Exchange Rate Economics: Theories and Evidence

The economics of the PPP puzzle 71
that to generate as much volatility by distance as generated by the border term,
the cities would have to be 75,000 miles apart. Engel and Rogers work therefore
confirms the findings of Wei and Parsley that a national border (which in this
context is a proxy for the nominal exchange rate),rather than distance,is the key
determinant of real exchange rate volatility.
3.1.2 Transactions costs and non-linear adjustment
Transportation costs have been used in another way to rationalise deviations from
PPP. In particular,Dumas (1992) has demonstrated that for markets which are
spatially separated,and feature proportional transactions costs,deviations from
PPP should follow a non-linear mean-reverting process,with the speed of mean
reversion depending on the magnitude of the deviation from PPP. The upshot
of this is that within the transaction band,as defined in (3.1),say,deviations are
long-lived and take a considerable time to mean revert: the real exchange rate is
observationally equivalent to a random walk. However,large deviations – those
that occur outside the band – will be rapidly extinguished and for them the observed
mean reversion should be very rapid. The existence of other factors,such as the
uncertainty of the permanence of the shock and the so-called sunk costs of the
activity of arbitrage may widen the bands over and above that associated with simple
trade restrictions (see,for example,Dixit 1989 and Krugman 1989). Essentially
the kind of non-linear estimators that researchers have applied to exchange rate
data may be thought of as separating observations which represent large deviations
from PPP from smaller observations and estimating separately the extent of mean
reversion for the two classes of observation.
Obstfeld and Taylor’s (1997) attempt to capture the kind of non-linear behaviour
imparted by transaction costs involves using the so-called Band Threshold
Autoregressive (B-TAR) model. If we reparametrise the standard AR1 model,
q t = βq t−1 + ε t as:
q t = λq t−1 + ε t ,(3.5)
where the series is now assumed to be demeaned (and also detrended in the work of
Obstfeld and Taylor,because the do not explicitly model the long-run systematic
trend in real exchange rates) and λ = (β − 1). Then the B-TAR is:
λ out (q t−1 − π)+ εt
out if q t−1 >π;
q t = λ in q t−1 + εt
in if π ≥ q t−1 ≥−π;
λ out (q t−1 + π)+ εt out if − π>q t−1 ;
(3.6)
where εt
out is N (0, σt
out ) 2 , εt
in is N (0, σt
out ) 2 , λ in = 0 and λ out is the convergence
speed outside the transaction points. So with a B-TAR,the equilibrium value for a
real exchange rate can be anywhere in the band [−π, +π] and does not necessarily
need to revert to zero (the real rate is demeaned). The methods of Tsay (1989)
are used to identify the best-fit TAR model and,in particular,one which properly