Palgrave Handbook of Econometrics: Applied Econometrics

366 The Long Swings Puzzle

It is now easy to see that theC 2 matrix has a similar reduced rank representation
toC 1 in theI( 1 )model, so it is straightforward to interpretα′⊥ 2

∑∑
εias a measure
of thes 2 second-order stochastic trends which load into the variablesxtwith the
weightsβ ̃⊥ 2.
From (8.22) we note that theC 1 matrix in theI( 2 )model cannot be given a
simple decomposition as it depends on both theC 2 matrix and the other model
parameters in a complex way. Johansen (2008) derives an analytical expression for
C 1 , essentially showing that:

C 1 =ω 0 α′+ω 1 α′⊥ 1 +ω 2 α′⊥ 2 , (8.24)

whereωiare complicated functions of the parameters of the model (not reproduced
here).
To summarize the basic structures of theI( 2 )model, Table 8.4 decomposes the
vectorxtinto the directions of(β,β⊥ 1 ,β⊥ 2 )and the directions of(α,α⊥ 1 ,α⊥ 2 ).
The left-hand side of the table illustrates theβ,β⊥directions, whereβ′xt+δ′xt
defines the stationary polynomially cointegrating relation, andβ′⊥ 1 xttheCI(2, 1)
relation that can only become stationary by differencing. Theβ,β⊥ 1 relations
define the two stationary cointegration relations between the differenced vari-
ables,τ′xt. Finally,β′⊥ 2 xt∼I( 2 )is a non-cointegrating relation, which can only
become stationary by differencing twice. The right-hand side of the table illus-
trates the corresponding decomposition into theα,α⊥directions, whereαdefines
the dynamic adjustment coefficients to the polynomially cointegrating relation,
whereasα⊥ 1 andα⊥ 2 define the first- and second-order stochastic trends as a linear
function of the VAR residuals.

8.6.2 Deterministic components

A correct specification of the deterministic components, such as trends, constant
and dummies, and how they enter the model, is mandatory for theI( 2 )analysis.
This is because the chosen specification is likely to strongly affect the reliability of
the model estimates and to change the asymptotic distribution of the rank test.
Because the typical smooth behavior of a stochasticI( 2 )trend sometimes can be
approximated with anI( 1 )stochastic trend around a broken linear deterministic
trend, one can in some cases avoid theI( 2 )analysis altogether by allowing for
sufficiently many breaks in the linear trend. Whether one specification is preferable

Table 8.4 Decomposing the data vector using theI(2) model

Theβ,β⊥decomposition ofxt Theα,α⊥decomposition

r= 1 [β′ 1 xt

︸︷︷︸

I( 1 )

+δ′ 1 xt

︸︷︷︸

I( 1 )

]∼I( 0 ) α 1 : short-run adjustment coefficients

s 1 = 1 β′⊥ 1 xt∼I( 1 ) α′⊥ 1

∑t

i= 1 εi:I(^1 )stochastic trend

p−s 2 = 2 τ′xt=(β,β⊥ 1 )′xt∼I( 0 )

s 2 = 1 β′⊥ 2 xt=τ′⊥xt∼I( 2 ) α′⊥ 2

∑t

s= 1

∑s

i= 1 εi:I(^2 )stochastic trend