Regressions of ex post changes in floating exchange rates on appropriate interest differentials typically imply that the high- interest rate currency tends to appreciate, the `forward discount puzzle.' Using data from the European Monetary System, we find that a large part of the forward discount puzzle vanishes for regimes of fixed exchange rates. That is, deviations from uncovered interest parity appear to vary in a way which is dependent upon the exchange rate regime. By using the many EMS realignments, we are also able to quantify the `peso problem.'

Andrew K. Rose

Robert P. Flood

English

Research Papers in Economics

NBER WORKING PAPER SERIES
FIXES: OF THE FORWARD
DISCOUNT PUZZLE
Robert P. Flood
Andrew K. Rose
Working Paper No. 4928
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
November 1994
We thank Kellett Hannah and the Federal Reserve Bank of San Francisco for assistance with the
data. This work was conducted in part while Rose was a consultant at the IMF Research
Department. This paper is part of NBER's research program in International Finance and
Macroeconomics. Any opinions expressed are those of the authors and not those of the National
Bureau of Economic Research.
© 1994 by Robert P. Flood and Andrew K. Rose. AU rights reserved. Short sections of text,
not to exceed two paragraphs, may be quoted without explicit permission provided that full
credit, including © notice, is given to the source.FIXES: OF THE FORWARD
DISCOUNT PUZZLE
ABSTRACT
NBER Working Paper #4928
November 1994
Regressions of ex post changes in floating exchange rates on appropriate interest
differentials typically imply that the high-interest rate currency tends to appreciate, the "forward
discount puzzle." Using data from the European Monetary System, we find that a largepart of
the forward discount puzzle vanishes for regimes of fixed exchange rates. That is, deviations
from uncovered interest parity appear to vary in a way which is dependentupon the exchange
rate regime. By using the many EMS realignments, we are also able to quantify the "peso
problem.'
Robert P. Flood
Research Department
International Monetary Fund
700 19th Street, NW
Washington, DC 20431
and NBER
Andrew K. Rose
Haas School of Business
University of California
Berkeley, CA 94720-1900
and NBERI: Introduction
It is well known thatdeviationsfrom uncovered interest parity (IJIP) are pervasive
and persistent.1/ In particular, currencies with high interest rates tend to appreciate
relative to those with lower interest rates, contrary to the hypothesis of UTP. Since foreign
exchange markets are among the deepest markets in the world, explaining UIP deviations is
an interesting and important task for international finance researchers.
In this short paper, we contribute to the literature in two ways. First, we test UIP
for fixed exchange rate currencies. Nearly all of the relevant literature has tested (and
rejected) UIP using data for floating exchange rates. A typical finding is that the slope
coefficient from a regression of the ex post change in the exchange rate on the appropriate
interest differential is usually negative, economically and statistically far below the UIP
value of +1.We find it easy to produce such results using daily data from the 1980s and
1990s pooled across a number of countries which are floating vis-a-vis the American
dollar. However, when we use fixed exchange rate data from the Exchange Rate
Mechanism (ERM) of the European Monetary System (EMS), we find that UIP fares much
better. Instead of being negative, the slope coefficient is typically around +.6, though still
significantly below its hypothesized value of unity.
Our second contribution stems from the fact that the ERM has experienced a number
of discrete exchange rate realignments (and other such events) since the EMS began in
1979. Many of these realignments were anticipated by the financial markets. Market
anticipations of an event which does not occur sufficiently frequently in the sample leads to
small-sample bias in UIP regressions. This bias is commonly referred to as the "peso
1/ Hodrick (1987), Froot and Thaler (1990), and Lewis (1993) provide recent surveys.2
problem". We attempt to quantify this bias by pooling our data across currencies, and
comparing regression results when the realignments are included in the sample to those
when the realignment observations are excluded. While there is no guarantee that the 54
realignments which the ERMexperiencedin our sample are enough to constitute a large
sample (especially as these events are not independent across country or time), our best
guess is that the peso problem leads to a bias of around -.35,thedifference between the
slope coefficient in the UIP regression when the latter is estimated with and without
realignments.
In section II of the paper, we describe our data set and empirical methodology. Our
empirical results are present and discussed in section III; a brief interpretation concludes.
II: Methodology and Data
The hypothesis of uncovered interest parity can be expressed as:
(l+i1) =(l+i*t)E%(S%÷b)ISt (1)
where: 1Lrepresentsthe return on a domestic asset at time t of maturityi" is the return
on a comparable foreign asset; S is the domestic currency price of a unit of foreign
exchange; and E(.) represents the expectations operator conditional upon information
available at t.
We follow the literature in taking natural logarithms and ignoring small cross terms
by considering only countries with "low' interest rates. Rearranging, we derive:3
-s)(j..j*)
—> S+ S =a+ /3(i..i*) + L (2)
where: s is the natural logarithm of s;is the forecasting error realized at t+ from a
forecast of the exchange rate made at time t; and a and j3 are regression coefficients.
Sinceis a forecasting error, it is assumed to be stationary and orthogonal to all
information available at time t, including interest rates; hence OLS is a consistent estimator
of(3.
Equation (2) has been used as the workhorse for most of the extant UIP literature.
The null hypothesis of UIP can be expressed as Ho: a=O, (3=1. Researchers have
typically estimated (3tobe significantly negative (3estimate,and a to be non-trivial.
In practice, we modify (2) in two slight ways. First, we pool together data from a
number of different countries, an admissible way of increasing the sample under the null
hypothesis. Second, we use data of daily frequency for exchange rate forecasts of usually
three months horizon. The fact thatis greater than unity induces to have a moving
average "overlapping observation" structure. We take account of this by estimating our
covariance matrices with the well-known Newey-West estimator, with an appropriate
number of off-diagonal bands.
We estimate (2) on two different data sets: one for flexible exchanges, one for fixed
exchange rates. Both data sets consist of daily observations on exchange and interest rates;
the data have been check and corrected for various errors.1/
1'Ourdata sets and programs are available upon receipt of a self-addressed ready-to-
mail package of four formatted high-density IBM diskettes.4
Throughout, we use 90-day euro-currency interest rate series, observed at 10:00 am
Swiss time. We convert both interest differentials and exchange rate changes to annualized
percentages.
Our floating exchange rate data set consists of bilateral dollar rates quoted at noon in
London from 1981 through early October 1994 (the sample period was chosen to match the
EMS data set relatively closely, subject to restrictions of data availability). We include the
following countries, which float more or less cleanly relative to the United States:
Australia; Canada; France; Germany; Japan; Switzerland; and the UK.jj
In Figure 1, the 90-day change in the (natural logarithm of the) exchange rate,
regressand of equation (2), is plotted (on the ordinate) against the 90-day interest
differential (regressor of (2), on the abscissa).
The figure, like our statistical work, pools
together the observations from all countries
and time periods. Figure 1 appears to be a
cloud of observations without any clear
pattern. There does not appear to be a clear
tendency for the observations to be sloped in
any particular way.
Urn nfl,,, Market: IO4*U%0 04,0
100
50
-50
-100
d
EoCrange RoteCnanges Agarnst
20
P.rC.MIOgeS
Interest
40
Ditferentials
Figure 1: Floating Rate Data
jj Canada smooths its exchange rate; the European exchange rates are linked through the
ERM. These reasons may lead one to believe that there should not be enormous
differences between the results from our different data sets. We view the issue as
debatable, and try to let the data speak for themselves.5
Our"fixed" exchange rate data set covers all members of the ERMfromits inception
in March 1979 through early March l994./ We treat Germany as the anchor of the
ERM, and measure all bilateral rates in DM terms. The other long-term members of the
ERM have been: Belgium-Luxembourg; Denmark; France; Ireland; Italy; and the
Netherlands. Portugal, Spain, and the UK were later entrants; Italy and the UK left the
ERM in mid-September 1992. The exchange rate data are cross-rates derived from dollar
rates, observed at 2:15 Swiss time by the BIS at the official "ecu fix". For this data set,
we have both 90- and 30-day euro-currency interest rates.
The ERM has experienced a large number of "events" which have affected European
exchange rates ex post. Counting events for different countries individually, there have
been 54realignmentsof bilateral DM central parities.2/ Many of these events were
anticipated by the financial markets, with varying degrees of accuracy. Comparing
estimates of /3 when (2) is estimated with and without the realignment observations is a
simple way to estimate the well-known small-sample peso problem bias. When we exclude
observations which include these events, we induce the bias. We hope that inclusion of
these observations should remove the peso problem. However, there is no guarantee that
our 54realignmentsare sufficient to constitute a large sample, since they are not
independent events./
1/ We recognize that the ERM was not, strictly speaking, a perfectly fixed exchange rate
system, since it allowed minor currency fluctuations even for "narrow band" members.
2/ There have also been three entrances into the ERM, two exits from the system, one
narrowing of a bandwidth, and a number of widenings of bandwidths. In practice, we
ignore the last events, since central DM parity were not changed, and check for the
sensitivity of our results with respect to the others. ISincewe either include or exclude allofthe realignments from the sample, we are
really providing a maximal estimate of the peso problem bias.6
Our pooled ERM data set is displayed in Figure 2. The top left panel displays the
30-day change in the exchange rate graphed against the corresponding 30-day interest rate
Figure 2: Fixed Rate Data
differential. Only observations which donot includean actual ERM realignment are
plotted. Immediately below is an exactly comparable graph which displays only
observations from the periods which doincludean ERM realignment. The graphs to the
right are analogues for the 90-day market.
Since the data in Figure 2 are sometimes crowded closely together, we also include
"box and whisker" plots for the marginal distributions. These are displayed above (for
No Pea1inrnertS
150
00
50
0
-50 -[
No Peahgr\meritS
60
30
+
0
0 20 40
One Month Market
Pe1iynnents
6b ob
-30
•
0 10 20
Three Month Market
PeaUgnments
30 40
60 150
100
50
0
-50
30
I
I
0
-30
0 &b eb ci ib 20 3b
Annualized Percentages
ExchangeRate Changes Against Interest DHfenentiais
407
interest differentials) and to the right (for exchange rate changes) of the joint distribution
scatterplot. The line in the middle of the box marks the median; the box covers the
interquartile range (i.e., from the 25th percentile to the 75th percentile). The whiskers
extend to upper and lower "adjacent values"; 150% of the interquartile range rolled back to
the nearest data point. Points beyond adjacent values are usually considered to be outliers.
For both the 30- and 90-day markets, the joint distributions for observations without
realignments resemble shapeless clouds. However, there are stronger indications of a
positive relationship between exchange rate changes and interest differentials for the
observations with realignments, particularly in the 90-day market. We now proceed to
investigate this hunch with more rigorous statistical analysis.
III: Results
We estimated (2) on our pooled sample of data. Our results are displayed in Table I.
The first lines present estimates from the data set of floating exchange rates. The
first line estimates (2) for the entire sample; the second adds country-specific intercepts to
the regression specification. Consistent with the findings of the literature, /3 is estimated to
be negative and significantly below its hypothesized value of unity, in both cases. A
negative estimate of 3 is a standard finding in the international finance literature, and
constitutes the "forward discount puzzle'. This finding implies that there is a non-trivial
correlation between ,thedisturbance in (2), and the interest differential.
The last lines present estimates from the EMS sample of fixed exchange rate
observations. There are four set of estimates: two each from the 30- and 90-day markets.
For each maturity, there are two sets of estimates. The first is estimated using the entire8
Table I: Estimates of (2)
Floating-Rate Sample
Three-Month Equation -.28
(.33)
Three-Month Equation, country-specific intercepts -1.07
(.44)
Fixed-Rate Sample
One-Month Equation, including realignments
One-Month Equaion, excluding realignments
data sample; the other excludes all observations which overlap EMS realignments.
In all cases, /3 is estimated to be positive with the EMS data, and significantly greater
than zero at conventional significance levels. This result stands in sharp contrast to our
estimates derived from floating rates, and is our first chief result. However, the point
estimates of /3arestill significantly below the hypothesized value of unity. Thus, there still
appears to be a non-trivial correlation between e and (ii*), but it varies by exchange rate
regime.
Our second main result stems from the fact that /3fallsby an economically and
statistically significant amount when realignments are excluded from the sample. Inducing
such sample selection leads /3 to fall by around .35,arough estimate of the peso problem
bias.
/3 cx
-1.08
(1.14)
n/a
N
19,971
19,971
Three-Month Equation, including realignments
Three-Month Equation, excluding realignments
22,828
21,859
.58 .31
(.11) (.43)
.18 .62
(.05) (.27)
.58 .20
(.09) (.39)
.25 .02
(.06) (.26)
cxand/3areOLS coefficient estimates; their standard errors (estimated with a Newey-
West covariance estimator) are in parentheses. N denotes the total sample size.
22,493
19,6109
We have checked for the robustness of our two results with respect to a number of
perturbations of our basic methodology. For instance, we checked to see if our results
depend on the inclusion of country- or year-specific intercept "fixed effect" terms. We
have also excluded the later ERM entrants, and, separately, all observations after 1989.
Finally, we have estimated 3 on a country-by-country basis. Our chief results
(significantly positiveestimates for fixed exchange rate data, which fall significantly if
realignment observations are excluded) are essentially robust to such changes.
IV: Conclusion
It is well known that countries with floating exchange rates and high interest rates
tend to experience appreciations. This deviation from uncovered interest panty, known as
the "forward discount puzzle' does not appear to characterize our fixed exchange rate data
set. In particular, high interest rates are associated with subsequent (though not
proportionate) currency depreciation in the EMS. Using the same data, we have also found
that excluding periods of realignment from a regression Df exchange rate changes on
interest differentials, leads to a change in the slope coefficient of about -.35. We take the
latter to be an estimate of the much discussed "peso problem" bias.
There remain significant deviations from uncovered interest parity, although these are
much smaller in the EMS regime of fixed exchange rates than in floating rate regimes. We
hope that the contrasting results across exchange rate regimes may enable others to pin
down some of the remaining deviations from UIP, especially given our quantification of the
peso-problem bias. Explanations of the forward discount bias which emphasize
heterogeneous beliefs and trading strategies on foreign exchange markets (either by central10
banks or by traders whoare not fully rational) which are regime-dependentseem
particularly plausible to us.1I
j/ McCallum(1994) has made some progress along these lines.11
References
Froot, Kenneth A. and Richard H. Thaler (1990) "Anomalies: Foreign Exchange' Journal
of Economic Perspectives 4-3, 179-192.
Hodrick, Robert J. (1987) The Empirical Evidence on the Efficiency of Forward and
Futures ForeignErchange Markets. Chur: Harwood.
Lewis, Karen K. (1993) "International Financial Markets" forthcoming in The Handbook of
International Economics, (Grossman and Rogoff, eds).
McCallum, Bennett T. (1994) "A Reconsideration of the Uncovered Interest Parity
Relationship" NBER Working Paper No. 4113, and forthcoming Journal of Monetaty
Economics.</ref_section>

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