460 Fractional Integration and Cointegration
- Excellent surveys can be found in Beran (1994), Baillie (1996), Doukhan, Oppenheim,
and Taqqu (2003) and Robinson (2003). - Much earlier, Hurst (1951) proposed the adjusted rescaled range, or R/S statistic. This spe-
cific estimator ofdcan be found in Mandelbrot and Wallis (1968) and its properties were
analyzed in Mandelbrot and Wallis (1969), Mandelbrot (1972, 1975) and Mandelbrot
and Taqqu (1979): see also Lo (1991) and Giraitiset al.(2003). - Multivariate methods of fractional integration (not involving cointegrating relation-
ships) have been examined by Gil-Alana (2003a, 2003b) and Nielsen (2004, 2005). - See also Hidalgo and Soulier (2004) and Hidalgo (2005) for recent developments in this
area. - The issue of fractional integration in the context of structural breaks has received
increasing attention in recent years. For a review, see Banerjee and Urga (2005). - Note that a normalization has been carried out in (10.5), the cointegration vector corre-
sponding to Engle and Granger’s (1987) definition being now(1,−ν)′. As demonstrated
by Phillips and Loretan (1991), (10.5)–(10.6) withγ=0,δ=1, represents “a typical
cointegrated system” in structural form. (10.5) could be regarded as a stochastic ver-
sion of the partial equilibriumyt−νxt, withu 1 t(−γ)representing deviations from this
equilibrium. (10.6) is a reduced form equation. - He did not analyze exactly the model (10.5)–(10.6), but a similar one whereytandxt
were covariance stationary long memory processes. - These conditions refer to the smoothness offand convergence rates of the estimates of
the nuisance parameters. - Chen and Hurvich (2003b) also propose estimators of the cointegrating parameter in a
system with deterministic trends, but their framework is much more specific than that
of Robinson and Iacone (2005), and their objective is different, because by means of
tapering and differentiating the data appropriately, they present a tapered NBLS which
is invariant with respect to deterministic polynomial trends in the series. - Kim and Phillips (2002) also provided a similar analysis to the one by Baillie and
Bollerslev (1994a), assuming also the memory of certain series of exchange rates to be
one.
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