352 The Long Swings Puzzle
length the specification of the deterministic components. Section 8.7 discusses an
estimation procedure based on maximum likelihood and shows how theI( 2 )struc-
ture can be linked to theI( 1 )model. Section 8.8 provides hypothetical scenarios
for the real exchange rate data. Section 8.9 presents the empirical results of the
pulling and pushing forces structured by theI( 2 )model, summarizes the puzzling
facts detected, and discusses what has been gained by this analysis compared to
theI( 1 )analysis. Section 8.10 concludes with a discussion of what the data were
able to tell when allowed to speak freely.
8.2 The VAR model
The baseline VAR(2) model in its unrestricted form is given by:
xt= 1 xt− 1 + 2 xt− 2 +Dt+εt, (8.1)with:
εt∼Np( 0 ,),t=1,...,T,wherex′t=[x1,t,x2,t,...xp,t]is a vector ofpstochastic variables andDtis a vector
of deterministic variables, such as a constant, trend and various dummy variables.
As the subsequent empirical VAR model has lag two, all results are given for the
VAR(2) model. A generalization to higher lags should be straightforward.
In terms of likelihood, an equivalent formulation of (8.1) is the vector equilib-
rium correction form:
xt= 1 xt− 1 +xt− 1 +Dt+εt, (8.2)where 1 =− 2 and=−(I− 1 − 2 ).
Alternatively, (8.1) can be formulated in acceleration rates, changes and levels:
^2 xt=xt− 1 +xt− 1 +Dt+εt, (8.3)where=−(I− 1 ). As long as all parameters are unrestricted, the VAR model is
no more than a convenient summary of the covariances of the data. As a result,
most VAR models are heavily overparameterized and insignificant parameters need
to be set to zero. The idea of general-to-specific modeling is to reduce the number
of parameters by significance testing, with the final aim of finding a parsimonious
parameterization with interpretable economic contents. Provided that the sim-
plification search is statistically valid, the final restricted model will reflect the full
information of the data. Thus, given the broad framework of a theory model, a cor-
rect CVAR analysis allows the data to speak freely about the underlying mechanisms
that have generated the data.
All three models are equivalent from a likelihood point of view, but (8.1) would
generally be chosen whenxtisI( 0 ), (8.2) whenxtisI( 1 ), and (8.3) whenxtisI( 2 ).