Session 2: Relative Prices
Jean Imbs
November 2009
Imbs (2009)
Relative Prices
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Relative Prices
Some De…nitions
Real Exchange Rates: The PPP Puzzle
PPP - Validity, Econometric Testing, Non-Linearities/Heterogeneity vs.
Price Discrimination
Border E¤ects
Nominal Exchange Rates:
Forecastability, Beating the Random Walk
Interest Parities - Covered, Uncovered. The Forward Discount Puzzle.
Risk Premia, Expectations, Limited Rationality
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De…nitions
Nominal Exchange Rate E : relative price of currencies. in two countries.
E.g. Euro/Dollar
The number quoted in Financial Times and The Economist.
Technically, this is the rate at which bank deposits in di¤erent currencies are
exchanged. Exchange rates for coins or notes slightly di¤erent as they
include transport costs.
Bilateral exchange rates are cross-rates, telling the price of one currency
relative to an other. 1 Euro = 1.25 Dollar
E¤ective exchange rates are weighted averages of all bilateral exchange
rates. Weights are chosen to re‡ect trade weights, i.e. share of total trade
accounted for by particular country. No obvious unit, so normalize to 100 in
an arbitrary year.
Useful to evaluate overall value of a currency over time.
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Real Exchange Rate
The Real Exchange Rate ε captures the cost of purchasing the same bundle
of goods in di¤erent countries. Connected to "competitiveness".
De…ne
ε=
E P
P
where: P denotes the domestic price of a bundle of goods, P is its foreign
counterpart, and E is labelled in units of domestic currency per unit of
foreign (e.g. £ /$)
An increase in ε means a gain in domestic competitiveness according to that
measure. A real depreciation.
Note that an increase in E means a nominal depreciation of the domestic
currency relative to the foreign currency.
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The First Model of Exchange Rates: PPP
PPP stands for Purchasing Power Parity. Conjectures there are arbitrage
forces that bring back to parity international relatives prices of (all?) goods.
Formally, the simplest form of PPP imposes
ε=1
Bundles of goods cost the same across borders. If they do not, buy where
cheap to sell where expensive - brings back parity.
This is also the strongest form of parity, or "Absolute PPP". Requires
instantaneous arbitrage, and in…nite amounts of resources taking advantage
of price di¤erences.
Related to "Law of One Price" (LOOP), which imposes parity at the
individual’s good level. Note that Absolute PPP is stronger than LOOP,
since it requires LOOP holds for all goods and the bundles of goods have
the same composition across borders. Not true with home bias and/or
non-traded goods.
Quite obvious LOOP or PPP do not hold at every point in time.
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-0.1
02
-0.2
-0.3
-0.4
-0.5
1995
1983
1971
1959
1947
1935
1923
1911
1899
1887
1875
1863
1851
1839
1827
1815
1803
1791
Long Run PPP: $/£
05
0.5
0.4
0.3
0.2
0.1
0
In the long run PPP appears to hold but can be long term
deviations in exchange rates from their PPP value
Long Run PPP: the Franc
Price LEVELS
In fact, Absolute PPP is very di¢ cult to test. To have a meaningful measure
of ε, need meaningful measures on the LEVELS of P and P
But P and P are typically price indices, such as CPI or PPI
Price indices are normalized to 100 (e.g.) on a given year. Their levels are
typically meaningless.
The graphs just shown conveyed the general notion that relative prices do
not explode out of proportion, and there are forces preventing explosive price
di¤erences in the long run. But they do not say to what LEVELS relative
prices converge back to.
For that, need data on price LEVELS. Not that much available: International
Price Comparisons (UN, every 5 years), Scanner’s data (but proprietary and
incomplete), Economist Intelligence Unit (but again incomplete)
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Relative PPP
Long run data on relative prices display some tendency to reversal. What
could be behind that? Suppose domestic in‡ation very high - much higher
than foreign one, and over a long time..
ε would fall following a long run trend - a real appreciation. Unless E
adjusts. How? A nominal depreciation will o¤set the increase in prices in
domestic currency.
Relative PPP implies high in‡ation economies see their nominal exchange
rate increase, i.e. depreciate. There is mean reversion in the RER.
That can be explored econometrically rigorously. Assume an autoregressive
process (of order one) drives real exchange rates, and let qt denote ln εt :
qt = α + βqt
1
+ ηt
β < 1 implies there is some reversion to the mean. What mean? Solve for a
time-invariant level of q :
α
q=
1 β
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Testing Relative PPP: Pitfalls
Low Power: it is di¢ cult to distinguish with certainty values of β that are
very close to 1 from β e¤ectively taking value 1.
Unfortunately, the di¤erence in economic terms is central. β = 0.99 still
means some mean reversion, if slow, and thus some predictability.
Way out: in the 90s, increase time dimension (as in previous graphs). Or use
panel approaches, i.e. introduce an international dimension. Estimation
becomes
qit = αi + βqit
1
+ η it
with i denoting a certain country’s exchange rate. Note that αi now
controls for the (unobserved) level of international price di¤erences, which
may be di¤erent for di¤erent countries. E.g. France-Germany as compared
with Japan-Sierra Leone. It is also controlling for the arbitrary
normalizations in price indices.
Consensus …nding: quite some certainty that β < 1. BUT it is so close to 1
that implied rate of mean reversion unconscionably slow.
Imbs (2009)
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The PPP Puzzle
In particular, so-called "half-life" of a given disturbance to international
relative prices between 3-5 years. In an AR(1) case, half-life H veri…es
βH =
1
ln 2
)H=
2
ln β
No closed form for AR(p). Typically use simulated impulse-response
function.
3-5 years to take advantage of international price di¤erences???? Rather
dense arbitragists...
First element of "PPP Puzzle". Possible to rationalize in model with
price stickiness (i.e. prices do not change for a while). But then (i)
required calibrated stickiness way too long (no more than 3-4 quarters
in the data), and (ii) impossible to reproduce the RER volatility we
observe in the data.
Imbs (2009)
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The PPP Puzzle
Three explanations:
Time Aggregation: frequency of observation is inadequate to observe
arbitrage
Non-Linearities: there are inaction bands, for instance because of trade
costs
Heterogeneity: some goods are easy to arbitrage, others are not.
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Time Aggregation
Suppose arbitrage occurs within weeks, but you only have (say) quarterly
data, which is an AVERAGE of higher frequency data.
No problem if frequency of observation is di¤erent - problem is
(moving) averaging.
Then the true model is
qt = α + βqt
1
+ ηt
but we only observe
qt =
1
(qs + qs +1 + ... + qs +P
P
1)
where P is the period of sampling. For instance, there are P = 12 weeks in
a quarter.
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Time Aggregation
The dynamics of qt are fundamentally di¤erent. For instance, suppose we
estimate
qt = α + β qt
1
+ ηt
Then, OLS estimator of β given by
β̂ =
cov qt , qt
varqt 1
1
and the corresponding half-life
H =
P ln 12
ln β
Now we seek to establish that H > H and
H explodes as P ! ∞.
Imbs (2009)
Relative Prices
H
H
increases with P and
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Time Aggregation
Now
β̂
=
=
cov (qs + qs +1 + ... + qs +P 1 , qs P + qs
var (qs + qs +1 + ... + qs +P
n (P )
d (P )
P +1
+ ... + qs
1)
1)
We know the true process is
qt = α + βqt
1
+ ηt
so that
cov (qt , qt
Imbs (2009)
k)
Relative Prices
= βk varq
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Time Aggregation
De…ne
N (P ) =
n (P )
= β + β2 + ... + βP
varq
+ β2 + β3 + ... + βP +1
+...
+ βP + βP +1 + ... + β2P
= β 1 + β + β2 + ... + βP
!2
1 βP
= β
1 β
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Relative Prices
1
1
2
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Time Aggregation
And
D (P ) =
d (p )
= D (P
varq
= D (P
1) + 1 + 2β + ... + 2βP
1) + 2
1 βP
1 β
1
1
since var (A + B + C ) = var (A) + var (B + C ) + 2cov (A, B + C ).
Notice D (1) = 1 and solve recursively to get
P (1
β2 )
D (P ) =
(1
2β 1
βP
β )2
So …nally,
β =
P 1
Imbs (2009)
2
β 1
βP
β2
2β 1
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βP
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Time Aggregation
The (biased) half life is given by
P ln 2
H =
ln
P (1 β2 ) 2β(1 βP )
β (1 βP )
2
The true half life, in turn, veri…es
H=
ln 2
)β=2
ln β
1
H
Replace:
P ln 2
H =
P 1 2
ln
2
Imbs (2009)
2
H
1
H
Relative Prices
21
1 2
1
H
P
H
1 2
P
H
2
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Time Aggregation
Finally:
H
=
H
P ln β
P (1 β ) 2β(1
ln
2
β (1 βP )
2
βP )
It is easy to show that
limP !∞
Imbs (2009)
H
= limP !∞
H
P ln β
ln
P (1 β2 )
β
Relative Prices
= limP !∞
P ln β
=∞
ln P
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Time Aggregation
The gap between β and β increases in P , i.e. a bias becomes larger as the
gap betwen observed and actual frequency increases. In other words, trying
to identify daily convergence using monthly (yearly!) data is particularly
biased.
It also increases in β: a positive bias is larger for very persistent processes.
In other words, if true mean reversion is somewhat slow, it will appear even
slower if the frequency of the data is not adequate.
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Non Linearities
There are "bands of inaction". Arbitrage will only occur provided price
di¤erences are high enough (to cover for transport costs).
This can explain permanent international di¤erences in prices. Can it also
explain slow convergence?
Consider the true model
qt
= α + β (qt 1 α) + η t if qt 1 > α
= qt 1 + η t if
α qt 1 α
=
α + β (qt 1 + α) + η t if qt 1 <
α
Here α captures the transport costs: price deviations have to exceed that
amount for reversal to occur (with speed captured by β). Rest of the time,
real exchange rate is e¤ectively *not* reverting to its mean.
Imbs (2009)
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Non Linearities
Suppose you ignore this non-linearity and estimate a linear setup instead
qt = α
+ β qt
1
+ ηt
How is that going to a¤ect your estimates of β?
The OLS estimate of β will be given by
β̂
=
=
cov qt , qt
varqt 1
E qt qt
1
1 IIN (qt 1
De…ne
θ IN = EIIN (qt
θ OUT = EIOUT (qt
1)
) + E qt qt
Eqt 21
var qt 1 jIIN (qt
varqt 1
1)
var qt
1 IOUT (qt 1
1
)
1 jIOUT (qt 1
varqt
)
)
1
Clearly, θ IN + θ OUT = 1
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Non Linearities
And we have
β̂
= EIIN (qt
1)
E qt qt 1 IIN (qt
Eqt 21
+EIOUT (qt
= θ IN
1)
E qt qt
1
)
1 IOUT (qt 1
Eqt 21
)
E qt qt 1 jIIN (qt 1 )
E qt 21 jIIN (qt 1 )
E qt qt 1 jIOUT (qt 1 )
E qt 21 jIOUT (qt 1 )
1 + θ OUT β
+θ OUT
= θ IN
Naive linear estimation is a weighted average of persistence within the band
(1) and persistence outside of it ( β). The weights θ are given by the
proportion of time spent in (θ IN ) and outside of the band (θ OUT ). The bias
will be larger the more time is spent within the band, i.e. the larger α.
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Non Linearities
In terms of half life:
H
Imbs (2009)
ln 2
ln 2 ln β̂
ln β̂ ln β̂
ln β̂
ln β
= H
ln ( β + θ IN (1 β))
=
=
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Heterogeneity
Some goods are easier to arbitrage than others. (gold or wheat vs. bed
mattresses or houses)
Speed of mean reversion should be allowed to di¤er, as in
qst = αs + βs qst
1
+ η st
where s denotes a good. Key is that βs di¤ers for di¤erent goods.
Suppose now βs = β + εs , i.e. each goods has a speed of mean reversion
that di¤ers additively from the "true" average β.
Then a naive estimation that ignores this heterogeneity rewrites (even if it
allows for FE):
qst = αs + β qst
1
+ (η st + εs qst
1)
Basic econometrics tells us this estimation will be biased, because its
residuals η st + εs qst 1 unavoidably depends on the regressor.
Instrumentation will not solve anything, since an instrument must be
correlated with the regressor qst 1 , but not with the residuals
η st + εs qst 1 . That is impossible.
Imbs (2009)
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Heterogeneity
Can say something about the direction of the bias. Suppose βFE is
the …xed e¤ects estimator. Can show that
p
lim
N !∞,T !∞
βFE
= β+∆
∆ =
N
∑ ( βs
β ) δs
i =1
δs
=
N
σ2s
/
∑
1 β2s i =1
σ2s
1 β2s
!
Necessary and su¢ cient condition for a positive bias is cov β̃, δ̃ .
∆ also increases in βs
persistence.
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β, i.e. the cross-sectional dispersion in
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Heterogeneity
The same is true of *aggregate* regression in the presence of
heterogeneity.
Consider country-level regression
qst
qt
= αs + βs qst
= ∑ ω s qst
1
+ η st
s
and ω are (CPI) expenditures weights.
Note that in reality, if qst follows an AR(1), the aggregate RER
veri…es
h
i
h
i
N
N
ΠN
1
β
L
q
=
ω
Π
1
β
L
η st
(
)
(
)
t
∑ s s =1
s =1
s
s
s =1
I.e an ARMA process - potentially of in…nite order. Any attempt to
account for heterogeneity estimating that will be an approximation.
Dead end.
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Heterogeneity
Now estimating
qt = α + β qt
1
+ ηt
will imply β = ∑s ω s βs and η t = ∑s ω s η st + ∑s ω s qst 1 . So that
the same pathology prevails.
Can sign that bias again. Consider the OLS estimate obtained from
aggregate data βOLS . By de…nition,
βOLS =
Eqt qt
Eqt2
1
Substitute disaggregated components of numerator and denominator,
to obtain:
βOLS
= β + ∆0
∆0 =
N
∑ ( βs
β ) γs
i =1
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Heterogeneity
And we have
γs =
ω s σ2s
+ ∑s 6=t ω1 s ωβt σβst
1 β2s
s t
ωb s σ2s
ω s ω t σst
∑ s 1 β2 + ∑ s 6 =t 1 βs βt
s
As before, bias positive for positive values of cov β̃, γ̃ .
Importantly, this is not *necessarily* requiring that high β sectors
have high CPI expenditures weights.
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Figure I
Check on Conditions for a Positive Aggregation Bias
Heterogeneity
Well known pathology. The way out is to estimate sector (good?) speci…c
βs , and then aggregate these up to the country level.
How to aggregate them? Several estimators are available: Mean Group
simply takes an arithmetic average, Random Coe¢ cient takes a weighted
average ,where weights re‡ect accuracy of estimation of each coe¢ cient.
Key insight is that aggregate estimates is fundamentally di¤erent from
aggregating data. Reasoning on aggregated data implicitly assumes all β are
the same for all goods forming the Real Exchange Rate.
β computed as a weighted average of βs is much smaller - at least on
European data. In fact, so small that half-life of mean reversion down to 10
months. Compatible with plausible nominal rigidities.
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So What to Make of PPP?
These results are not uncontroversial. For instance, debates exist about the
standard errors around these estimates.
Two remaining puzzles:
The Border E¤ect: relative prices are much, much more volatile around
borders, than within the same country.
The Traded vs Non Traded decomposition: it would make sense that
prices be di¤erent between countries because of non-traded goods. A
house in London is not a house in Tokyo or Dacca. Surprisingly
however, virtually all of the persistence in international relative prices
arises in (what is usually construed as) traded goods.
Both puzzles point to the alternative view that there is "pricing to market",
i.e. market power on the part of international …rms, who price discriminate
across di¤erent (international) markets.
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The "Engel" Puzzle
Consider a two-country model with both traded and non-traded
goods. The (log) ideal price indices are:
= (1
= (1
pt
pt
α) ptT + αptN
β) pt T + βpt N
The RER is
qt
= s t + pt
=
st + ptT
pt
ptT
+ β pt N
pt T
α ptN
ptT
The …rst term pertains to arbitrage, presumably.
The second and third ones pertain to the within-country relative price
of traded vs. non traded goods - and whether that di¤erence is the
same across countries.
Engel performs this decomposition and …nds the bulk of volatility in q
does NOT come from the former.
Imbs (2009)
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So What to Make of PPP?
Most prominent approach is to compute the degree of "exchange rate
pass-through", i.e. how much changes in the nominal exchange rate are
passed into …nal prices. Imperfect pass-through requires some market power,
since it means …rm can choose its level of pro…t.
Enormous empirical industry in current international macroeconomics.
Seems to be some (very recent) evidence that all three re…nements (time
aggregation, non-linearities, heterogeneity) are present in estimates of border
e¤ects, traded vs. non-traded decomposition and exchange rate
pass-through.
Reversion to price parity still a powerful force in determining international
relative prices. Especially in disaggregated (goods? sectors?) data.
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