Katarina Juselius 357This is a fairly pragmatic way of classifying data that allow a variable to be treated
asI( 1 )in one sample andI( 0 )or evenI( 2 )in another. The idea is that, in a general
equilibrium world, a persistent departure from a steady-state value of a variable
or a relation should generate a similar persistent movement somewhere else in
the economy. For example, if the Fisher parity holds as a stationary relation (sta-
tionary real interest rates) and we find that inflationary shocks have been very
persistent, then we should expect interest rate shocks to have a similar persistence.
Thus empirical persistence is a powerful property that can be used to investigate
whether our prior hypothesis (the Fisher parity) is empirically relevant, and if not,
which other variables have been co-moving in a similar manner, giving rise to new
hypotheses.
From the outset, many economists would consider the idea that economic vari-
ables areI( 2 )highly problematic. The argument is often that all inference on
long-run values (the steady-state value a variable converges to when the errors
are switched off) would lead to meaningless results. This is a valid argument pro-
vided one can argue that the order of integration is a structural parameter, which
often seems doubtful. Nonetheless, there are cases when a structural interpretation
is warranted. For example, Frydmanet al. (2008) show that speculative behavior
based on IKE is consistent with nearI( 2 )behavior; arbitrage theory suggests that a
nominal market interest rate should be a martingale difference process, i.e., approx-
imately a unit root process. Of course, in such cases a structural unit root should
be invariant to the choice of sample period.
8.4 ModelingI( 2 )data with theI( 1 )model: does it work?
It often happens thatI( 2 )data are analyzed as if they wereI( 1 )because theI( 2 )
possibility was never checked, or one might have realized that the data exhibit
I( 2 )features but decided to ignore these signals in the data. For this reason, it is
of some interest to ask whether the findings from suchI( 1 )analyses are totally
useless, misleading, or can be trusted to some extent.
Before answering these questions, it is useful to examine the so-calledR-model,
in which short-run effects have been concentrated out. We consider first the simple
VAR(2) model:
xt= 1 xt− 1 +αβ′xt− 1 +μ 0 +εt
εt∼Np( 0 ,),t=1,...,T, (8.10)and the correspondingR-model:
R 0 t=αβ′R 1 t+εt, (8.11)whereR 0 tandR 1 tare found by concentrating out the lagged short-run effects,
xt− 1 :
xt=Bˆ 1 xt− 1 +μˆ 0 +R 0 t, (8.12)
and:
xt− 1 =Bˆ 2 xt− 1 +μˆ 0 +R 1 t. (8.13)