454 Fractional Integration and Cointegration
to this type of cointegration, denoted stationary cointegration, appears in Robin-
son (1994b). Robinson and Marinucci (2003) investigated further this situation,
indicating that the phenomenon of cointegration between stationary variables
had recently emerged in finance, and emphasizing the difficulty of distinguish-
ing between a unit root process and a stationary long memory process with an
autoregressive part having a root near the unit circle.
Andersenet al.(2001) examined “realized” daily equity return volatilities and
correlations obtained from high-frequency transaction prices on individual stocks
in the Dow Jones Industrial Average. They provided evidence of long memory
for certain time series of logarithmic standard deviations and correlations, and
stressed the evidence of comovements in volatility across assets. Christensen and
Nielsen (2006) make a similar point and argue for the existence of stationary
cointegration between the volatility implied in option prices and the subsequent
realized return volatility of the underlying asset, since, in their view, the observ-
ables (log-volatilities) were integrated of order between 0.35 and 0.40, whereas the
cointegrating error seemed weakly dependent. By using an NBLS estimator, they
obtained a much higher value for the estimate of the slope of their cointegration
relation than that obtained in similar work by Christensen and Prabhala (1998),
who used OLS, which, as shown by Robinson (1994b), is inconsistent in the case
of stationary cointegration.
Brunetti and Gilbert (2000) proposed a bivariate cointegrated fractional volatility
(FIGARCH) model, and applied it to the volatility (measured in terms of squared
and absolute returns) of the New York NYMEX and London IPE crude oil markets.
They concluded that both processes were highly persistent, with a common degree
of fractional integration (around 0.4), and were fractionally cointegrated. Using
similar series, Robinson and Yajima (2002) analyzed stationary cointegration in the
context of testing for cointegration rank, finding support for this type of behavior
in spot closing prices of crude oil.
Beltratti and Morana (2006) also provided evidence of fractional cointegra-
tion in stock market volatility. They analyzed the relationship between S&P 500
returns volatility and that of some macroeconomic variables over 1970–2001 using
monthly data and allowing for both long memory and structural breaks. They
found evidence of long memory and structural change in the volatility, the break
possibly being related to break processes in the volatility of the macroeconomic
factors, and carried out a fractional cointegration analysis on break-free processes.
They found a common memory parameter of 0.25 for the series and concluded in
favor of the existence of three cointegrating relations among the variables using
the cointegrating test of Robinson and Yajima (2002).
Finally, Caporale and Gil-Alana (2004b) tested the present value model by check-
ing for cointegration between stock prices and dividends using annual data for the
period 1871–1995 (updating the series employed by Campbell and Shiller, 1987).
They provided evidence of the unit root nature of these series and, by applying
Robinson’s (1994a) test to OLS residuals, concluded that the series were fraction-
ally cointegrated with long memory cointegrating error and mixed evidence about
the type of cointegrating relation (weak or strong) which characterizes the data.