Testing Hypotheses in an I(2) Model with Applications to the Persistent Long Swings in the Dmk/$ Rate



paper, we examine more broadly whether the estimated I(2) model is consistent with
the data.

Onr aim is to demonstrate that by structuring the data with the help of the cointe-
grated
I(2) model, one can achieve a better understanding of the empirical regularities
underlying the persistent swings in nominal exchange rates that are typical in periods
of floating currencies. We shall argue that the cointegrated
I(2) model is well designed
to study empirical problems characterized by different levels of persistent behavior, as
it allows us to study highly persistent
I(2), persistent I(1), and transitory I(0) behav-
ior in one model. By structuring the data in this way, we are able to present a number
of ‘sophisticated stylized facts’ that any theory model should replicate in order to claim
empirical relevance.

The econometric theory of this paper builds on Johansen (1992, 1995, 1997, 2006),
Kongsted, Rahbek and Jprgensen (1999), Paruolo (2000), Paruolo (2002), Nielsen and
Rahbek (2007). There are several studies in the literature that have found nominal
variables, such as exchange rates, goods prices, and money supplies, to be well approxi-
mated as I(2).4 Despite this evidence, however, there is a resistance among economists
to consider economic data to be
I(2). This resistance can be traced to the fact that the
popular REH macro models imply time series that are at most integrated of order 1.
Consequently, macro economists make use of the I(1) framework, often without testing
whether this framework provides an appropriate structuring of the data.

The common practice of ignoring trends in data that exhibit two roots near the
unit circle may lead economists to draw erroneous inferences from their “statistical”
analyses. Instead of forcing such data into an I(1) framework, it would be more useful
to construct economic models that are consistent with I(2) behavior. Indeed, FGJ
show that, under plausible assumptions, the IKE model of swings in Frydman and
Goldberg (2007) implies near I(2) behavior for exchange rates, relative goods prices,
and interest rate spreads. Thus, the finding in FGJ and in the present study that
the I(2) hypothesis cannot be rejected for these variables indicates a rejection of the
monetary model under REH in favor of its IKE counterpart.

The organization of the paper is as follows: Section 2 gives the theoretical back-
ground for the
I(2) analysis with particular attention to the role of deterministic com-
ponents in the model
. Section 3 discusses the Maximum Likelihood parametrization
of the
I(2) model and section 4 shows how to test structural hypotheses in that model.
Section 5 provides an ocular inspection of the data, which suggests that the nominal ex-
change rate, goods prices, and interest rate spreads should be modeled as I(2) variables.
Section 6 estimates an unrestricted VAR model with particular attention to problem of
specifying its deterministic component. Section 7 discusses the choice of rank indices
in the
I(2) model. Section 8 reports a number of test results based on non-identifying
hypotheses as a general description of the properties of the data. Finally, Section 9
reports an overidentified long-run structure and describes the dynamic adjustment of
the international transmission mechanisms between Germany and the USA from the

4See Johansen (1992), Juselius (1994), Kongsted (2003, 2005), Kongsted and Nielsen (2004), and
Bacchiocchi and Fanelli (2005).



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