Exchange Rate Economics: Theories and Evidence

58 Purchasing power parity and the PPP puzzle
this is straightforward – if the underlying process driving the real exchange rate
is mean-reverting the variance of the series would decrease as k becomes larger.
Alternatively,if V k turns out to be greater than one the real exchange rate would
exhibit ‘super-persistence’.
Huizinga (1987) calculates the variance ratio test for 10 (industrial) currencies
and 120 months of adjustment,and finds that the average V k implies a permanent
real exchange rate component of around 60%,with the remaining 40% being
transitory; however,on the basis of standard errors,constructed using the T 1/2
formula,none of the estimated variance ratios are significantly below one.
Glen (1992) and MacDonald (1995a) demonstrate that on using Lo and
MacKinlay (1988) standard errors,which are robust to serially correlated and
heterogeneous errors,significant rejections of a unitary variance ratio may be
obtained,but that the extent of any mean reversion is still painfully slow. For
example,on the basis of WPI constructed real exchange rates,MacDonald finds
that the Swiss franc,pound sterling and Japanese yen all have variance ratios which
are approximately 0.5 after 12 years (and these values are significantly less than
unity). So on a single currency basis for the recent float the evidence from mean
reversion in real exchange rates suggests that adjustment to PPP is painfully slow.
2.4 Econometric and statistical issues in unit root
based tests and in calculating the half-life
2.4.1 The power of unit root tests and the span of the data
One natural way of increasing the power of unit root tests is to increase the span
of the data. Intuitively,what this does is to give the real exchange rate more
time to return to its mean value,thereby giving it greater opportunity to reject
the null of non-stationarity. In increasing the span,it is insufficient,as Shiller
and Perron (1985) have indicated,to merely increase the observational frequency,
from,say,quarterly to monthly data for a particular sample period. Rather,what
is important for power purposes is to increase the span of the data using long runs
of low frequency,or annual,data. For example,moving from,say,approximately
30 years of annual data for the post-Bretton Woods period to 30 years of monthly
data for the same period is unlikely to increase the lower frequency information
necessary to overturn the null of no cointegration or the null of a unit root.
For example,assume the estimated value of ρ is 0.85,and its estimated asymptotic
standard error is [(1 − ρ 2 )/X ] 1/2 ,where X equals the total number of
observations. With 30 years of annual data the standard error would be approximately
0.1,with an implied t-ratio which is insufficient to reject the null of a unit
root (i.e. the t-ratio for the hypothesis ρ = 1 is 1.5). However,with 100 annual
observations the standard error falls to 0.05,implying a t-ratio for the hypothesis
ρ = 1 of 6.8. Defining X = N · T ,where T denotes the time series dimension and
N denotes the number of cross-sectional units,then this example makes clear that
expanding the span in a time series dimension increases the likelihood of rejecting