator of the cointegration space and the loading matrix for the common factors for fractionally cointegrated long memory processes. A Monte Carlo simulation exercise reveals that the proposed estimator has already good properties with relatively small sample sizes.

Acknowledgements

The paper was presented at ESEM 2003, Stockholm, 19–24 August 2003. The author is grateful to A. Beltratti, J. Rodrigues and to conference participants, especially to R.T. Baillie, J. Davidson and J. Hualde, for comments and discussion.

Notes

Robinson, and Yajima (2002) have demonstrated this result as the frequency tends to zero. Robinson and Marinucci (1998) have also shown that this result holds for the series in levels. The two results are related since for the I(d) vector process x _t f _x (ω) ∼ ω ^{−2 d} G ω → 0⁺ and

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 The study assumes that all the long memory processes share the same order of integration d, ad the I(d)–I(0) case is considered. Hence, the definition of fractional cointegration follows Engle and Granger (1987), namely p long memory processes are fractionally cointegrated if there exists at least one linear combination which is I(0), i.e. x ₁,  _t , x ₂, _t , … , x _p, t  ∼ CI(d, d) d ∈ (0, 0.5). While for long memory processes other definitions of cointegration can be envisaged (Robinson and Yajima, 2002), the case considered in this paper is likely to be of empirical relevance, and is an important benchmark in any case. However, it is shown that the results extend also to the general case CI(d, b) b > 0 d − b > 0, i.e. where the cointegrating residuals still show long memory, although their degree of integration is lower than the one of the actual series.

Note that the same results hold for the case in which the u vector is I(b) b > 0 d − b > 0, since Δ ^d u ∼ I(b − d).

 The reduced rank of the spectral matrix for the differenced series was firstly noted by Phillips (1986) and Phillips and Ouliaris (1988) for the I(1) case. Phillips and Ouliaris (1988) have proposed a cointegrating rank test based on the number of non-zero eigenvalues of the spectral density matrix at the zero frequency, which provides the number of common trends k, and therefore the number of cointegration relationships r = p − k. Robinson and Marinucci (1998) and Robinson and Yajima (2002) have shown that a similar result holds for the I(d) case (0<d<0.5) for the series in levels as the frequency tends to zero, i.e. given the I(d) vector process x _t , f _x (ω) ∼ ω ^{−2 d} G ω → 0⁺, where G = (ΘΘ′/2π). Differently from what is done in these latter papers, by fractionally differencing one works with I(0) series, and therefore with well defined spectral density functions at the zero frequency. This allows one to establish results exactly at the zero frequency. A fractional cointegrating rank test, based on the number of non-zero eigenvalues of the matrix G, has been suggested by Robinson and Yajima (2002), along the lines of Phillips and Ouliaris (1988).

 Note that this decomposition is always possible since f(0) is positive semidefinite, so that the non-null eigenvalues are real and positive.

 Since for an I(d) vector process x _t f _x (ω) ∼ ω ^{−2 d} G ω → 0⁺ the eigenvectors of f _x (ω) ω → 0⁺ are the same as the eigenvectors of G, f(0), or ΘΘ′. Therefore, (Λ ^{1 / 2} Q′)* _i x′ are the orthogonal linear combinations of the series in levels characterized by the largest (long-run) variances. These linear combinations bear the interpretation of long-run principal components.

Phillips (1986) and Phillips and Ouliaris (1998) have previously shown that the cointegrating vectors are the eigenvectors of the spectral matrix of the innovations at the zero frequency associated with the zero roots for the CI(1, 0) case, following a different approach from the one discussed in the paper.

Note that the imposition of the identification conditions is always possible. In the case where the upper submatrix of dimension k is singular, the identify matrix can be positioned differently in the factor loading matrix, without any consequences for the identification of the model.

 Note that a lower triangular ρ matrix does not imply any recursive structure for the factor loading matrix, since the way the common trends affect the vector x _t is determined by the identification condition imposed on Q*. See Warne (1993).

 The u _t vector is I(b) when the cointegrating residuals are I(b) or when the largest order of fractional integration of the cointegrating residuals is I(b). Note in fact that the u _t vector is computed as a linear combination of the cointegrating residuals.

 See Morana (2004b) for additional results and a proof of consistency of the estimator of the cointegration space.

 For reasons of space we only report results for d = 0.45 and T = 100. See the preprint version of this paper (Morana 2004b) for additional results.