Exchange Rate Economics: Theories and Evidence

Purchasing power parity and the PPP puzzle 65
examine the relative prices of 100 identical goods sold in 25 countries by IKEA.
They report significant common currency price divergences across countries
for a given product and across products for a given country pair and they
interpret this as reflecting pricing to market. Haskel and Wolf also report evidence
of significant mean reversion for deviations from the LOOP,although
such mean reversion is relatively slow (they obtain a mean reversion coefficient
of 0.89).
Imbs et al. (2002) argue that differentiated goods prices mean-revert at different
rates and aggregating across goods will introduce a positive bias into aggregate
half-lives. This may be seen by assuming the relative price of individual traded
goods, i,follows an AR1 process:
q T it = ρ i q T t−1 + β i + ε it ,(2.37)
where ε it is ∼ iid, E(ε it ) = 0 and E(ε 2 it ) = σ 2
i
and the slope coefficients vary across
sectors according to:
ρ i = ρ + η i ,(2.38)
where η i is the sectoral specific component and has mean zero and a finite variance.
Ims et al. assume that each sector receives equal weight in the aggregate price index
NT ,T
in all countries and that the relative price term, qt
,is zero. This kind of set up
can be used to address two related questions: what happens when the autoregressive
parameter is constrained to be equal in a panel context and what happens when
relative sectoral prices are aggregated into the real exchange rate? In terms of the
former question,consider the following panel equation where a common slope is
imposed for all sectors:
q it = ρq it−1 + β i + v it ,(2.39)
and v it = ε it +η i q it−1 which indicates that error term includes lagged relative prices
and will,as a result,be correlated with the regressor (instrumental variables will
not be able to address this issue because any useful instruments must be correlated
with q it−1 and therefore also with the error term). Imbs et al. demonstrate that the
bias of the pooled estimator can be expressed as:
ˆρ − ρ = E ( η i /(1 − ρi 2 ))
E ( 1/(1 − ρi 2 (2.40)
))
where ˆρ denotes the probability limit of the fixed-effects estimator of ρ. This bias
will be zero in the absence of heterogeneity and unambiguously positive when
0 < ˆρ i < 1. They also demonstrate that the magnitude of the bias is increasing
with the degree of sectoral heterogeneity. Estimates of half-lives generated from
the kind of panel studies referred to earlier will overstate the half-life of the real
exchange rate and this will be especially so if mean reversion speeds are highly
heterogeneous across goods.