European Economic Review 43 (1999) 1531}1567
Noisy signals in target zone regimes:
Theory and Monte Carlo experiments
Steinar Holden*, Dag Kolsrud
Department of Economics, University of Oslo, Box 1095, Blindern, 0317 Oslo, Norway
Received 1 January 1996; accepted 14 January 1998
Abstract
Previous empirical evidence indicates that uncovered interest parity (UIP) does not
hold for target zone exchange rates, like those in the European Monetary System and in
the Nordic countries. We explore a target zone model where the market infers the
probability of a realignment of the band on the basis of a noisy signal. We show
theoretically and through Monte Carlo simulations that if the market overrates the
information content in the signal, then this may explain the empirical results obtained
from testing UIP for target zone exchange rates. 1999 Elsevier Science B.V. All rights
reserved.
JEL classixcation: E34; G14; D84; C15
Keywords: Uncovered interest parity (UIP); Target zone; Realignment; Forward discount
1. Introduction
In a world of fairly free capital mobility, investors are free to choose where
and in which currency to invest. In the absence of any risk premium, the
possibility of arbitrage then implies that uncovered interest parity (UIP) should
hold: The interest rate di!erential between investments in two di!erent currencies should re#ect the expected change in the exchange rate between these
currencies. One implication of UIP that has been subject to testing in a large
* Corresponding author. Tel.: 47 22 85 51 56; fax: 47 22 85 50 35; e-mail: steinar.holden@econ.uio.no.
0014-2921/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 0 1 4 - 2 9 2 1 ( 9 8 ) 0 0 0 4 4 - 0
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
literature, for many countries and time periods, is that the interest rate di!erential should be an unbiased predictor of the change in the exchange rate. Usually,
this test is performed by regressing the actual change in the exchange rate on the
interest rate di!erential, and then seeing whether the b-coe$cient of the interest
rate di!erential is equal to unity, as implied by UIP.
The evidence is however disappointing for UIP. The interest rate di!erential
(the forward discount) is almost always found to be a biased predictor of the
change in the exchange rate, as the coe$cient is generally below unity, and quite
often even negative (Froot and Thaler, 1990; Engel, 1995). Possible explanations
for these "ndings can roughly be divided into three groups. One possible
explanation is based on the idea that investors are risk averse, inducing the
existence of a risk premium in the interest rate di!erential. If the risk premium
varies over time, this might lead to a bias in the b-coe$cient. The second type of
explanation is that agents do not have rational expectations, with various
possibilities for the type of expectations that agents make. The third type of
explanation is that UIP in fact holds, but the empirical "ndings are misleading
due to small samples inducing random expectational errors. One example is the
well-known peso problem, which is named after the period 1955}1976 when
Mexico "xed the peso at a constant rate against the US dollar (Krasker, 1980).
Yet the Mexican interest rates were higher than the US interest rates, re#ecting
the probability, as seen by the market, that a devaluation of the peso would take
place. In a limited sample, no devaluation needs take place. There will be
a systematic expectational error in the sample, but this does not necessarily
involve a violation of the rational expectations hypothesis.
This paper investigates the hypothesis of UIP for countries with target zone
exchange rates, that is, where the government or central bank has made an
explicit commitment to keep the exchange rate between an upper and a lower
bound. The bias in the interest rate di!erential seems to exist for all types of
exchange rate regimes. (The average b-estimate for target zone regimes is about
0.3}0.4, cf. references in Section 3 below). But the recent literature on target zone
regimes, following Krugman (1991), has shown that the existence of a target
zone has important e!ects on the relationship between interest rate di!erentials
and changes in the exchange rate. It seems natural, therefore, to treat the UIP
hypothesis within a target zone as a separate issue. (Recently, Flood and Rose
(1996) emphasize the di!erence in results in tests of UIP for "xed and #oating
exchange rate regimes.)
We explore this issue at two levels. Based on the recent literature on target
zones, we set up a simple theoretical model of a target zone regime, and use this
for Monte Carlo simulations. The aim of this exercise is to explore the small
sample properties of target zone models. As pointed out by Krasker (1980), the
non-normality of the errors makes standard inference invalid. Thus, without
Monte Carlo simulations, we cannot know whether an average b-estimate of
0.3}0.4 is due to non-rational expectations, or whether it can be due to random
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1533
expectational errors. We "nd that the empirical bias in the present and other
studies is unlikely to be explained within the target zone model we simulate. This
indicates that the rejection of UIP must be sought in the existence of a timevarying risk premium or in a violation of the rational expectations hypothesis.
We then proceed to suggest a possible explanation for the rejection of UIP.
According to Froot and Thaler (1990), evidence &suggest that the bias is entirely
due to expectational errors and that none is due to time-varying risk'. Furthermore, Svensson (1992) argue that the risk premium is likely to be very small in
a target zone with narrow bands and moderate devaluation risk, in which case
a time-varying risk premium is unlikely to be the cause of the bias in the
b-coe$cient. Although we do not claim that the case is settled, we at least feel
that this justi"es an attempt to look for explanations based on a violation of
rational expectations.
We investigate a model where the market does not know the &true' probability
of a realignment of the exchange rate, but derives this probability on the basis of
a noisy signal that consists of the true probability of a realignment and a random noise term. Within a rational expectations setting, the market would know
the variances of the two components of the signal, and there would be no bias in
the b-coe$cient. However, it seems di$cult to justify that the market should
know these variances. Previous research shows that learning may converge to
rational expectations (see Bray and Kreps (1987) for a survey of the literature),
but learning seems exceedingly di$cult in this situation. In contrast to most
learning models, the market does not obtain direct observations of the process it
is to learn, as the true probability of a devaluation is not observable, even
ex post. The market only obtains indirect information about the process, by
inference from the relationship between the signal and observations of realignments. Over, say, a ten year period, the market would have only a limited
number of observations of realignments, and it would not be possible to form
precise estimates of the variances of the true probability and the noise on the
basis of this information.
A possible interpretation of the idea that the market does not know the true
probability of a realignment (based upon Drazen and Masson (1994) and
Holden and Vik+ren (1997)) is that the central bank has chosen an explicit
decision criterion and thus will realign if the state of the economy satis"es this
criterion. In this case there will be a true probability of a realignment depending
on the likelihood of the state of the economy satisfying the criterion. However,
the agents in the market do not know the decision criterion used by the central
bank, and the occasional observations of realignments do not provide su$cient
information to identify it.
We then analyse the consequences of the market forming its expectations on
the basis of wrong estimates of the variances of true probability and noise. It
turns out that if the market overrates the variance of the true probability, in
essence, the market overrates the information content in the signal, then this will
1534
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
lead to a downward bias in the b-coe$cient. On the other hand, there will be an
upward bias in the b-coe$cient if the market underrates the information content
in the signal.
Our "ndings can be given two di!erent interpretations. The "rst concerns
how agents treat information. On the premise that private agents cannot know
the true information content in the signal, we argue that the downward bias in
the empirical b-estimates is evidence in favour of the hypothesis that agents
overrate the information content in the signal. This "nding is of independent
interest. Almost all economic behaviour is undertaken in an uncertain environment, and how agents treat the information they receive clearly a!ects their
behaviour. With this view it is important to shed light on how agents interpret
and use new information. Our "nding of an overrating of the information
content of a signal is consistent with previous research suggesting that security
analysts overreact, cf. de Bondt and Thaler (1990) and the references therein.
The second and main interpretation of our "ndings is that it provides an
explanation for the downward bias in the empirical b-estimates. We suggest that
the bias might be due to agents overrating the information content in the signal.
We explore this explanation by use of Monte Carlo simulations, where we
compare the simulation results in a model where agents overrate the information content in a signal with the results from previous empirical studies.
Our analysis of a violation of the rational expectations hypothesis is related to
several recent papers. Roberts (1995) investigates a Mundell-Fleming model
where the agents do not fully know the parameter values of the model. Kandel
and Pearson (1995) present evidence indicating that agents interpret public
signals di!erently because they use di!erent likelihood functions.
This paper is organized as follows. In Section 2, we present the basic theoretical model. Section 3 provides the results of Monte Carlo experiments based on
the model presented in Section 2. Section 4 concludes.
2. The model
Target zone models have received considerable attention over the last years
(Krugman, 1991; Bertola and Svensson, 1993; Mundaca, 1991). We have a much
more restrictive purpose than this literature, namely to provide a framework for
simulations and estimations of the b-coe$cient of the interest rate di!erential.
Thus, we will sidestep many of the issues discussed in this literature, and base
our model on an important "nding of Bertola and Svensson (1993), that the
exchange rate displays mean reversion within the band. However, we believe
that much of the intuition that we obtain in this speci"c model also holds under
less restrictive assumptions.
Let s denote the logarithm of the exchange rate at the beginning of period t,
R
measured as units of home currency per unit of foreign currency (or per unit of
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1535
a basket of foreign currencies). The logarithm of the central parity is denoted by
c and x measures the deviation of the actual exchange rate from the central
R
R
parity (`the exchange rate within the banda). We may then write the exchange
rate as the sum
s "c #x .
R
R
R
Following Rose and Svensson (1991) and Lindberg et al. (1993) we assume that
the mean reversion e!ect within the band can be approximated by a linear
relationship, so that in periods where there is no realignment of the central
parity, the change in the exchange rate is
Dx "x !x "k !kx #u ,
R>
R>
R
R
R>
E(u "x )"0, var(u )"p,
R> R
R>
S
(1)
where 0(k(1. To simplify the theoretical exposition, the upper and lower
bounds of the exchange rate band are not explicitly included, but these bounds
will be incorporated in the simulations. (The mean reversion e!ect is of course
a consequence of the bounds, cf. Bertola and Svensson (1993).)
It is convenient to introduce D as the net impact of a realignment (measured
R
in absolute value). It equals the total change in the exchange rate in period t,
denoted by Ds , minus the change in the exchange rate that would have
R>
occurred if no realignment had taken place in period t, given from (1). D is
R
measured in absolute value, so that, on de"ning a dummy variable d which is
R
1 in periods of devaluation, !1 in periods of revaluation and zero otherwise,
the change in the exchange rate (in periods with and without a realignment) is
Ds "d D #k !kx #u , D '0, d 3+!1, 0, 1,.
R>
R R
R
R>
R
R
(2)
As seen from the last day of one period, we assume that the event that
a realignment takes place during the following period can be seen as a stochastic
variable, with a well-de"ned probability. Furthermore, we assume that in each
period there is either a positive probability of a revaluation or a positive
probability of a devaluation. This is determined by a stochastic variable n with
R
normal distribution that is compressed so that the support is [n*, n3]. It is
de"ned by
Pr(d "1)"n
R
R
if n '0,
R
Pr(d "!1)"!n if n (0,
R
R
R
Pr(d "0)"1!"n ".
R
R
Below we shall for simplicity refer to n as the probability of a realignment:
R
(3)
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Empirical evidence indicates that the probability of a realignment is correlated with the position of the exchange rate within the band (cf. Holden and
Vik+ren, 1992), so that a devaluation is more likely when the exchange rate is
weak (i.e. n and x are positively correlated). To capture this relationship in
R
R
a simple fashion, we assume that
n "n#rx #e , E(e )"0, var(e )"p,
R
R
R
R
R
C
cov(n , x )"cov((rx ), x )"r var(x )"p 50.
R R
R R
R
L V
(To ensure that n lies within the interval [!1,1] there must be bounds on x , cf.
R
R
Section 3 below). We have var(n "x )"p and nC(x )"E(n "x )"n#rx . FurR R
C
R
R R
R
thermore, we assume that D "D for all t. Assuming n and D to be time
R
invariant is not empirically correct, as there appears to be autocorrelation in
empirical realignment probabilities, even controlling for the e!ect of x , and
R
there certainly are realignments of various sizes. We return to this issue in
Section 3 below.
The assumption that there exists a well-de"ned probability of a realignment
can be justi"ed within the models proposed by Drazen and Masson (1994) and
Holden and Vik+ren (1997). In these models there is a central bank with
a speci"ed preference function, and a realignment is chosen if it yields higher
payo! to the central bank than maintaining a "xed parity. The central bank thus
has an explicit decision criterion for whether to realign the currency. As seen
from the beginning of the month, the probability of a realignment is the
probability that the economy evolves so that the decision criterion indicates that
the currency is realigned. Within the models of Drazen and Masson and Holden
and Vik+ren, the agents in the market have imperfect knowledge about the
decision criterion used by the central bank. The lack of perfect knowledge can
apply to the preferences of the central bank and/or how the central bank
evaluates a highly complex economic situation. The imperfect knowledge
of the market implies that the market does not know the true probability of a
realignment.
To simplify the exposition, we choose a simpler framework than the one
suggested by Drazen and Masson, which nevertheless captures the main elements. We assume that the market in each period receives a signal w which
R
consists of the true probability, n , and noise, v :
R
R
w "n #l "n#rx #e #v .
R
R
R
R
R
R
(4)
The signal captures the information about the state of the economy and the
preferences of the central bank that is available to the market; the noise re#ects
that this information is not perfect.
In the present setting rational expectations entails that the market knows the
structure of the model, and all the parameters, and derives expectations on the
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1537
basis of inference from the model. This implies that there is no systematic bias in
the noise component of the signal, so that
E(v "x , n )"0,
R R R
(5)
var(v )"pNvar(w )"p "p#p.
R
T
R
U
C
T
Thus, under rational expectations, the market's information set can be speci"ed
as
I "+x , w , k, k , p, p , n, r, p, p, D,.
R
R R
S VL
C T
However, we shall argue that rational expectations require much too strong
assumptions in the present setting. As observed in the Introduction, the true
probability of a realignment is not observable even ex post. Thus, the market
cannot observe to what extent variation in the signal is due to variation in the
true probability or variation in the noise term. The market only obtains indirect
information about this, by inference from the relationship between the signal
and observations of realignments. One would expect the learning process of the
market to be extremely slow, since realignments occur infrequently and the
event that a realignment occurs gives, in any case, only limited information
about the true probability of a realignment. Moreover, the fact that policy and
other parameters of the model are likely to change occasionally will further
inhibit learning. In other words, deviations from rational expectations will be
di$cult to detect.
To investigate the implications of this reasoning, we set up an alternative
information set where the market have exogenous, and possibly incorrect,
estimates of the variances of the true probability and the noise term. In order to
focus on this particular aspect, we assume that the market knows all the other
parameters in the model. (The other parameters are more directly related to
observable variables, and thus easier to learn; in Appendix A we also consider
the e!ect of other deviations from the rational expectations information set).
The alternative information set is thus
J "+x , w , k, k , p, p , n, r, q, q, D,,
R
R R
S VL
L T
where q and q denote the market's estimate of the variance in n (conditional
L
T
R
on x ) and v . We assume that the market treats q and q as certain, where q#
R
R
L
T
L
q"p .
T
U
To ensure a signal in the probability interval [!1, 1] we assume that the signal noise v has
R
a normal distribution that is compressed to its support [v*, v3], where !1!n*(x )4v*(
R
v341!n3(x ) for all x . In the simulations in Section 3 the support is so wide relative to the
R
R
variance of the noise that the bounds are rarely binding.
An explicit modelling of the learning process is a challenging issue for future research, but
outside the scope of the precent study. See Lewis (1989) for an interesting analysis in a related model
of the learning process of the market after a policy change.
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Regardless of information set, the market makes an estimate of the probability of a realignment, which we will refer to as the subjective probability of
a realignment on the basis of the signal. Under information set I we denote the
R
subjective probability of a realignment by p , which is
R
p ,E(n "I )"E(w !v "I )"w !E(v "I ).
(6)
R
R R
R
R R
R
R R
In forming expectations about v on the basis of the signal w , we assume that
R
R
the market treats the conditional expectation of v given I as a linear function
R
R
of w , i.e.
R
E(v "I )"a#bw .
R R
R
Under this assumption it can be shown (cf. Appendix B) that
cov (v , w )
R R
b"
cov (v , w )
R R (w !E(w "I)),
var (w )
NE(v "I )"
(7)
R
R R
R
R R
var (w )
a"E(v )!bE(w "I)
R
R
R R
where I"I !+w ,. Using Eq. (4) we get E(w "I )"E(n "x )"nC(x ). SubstitutR
R
R
R R
R R
R
ing out for Eq. (7) in Eq. (6), using the decomposition Eq. (5) and
cov (v , w )"p, yields
R R
T
p
p
T nC(x )#
C
p ,E(n "I )"
w.
(8)
R
R R
R
p#p
p#p R
C
T
C
T
Eq. (8) shows that the subjective probability p is a weighted average of the
R
signal w and the ex ante expectation of the true probability nC(x ). The weight of
R
R
the signal is decreasing in the ratio of the variance of the noise to the variance of
the true probability. Thus, if the variance of the noise is small compared to the
variance of the true probability, then the signal is fairly accurate, and the signal
should have a large weight in the subjective probability Eq. (8), (see Johansen
(1978), Chapter 8.9, for a similar argument).
Under the alternative information set J , denoting the subjective probability
R
of a realignment for q , we have correspondingly:
R
q
q
L w.
T nC(x )#
q ,E(n "J )"
(9)
R
R R
R
q#q R
q#q
L
T
L
T
On comparing Eqs. (8) and (9), we observe that if the market overrates the share
of the variability in the signal w that derives from variation in the true
R
probability of a realignment, i.e. q'p, then this will cause the subjective
e
L
If v and n were not bounded, the conditional expectation of v given w would indeed take this
R
R
R
R
form. Our assumption is justi"ed by the fact that the bounds are rarely binding, cf. Section 3.
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1539
probability of a realignment q to vary more with the signal than is warranted. In
R
other words, the market overrates the information content in the signal.
If investors are risk-neutral, equilibrium implies that the expected returns on
investments in home and foreign money markets are equal, i.e. uncovered
interest parity (UIP) holds. Taking expectations of (2) under the respective
information sets, we obtain
d'"E(Ds "I )
E(n "I )D
p D,
R
R> R "E(Dx )#
R R
"k !kx # R
R>
R
d("E(Ds "J )
E(n "J )D
q D,
R
R> R
R R
R
(10)
where we have de"ned d "i!i as the di!erence between the nominal interest
R
R
R
rate in the domestic and the foreign money markets.
The most popular approach to testing UIP has been the regression
Ds "a#bd #g , d 3+d', d(,,
R>
R
R>
R
R R
(11)
where e
is an error term. Under UIP, g
is serially uncorrelated with zero
R>
R>
expectation; otherwise the market could improve upon its prediction of Ds
R>
that is re#ected in d , cf. Eq. (10). Under I the expectation of the coe$cient on
R
R
the interest rate di!erential is
cov (Ds , d')
R> R .
E(bK )"
var (d')
R
(12)
As shown in the appendix, we "nd that the expectation of the b coe$cient in this
case is unity:
Dpp/(p#p)#kvar (x )!2Dkp
C C C
T
R
LV"1.
E(bK "I )"
R
Dpp/(p#p)#kvar (x )!2Dkp
C C C
T
R
LV
(13)
The intuition is that although the market does not know the true probability of
a realignment, it puts correct emphasis on the signal in deriving the subjective
probability of a realignment. Thus, although the correlation between the interest
rate di!erential and the actual change in the exchange rate is lower than if the
market were to know the true probability of a realignment, the interest rate
di!erential will also vary less, and on average these two e!ects will cancel out.
Using the same procedure under the alternative information set J , and
R
Eqs. (2), (9) and (10), we obtain
Dpq/(p#p)#kvar (x )!2Dkp
C L C
T
R
LV"1.
E(bK "J )"
R
Dqq/(p#p)#kvar (x )!2Dkp
L L C
T
R
LV
(14)
If q'p, that is, if the market overrates the information content in the signal,
L
C
then it is clear from Eq. (14) that E(bK "J )(1. Likewise, E(bK "J )'1 if q(p. To
R
R
L
C
1540
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
obtain some intuition concerning the possible values of E(bK "J ), consider the
R
limit case where k"0 (i.e. no mean reversion). Then E(bK "J )"p/q, that is, the
R
C L
ratio of the market's estimate of the variance of the probability of realignment to
the true variance of this probability. For k'0, E(bK "J ) lies in the interval
R
(p/q, 1) if k var (x )'2Dkp . In the Monte Carlo simulations below we
C L
R
LV
explore the case where q'p further.
L
C
The main message of the paper is to suggest a possible explanation of the
downward bias in the empirical b-coe$cient, namely that the market overrates
the information content in the signal. Thus, in the Monte Carlo simulations
below it is this case we explore further. However, before turning to the simulations, we shall make a remark on a di!erent interpretation of the results above.
Let us start from the plausible premise that the market cannot know the true
information content in the signal. By assumption, the market knows the average
probability of a realignment, n, but it does not know how much the true
probability varies over time (the variance of n ). By chance, the market may of
R
course guess correctly on the variance of n , implying that it guesses correctly on
R
the information content in the signal. In practice though, we must expect that
the market either overrates or underrates it. This section has shown one way of
providing evidence on this issue; overrating leads to a downward bias in the
b-coe$cient, underrating to an upward bias. The downward bias that prevails in
empirical b-estimates thus constitutes clear evidence in favour of the hypothesis
that the market overrates the information content in the signal it receives on
future exchange rate movements.
3. Monte Carlo experiments
In this section we present Monte Carlo experiments based on a parameterization of the theoretical model in Section 2. The simulation model consists of
three parts: (1) the movement of the exchange rate within the band x , (2) the
R
realignment probability n , and realizations of realignments d (1 and 2 then
R
R
decides s ), and (3) the signal w , and the subjective realignment probabilities
R
R
p and q . The interest rate di!erentials can then be derived on the basis of UIP.
R
R
The simulated samples of observations of exchange rate changes Ds and interest
R
rate di!erentials d can be used to obtain synthetic b-estimates. The "nite sample
R
distributions of the b-estimator under various assumptions regarding the market's information set can be compared with the empirical b-estimates. The
Monte Carlo experiments where the rational expectations hypothesis is assumed to hold will indicate to what extent the bias in the empirical estimates
may be explained by a "nite sample bias. The experiments based on the
relaxation of rational expectations suggested in Section 2 (that the market
overrates the information content in the signal), may reveal whether this hypothesis is consistent with the empirical "ndings.
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1541
The parameter values of the simulation model are chosen so as to make
estimates and sample statistics of the simulated data come close to their
empirical counterparts in the target zone models of the major Nordic countries,
based on monthly data from Denmark, Finland, Norway and Sweden for the
period 1978/79}1992. Fig. 1 displays the exchange rate data for the four major
Nordic countries.
The observation periods, the same as used by Holden and Vik+ren (1994),
represent the periods from the time these countries adopted a new "xed exchange rate regime (Denmark a member of the EMS; the other countries
unilateral currency baskets) until Finland, Sweden and Norway let their
currencies #oat. For Denmark the observation periods were 1979(3)}1992(12),
for Norway 1979(1)}1992(12), for Finland 1978(1)}1992(9) and for Sweden
1978(1)}1992(11). We calibrate the simulation model to make certain statistical
characteristics consistent with empirical "ndings. The robustness of the simulation speci"c results is assured by sensitivity analysis with respect to the calibrated parameter values of the model.
In the following we use the term empirical to denote real world data and
results based upon observations, while the term synthetic denotes simulated
data and results based upon arti"cal data. The computer programs were written
in the Mathematica2+ programming language, and executed on a Unix workstation, (the programs are available from the authors upon request).
3.1. The exchange rate within the currency band
Table 1 shows empirical estimation results for the exchange rate within the
band, least squares (OLS), instrumental variable (IV) and generalized methods
of moments (GMM). The estimation methods give very similar results. Thus, we
use the simplest and most common method in this setting, OLS, for calibration
and sensitivity analysis. In the basic model for our simulations we set the
parameter values equal to the mean of the empirical estimates presented in the
table. To test the robustness of the simulated results, we let the minimum and
maximum empirical estimates span intervals within which the parameter values
are varied for the purpose of a sensitivity analysis. Table 1 provides
kK 3+0.02,2, 0.2,2, 0.33,, kK 3+!0.08,2, 0,2, 0.2,,
pL 3+0.15,2, 0.35,2, 0.55,,
(15)
S
where the middle numbers approximate the mean empirical estimates (the mean
of k is calculated without the large estimates for Denmark).
We implement the bounded AR(1) model of the exchange rate within the
band:
x "max[x*, min(x3, k #(1!k)x #u )], u &IN(0, p),
R>
R
R>
R>
S
t"1,2,2, ¹,
(16)
Fig. 1. Monthly exchange rates for the four Nordic countries; 180 observations from 1978(1) to 1992(12) measured in percent deviation from central
parities. The vertical lines mark when realignments of the currency band (mostly devaluations) took place. The horizontal lines are the bounds of the
target zones. The shaded areas mask observations outside our sample periods.
1542
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
0.16
(0.04)
0.19
(0.05)
1.39
0.38
0.09
0.18
(0.04)
0.20
(0.05)
1.39
0.37
0.11
(0.05)/(0.05)
1.39/1.39
0.36/0.36
0.11/0.11
1.33/0.28
0.16/0.19
(0.04)/(0.04)
0.16/0.18
(0.04)
1.15
0.15
0.01
0.03
(0.02)
!0.02
(0.04)
1.14
0.16
0.002
0.02
(0.03)
0.00
IV
(0.04)/(0.04)
1.15/1.14
0.15/0.15
0.01/0.01
1.92/1.31
0.03/0.02
(0.02)/(0.02)
0.00/0.01
GMM
(0.06)
1.08
0.53
0.14
0.28
(0.06)
!0.05
OLS
Norway
(0.06)
1.09
0.55
0.12
0.26
(0.06)
!0.04
IV
OLS
(0.05)/(0.05)
1.09/1.09
0.53/0.53
0.14/0.14
1.80/4.98
(0.03)
0.58
0.17
0.10
0.27/0.33
0.18
(0.08)/(0.05)
(0.04)
!0.08/!0.06 !0.03
GMM
Sweden
(0.03)
0.59
0.17
0.07
0.15
(0.04)
!0.02
IV
0.17/0.16
(0.05)/(0.04)
!0.02/!0
.02
(0.03)/(0.03)
0.59/0.59
0.17/0.17
0.10/0.10
0.10/5.18
GMM
Notes. The numbers in parentheses are the standard errors of the estimates. The IV estimator uses x
as an instrument for x in a least-squares regression
R\
R
of x , assuming MA(1) innovations u . The two GMM estimates are separated by the slash. The "rst GMM estimator uses 1, x , x ,x
as four
R>
R>
R\ R\ R\
instruments orthogonal to the innovation u , while the second GMM estimator uses 1, x , x , x , x as "ve instruments, where x , x , x denote the
R>
R\ R R R
exchange rates of the other three Nordic countries. Both estimators use the Newey}West heteroscedasticity/autocorrelation consistent covariance matrix
with one lag. R is the correlation coe$cient. J is a test of overidentifying restrictions on the GMM estimation. It is asymptotically s distributed with two
and three degrees of freedom, respectively. The critical values of the tests (0.95) are 5.99/7.82.
pL V
pL S
R
J
kK
kK
OLS
GMM
OLS
IV
Finland
Denmark
Table 1
Empirical estimates and sample statistics of the AR(1) exchange rate model (1), using di!erent estimators and samples of 158}175 monthly observations for
the Nordic countries in the period 1978}1992, cf. Fig. 1 for plots of the empirical time series. Standard deviations of the estimates are in parentheses.
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1543
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
where the innovations u
are independent and normally distributed (IN). The
R>
results above motivate the following parameter values
k"0.15, k "0, p"0.6,
S
x*"!2.5, x3"2.5, ¹"160,
(17)
for the generation of synthetic time series data. Fig. 2 shows the sample distributions of the exchange rate parameters and sample statistics estimated on synthetic data generated by the model (16)}(17). By using the parameter values (17)
as input to the model simulations, the mean synthetic estimates correspond
closely to the mean of the Nordic empirical estimates. In particular, the bounds
x* and x3 of the band compress the innovations and induce a mean reversion
e!ect that biases the estimate of k close to the mean empirical value of 0.2. The
length of the series (¹"160) approximates the Nordic series.
Table 2 shows the results of a partial sensitivity analysis of the estimates in
Fig. 2 (based on simulated data) with respect to the values of the exchange rate
parameters (17) put into the simulations. One single parameter at the time is
changed from its input value to the minimum and maximum value of the
corresponding empirical estimates (15) before repeating the regression on regenerated synthetic data. In addition we vary the width of the exchange rate band
(x3!x*) and the length of the synthetic data series, i.e. the size ¹ of the
&observation' samples. Finally, we let all the model parameters be independent
stochastic variables with a normal distribution and 10% standard deviations.
The partial sensitivity analysis ensures that we also undertake simulations with
models that are closer to each of the Nordic countries, and not only close to the
&average' Nordic country. The results are mostly quite intuitive, as can be seen
from Table 2. The partial sensitivity analysis shows that by changing the model
parameters within the ranges of the empirical estimates, the synthetic estimates
change within the empirical ranges. In this sense, we conclude that the exchange
rate model is robust and statistically consistent with the empirical "ndings.
3.2. The realignment probability and realizations
The probability of a realignment in period t is implemented by the equation
n "n#rx #max[e*, min(e3, e )], e &IN(0, p), t"1, 2,2, ¹,
R
R
R
R
C
(18)
Note that we do not try to capture all the characteristics of the evolution of the exchange rate
within the band (in which case a more sophisticated dynamic model would be called for, cf. e.g.
Pesaran and Samiei (1992)). Our more modest aim is to calibrate our simple model to share certain
statistical properties of the &average' Nordic exchange rate.
This e!ect adds to a "nite sample bias in estimates of autocorrelation (0.022 in our model, cf.
Mariott and Pope (1954)).
Fig. 2. Setting the parameters of the exchange rate model to k"0.15, k "0 and p"0.6, we get the following "nite sample (¹"160
S
observations) distributions of 10.000 synthetic OLS estimates of the exchange rate model parameters and sample statistics. The mean synthetic
estimates, denoted by a tilde, are close to the mean empirical estimates of the Nordic countries: kM "0.2, kM "0, pN "0.35, pN "1.05, R"0.085.
S
V
Mean standard deviations of the estimates (not of their means) are in parentheses.
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1545
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Table 2
Partial sensitivity analysis of the basic exchange rate model: Consequences for the mean value and
mean standard deviation of 2000 synthetic parameter estimates from partial changes to the
parameters of the data generating model (DGM). In each alternative only one single parameter of
the basic model is changed at a time. The mean standard deviations of the parameter estimates (not
of their means) are in parentheses
DGM
Basic model
Mean and (standard deviations) of 2000 estimates
Partial
change
Basic model,
No change
Eqs. (16) and (17)
k"0.15
0.25
0.05
k "0
0.2
!0.1
p"0.6
S
0.7
0.5
x*, x3"$2.25 $3.0
$1.5
No bounds
¹"160
¹"1000
120
60
Stochastic
(normal)
parameters
Mean empirical
estimates
kI
(pJ )
I
kI
(pJ )
IŠ
pJ S
(pJ )
NS
pJ V
(pJ )
NV
0.188
(0.043)
0.276
(0.054)
0.116
(0.034)
0.226
(0.056)
0.198
(0.047)
0.201
(0.043)
0.179
(0.045)
0.176
(0.045)
0.234
(0.045)
0.173
(0.047)
0.171
(0.015)
0.196
(0.053)
0.223
(0.084)
0.209
(0.059)
0.000
(0.055)
!0.002
(0.052)
0.000
(0.065)
0.240
(0.095)
!0.116
(0.066)
!0.002
(0.063)
!0.001
(0.046)
!0.002
(0.056)
!0.002
(0.050)
!0.002
(0.056)
!0.000
(0.019)
!0.000
(0.068)
0.005
(0.111)
0.004
(0.139)
0.340
(0.036)
0.351
(0.038)
0.309
(0.035)
0.299
(0.037)
0.329
(0.036)
0.443
(0.046)
0.243
(0.026)
0.353
(0.039)
0.295
(0.031)
0.356
(0.040)
0.340
(0.014)
0.337
(0.042)
0.335
(0.061)
0.336
(0.075)
1.040
(0.210)
0.761
(0.141)
1.508
(0.349)
0.789
(0.195)
0.969
(0.209)
1.269
(0.227)
0.787
(0.183)
1.166
(0.290)
0.733
(0.105)
1.200
(0.327)
1.093
(0.086)
1.012
(0.238)
0.937
(0.312)
1.092
(0.354)
0.094
(0.22)
0.138
(0.27)
0.058
(0.018)
0.113
(0.028)
0.099
(0.024)
0.101
(0.022)
0.090
(0.023)
0.088
(0.023)
0.117
(0.023)
0.087
(0.024)
0.086
(0.008)
0.098
(0.028)
0.112
(0.044)
0.128
(0.058)
0.35
1.05
0.085
0.2
0
RI (pJ )
0
where e* is a lower and e3 is an upper bound on the random part of the
realignment probability (to keep n 3 [!1, 1]), and x is generated by the basic
R
R
model (16)}(17). In the simulations, a realignment occurs according to a draw
from a binomic distribution with probability "n " (a devaluation if n '0, a reR
R
valuation if n (0).
R
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1547
We do not have the same empirical footing when deciding on what values
to set for the parameters of the realignment probability model (18). The realignment probability is not observable, and cannot be derived from the observed
realignments either. Hence, the following values are arbitrarily chosen on the
grounds that they realize a number of realignments that is consistent with the
empirical "ndings. To ensure that there are considerably more devaluations
than revaluations, the constant term n is positive so that E(n )'0. We use the
R
values
n"0.028, r"0.035, p"0.02, e*"!0.2, e3"0.2,
C
¹"160,
(19)
for the generation of the time series data.
The distribution of the simulated numbers of devaluations and the distribution of the simulated numbers of revaluations are both depicted in Fig. 3, along
with the "nite sample mean and variance of the realignment probability, and its
covariance with the exchange rate. The mean number of realignments correspond closely to the mean empirical numbers in our sample: 5.25 devaluations
and 0.75 revaluations.
We do not perform a partial sensitivity analysis of the realignment probability
model with respect to the parameter values, because the parameters (19) are
tuned to the model (18) to get the number of realignments close to the empirical
means. However, we look at a restricted model of the realignment probability,
where the probability of a realignment is independent of the position of the
exchange rate in the band:
n "0.028#e ,
e &IN(0, 0.035), t"1, 2,2, ¹.
R
R
R
We have increased the variance of the innovations e relative to the basic model
R
(19), in order to get the required number of realignments. Finally, we include
a model where all the parameters in (19) are independent stochastic variables
with normal distributions and 10% standard deviations. In all models the
innovations are bounded to ensure a realignment probability below unity in
absolute value. Table 3 shows the results for the alternative models.
Our speci"cation of the probability of a realignment does not capture the
autocorrelation that exists in empirical realignment probabilities (which in part
re#ects autocorrelation in macroeconomic variables like unemployment rates
and trade de"cits). Thus, in empirical samples observations with high probability and high interest rate di!erentials are likely to be grouped, whereas in our
synthetic data they will be randomly distributed. However, for our purposes it is
not necessary to include this property. The aim of our model is to provide
synthetic data so as to obtain a distribution of OLS-estimates of the b-coe$cient. When using OLS, as most empirical b-estimates are based on, the ordering
of the observations does not a!ect the estimates. Thus, the important issue is the
Fig. 3. The distributions of 10.000 simulated numbers of realignments, the simulated sample mean and variance of the realignment probability and
"nally the simulated covariance of the realignment probability and the exchange rate. Standard deviations of the estimates (not of their means) are in
parentheses.
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1549
Table 3
Finite sample properties of alternative realignment probability models. The standard deviations of
the estimates (not of their means) are in parentheses
Model
Mean and (standard deviations) of 2000 estimates
C
" (pJ )
I
C
0 (pJ )
I
nJ (pJ )
R NL
pJ (pJ )
L NL
pJ (pJ )
VL NVL
Basic model,
Eqs. (18) and (19)
5.505
(2.491)
0.996
(1.068)
0.028
(0.010)
0.0017
(0.0003)
0.036
(0.008)
r"0
5.130
(2.165)
0.661
(0.831)
0.028
(0.003)
0.0012
(0.0001)
!0.000
(0.003)
Stochastic (normal)
parameters
5.732
(3.422)
1.275
(1.466)
0.029
(0.021)
0.0018
(0.0006)
0.038
(0.013)
Mean empirical
estimate
5.25
0.75
distribution of realignment probabilities, on which our synthetic data match
their empirical counterparts.
This argument hinges on there being no autocorrelation in the regression
residual in Eq. (11), as the distribution of the b-estimate is sensitive to combined
autocorrelation in the interest rate di!erential and in the regression residual.
However, as mentioned above, UIP implies that there is no autocorrelation in
the regression residual. Autocorrelation in the residual would thus involve
a rejection of UIP. When we test UIP on the basis of the b-estimates, we cannot
allow for violations of UIP in other respects. Moreover, autocorrelation in the
regression residuals in Eq. (11) appears to be a less severe problem than the bias
in the b-coe$cient. The most extensive study of UIP in European countries that
we know of, Bernardsen (1997), reports signi"cant autocorrelation on monthly
observations for two of ten countries.
3.3. The signal in the market and the subjective realignment probabilities
The signal is modelled as a perturbation of the true realignment probability
by additive &noise':
w "n #max[v*, min(v3, v )],
R
R
R
t"1, 2,2, ¹,
where the following values are used
c"2Np"2p"2 ) 0.02,
T
C
v &IN(0, p), p"cp,
R
T T
C
(20)
v*"!0.3, v3"0.3, ¹"160,
(21)
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S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
and n is generated by the basic model (18)}(19). Note that v is compressed at the
R
R
bounds v* and v3 (very rarely binding) to ensure that the signal w lies within
R
[!1, 1]. The important parameter in (21) is the ratio between the conditional
variance of the signal relative to the conditional variance of the true realignment
probability, c"p/p, which we have arbitrarily set equal to two. This ratio is
T C
chosen without any empirial foundation. We are, of course, in the same position
that we suggest the market is in, that we do not know how much noise there is in
the information that the market has to its disposal.
The rational expectation hypothesis implies that the market uses an information set that contains all the true parameter values: I "+w , n, r, D, k , k, x ,
R
R
R
p, p, p ,. Knowing that the conditional variance of the signal is twice the
S C T
conditional variance of the realignment probability, the market forms its subjective probability of a realignment as (cf. Eq. (8)),
p
p
T E (n )#
C
p "E(n "I )"
w
R
R R
R
R
p#p
p#p R
C
T
C
T
c
1
2
1
"
E (n )#
w " E (n )# w
R
R
R
R
R
1#c
1#c
3
3 R
1
1
"E (n )#
(e #v )"E (n )# (e #v ),
R R
R
R R
R
1#c R
3 R
(22)
where the bounds are ignored as they are rarely binding.
In the alternative information set J "+w , n, r, D, k , k, x , p, c, q,, we
R
R
R S C T
relax the rational expectations hypothesis by assuming that the agents in the
market do not know the true information content of the signal. As argued in
Section 2, the relative sizes of the unknown variances p and p cannot be
e
inferred from occasional realignments. We assume that the market overrates the
information content in the signal and set the ratio of the subjective variances
q/q"jc, where j(1 re#ects the degree of overrating of the true ratio c"2.
T C
(We do not investigate the consequences of underrating. As seen from Section 2
above this results in b-estimates above unity, which is of little interest from an
empirical point of view.) We arbitrarily choose j"1/4, which implies an
incorrect weighting of the signal by 2/3 rather than the correct weight 1/3.
Hence,
q
q
T E (n )#
C
q "E(n "J )"
w
R
R R
R
R
q#q
q#q R
C
T
C
T
jc
1
1
2
"
E (n )#
w " E (n )# w
1#jc R R
1#jc R 3 R R
3 R
1
2
"E (n )#
(e #v )"E (n )# (e #v ),
R R
R
R R
R
1#jc R
3 R
(23)
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1551
again not accounting for the probability bounds. The "rst row in Table 4 shows
the "nite sample moments of the subjective realignment probabilitites (22) and
(23). The results depend on the noise level in the signal w relative to the true
R
realignment probability n , i.e. c, and the weigthing between the expectations
R
E (n ) and the signal based on the subjective perception of the latter ratio, i.e. j.
R R
To check upon this two-parameter dependence we allow for both twice and half
the overrating of the signal, and twice and half the information/noise ratio of the
signal. These changes to the basic model (20)}(23) are all partial, in the sense that
only one of the two parameters (c, j) is changed at the time, to yield (c, j/2),
(c, 2j), (2c, j) and (c/2, j), accordingly. The results of these changes are displayed
in the second to "fth row in Table 4. We see that the di!erent information
contents in the signals (c) and the overrating (j) do not make much di!erence to
the mean and variance of the subjective realignment probabilities. But we shall
see that these small di!erences make large di!erences for the b-estimates.
3.4. The interest rate diwerential
The interest rate di!erential is an implementation of equations (10) with one
di!erence. Since x "x is distributed as IN(k #(1!k)x , p) and the series are
R> R
R S
Table 4
Expected value and variance of the subjective realignment probabilities p and q based on the two
R
R
di!erent information sets I and J , respectively ("rst row). To check the partial sensitivity of the
R
R
results, more or less overrating of the information content in the signal (second and third row) and
di!erent levels of noise in the signal (fourth and "fth row) are applied. Standard deviations of the
2000 synthetic estimates (not their means) are in parentheses. The mean empirical estimate of the
realignment probability the Nordic countries are 0.036 when Denmark is excluded, and 0.075 when
Denmark is included, cf. Table 2 in Holden and Vik+ren (1994)
p (pJ )
N NN
qJ (pJ )
O
p (pJ )
O NO
Model
pJ (pJ )
N
Basic model (22)}(23)
0.02806
0.00140
0.2809
0.00180
(0.00944) (0.00027) (0.00959) (0.00030)
c"2, j"1/8Np "E (n )#w , q "E (n )#w
R R R
R R R R
R
0.02806
0.00140
0.02810
0.00203
(0.00944) (0.00027) (0.00966) (0.00032)
c"2, j"1/2Np "E (n )#w , q "E (n )#w
R R R
R R R R
R
0.02806
0.00140
0.02808
0.00157
(0.00944) (0.00027) (0.00951) (0.00028)
c"4, j"1/4Np "E (n )#w , q "E (n )#w
R R R
R R R R
R
0.02805
0.00135
0.02809
0.00177
(0.00942) (0.00027) (0.00957) (0.00030)
c"1, j"1/4Np "E (n )#w , q "E (n )#w
R R R
R R R R
R
0.02806
0.00147
0.02809
0.00178
(0.00947) (0.00027) (0.00958) (0.00030)
1552
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
not allowed to exceed the limits x3 and x* of the target zone, we get
"E(x #1"x )"("k #(1!k)x " due to the bounded innovations. The expected
R
R
R
value of the exchange rate in the next period is approximated by numerical
integration, cf. Appendix D. The implemented equation under the two information sets is thus given by (10), where the integral replaces E(*x ). The size of
R>
the change in the central parity is D"6 (per cent), which is close to the average
size of the devaluations for the Nordic countries (which is 6.6 per cent, cf.
Holden and Vik+ren (1994), Table 2). The "rst row of Table 5 gives the means
and variances of 2000 synthetic interest rate di!erentials.
The reason that the mean values do not correspond exactly to their empirical
counterparts is that for Denmark and Norway the empirical interest rate
di!erentials have been higher than what have been warranted by the empirical
changes (devaluations) in the nominal exchange rate, leading to an excess return
of investments in Danish and Norwegian kroner. (This is the basis for the
rejection of UIP in Holden and Vik+ren, (1994)). We have chosen to calibrate the
model to the actual number of realignments, rather than the actual interest rate
Table 5
The means and standard deviations (in parentheses) of 2000 simulated interest rate di!erentials in
the basic model ("rst row) and in alternative models. To check the sensitivity of the results, di!erent
partial changes to the parameters of the basic model are applied (U denotes a uniform distribution
on the interval [2, 10])
Model
Mean and (standard deviations) of 2000 estimates
dI ' (pJ ')
B
pJ ' (pJ ')
B NB
dI ( (pJ ()
B
pJ ( (pJ ()
B NB
Basic model, D"6
0.1684
(0.0130)
0.0070
(0.0008)
0.1686
(0.0160)
0.0213
(0.0024)
D"8
0.2244
(0.0313)
0.0219
(0.0030)
0.2246
(0.0337)
0.0473
(0.0056)
D"4
0.1121
(0.0083)
0.0033
(0.0006)
0.1123
(0.0102)
0.0096
(0.0012)
D &N(6, 1.5)
R
0.1687
(0.0130)
0.0068
(0.0008)
0.1688
(0.0163)
0.0213
(0.0024)
D &;(2,10)
R
0.1627
(0.0128)
0.0069
(0.0008)
0.1687
(0.0160)
0.0212
(0.0024)
Stochastic (normal) parameters
0.1697
(0.1138)
0.0077
(0.0035)
0.1697
(0.1143)
0.0224
(0.0066)
Mean empirical estimate
0.235
0.025
0.235
0.025
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1553
di!erentials, but this is unlikely to have much e!ect on the b-coe$cient (it will
however a!ect the constant term in the regression). For a sensitivity analysis we
include alternative constant realignment sizes (D"4 or 8), and also let DR be
a normal and a uniform stochastic variable, cf. Table 5.
3.5. The xnite sample distribution of the b-estimates
Fig. 4 shows the results of regressing the total change in the exchange rate on
the interest rate di!erential, cf. (10)}(11), yielding the distributions of the synthetic b-estimates (upper row of four plots) and their corresponding t-statistic:
t(bK )"(bK !1)/pL @, which measures how many standard deviations the b-estimate
deviates from the theoretical UIP value of unity (lower row of four plots). From
left to right it is assumed that market's expectations re#ect the true realignment
probability nR, the subjective expectations pR and qR (i.e. the two di!erent
information sets IR and JR), and the signal wR only. The vertical lines in all
histograms mark eight empirical OLS estimates from the literature. de Grauwe
(1989) examines the relevance of UIP for four EMS currencies against the
German mark, using OLS on monthly data for the period 1979 to 1988. He
obtains estimates of b (t-statistic in parentheses) of 0.96 (!0.125) for French
francs, 0.65 (!1.84) for Italian lira, 0.61 (!1.39) for Belgian francs and !0.49
(!1.96) for Dutch guilders. In an earlier version of this paper (Holden et al.
(1993)), we tested UIP for the Nordic currencies using OLS on monthly data for
the period 1978/79 to 1990. We then obtained estimates of 0.48 (!1.73) for
Danish kroner, 0.72 (!0.80) for Finnish mark, 0.41 (!1.59) for Norwegian
kroner and !0.48 (!1.97) for Swedish kroner. The mean b-estimate of all eight
currencies is 0.36. (After our simulations were undertaken, Bernhardsen (1997)
reports a b-coe$cient of 0.10 for Austria for the period March 1979 to February
1995, consistent with our view that there is a bias in empirical b-estimates.)
Before comparing the empirical and synthetic estimates, note from Fig. 4 that
the e!ect of the various assumptions concerning the market's expectations is
consistent with the theoretical predictions. If the market does not overrate the
information content, the mean b' is virtually the same as when using the true
realignment probability n, but the variance of the estimates is much larger. As
predicted by the theoretical model in Section 2 the mean value drops considerably to b("0.53 when the market overrates the information content in the
signal.
Then we compare the empirical estimates with the simulated distributions.
Under rational expectations, seven of the eight empirical b- and t-estimates are
NesseH n (1994) examines UIP for the Nordic countries over slightly shorter time periods than
Holden et al. (1993), and obtain b-estimates closer to zero (mean value equal to !0.01).
Fig. 4. The "nite sample distributions of 10.000 synthetic OLS-estimates of the b-coe$cient (upper plots) and its t-value (bK !1)/pL (lower plots), in
@
a regression of the change in the exchange rate on the (subjective) expectations, cf. Eq. (11). The eight vertical lines in each of the four upper plots
mark the empirical estimates referred to in the text. The eight lines in the four lower plots mark the t-values of the eight empirical estimates. In each of
the lower plots the Students t-distribution (centered on zero) is superimposed on the sample distributions to visualize any di!erences. The di!erent
sample distributions result (from left to right) when the market uses the true realignment probability n , the information sets I and J re#ecting correct
R
(p ) and incorrect weighting (q ), respectively, between signal and expectations, and "nally the noisy signal w only. The "gure just below the horisontal
R
R
R
axis in the plots are the median synthetic estimate. The mean synthetic estimate and the parenthesized mean standard deviation of the estimates are
shown below the plots.
1554
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1555
below the median (and the mean) of the synthetic distributions, according to the
two leftmost plots in each row of Fig. 4. This suggests that the simulated model
with rational expectations is not an acceptable representation of the empirical
data generating process. Relaxing the rational expectations hypothesis, the
synthetic distributions become more consistent with the empirical estimates, as
shown by the four rightmost plots in Fig. 4.
We shall analyse the issue of consistency by use of a more formal statistical
method, and test whether the empirical and the synthetic b- and t-estimates can
be viewed as two samples independently drawn from identical distributions.
According to Press et al. (1986), a generally accept such test for continuous
variables is the Kolmogorov}Smirnov (K}S) test, which measures the overall
di!erence between the synthetic and the empirical samples by the maximum
value of the absolute di!erence between the cumulative distributions of the two
samples. The distribution of the K}S statistic can be approximated under the
null hypothesis that the two samples come from identical distributions, thus
giving the signi"cance of any observed discrepancy between the two cumulative
distributions.
The K}S statistic (P column) in Table 6 shows that the hypothesis that the
@
empirical b-estimates and the rational expectation estimates in the simulation
model (models n and I ) come from identical distributions can be rejected at
R
R
10 per cent level, but not at 5 per cent (P "0.06 and 0.07, respectively). As for
@
the b-estimates of the non-rational expectation models (w and J ), the K}S
R
R
statistics of 0.43 and 0.58 are far from rejection of identical distributions.
Turning to the K}S statistic on the hypothesis that the empirical and synthetic
t-values are drawn from identical distributions, the rational expectation models
are rejected at 1 per cent level. Model J is rejected at 10 per cent level, while
R
model w is still far from rejection.
R
Table 6
Descriptive statistics of 10.000 synthetic and 8 empirical estimates. A tilde denotes the mean value of the
synthetic estimates. Subscript 0.5 denotes the median. C denotes the smallest probability intervall (in
steps 0.99, 0.95, 0.9, 0.8, 0.7, 0.6, 0.5) that contains all the empirical estimates. C denotes the number of
empirical estimates below and above the median of the synthetic estimates. P is the signi"cance level of
the Kolmogorov Smirnov statistic
Model
n
R
I
R
J
R
w
R
Statistics of the b-estimates
Statistics of the t-estimates
bI
pJ
@
b
C
@
C
P
@
0.961
0.923
0.533
0.360
0.917
1.357
0.762
0.517
0.896
0.833
0.491
0.335
0.95
0.80
0.90
0.95
7#1
7#1
4#4
2#6
0.06
0.07
0.58
0.43
tI
!0.158
!0.164
!0.759
!1.429
pJ
R
1.160
1.123
1.142
1.185
t
!0.127
!0.149
!0.739
!1.396
C
R
C
0.90
0.90
0.80
0.80
7#1 (0.01
7#1 (0.01
7#1
0.06
5#3
0.42
P
R
1556
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
There are two reasons that one should be cautious when interpreting
the results of the K}S statistic. First, the sample of eight empirical estimates
is very small. The signi"cance of K}S statistic is based on an approximation
formula that becomes asymptotically accurate, and Press et al. write that
a sample size of 20 su$ces in practice. Secondly, the empirical estimates
are likely to be correlated, while the K}S formula requires independent
draws. If the empirical estimates are uncorrelated we can to a certain degree
circumvent the problem of too few observations, by using the simple method of
comparing the empirical estimates with the median of the synthetic estimates. As
noted above, seven out of eight empirical estimates are below the median values
of the synthetic estimates based on rational expectations, cf. the C column in
Table 6. The probability of seven (or more) of eight estimates are below the
median is 9;(1/2)"9/256+0.035; thus the hypothesis of identical distributions would be rejected at 5 per cent level. The problem of potensial correlation
between the empirical estimates can however not be circumvented. In spite of
this caveat, we conclude that it seems unlikely that the empirical estimates are
generated within the rational expectation models that we have simulated. On
the other hand, the empirical b-estimates do not seem inconsistent with the
simulation models where the market overrates the information content in the
signal.
The remainder of this subsection contains a sensitivity analysis with respect to
the distribution of the b-estimates. Table 7 displays the means and standard
deviations of the b-estimates and their t-statistics for all the partial changes to
the basic model. In addition it shows the widths of the probability intervals that
contain the empirical estimates, the number of estimates on either side of the
median, and the signi"cance level of the K}S statistic, i.e. the C, C and
P columns of Table 6. The results are quite similar to those of the basic model in
Table 6, but with a slight tendency towards less signi"cant K}S statistics in all
model versions.
Let us review the panels of Table 7. Panel 1 shows how the partial changes to
the parameters of the exchange rate model a!ect the mean b-estimates and their
t-statistics. The values are all fairly similar, and to show that the b-estimates are
not very sensitive with respect to the parameters values of the implemented
model. The &No bounds' row is equivalent to a very, very wide band (which does
get realigned like the other bands). Panel 2 shows that breaking the correlation
To save space we do not display frequency distributions like in Fig. 4 for any alternative model.
Plots (available upon request from the authors) show that the changes imply mainly to the locations
and widths, and not much to the &shapes' (asymmetries) of the distributions (e.g. all distributions of
the t-statistic remain more or less left/downward skewed).
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1557
Table 7
Sample statistics calculated in sets of 2000 b-estimates and their t-statistics: t "(bK !1)/pL . The statistics
@
@
are (i) mean over standard deviation, (ii) probability interval containing all empirical estimates (C) over
the number of estimates below and above the synthetic median (C), and (iii) the signi"cance level of the
Kolmogorov}Smirnov statistic (P), cf. Table 6. The various panels show the e!ects on the statistics of
partial changes in a single model parameter value. Note that it is the mean standard deviations of the b- and
t-estimates (not of their means) that are in parentheses
Change
Panel 1: E!ects of partial changes to the parameters
bI
bI
bI
bI
tI
L
'
(
U
L
(pJ ')
(pJ ()
(pJ U)
(pJ L)
(pJ L)
@
@
@
R
@
C
C
C
C
C
@L
@'
@(
@U
RL
(C K )
(C K )
(C K )
(C K )
(C K )
@
@
@
@
R
PL
P'
P(
PU
PL
@
@
@
@
R
to the exchange rate model x
R
tI
tI
tI
'
(
U
(pJ ')
(pJ ()
(pJ U)
R
R
R
C
C
C
'
(
R
R
RU
(C K )
(C K )
(C K )
R
R
R
P'
P(
PU
R
R
R
k"0.25
1.030
(0.809)
0.99
(8#0)
0.02
1.115
(1.304)
0.80
(8#0)
0.01
0.591
(0.721)
0.90
(4#4)
0.43
0.389
(0.493)
0.95
(2#6)
0.39
0.009
(1.012)
0.99
(8#0)
(0.01
!0.109
(1.038)
0.99
(8#0)
(0.01
!0.611 !1.337
(1.033)
(1.080)
0.90
0.80
(7#1)
(6#2)
0.02
0.57
k"0.05
0.926
(0.664)
0.99
(7#1)
0.05
0.890
(0.747)
0.95
(7#1)
0.08
0.707
(0.606)
0.99
(6#2)
0.29
0.527
(0.478)
0.99
(4#4)
0.75
!0.247
(1.217)
0.90
(7#1)
0.01
!0.266
(1.168)
0.90
(7#1)
0.01
!0.649 !1.216
(1.209)
(1.253)
0.80
0.70
(7#1)
(6#2)
0.04
0.37
k "0.2
1.001
(1.006)
0.90
(8#0)
0.03
1.023
(1.584)
0.70
(7#1)
0.04
0.568
(0.876)
0.80
(4#4)
0.37
0.380
(0.595)
0.90
(2#6)
0.55
!0.033
(0.948)
0.99
(8#0)
(0.01
!0.009
(0.864)
'1
(6#0)
(0.01
!0.497 !1.048
(0.951)
(0.983)
0.90
0.70
(7#1)
(6#2)
0.01
0.22
k "!0.1
0.987
(0.870)
0.95
(7#1)
0.05
0.974
(1.349)
0.80
(7#1)
0.05
0.546
(0.735)
0.90
(4#4)
0.65
0.366
(0.496)
0.95
(2#6)
0.44
!0.135
(1.186)
0.90
(8#0)
(0.01
!0.126
(1.157)
0.90
(8#0)
(0.01
!0.785 !1.511
(1.177)
(1.231)
0.70
0.80
(7#1)
(5#3)
0.07
0.33
p"0.7
S
0.987
(0.944)
0.95
(7#1)
0.04
0.969
(1.443)
0.70
(7#1)
0.05
0.549
(0.798)
0.90
(4#4)
0.47
0.370
(0.541)
0.95
(2#6)
0.46
!0.093
(1.104)
0.95
(8#0)
(0.01
!0.092
(1.039)
0.95
(8#0)
(0.01
!0.656 !1.286
(1.082)
(1.120)
0.80
0.70
(7#1)
(6#2)
0.04
0.49
p"0.5
S
0.987
(0.869)
0.95
(7#1)
0.05
0.964
(1.318)
0.80
(7#1)
0.06
0.559
(0.736)
0.90
(4#4)
0.59
0.376
(0.498)
0.95
(2#6)
0.47
!0.140
(1.177)
0.90
(8#0)
(0.01
!0.136
(1.122)
0.90
(8#0)
(0.01
!0.757 !1.465
(1.163)
(1.220)
0.80
0.80
(7#1)
(5#3)
0.07
0.36
x*, x3"$3.0 0.973
(0.897)
0.95
(7#1)
0.05
0.392
(0.524)
0.80
(7#1)
0.05
0.929
(1.302)
0.90
(4#4)
0.65
0.577
(0.766)
0.95
(2#6)
0.44
!0.141
(1.161)
0.90
(8#0)
(0.01
!1.339
(1.180)
0.90
(8#0)
(0.01
!0.154 !0.694
(1.135)
(1.148)
0.70
0.80
(7#1)
(5#3)
0.07
0.33
1558
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Table 7
Continued
x*, x3"$1.5 1.019
(0.847)
0.95
(8#0)
0.03
1.061
(1.365)
0.80
(8#0)
0.03
0.552
(0.725)
0.90
(4#4)
0.55
0.368
(0.490)
0.95
(2#6)
0.49
!0.044
(1.053)
0.95
(8#0)
(0.01
0.011
(0.973)
0.99
(8#0)
(0.01
!0.687 !1.408
(1.050)
(1.119)
0.80
0.80
(7#1)
(5#3)
0.04
0.47
No bounds
0.912
(1.276)
0.80
(7#1)
0.08
0.582
(0.762)
0.90
(4#4)
0.48
0.397
(0.524)
0.95
(2#6)
0.53
!0.147
(1.169)
0.90
(8#0)
(0.01
!0.166
(1.150)
0.90
(7#1)
(0.01
!0.692 !1.333
(1.155)
(1.184)
0.80
0.80
(7#1)
(6#2)
0.05
0.54
0.968
(0.914)
0.95
(7#1)
0.05
Panel 2: E!ects of partial changes to a parameter of the realignment probability model n . The entry '1
R
means that some empirical estimates are outside the interval spanned by the synthetic sample, hence the
empirical estimates within the synthetic distribution do not sum to 8 (like e.g. (6#0))
r"0
1.029
(0.409)
'1
(6#0)
(0.01
1.070
(0.504)
'1
(6#0)
(0.01
0.701
(0.358)
'1
(4#2)
0.35
0.475
(0.267)
'1
(1#5)
0.70
!0.029
(1.129)
0.95
(8#0)
(0.01
0.083
(1.039)
0.99
(8#0)
(0.01
!0.977 !2.231
(1.136)
(1.279)
0.70
0.95
(6#2)
(0#8)
0.14
0.01
Panel 3: E!ects of changing either the degree of overrating (j) the information content in the signal w , or
R
the information content itself (c)
c"2, j"1/8
0.976
(0.905)
0.95
(7#1)
0.05
0.953
(1.364)
0.80
(7#1)
0.05
0.464
(0.642)
0.95
(3#5)
0.77
0.373
(0.518)
0.95
(2#6)
0.48
!0.133
(1.163)
0.90
(8#0)
(0.01
!0.132
(1.128)
0.90
(8#0)
(0.01
!0.986 !1.389
(1.157)
(1.184)
0.60
0.80
(6#2)
(6#2)
0.15
0.49
c"2, j"1/2
0.976
(0.905)
0.95
(7#1)
0.05
0.953
(1.364)
0.80
(7#1)
0.05
0.711
(0.987)
0.90
(5#3)
0.21
0.373
(0.518)
0.95
(2#6)
0.48
!0.133
(1.163)
0.90
(8#0)
(0.01
!0.132
(1.128)
0.90
(8#0)
(0.01
!0.408 !1.389
(1.131)
(1.184)
0.90
0.80
(7#1)
(6#2)
0.01
0.49
c"4, j"1/4
0.976
(0.905)
0.95
(7#1)
0.05
0.918
(1.594)
0.70
(7#1)
0.07
0.465
(0.776)
0.80
(3#5)
0.67
0.232
(0.399)
0.95
(2#6)
0.10
!0.133
(1.163)
0.90
(8#0)
(0.01
!0.152
(1.124)
0.90
(7#1)
(0.01
!0.820 !2.124
(1.133)
(1.223)
0.70
0.95
(7#1)
(0#8)
0.08
0.02
c"1, j"1/4
0.976
(0.905)
0.95
(7#1)
0.05
0.972
(1.184)
0.90
(7#1)
0.04
0.663
(0.786)
0.95
(5#3)
0.36
0.539
(0.637)
0.95
(4#4)
0.72
!0.133
(1.163)
0.90
(8#0)
(0.01
!0.122
(1.135)
0.90
(8#0)
(0.01
!0.561 !0.881
(1.150)
(1.168)
0.80
0.70
(7#1) (6#2)
0.03
0.09
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Table 7
Continued
Panel 4: E!ects of partial changes to the parameters of the interest rate di!erential d
1559
R
D"20
0.968
(0.566)
'1
(5#1)
0.03
0.965
(0.626)
'1
(5#1)
0.04
0.767
(0.516)
'1
(5#1)
0.21
0.572
(0.412)
'1
(2#4)
0.71
!0.275
(1.495)
0.90
(7#1)
0.01
!0.255
(1.422)
0.90
(7#1)
0.01
!0.732 !1.423
(1.503)
(1.613)
0.70
0.70
(7#1)
(6#2)
0.06
0.42
D"8
0.956
(0.790)
0.99
(7#1)
0.06
0.924
(1.021)
0.90
(7#1)
0.08
0.615
(0.681)
0.95
(4#4)
0.51
0.422
(0.486)
0.99
(2#6)
0.60
!0.192
(1.274)
0.90
(8#0)
(0.01
!0.200
(1.240)
0.90
(7#1)
(0.01
!0.747 !1.438
(1.257)
(1.307)
0.70
0.70
(7#1)
(5#3)
0.06
0.40
D"4
1.052
(0.924)
0.95
(8#0)
0.02
1.163
(1.424)
0.80
(8#0)
0.01
0.611
(0.803)
0.90
(5#3)
0.37
0.397
(0.553)
0.90
(2#6)
0.65
0.041
(1.010)
0.95
(8#0)
(0.01
0.120
(0.994)
0.99
(8#0)
(0.01
!0.498 !1.127
(1.000)
(1.023)
0.90
0.70
(7#1)
(6#2)
0.01
0.28
D&N(6, 1.5) 0.995
(0.946)
0.95
(7#1)
0.04
0.895
(1.388)
0.80
(7#1)
0.08
0.517
(0.773)
0.90
(3#5)
0.66
0.349
(0.523)
0.95
(2#6)
0.35
!0.122
(1.166)
0.90
(8#0)
(0.01
0.183
(1.081)
0.90
(7#1)
(0.01
!0.760 !1.414
(1.118)
(1.168)
0.80
0.80
(7#1)
(6#2)
0.05
0.44
D&U(2, 10)
0.936
(1.459)
0.70
(7#1)
0.07
0.533
(0.820)
0.90
(4#4)
0.56
0.357
(0.556)
0.95
(2#6)
0.40
!0.119
(1.151)
0.90
(8#0)
(0.01
!0.168
(1.096)
0.90
(7#1)
(0.01
!0.733 !1.369
(1.150)
(1.211)
0.80
0.80
(7#1)
(6#2)
0.06
0.47
1.010
(0.971)
0.95
(7#1)
0.05
Panel 5: E!ects of changing the length ¹ of the synthetic data series. The entry '1 marks when
the synthetic sample does not contain all of the empirical estimates
¹"10.000
1.001
(0.112)
'1
(2#0)
0
0.993
(0.171)
'1
(4#0)
0
0.551
(0.97)
'1
(2#3)
0.44
0.365
(0.066)
'1
(0#2)
0.01
0.002
(1.105)
0.95
(8#0)
(0.01
!0.053
(1.094)
0.95
(8#0)
(0.01
!4.997
(1.112)
'1
(0#0)
0
!10.266
(1.143)
'1
(0#0)
0
¹"1.000
0.980
(0.362)
'1
(5#1)
(0.01
0.974
(0.538)
'1
(5#1)
0.02
0.545
(0.304)
'1
(2#4)
0.70
0.362
(0.206)
'1
(0#5)
0.20
!0.063
(1.116)
0.95
(8#0)
(0.01
!0.068
(1.070)
0.95
(8#0)
(0.01
!1.613
(1.092)
0.90
(4#4)
0.23
!3.283
(1.125)
'1
(0#7)
(0.01
1560
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
Table 7
Continued
¹"120
0.972
(1.063)
0.90
(7#1)
0.05
0.859
(1.562)
0.60
(7#1)
0.08
0.507
(0.867)
0.80
(3#5)
0.54
0.344
(0.587)
0.90
(2#6)
0.41
!0.166
(1.190)
0.90
(8#0)
(0.01
!0.214
(1.097)
0.90
(7#1)
(0.01
!0.713 !1.290
(1.141)
(1.197)
0.80
0.70
(7#1)
(6#2)
0.05
0.43
¹"60
0.965
(1.531)
0.70
(7#1)
0.07
0.735
(2.259)
0.50
(3#5)
0.13
0.454
(1.263)
0.60
(2#6)
0.39
0.313
(0.857)
0.70
(2#6)
0.34
!0.241
(1.231)
0.90
(7#1)
0.01
!0.300
(1.101)
0.90
(7#1)
0.01
!0.648 !1.069
(1.189)
(1.271)
0.80
0.60
(7#1)
(6#2)
0.04
0.23
Panel 6: E!ects of all parameters being stochastic (variables with about 10% standard deviations)
Stochastic
parameters
0.996
(0.914)
0.95
(7#1)
0.04
1.010
(1.470)
0.80
(7#1)
0.04
0.558
(0.813)
0.90
(4#4)
0.52
0.367
(0.549)
0.90
(2#6)
0.47
!0.090
(1.101)
0.95
(8#0)
(0.01
!0.077
(1.063)
0.95
(8#0)
(0.01
!0.691 !1.359
(1.120)
(1.175)
0.80
0.80
(7#1)
(6#2)
0.04
0.50
between the realignment probability and the exchange rate within the band
lowers the e!ect of overrating the signal, and the standard deviations drop. As
a consequence the synthetic b-distributions do not contain all the empirical
estimates, and all the alternative simulation models are less consistent with the
empirical estimates.
Panel 3 shows that when the market has rational expectations (columns 1, 2,
5 and 6) the noise level in the signal is of no consequence; all the models are
rejected at 7 per cent level. Under non-rational expectations (columns 3 and 4),
the lower the information content in the signal (c increases) or the more the
market overrates it (j decreases), the lower the b-estimates. More speci"cally,
when we vary the degree of overrating, j, from no overrating (information set I )
R
via three intermediate values, j3+1/2, 1/4, 1/8,, to complete overrating
(signal w ), the mean synthetic b-estimate varies monotonously from 0.95 via
R
0.71, 0.53 (in Table 6), 0.46 to 0.37. When we vary the information content in the
signal, with values c3+4, 2, 1,, the mean synthetic b-estimate again varies
monotonously, from 0.47, 0.53 (in Table 6) to 0.66. The t-values, the containing
intervals and the K}S signi"cances follow the pattern of Table 6. We note that
the problem with a low value of P when the market uses information set J is
R
R
} to a certain degree } depending on the weighting between the expected
realignment probability and the signal in the market, relative to the information
content in the signal. The K}S statistic on the t-values does not reject the
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1561
J model when j is 1/8, i.e. the case with the greatest overrating (P "0.14).
R
R
This implies a considerable overrating of the information content, as the
market then puts a weight of 4/5 on the signal, while the correct weight would
have been 1/3, cf. the expressions for p and q in Table 4. This extent of
R
R
overrating does not seem realistic if the single reason for overrating is that the
private agents do not know the true information content, but it is more plausible
if agents have an incentive to overreact due to concern for their reputation, cf.
the concluding remarks below. Yet we are reluctant to put too much weight on
the K}S statistics for the t-values. The inconsistencies between the empirical
t-estimates and the synthetic t-distributions might also re#ect that our simulation model is too restrictive (in particular the assumptions of linearity) to
capture the higher order moments of the exchange rate and the interest rate
di!erential.
Panel 4 shows that the results are neither sensitive to the size of the realignments nor the realignment tosize being a random variable. Panel 5 shows that
there is substantial small sample dispersion in the b-estimates, as re#ected by the
parenthesized standard deviations and the t-statistics. Panel 6 shows that
allowing all model parameters to be stochastic variables, normally distributed
about the basic model values with approximately 10% standard deviation, give
results that are virtually identical to the basic model, cf. Table 6. We conclude
that generally the pattern of Table 6 is maintained by the alternatives in
Table 7.
4. Concluding remarks
This paper investigates the relevance of uncovered interest parity (UIP) for
target zone exchange rates like those in the European Monetary System and in
the Nordic countries during the 1980s. Previous literature has found that the
interest rate di!erential is a biased predictor of the upcoming change in the
exchange rate, and has thus rejected UIP. This paper presents Monte Carlo
simulations of a simple target zone model which indicate that the overall
empirical evidence of a bias in the interest rate di!erential is so large that it
seems unlikely that the rejection of UIP is due to peculiar small sample
properties of target zone models. A caveat to this conclusion is that the empirical
b-estimates probably are correlated (but where we do not know how much they
are correlated), in which case it is di$cult to draw "rm conclusions. A rejection
of UIP involves either a rejection of the assumption of zero (or time-invariant)
risk premium or a rejection of the rational expectations hypothesis.
In setting up the simulation model and choosing parameter values, much care
was taken to ensure that the simulated data was as similar as possible to the
historical data of the Nordic countries presented in Holden and Vik+ren (1994).
Extensive sensitivity analysis indicates that the results are robust with respect to
1562
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
the chosen parameter values. It is our hope that the results from the Monte
Carlo simulations in this paper will prove useful for the interpretation of future
empirical work on UIP in target zones.
We then explore a model where the market observes a signal that consists of
the probability of a realignment plus random noise, and where rational expectations do not hold. More precisely, we consider the case where the market
overrates the information content in the signal. We show theoretically and by
use of Monte Carlo simulations that this alternative speci"cation may explain
the downward bias in the empirical b-estimates that is found when testing UIP
for target zone exchange rates.
To many economists, any violation of the rational expectations hypothesis
will be viewed as ad hoc. But agents are not born with precise knowledge about
the world (the parameters in the model) } this has to be learned. In many
situations, it may seem plausible to assume that this learning process leads to
rational expectations. In the present situation, we argue that there is no way that
the market can learn the true information content in the signal (that is, the true
probability of a realignment), because the true information content (the true
probability of a realignment) is not observable even ex post.
The relationship between the empirical b-estimates and rating of information
content may also be given a di!erent interpretation. Instead of attempting to
explain empirical b-estimates we may try to "nd out how the market rates the
information content in signals that it receives. Viewed this way, the downward
bias in empirical b-estimates constitutes clear evidence in favour of the hypothesis that the market overrates the information content. This accords with recent
research suggesting that agents in stock markets overreact, cf. de Bondt and
Thaler (1990). There is of course no de"nite proof of overrating, as there may
exist an entirely di!erent mechanism that causes a strong downward bias in the
b-estimates. But in spite of extensive research in the literature there is no such
mechanism that is generally accepted. If the market were to underrate the
information content in the signal, this would involve an upward bias in the
b-coe$cient (as shown in Section 2), which would make it even harder to
explain the downward bias that prevails in empirical b-estimates.
Our suggested explanation clearly requires some motivation for why the
market might overrate the information content in signals. At the super"cial
level, overrating of information content is clearly consistent with the view of
many observers (which is consistent with empirical research, cf. above) that the
market often overreacts to rumours and sentiment (cf. Blinder's Law of Speculative Markets: The markets normally get the sign right, but exaggerate the
magnitude by a factor between three and ten, Blinder, 1997). Recent research on
&herd behaviour' of economic agents provides a theoretical foundation for this
view; Scharfstein and Stein (1990) show that if managers are concerned about
their reputation, they will under certain circumstances simply mimic the behaviour of other managers, thereby ignoring their own private information. In our
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1563
exchange rate setting, a manager might react to rumours of a devaluation even if
he believes that the rumours are exaggerated, because if he does not react and
there is a devaluation, it would be easy to blame him afterwards. In fact, a recent
court decision in the U.S. (Indiana) is a good illustration of this argument. The
manager of a grain cooperative failed to hedge against the risk of falling prices,
in spite of a worried accountant's advice to do so. When the prices fell, the
shareholders sued the manager and four directors, and the courts supported the
shareholders' view (the Economist, 13 March 1993).
Our two arguments for an overrating of information content } that the agents
cannot know the true information content and that the agents may have an
incentive to overreact } are not directly related; the concern for reputation and
fear of blame may provide an incentive to overreact even if the agents were to
know the true information content. Yet we believe that the arguments are
complementary; it seems plausible (but perhaps speculative) that the concern for
reputation is given more weight in a situation where the agent believes but does
not know that the rumours are exaggerated, than in a situation where the agent
knows that the rumours are exaggerated.
Acknowledgements
The project was initiated while the authors worked in the Norges Bank. The
second author continued his contribution when he worked in Statistics Norway.
We wish to thank Birger Vik+ren for the collaboration in the earlier stages of
this project. Previous versions have bene"tted from comments from Sigbj+rn
Atle Berg, Gabriela Mundaca, Asbj+rn R+dseth and two anonymous referees.
We are also grateful to Tore Schweder and in particular Harald Goldstein for
advice on some of the statistical issues. Anders Vredin has generously provided
data for Finland and Sweden.
Appendix A
In this appendix we consider the consequences of allowing for the market to
have wrong estimates of more of the parameters in the model. We still assume
that the market treats all estimates as certain, and we also assume that
q#q"p , as p can be &fairly rapidly' learned from observing w . The
L
T
U
U
R
market's information set is now assumed to be
J "+x , k(, k( , q, p , n( , r, q, q, D(,,
R
R
S VL L T
To simplify formulas below we assume that the market knows the e!ect of x on n .
R
R
1564
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
and the market's subjective probability of a realignment is
q
q
T n((x )#
L w.
q "E(n "J )"
R
R R
R
q#q
q#q R
L
T
L
T
The interest rate di!erential is
E(Ds "J )"k( !k(x #q D("d(,
R> R
R
R
R
and the expectation of the coe$cient of the interest rate di!erential is
DpD(p/(p#p)#kk(var(x )!2Dkp
L
L L
T
R
LV .
E(bK )"
D(qD(p/(p#p)#(k()var(x )!2Dkp
L
L L
T
R
LV
(24)
Expression (24) shows that there are several possible causes for
E(bK )O1: D('D, q'p or k('k would all lead to E(bK )(1. However, neiL
L
ther n(On' nor k( Ok' would a!ect the expectation of the b-coe$cient, and
k("0 would not by itself (i.e. if all other parameters were correct) cause
E(bK )O1.
Appendix B
We are grateful to Harald Goldstein for providing the following proof:
Proof of expression (7). Assume that E(v "I )"a#bw , and that var (v ) and
R R
R
R
var (w ) exist. Then
R
E(v )"E [E(v "I )]"E (a#bw )"a#bE(w ).
R
U
R R
U
R
R
It follows that
a"E(v )!bE(w ),
R
R
E[(v !E(v ))"I ]"bw !(a#bE(w ))"b(w !E(w )).
R
R R
R
R
R
R
If we allowed for time variation in the expected realignment size, then n(On' would a!ect E(bK ).
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
1565
From this we get
cov (v , w )"E [(v !E(v )) (w !E(w ))]
R R
R
R
R
R
"E (E[(v !E(v )) (w !E(w ))]"I )
U
R
R
R
R
R
"E[E +((v !E(v ))"I ,(w !E(w ))]
U R
R R R
R
P
"E[b(w !E(w )) (w !E(w ))]"b var (w )
R
R
R
R
R
b"cov (v , w )/var(w ).
R R
R
Appendix C
Derivation of formula (13). Using Eqs. (2) and (10) we obtain
cov (Ds , d')"cov (k !kx #d D, k !kx #p D)
R> R
R
R
R
R
"Dcov (d , p )#k var (x )!Dk(cov (d , x )#cov (p ,x )),
R R
R
R R
R R
as we assume cov(u , p )"cov(v , x )"0. Substituting out for p , using Eq. (8),
R R
R R
R
and exploiting that cov(d , n )"p and cov(d , v )"0, we obtain
R R
L
R R
p
L #k var(x )!2Dkp .
cov(Ds , d')"Dp
R> R
L p#p
R
LV
L
T
(25)
Note that x has the same e!ect on n and p , so that cov(d , x )
R
R
R
R R
"cov(p , x )"p . Correspondingly, using Eqs. (2), (8) and (10) we obtain
R R
LV
var(d')"var(k !kx #p D)
R
R
R
"D
p
L
(p#p)#k var(x )!Dkp .
L
T
R
LV
p#p
L
T
(26)
Substituting out for Eqs. (25) and (26) in Eq. (12), we "nd Eq. (13).
Appendix D
Using u(z)"(p (n)\ exp(!(z!k !(1!k)x )/p) as a shorthand notaS
R
S
tion for the normal density, we compute the expected (bounded) change in the
1566
S. Holden, D. Kolsrud / European Economic Review 43 (1999) 1531}1567
exchange rate within the currency band,
E(Dx )"x*
R>
V*
V*\NS
u(z) dz#
V3
V3>NS
zu(z) dz#x3
u(z) dz,
V*
V3
by numerical integration, with the parameter values given by Eq. (17).
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