Exchange Rate Economics: Theories and Evidence

Purchasing power parity and the PPP puzzle 57
PPP would now seem to be widely accepted in the literature,it is important to note
that the implied mean reversion from the studies discussed in this section is often
painfully slow.
2.3.3 Unit root based tests of PPP
Most other recent tests of the PPP proposition have involved an examination of the
time series properties of the real exchange rate. In order to test if the autoregressive
parameter in an estimated version of equation (2.10) is significantly different from
unity,a number of researchers (see,inter alia,Roll 1979; Darby 1980; MacDonald
1985; Enders 1988; Mark 1990) have used an augmented Dickey–Fuller (ADF)
statistic,or a variant of this test,to test the unit root hypothesis for the recent
floating period. A version of an ADF statistic for the real exchange rate is given as:
n−1
∑
q t = γ 0 + γ 1 t + γ 2 q t−1 + β j q t−j + ν t ,
j=1
where n is the lag length from a levels autoregression of the real exchange rate. As
is standard in this kind of test,evidence of significant mean reversion is captured
by a significantly negative value of γ 2 . However,in practice the estimated value of
γ 2 is insignificantly different from zero implying that the autoregressive coefficient
in (2.8) is statistically indistinguishable from unity. This,therefore,has been taken
by some (see Darby 1980,for example) as evidence in favour of the EMPPP
discussed in Section 2.1. However,as Campbell and Perron (1991),and others,
have noted univariate unit root tests have relatively low power to reject the null
when it is in fact false,especially when the autoregressive component in (2.8) is close
to unity.
Alternative tests for a unit root have therefore been adopted in a bid to overturn
this result. The variance ratio test,popularised by Cochrane (1988),is potentially
a more powerful way of assessing the unit root characteristics of the data,since it
captures the long autocorrelations which are unlikely to be captured in standard
ADF tests,and which will be important for producing mean reversion. Under the
null hypothesis that the real exchange rate follows a random walk,the variance of
the kth difference should equal k times the first difference. That is,
Var(q t − q t−k ) = kVar(q t − q t−1 ).
On rearranging this expression we have:
V k = (1/k) ·[(Var(q t − q t−k )) · (Var(q t − q t−1 )) −1 ]=1,(2.28)
where V k denotes the variance ratio,based on lag k. So a finding that an estimated
value of V k equals unity would imply that the real exchange rate follows
a random walk. However,if V k turns out to be less than unity this would imply
that the real exchange rate was stationary and mean reverting. The intuition for