Palgrave Handbook of Econometrics: Applied Econometrics

444 Fractional Integration and Cointegration

ensures thatxthas finite variance, and implies thatxt=0,t≤0. This restriction
is unnecessary ifγ< 1 /2 because, in that case,yt−νxtis covariance stationary
without it and “asymptotically covariance stationary” with it, but it is imposed
for the sake of a uniform treatment, implying thatyt=0,t≤0. Under (10.5)–
(10.6),xtisI(δ),asisytby construction, while the cointegrating erroryt−νxtis
I(γ ). Model (10.5)–(10.6) reduces to the bivariate version of Phillips’ triangular form
whenγ=0 andδ=1, which is one of the most popular models displayingCI(1, 1)
cointegration considered in both the empirical and theoretical literatures. This
model allows greater flexibility in representing equilibrium relationships between
economic variables than the traditionalCI(1, 1)prescription. On the one hand, it
is plausible that there exists long-run co-movements between non-stationary series
which are not preciselyI( 1 ). On the other hand, there is usually noa priorirea-
son for restricting analysis to justI( 0 )cointegrating errors, as the convergence to
equilibrium of any cointegrating relation could be much slower than the adjust-
ment implied by, for example, a finite ARMA cointegrating error. Furthermore,
we could also consider cointegration among (asymptotically) stationary variables,
with some linear combinations producing cointegrating errors characterized by
having weaker memory than that of the observed series. Also, it could be that the
cointegrating error is purely non-stationary but mean-reverting, so that a certain
long-run equilibrium among non-mean-reverting observables holds.^8
There are various directions in which model (10.5)–(10.6) has been generalized.
Robinson and Iacone (2005), still within a bivariate framework, allow for deter-
ministic components, extending (10.5)–(10.6) to:

yt=νxt+

∑p^1

j= 1

μ 1 jtφ^1 j−^1 /^2 +u 1 t(−γ), (10.7)

xt=

p 2

∑

j= 1

μ 2 jtφ^2 j−^1 /^2 +u 2 t(−δ), (10.8)

whereδ>max(γ, 0.5), and theφijare real numbers satisfyingφ 11 >···>φ 1 p
1

0;
φ 21 >···>φ 2 p
2
0, noting that an intercept appears in (10.7)–(10.8) when

φ 1 j=φ 2 j= 1 /2, while integer powers are also possible.
Kim and Phillips (2002) proposed a multivariate version of (10.5)–(10.6), employ-
ing the Type I definition of fractionally integrated processes instead, so that:

y ̃t=νx ̃t+v 1 (γ )t, t≥1, (10.9)

x ̃t=v 21 (δ)+···+v 2 (δ)t, t≥1, (10.10)

wherev 1 (γt)andv 2 (δt)are jointly stationary Type I fractionally integrated processes
of ordersγandδ−1, respectively, with|γ|<0.5, 0.5<δ<1.5,y ̃tandx ̃tarep× 1
andq×1 vectors, respectively, andνis ap×qmatrix of cointegrating parameters.

Note that, whenp=q=1 andγ=0,δ=1,

(

v 1 (γt),v 2 (δt)

)′

≡

(

u 1 t,u 2 t

)′
implies that
(
x ̃t,y ̃t

)′

=

(

xt,yt

)′

, but more generally this is not the case. Model (10.9)–(10.10) is