356 The Long Swings Puzzle
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+++–0.20.00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.20.4Figure 8.2 A cross-plot of US–German relative prices and the Dmk–$ rate for the period
1975:4–1998:12
pppt∼I( 0 )were correct, then the cross-plots should be randomly scattered around
the 45◦line defining the equilibrium positionppt=s12,t. Obviously, the cross-plots
measuring the deviation fromppp, i.e.,β′xt=ppt−s12,t−β 0 , are systematically
scattered either above or below the 45◦line. Thus the reality behind the observed
real exchange rate looks very different from the assumed stationary PPP illustrating
the puzzle. The non-stationarity of real exchange rates has been demonstrated in a
number of studies (see Froot and Rogoff, 1995, and MacDonald, 1995, for surveys;
Cheung and Lai, 1993; Juselius, 1995; Johansen and Juselius, 1992).
8.3.3 Approximating persistent behavior withI( 1 )orI( 2 )
The above ocular analysis showed that the long swings puzzle is essentially a ques-
tion of why nominal exchange rates have so persistently moved away from relative
prices. The previous sub-section suggested that the cointegrated VAR model should
be used to structure such data by the pulling and pushing forces. Section 8.3
defined theI( 1 )andI( 2 )models as reduced rank parameter restrictions on theI( 0 )
model, providing us with an empirically strong procedure for addressing behav-
ioral macroeconomic problems. This is because the reduced rank parameterization
of the CVAR allows us to group together components of similar persistence over
the sample period. The characterization of the data intoempirically I( 0 ),I( 1 )and
I( 2 )components is a powerful organizing principle, allowing us to structure the
data into long-run, medium-run and short-run behavior. An additional advantage
is that inference is likely to become more robust than otherwise. For example,
treating a near unit root as stationary tends to invalidate certain inferences based
on theχ^2 ,Fandtdistributions unless we have a very long sample.^2