Palgrave Handbook of Econometrics: Applied Econometrics

446 Fractional Integration and Cointegration

Finally, Johansen (2008) proposes a new vector autoregressive model:

dzt=αβ′d−bLbzt+

∑k

i= 1

idLibzt+εt, (10.14)

whereLb= 1 −bis a particularly useful lag operator with 0<b≤d. Here, if there is
a unit root in an associated characteristic polynomial, (10.14) generates a fractional
processztof orderd, for which ther×1 vectorβ′ztis fractional of orderd−b.

10.3.2 Estimation methods for fractional cointegration

Depending on the cointegrating model considered, different estimation techniques
have been proposed. For simplicity, we deal initially with the estimation ofνin
the bivariate system (10.5)–(10.6), although the estimation methods below can be
straightforwardly generalized to cover multivariate situations where the observables
and cointegrating errors still depend on two integration orders,δandγ, respectively.
The most obvious proposal is to estimateνin (10.5)–(10.6) by the ordinary least
squares (OLS) estimator:

νˆOLS=

∑n

t= 1

xtyt

∑n

t= 1

x^2 t

. (10.15)

Here, in the standard cointegrating setting, withγ=0 andδ=1, it has been shown
(see, for example, Phillips and Durlauf, 1986) thatνˆOLSisn-consistent with non-
standard asymptotic distribution, in general. In fractional settings, the properties
of OLS could be very different from those in this standard framework. For example,
Robinson (1994b) showed the inconsistency ofνˆOLSwhenδ<0.5, which has been
termed stationary cointegration (with special importance in finance, see section
10.3.3).^9 When the observables are purely non-stationary (so thatδ≥0.5), consis-
tency ofˆνOLSis retained, but its rate of convergence and asymptotic distribution
depends crucially onγandδ. In particular, Robinson and Marinucci (2001) showed,
for a model slightly more general than (10.5)–(10.6), that ifδ≥0.5,γ≥0, the rate

of convergence of OLS isnmin(^2 δ−1,δ−γ), except whenγ>0 andγ+δ=1, where
OLS isnδ−γ/logn-consistent. In all these cases OLS has non-standard limiting distri-
butions in general. An alternative method of estimatingνwas developed from the
following observation. Equation (10.15) is obviously a time-domain representation
of the estimate, but it can easily be shown that:

νˆOLS=

n∑− 1

j= 0

Ixy

(

λj

)

n∑− 1

j= 0

Ix

(

λj

), (10.16)

whereλj = 2 πj/n,j =1,...,n, are the Fourier frequencies, and for arbitrary
sequencesξt,ζt, (possibly the same asξt), we define the discrete Fourier transform