Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach

Abstract

We propose to extend the cointegration rank determination procedure of Robinson and Yajima [2002. Determination of cointegrating rank in fractional systems. Journal of Econometrics 106, 217-242] to accommodate both (asymptotically) stationary and nonstationary fractionally integrated processes as the common stochastic trends and cointegrating errors by applying the exact local Whittle analysis of Shimotsu and Phillips [2005. Exact local Whittle estimation of fractional integration. Annals of Statistics 33, . The proposed method estimates the cointegrating rank by examining the rank of the spectral density matrix of the dth differenced process around the origin, where the fractional integration order, d, is estimated by the exact local Whittle estimator. Similar to other semiparametric methods, the approach advocated here only requires information about the behavior of the spectral density matrix around the origin, but it relies on a choice of (multiple) bandwidth(s) and threshold parameters. It does not require estimating the cointegrating vector(s) and is easier to implement than regression-based approaches, but it only provides a consistent estimate of the cointegration rank, and formal tests of the cointegration rank or levels of confidence are not available except for the special case of no cointegration. We apply the proposed methodology to the analysis of exchange rate dynamics among a system of seven exchange rates. Contrary to both fractional and integer-based parametric approaches, which indicate at most one cointegrating relation, our results suggest three or possibly four cointegrating relations in the data. r

Nonparametric cointegration analysis of fractional systems with unknown integration orders

“…Of course, this aspect is a consequence of the nonparametric nature of the variance ratio test statistic, and is empirically important because misspeci…ed short-run dynamics leads to inconsistent estimation of the remainder of the model and hence to erroneous inferences on the cointegration rank. There is also no need to specify a bandwidth as in the nonparametric approach of Bierens (1997) or the recent semiparametric frequency domain approaches of Robinson & Yajima (2002), Chen & Hurvich (2003), and Nielsen & Shimotsu (2007). The approach proposed here is based on a family of test statistics indexed by a parameter d 1 de…ned below.…”

“…However, no formal testing procedure is available to …nd the rank (the testing procedure is only valid for r = 0), and the model selection procedure requires several bandwidth parameters to be chosen by the researcher which may induce additional uncertainty about the results obtained. Robinson & Yajima (2002) considered stationary variables with d < 1=2, and their approach was extended to accommodate both stationary and nonstationary data by Chen & Hurvich (2003, 2006 and Nielsen & Shimotsu (2007). In this section, an alternative and completely nonparametric approach to cointegration rank testing is analyzed.…”

“…There is also no need to specify a bandwidth as in the nonparametric approach of Bierens (1997) or the semiparametric frequency domain approaches of Robinson & Yajima (2002), Chen & Hurvich (2003), and Nielsen & Shimotsu (2007). All these parameters are tuning parameters which change the test statistics and possibly also the inference drawn from the data since the asymptotic distribution theory does not re ‡ect the particular choice of tuning parameters.…”

Semiparametric inference in multivariate fractionally cointegrated systems

“…In practice the presence and extent of cointegration has to be determined from the data, a difficult task in itself. As already mentioned, Hualde (2008) proposed an algorithm for choosing these values, which is based on the tests for equality of integration orders and cointegration proposed by Robinson and Yajima (2002), and extended by Nielsen and Shimotsu (2007) to cover nonstationary variables. A serious drawback of these procedures is the need (in addition to a bandwidth like our b) for user-chosen tuning numbers in both steps of the procedure.…”

A Wald test for the cointegration rank in nonstationary fractional systems

“…For stationary series, Robinson and Yajima (2002) analyzed testing procedures based on the eigenvalues of the estimated and normalized spectral density matrix around frequency zero after a preliminary step to partition the vector series into subsets with identical differencing parameters. The restriction imposed by cointegration on the spectral density matrix at zero frequency was also investigated by Nielsen (2004b) and by Nielsen and Shimotsu (2007) using alternative semiparametric memory estimates. A different route was explored by Chen and Hurvich (2003), who proposed to estimate the cointegrating relationships by the eigenvectors corresponding to the smallest eigenvalues of an averaged periodogram matrix of tapered, differenced observations.…”