Palgrave Handbook of Econometrics: Applied Econometrics

Katarina Juselius 371

of the unrestricted VAR model (8.3) can be decomposed as:

xt− 1 +xt− 1 =(β)β′xt− 1

︸ ︷︷ ︸

I( 0 )

+(αα′β⊥ 1 +α⊥ 1 )β′⊥ 1 xt− 1

︸ ︷︷ ︸

I( 0 )

+(αα′β⊥ 2 )β′⊥ 2 xt− 1

︸ ︷︷ ︸

I( 1 )

+αβ′xt− 1

︸ ︷︷ ︸

I( 1 )

, (8.32)

whereβ =β(β′β)−^1 andαis similarly defined. The decomposition describes
three types of linear relations between the growth rates,β′xt− 1 ,β′⊥ 1
xt− 1 and
β′⊥ 2 xt− 1 , of which the first two defineI( 0 )relations, and the third anI( 1 )
relation. The coefficients in soft brackets define the corresponding adjustment
coefficients.
Sinceβ′⊥ 2 xt− 1 isI( 1 ), it needs to be combined with anotherI( 1 )variable to
become stationary. An obvious candidate for this isβ′xt− 1. It is now easy to see
how the parameterization in (8.3) relates to the one in (8.31):

α(β′xt− 1 +(α′β⊥ 2 )β′⊥ 2 xt− 1 )=α(β′xt− 1 +δ′xt− 1 ). (8.33)

Finally, whenr>s 2 , the long-run matrixcan be expressed as the sum of the two
levels components:

=α 0 β′ 0 +α 1 β′ 1 ,

whereβ′ 0 xt− 1 definesr−s 2 directly stationaryCI(2, 2)relations, whereasβ′ 1 xt− 1
definess 2 non-stationaryCI(2, 1)cointegrating relations, which needs to be com-
bined with the differenced process to become stationary through polynomial
cointegration.
Thus, theI( 2 )model can distinguish between theCI(2, 1)relations between lev-
els {β′xt,β′⊥ 1 xt}, theCI(1, 1)relations between levels and differences {β′xt− 1 +
δ′xt}, and finally theCI(1, 1)relations between differences{τ′xt}. As a con-
sequence, when discussing the economic interpretation of these components,
the generic concept of a “long-run” equilibrium relation needs to be modified