In this paper we present new results for the frequency domain principal components estimator of the cointegration space for stationary long memory processes of Morana [Appl. Econ. Lett. 11 (2004) 837], concerning asymptotic properties, identification of the cointegration space and the linkage with the frequency domain least-squares estimator. An application of the approach to stock market volatility data shows that the methodology can effectively be employed for the modelling of long-run relationships, which could not be handled using the standard I(1)I(1)–I(0)I(0) cointegration approach.
Recently several contributions to the estimation of fractionally cointegrated processes have been made in the literature. Contributions have been concerned with both the estimation of single and multiple cointegrating vectors for mean reverting and non-mean reverting fractionally integrated processes and testing for cointegration. As far as the stationary long memory case is concerned, the narrow band frequency domain least-squares estimator (FDLS) has initially been proposed by Robinson [1] for the estimation of single cointegrating vectors, with cointegrating residuals possibly also showing long memory. The estimator has been shown to have superior asymptotic properties relative to the ordinary least-squares estimator (OLS), being consistent also in the presence of correlation between the cointegrating residuals and the regressors. Asymptotic normality of the estimator has been proved by Christensen and Nielsen [2] for the case of a collective memory of the regressor and the cointegrating residuals below the long memory stationary boundary. As an alternative to the FDLS estimator, Hualde and Robinson [3] have proposed a root-n consistent estimator (RNC) with Gaussian limiting distribution in the case of weak cointegration, which amounts to OLS estimation of the cointegrating vector using a dynamic relation for the variables of interest, close to the error correction representation. A different approach has been proposed by Nielsen [4], consisting of two-step local Whittle joint estimation (2SLW) of the order of fractional differencing (of the actual series and the cointegrating residuals) and of the cointegrating vector. The proposed estimator is consistent and asymptotically normal, and relatively asymptotically more efficient than the FDLS estimator, although showing the same rate of convergence of this latter estimator. On the other hand, Morana [5] has proposed system estimation for the case of multiple cointegrating vectors, based on a frequency domain principal components approach (FDPC), implemented through the eigenvalue–eigenvector decomposition of the long-run variance–covariance matrix of the fractionally differenced processes. An approach to identify and estimate the common long memory factor structure and to implement a persistent–non-persistent decomposition has also been proposed by Morana [5], and existing theoretical results have been employed to demonstrate the consistency of the estimator of the common long memory factor structure. Monte Carlo simulations comparing the small sample performance of the FDPC, FDLS, 2SLW and RNC estimators for the stationary long memory case are available in Refs. [5] (FDPC and FDLS), [4] (2SLW), and [3] (RNC and OLS). The results suggest that all the estimators are unbiased when the regressor and the cointegrating residuals are not contemporaneously correlated, while the FDPC, FDLS, RNC, and OLS estimators are biased when this property does not hold. In particular, the bias has been found to increase with the degree of correlation between the regressor and the cointegrating residual, and to fall as the strength of cointegration increases, i.e., as the collective memory decreases. Yet, fully unbiased estimation can be achieved by the FDPC and FDLS estimators by applying bias correction factors computed as suggested in Ref. [5]. On the other hand, the performance of the 2SLW estimator when the regressors and the cointegrating residuals are correlated has not yet been assessed. Finally, the performance of the RNC estimator has been contrasted with that of the OLS estimator. Moreover, while the feasible RNC estimator tends to perform better in terms of bias than the OLS estimator for samples larger than hundred observations, in terms of Monte Carlo standard deviation the RNC estimator tends to perform very poorly, while the FDLS estimator tends to be more efficient than the FDPC estimator. 1
In this paper, we present further results concerning the FDPC estimator, establishing the linkage between the FDPC and FDLS estimators in the case of a known single cointegrating vector. This also allows us to further discuss the asymptotic properties of the FDPC estimator of the cointegration space, establishing its consistency also in the presence of correlation between the regressor and the cointegrating residual. Moreover, from an empirical point of view we present an application to stock market volatility data. The application shows that the methodology can effectively be employed for the modelling of long-run relationships relating stationary long memory processes, which could not be handled using the standard I(1)I(1)–I(0)I(0) cointegration approach. The empirical exercise is also important since so far most applications have been concerned with the non-stationary long memory case, exception being Morana [6] and Beltratti and Morana [7].
The plan of the paper is as follows. In Section 2 we present the frequency domain principal components approach to the estimation of the cointegration space and the common long memory factor structure of Morana [5], presenting new results concerning asymptotic properties, identification of the cointegration space and the linkage with the frequency domain least-squares estimator. Finally, in Section 3 we present the empirical application.
