Palgrave Handbook of Econometrics: Applied Econometrics

Katarina Juselius 353

The hypothesis thatxtisI( 1 )is formulated as a reduced rank restriction on the
matrix:

=αβ′, whereα,βarep×r, (8.4)

and thatxtisI( 2 )as an additional reduced rank restriction on the transformed
matrix:

α′⊥β⊥=ξη′, whereξ,ηare (p−r)×s 1 , (8.5)

whereβ⊥,α⊥are the orthogonal complements ofβandα. The first reduced rank
condition is formulated on the variables in levels, the second on the variables in
differences. Condition (8.4) tells us that the variables contain stochastic trends
(unit roots) that can be canceled by linear combinations. Condition (8.5) tells us
that the differenced process also contains unit roots when data areI( 2 ). However,
in this case the linear combinations that cancel these roots are more complicated.
Thus, whenxt∼I( 2 ), and hencext∼I( 1 ), it is not sufficient to impose the
reduced rank restriction on the matrixto get rid of all (near) unit roots in the
model. This is becausextis also a unit root process and lowering the value ofr
does not remove the unit roots belonging to=−

(

I− 1

)

. Therefore, even though
the rank of=αβ′has been correctly determined, there will remain additional
unit roots in the VAR model when the data areI( 2 ). As will be demonstrated below,
this provides a good diagnostic tool for detectingI( 2 )problems in the VAR analysis.
Inverting the VAR model gives us the moving average (MA) form. Under the
reduced rank of (8.4) and the full rank of (8.5), the MA form is given by:

xt=C

∑t

i= 1

(εi+Di)+C∗(L)(εt+Dt)+A, (8.6)

whereC∗(L)is a lag polynomial describing the impulse response functions of the
empirical shocks to the system,Ais a function of the initial valuesx 0 ,x− 1 ,x− 2 ,
andCis of reduced rankp−r:

C=β⊥(α′⊥β⊥)−^1 α′⊥=β ̃⊥α⊥′, (8.7)

withβ ̃⊥=β⊥(α′⊥β⊥)−^1.
Inverting the VAR under the reduced rank of both (8.4) and (8.5) will be discussed
in section 8.6.

8.3 The persistent movements in real exchange rate data

Parity conditions are central to international finance and, more specifically, to
many open economy macro-models, such as the Dornbusch (1976) sticky-price
overshooting model with rational expectations (RE). One important implication
of this model and its modifications is that PPP should hold as an equilibrium cointe-
grating relationship (see Frydmanet al., 2008, and references therein). The idea here
is to formulate the PPP condition under a currency float into testable hypotheses