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An arbitrage free trilateral target zone model
Pergamon
Journal oflnternationalMoney and Finance, Vol. 15, No. 1 pp. 117-134, 1996 Copyright @ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0261-5606(95)00050-X
0261-5606/96 s15.00+ 0.00
An arbitrage free trilateral target zone model B J O R N N JORGENSEN*
J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208-2002, USA AND HANS O L E hE MIKKELSEN
School of Business Administration, Universityof Southern California, Los Angeles, CA 90089-1421, USA This paper proposes an arbitrage free trilateral model of a credible target zone regime with bands on each bilateral exchange rate. The no arbitrage condition reduces the system to only two dimensions. Any two rates must obey their own boundaries but, in addition, the free rein for movements is restricted by the band of the redundant rate. Therefore, target zone models defined in bilateral settings do not apply to general systems with a cobweb of bilateral bands, such as the European Monetary System. Since the model has no known analytical solution it is estimated by the method of simulated moments. (JEL F31).
Following the pioneering work of Krugman (1991), there has been a considerable amount of research directed towards the behavior of bilateral foreign exchange rates in a target zone regime. In the presence of publicly announced credible restrictions on the fluctuations of an exchange rate around a central parity rate, several non-trivial problems arise which are absent under freely floating rates. Basically, Krugman's model is a classical monetary model with imposed bands, giving rise to a less than proportional S-shaped relation between the logarithmic exchange rate and some fundamental determinants rather than the proportional relation arising in a floating regime. Hence a credible target zone has a stabilizing effect on the exchange rate relative to the situation under free float. The exchange rate dynamics in Krugman's model are * Most of this paper was written while the second author was affiliated with Aarhus University and visiting Northwestern University. We thank Robert Hodrick for helpful discussions and suggestions. Comments from participants at the 1993 Econometric Society European Meeting, the 1993 meeting of the European Finance Association, seminar participants at the University of Aarhus, and an anonymous referee are acknowledged. Any remaining errors are the responsibility of the authors. 117
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
driven by fundamentals that are regulated Brownian motions. Froot and Obstfeld (1991a) extend the analysis to allow for a drift in the fundamentals and Froot and Obstfeld (1991b) consider mean reverting fundamentals. Due to technical obstacles outlined below, this paper takes an approach quite different from these models and postulates a consistent, but admittedly ad hoc, stochastic behavior for three exchange rates in a target zone. Besides the behavior in a credible target zone, the case of imperfect credibility is of interest 1. The success of a publicly announced target zone system depends critically on the credibility of governments' and central banks' commitment to enforce the regime. If credibility is low, some of the currencies involved are likely to be subject to speculative attacks, forcing them to realign or eventually pushing them out of the regime. Credibility issues, of course, are complex since they involve numerous factors such as the number of countries participating in the zone, differences in economic development, and whether the bilateral exchange rate is defended by both countries. However, it could be quantified by realignment expectations as in Rose and Svensson (1991), but empirical results are sensitive to the classification into expected movements within the band and expected realignments. The exchange rate mechanism (ERM) in the European Monetary System (EMS) is an example of a system with a relatively large number of participating countries, where each bilateral exchange rate is defended by both countries. The non-EMS Scandinavian countries recently imposed unilaterally enforced exchange rate bands on their currencies against the European currency unit (ECU). The fact that their exchange rate bands were unilateral and therefore not enforced by the EMS central banks may have caused a credibility problem contributing to their collapse. To the authors' knowledge, all previous target zone models are defined in the context of a two-country target zone, where only one band for the single bilateral exchange rate matters. In a more general target zone system such as the ERM in the EMS several exchange rates are linked together in a cobweb of bands on bilateral exchange rates. Here, the behavior of exchange rates located well inside their own bands may be restricted by the fact that other exchange rates are close to their bands. To see this, consider a target zone on the three bilateral exchange rates existing between three countries. Since only two rates are independent due to triangular arbitrage, one may wish to model these two by, say, the Krugrnan model. However, since the band on the third exchange rate must also be maintained, the movements of the two exchange rates must be further constrained. As an example, the third exchange rate can be close to its band while the other two lie well inside their respective bands, and hence the two rates must move in a way that keeps the third rate inside its band although none of their own bands are binding. In a general N-country target zone system there are ( N - 1)5//2 bilateral exchange rates and bands, of which at most N - 1 are independent. Therefore, exchange rates have to move in accordance with a large number of restrictions in addition to their own bands. 2 Intuitively, the free rein for the movements of the independent exchange rates will be inversely proportional to the number of other exchange rates being close to their bands. 118
Trilateral target zone model: B N Jorgensen and H 0 AS Mikkelsen
If a target zone system starts out in a situation where all exchange rates are at their parities, differences in monetary policy or general economic development between the participating countries may over time trigger a situation in which several exchange rates are close to their bands. Such a situation will severely reduce the flexibility of all other exchange rates because of the above mentioned triangular no-arbitrage requirements. Since the target zone is viewed as a system, these other rates can still fluctuate but the movement of individual exchange rates is restricted conditional on values for the other rates in the system. The higher the number of countries, the more flexibility is lost. Besides drastic and possibly unpleasant policy changes through, say, increased interest rate differentials, this problem could be resolved by occasional realignments of parities. The inverse variability of the exchange rates in a target zone may then be a good predictor of future realignments. Using a three-country economy, this paper attempts to account for the above mentioned restrictions and interdependencies between exchange rates in a fully credible target zone. The model is then estimated on the exchange rates between three large integrated EMS economies: Germany, France, and the Netherlands. The sample period is restricted to a period with no realignments; therefore the results are open to the criticism of the so-called Peso problem. 3 Since analytical expressions for the asymptotic distributions of the exchange rate processes and their moments cannot be derived, standard estimation procedures do not apply. However, the method of simulated moments (MSM) seems well suited to handle this problem, see Smith and Spencer (1992) and the references cited therein. The idea is that, even though theoretical moments cannot be derived, the process can be simulated and the simulated moments used instead. Lindberg and S6derlind (1991) use MSM to estimate the basic two-country target zone model with infinitesimal interventions at the margins, and Lindberg and S6derlind (1992) consider the case of mean-reverting fundamentals. The paper proceeds as follows. Section I proposes a continuous time model with mean reversion and demonstrates the well known implication of no arbitrage of reducing the number of linearly independent dimensions. Based on constraints on the model consistent with the regulatory environment in the ERM imposed in Section II, Sections III and IV estimate the model using MSM and compare the results of simulations with the estimated model to the actual data. Finally, Section V concludes the paper. I. A trilateral model It is well known from continuous time finance that not all stochastic processes are permissible descriptions of prices. 4 This was a recurring problem in the development of the theory of the term structure of interest rates. Two general problems would arise: first the price process itself might allow for intertemporal arbitrage and, second the internal consistency requirements among the available assets could overly strain the model. The model of the term structure of interest rates due to Cox et al. (1985) is internally consistent, in that it does not permit such arbitrage opportunities. The goal of this paper is similar, 119
Trilateraltargetzone model:B N Jorgensenand H 0 AF,Mikkelsen namely to identify a system of stochastic differential equations (SDE) that characterizes arbitrage free time-series behavior of three exchange rates in a target zone regime. Assume that investors in all countries are allowed to hold foreign currency that does not yield interest. The first problem alluded to above could then occur if one exchange rate hits a time-invariant upper or lower bound of its band since the direction of the exchange rate process is then predictable. There are many potential solutions to this problem. One approach is to assume a stochastic band or a non-stochastic, widening band. Another approach is a Markov-switching model, where the exchange rate is affected by stochastic discretionary shocks that force the country to either change the band or leave the target zone regime. Since these approaches are not consistent with the credible regime considered in this paper, the next section assumes that any exchange rate can get arbitrarily close to its band without ever touching it. 5 This property is consistent with the Krugrnan model. There, however, the exchange rate is a monotone function of a regulated fundamental and nothing prevents this fundamental from hitting its time-invariant upper or lower band, where the exchange rate will be perfectly predictable. The remainder of this section performs a basic inspection of the consequences of the no arbitrage assumption without specific reference to target zone models. In a fully competitive market without transactions costs, no arbitrage opportunities can exist. The high liquidity, implying low bid-ask spreads in the foreign exchange markets, makes this a reasonable assumption. One fundamental property of the three spot exchange rates existing between three countries is the triangular no arbitrage condition that must hold for any time t:
(l)
S12(t) -- S13(t) = S32(t),
where lower case letters denote the logarithm of the nominal amounts, i.e., sjj(t) denotes the log of the spot exchange rate between country i and j prevailing at time t expressed in units of currency i per unit of currency j. For notational purposes define s 1 = $12, s 2 S13, and s 3 $32 and suppress the time index. Written in SDE form, a general error correction model of the exchange rates would be =
[ dS1 (2)
Ids2 Lds3
=
K1 (Jt/~l--S1) ] X2 ( lU,2- S2) d t + K3 ( ~[Z3-- S3 )
=
[ dzl 1
vldz /' [dg3J
where /xi is the long-run level of exchange rate i, r i is the speed of adjustment of deflations from the long-run level, Z i are three independent standard Brownian motions, and V is a 3 x 3 matrix that can vary with the current levels of the exchange rates. When the terms standard deflation, variance and covariance are applied to dsi they refer to the stochastic component of ds i. Thus, V is the Cholesky decomposition of the local covariance matrix, VV'. The model would be the three-dimensional Ornstein-Uhlenbeck process in the special case where V = trL but in this paper V will invariably depend on the current exchange rates.
120
Trilateral targetzone model: B N Jorgensen and H 0 A~ Mikkelsen
From (2} the drift of an exchange rate process is a constant proportion of its deviation from a natural level, so that the expected future rate is linear in the current rate. In the Krugman model this drift equals the interest rate differential between two countries and is approximately linearly decreasing in the fundamental determinants of the exchange rate for narrow target zones (Svensson, 1991). Furthermore, since the exchange rate reacts less than proportionally (S-shaped) to these fundamentals the absolute drift of the exchange rate is an increasing function of the current rate. For the Krugman model extended to mean-reverting fundamentals (intra-marginal interventions), the nonlinear relationship between the exchange rate and the fundamentals is much less pronounced because of a smaller probability of getting close to the band. Therefore, the drift of the exchange rate is approximately linear in the current level, see Lindberg and S6derlind (1992). This suggests that the drift in (2) is an appropriate description for the situation of intra-marginal interventions. The three dimensional system in (2) can be reduced using the no-arbitrage condition from (1). First, the no-arbitrage condition must hold for the long-run levels (3)
11/'1 -- ~[/'2 = ~U'3"
Since the no-arbitrage condition holds for all realizations of the underlying stochastic processes, the condition applies to the drift term, i.e., ~1 (/Xl - s l ) (4)
[1 - 1 -11
K 2 (~t/,2--S 2)
=0.
K 3 (ill, 3 - $ 3 )
Using (1) and (3) in (4) and rearranging terms yields <5)
( K 3 -- K1) (S 1 -- ~/q) -t- ( K 2 -- /I;3) (S 2 -- IA~2) = 0
that must hold for any possible pair of values, (s 1, s2). This is only possible if (6)
K= K1= K2 = K3.
Having analyzed the drift term of the three exchange rate processes, we apply the no-arbitrage arguments to the stochastic term (7)
dZ1 [1 - 1 - 1 1 V [dZ 2 =0, [dZ3
which also must hold independent of the realizations of dZ i and therefore reduces to the three equations (8)
[1-1
- l l Z = [O 0 0].
Obviously, V is not of full rank, so the third column is arbitrarily chosen to consist of zeros. Therefore, only two independent stochastic processes are needed to describe the dynamics of three exchange rates. This is not to say that there cannot be three independent, country-specific stochastic processes, rather, 121
Trilateral target zone model: B N Jorgensen and H 0 dE Mikkelsen
it implies that at most two components can be identified from three exchange rate series. Further reduction is possible by applying the no-arbitrage condition on the elements of the covariance matrix (9)
VAR[ds3] = VAR[ds 1 - ds2] = V i R [ d s l ] + V i R [ d s 2 ] - 2COV[dSl, ds 2 ]
or, alternatively stated with o-i as the local standard deviation of exchange rate i, the correlation between exchange rate 1 and 2 is o-? +
(10)
P12 =
-
2O-lO,2
The model can therefore be rewritten as the no-arbitrage condition in (1) and (11)
Lds2 ]
r
/z2 - s2
where (12)
,
o ]
g =
012 o-2 o - 2 ~ is the Cholesky decomposition of the covariance matrix, (13)
EE, = [0-~ [ P12 O"10"2
/9120"10"2] 0.2 J"
Note, that the volatility of the third exchange rate does affect the two others through the correlation ,o12 from equation (10). This section repeatedly applied the basic principle of no arbitrage (equation (1)) to a trilateral exchange rate model with mean reversion and reduced the rank of the SDE by one. The next section will impose a target zone regime on the above model.
II. Target zone regime We assume that the three countries participate in a fully credible target zone, where each bilateral exchange rate is allowed to fluctuate within a symmetric band around some parity rates. These parities are set in accordance with the no arbitrage condition and, without loss of generality, are normalized to zero. Henceforth, s i is interpreted as the log deviation from the central parity. Any target zone with these properties involves the constraints (14)
Isil <
~7, Vi,
where 100r/ approximately equals the maximal allowed percentage deviation from the parity rate. Obviously, the drift terms must tend to relax any restrictions arising from one or two exchange rates approaching their bands since otherwise the exchange rates would either leave the band or be absorbed
122
Trilateral target zone model." B N Jorgensen and H 0 A~ Mikkelsen
by the barriers. This translates into the following two conditions on the drift term (15)
I il < 71, ¥ i
(16)
r > O.
From the no-arbitrage condition in (1), at most two exchange rates can be linearly independent, in the sense that the third is given by the other two. Off hand, one might model the exchange rates between three countries by considering only two of the exchange rates explicitly such that the possible exchange rate combinations seem bounded within the full square (17)
{[sa $2 $3 ] ~.,~3: [Sl[ ~ 77 A 1S2[ ~ '/7 AS 3 =-S 1 --$2}
depicted on Figure 1. However, since the fluctuations in the two exchange rates should maintain the band of the third exchange rate, (18>
[s3l=ls 1 - s 2 1 < ~/
follows from (1) and (14). Thus, in the case of a trilateral target zone regime, possible exchange rates are really confined to (19> {[Sl s2 s3 ] ~ 3 :
iSll < r/Als2l< n A _ 7 1 + s 2 < s l < Tl+s 2 As 3 = s l _ s 2 } .
Consider first the hypotenuses of triangles A and B in Figure 1. On these lines, the third exchange rate is always on its band. This implies that exchange rate combinations within the square but outside the shaded area do not have the third exchange rate adhering to its boundaries. When the third exchange rate is on its limit, the local variance of this process must be zero. If this were not the Ca
11
A
SI
B
FIGURE1. Restrictions in a trilateral target zone. 123
Trilateral target zone model: B N Jorgensen and H 0 AE Mikkelsen
case, that exchange rate could move out of its band and out of the shaded region with a strictly positive probability and, therefore, the two other exchange rates must be perfectly correlated. In this case, equation (10) reduces to (20) 2o'lo"2 = o'1z + o"2 implying that o'1 0"2" Intuitively, the identical variances make sense because there is no asymmetry in our formulation of a target zone. Of course, this argument holds for all exchange rates. For any of the six vertices of the shaded hexagon, two exchange rates are on their band simultaneously. By the same no arbitrage argument, the last exchange rate cannot have any variability component even though it is on its parity. From the above discussion two properties must characterize any model of a credible trilateral target zone. First, the local variance of any exchange rate must approach zero as the exchange rate approaches its band. This is ensured by forcing the two others to approach perfect correlation. Second, when two exchange rates approach their bands, the local variance of the third should approach zero. Therefore, a natural but indeed arbitrary local standard deviation of exchange rate i is chosen as follows =
Ear (21)
O"1 = g a p
(22)
a i =
Vj--/=i
2 ' where p is a non-negative parameter and the normalized weights, a i,
,77-s rli2
,
ensure that the exchange rates possess all properties of a target zone system as discussed above. Since in certain cases 0"1 = o'2, the variance scaling parameter, 8, has to be common across exchange rates. To analyze the above standard deviation, consider the ith exchange rate and fix the other two rates at zero corresponding to a bilateral target zone. Figure 2 depicts the standard deviation as a function of the deviation from the central parity rate for three different values of p and with the scaling parameter, 8, equal to unity. When s --0, the exchange rate volatility is maximized at unity and o. is locally insensitive to s. Furthermore, as s ---> + ~/(~/= 2.25 percent in the figure) the standard deviation tends to zero. The manner in which o. approaches zero is controlled by the parameter p. For p = 1 the reaction of the standard deviation to s is moderate with a rate of decay that increases linearly with the absolute value of s. As p decreases, e . g . to 0.1, the standard deviation gets less sensitive for small departures from the parity but declines rapidly as the exchange rate gets close to the bands. When p is increased to, say, 10, the standard deviation displays a rapid decay to zero. It follows that the chosen functional form is flexible enough to mirror a variety of patterns for the standard deviation between the extremes of a collapse to zero only at the bands (as p ~ 0) and a collapse immediately as the exchange rate tends to leave the 124
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
central parity (for p ~ ~). The corresponding standard deviation in the Krugman model with interventions only at the margin is given by 8(~s//af), where f is a regulated Brownian motion process with diffusion coefficient 8, see Svensson (1991, 1992). The derivative of s with respect to f is a function which is qualitatively similar to the a P function defined above giving rise to an inverted U-shaped relationship similar to Figure 2 for p approximately equal to unity, as in Figure 1 of Svensson (1992). For the Krugman model with mean-reverting fundamentals c~s/af is almost linear except perhaps for values of s close to the band. If the long-run level of the exchange rate was zero, the standard deviation would again be maximized (cTs/~f= 1) for s = 0 and be zero (c~s/c~f=0) at s = + ~7. Therefore the relationship between an exchange rate and its standard deviation would be shaped like an inverted V if drawn in Figure 2, resulting in a less volatile exchange rate since shocks are partly neutralized within the band. Such a shape can easily be mimicked by (21) with p above unity, say, 1.25. It follows that the selected parameterization is sufficiently flexible to capture a variety of policy scenarios including marginal as well as intra-marginal intervention. Thus, p is labelled the policy parameter. The summation in (21) ensures that the standard deviation of an exchange rate approaches zero as the remaining two rates both get close to their bands. In Krugman's model, the exchange rate had an S-curved relation to the O
SS
/
o
:
i
~%
:
i
/ o
,
:
\
: 0
\
: ,: •"
0 0
".
,.,,"
-3
-2
%,...,
-1
0
1
2
3
s F]GURE 2. ~ as a ~ n c t J o n o f p=O.1
w i t h other e x c h ~ g e rates ~ ¢ d at zero. K E Y : ~ p = l . O ................ p = l O . O
si
125
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
underlying fundamental. Here, since there are two independent forcing processes, the reaction function is S-curved in both dimensions. In Figure 3, the x- and y-axes represent forcing processes and the z-axis is the corresponding exchange rate levels giving rise to a snake-shaped relation. If exchange rates were freely floating the reaction function would be depicted as a hyperplane. The main empirical implication of this section is an issue of model (mis)specification. A test of e.g. Krugman's two country model to a target zone that has more than two exchange rates must take into consideration restrictions imposed by no arbitrage from non-modelled exchange rates. The subsequent application of the above model to the EMS is subject to such critique since more than three countries participate in the target zone. IlL MSM estimation The data used in this section are the three bilateral exchange rates between Deutsche-mark (DM), Dutch guilder (NG), and French franc (FF) from Citibase. In particular D M / N G is chosen for s 1, D M / F F for s 2, and F F / N G for s 3. Data points are sampled daily as the average of spot buy and sell
eq
Q
FIGURE 3. Reaction of exchange rates to underlying processes: The snake effect. Note: This figure was generated from an approximate numerical integration of the trilateral model with a given parameterization. 126
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
quotations during the period from January 13, 1987 until January 4, 1990. This period was chosen because it is the longest interval available without any realignments during the ERM, yielding a total of 748 observations on each exchange rate. 6 In order to make the data consistent with the model, units are chosen so that logarithmic exchange rates have a central parity of zero. The resulting exchange rates are then multiplied by 100 to give percentage deviations from the parity rate. Figure 4 shows the time series behavior of the data. 7 The absence of analytical asymptotic distributional results for the exchange rate processes renders the use of traditional estimation techniques infeasible. The recently developed method of simulated moments (MSM) provides a convenient tool to overcome this problem. 8 The philosophy underlying this approach is that even though true moments are unknown, the model can always be simulated and simulation moments calculated. Given an N x 2 matrix, dZ, of simulated standard normal random variables, the parameter estimates can be chosen as those minimizing some distance between the resulting simulated moments and the actual moments of the data. This paper follows Lindberg and SiAderlind (1991) and Smith and Spencer (1992) in estimating an exactly identified system in which the number of moments to be matched equals the number of parameters to be estimated. The estimation procedure in these papers is restricted MSM where one parameter
i
I
:
•
~'~¢~"
'
i t~,":"
;~.
"~."
S
•
'
~
.
I
h"
l".~',tr~
!
'
'~r~
r
S
? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
0
L
I
I
I
I
I
I
I
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
Actual
observation
FIGURE 4. Actual series from January 13, 1987 through January 4, 1990. KEY: F F / N G ................ D M / N G _ _ DM/FF Margins 127
Trilateral target zone model." B N Jorgensen and H 0 AE Mikkelsen
is estimated conditional on arbitrary values for the others, while Lindberg and S6derlind (1992) uses a multidimensional grid. In the present analysis a suitable multidimensional minimization algorithm is used to simultaneously chose the parameters that minimize the following quadratic loss function: 5
(23)
Loss= ~
(Msi-MAi) 2,
i=1
where Msi is the ith simulated sample moment, and MAi is the ith actual sample moment. Note that in this exactly identified system parameters may be chosen that bring the loss function arbitrarily close to zero. Also note that the credible target zone structure ensures the existence of all moments. The above procedure is then repeated for R = 100 times (i.e. for R different random draws of forcing processes, dZ). The final parameter estimates are given as the arithmetic average from these R replications. Monte Carlo standard deviations are reported to allow evaluation of the statistical significance of the estimates. Table 1 reports results from MSM estimation of the trilateral target zone model with N = 2000. The moments used are the mean of s I and s 2, the covariance between s 1 and s 3 and the variance of the first difference of s 2 and s 3. All parameters appear to be estimated with sufficient precision, i.e. standard errors are very small compared to parameter estimates, except for/.t 1 which is not significantly different from zero. The loss function was essentially always equal to zero, with a mean value of 3.4.10 -7. The values o f / z 1 and /z2 reveal the difference between the positions of D M / N G and D M / F F within the band, with the latter being relatively weak during the sample period. By the no arbitrage argument, the estimated value o f / z 3 = / z 1 -/~2 = 0.6869 corresponds to the relative position of F F / N G in its band. The estimate for r indicates that 12.5 percent of a given deviation from the long-run level is discounted away in each step which seems somewhat higher than would be expected from inspection of the data. Finally, the estimate of p is close to but significantly above 1. Overall, the estimated model appears to catch some of the characteristics of the given exchange rates. This study encounters the aliasing problem that numerous continuous time models can match the discrete data. Also, an approximation error arises from the discrete time simulation of the underlying model. Under usual regularity conditions convergence of the traditional GMM estimator is assured, see Hansen and Scheinkman (1992), and with additional regularity conditions related to the approximation error there is no reason to suspect problems in TABLE 1. M S M estimates of the trilateral target zone model.
Estimate Std. Dev.
r
/x 1
/~2
8
p
0.1250 (0.0189)
0.0034 (0.0225)
- 0.6835 (0.0216)
0.1413 (0.0019)
1.0824 (0.0003)
Note: T h e table reports the m e a n and standard deviation across 100 replications.
128
Trilateral target zone model: B N Jorgensen and H 0 AE Mikkelsen
the convergence of the MSM estimator. Since the data have been time-aggregated on a daily basis, the quantitative estimates presented above may not correspond exactly to the parameters in the true, underlying continuous time process. In particular, since a discrete time interval contains (infinitely) many continuous time differentials, the true r and 8 will be smaller than the estimates.
IV. Properties of the estimated model The previous section estimated the model by exact matching of sample and simulation moments, and the parameter estimates appear to be fairly precise in terms of standard deviation relative to the mean. In order to further examine the precision of the estimates, it is of interest to investigate how well the estimated model performs on a large number of new random forcing processes with fixed parameters. The results from this exercise will reveal the stability of the model and show how well moments that are not used in estimation are matched. In particular, 2000 observations from the estimated model, as it appears in Table 1, are simulated 1000 times and various moments and statistics are calculated as well as for the actual data. Table 2 compares the first four moments of the actual data to the simulated data in logarithmic levels. It is evident that the first order moments are well matched, consistent with what one would expect, given that they are matched
TABLE 2. Moments for actual and simulated exchange rates. DM/NG Actual
Simulated
DM/FF Actual
FF/NG
Simulated
Actual
Simulated
Mean
0.005
0.003* (0.022) [0.457]
- 0.680
- 0.680* (0.021) [0.468]
0.685
0.685 (0.021) [0.501]
Variance
0.043
0.065 (0.006) [1.000]
0.730
0.060 (0.005) [0.0001
0.752
0.060 (0.005) [0.0001
Skewness
0.877
- 0.004 (0.111) [0.000]
0.490
0.328 (0.120) [0.090]
Kurtosis
3.172
2.866 (0.176) [0.047]
1.977
3.037 (0.261) [1.000]
-
0.609
2.058
- 0.330 (0.119) [0.986] 3.039 (0.272) [1.000]
Note: Simulated moments are calculated as the average across 1000 replications of the
estimated model with 2000 observations, parentheses denote Monte Carlo standard deviations and square brackets the fraction of simulated models with a larger moment than the actual data. An * indicates a moment used in estimation.
129
Trilateral target zone model: B N Jorgensen and H 0 t E Mikkelsen
in estimation. For both variance and kurtosis, only D M / N G is within reasonable distance to the actual values. In contrast, the skewness for D M / N G is far away from the true value. Table 3 reports the actual and simulated covariances of the level processes. Again, since the covariance between D M / N G and F F / N G is used in estimation, it is well matched in the simulations. The covariance between D M / N G and D M / F F seems reasonable while the covariance between D M / F F and F F / N G is at odds with the actual data. Table 4 reports the first four moments, the Jarque-Bera test for normality and ARCH tests for the first difference of the actual and simulated series. Similar to previous results, the moments that are used in estimation are well matched while the results for the others are mixed. Interestingly, the Jarque-Bera test rejects the null of normality in more than 50 percent of the simulations for D M / F F and F F / N G at the 5 percent significance level. Though the statistic never reaches the same absolute value as for the actual series, this is interpreted as evidence in favor of the model's ability to generate non-normality as observed in actual data. The ARCH statistics are somewhat disappointing in that the rejection rate under the null of no ARCH effects is only 10 percent (at the 5 percent significance level). On the other hand, rejection occurs more frequently than allowed by the nominal size of the test, but whether this can be attributed to finite sample size distortions of the test or simply low power is beyond the scope of this paper. Another feature of the simulations is that the model is not capable of generating sufficient asymmetry across the three exchange rates. As an example, the three simulated covariances in Table 3 are all of equal magnitude but the covariances for the actual data are very different. Figure 5 shows simulations of the estimated model of equal length as the actual time series and starting at the same initial values. Casual inspection of the graphs seems to indicate that there is a more pronounced mean reversion in the simulated series than in the actual data. This might indicate why some moments in Tables 2, 3 and 4 are not well matched.
TABLE 3. Cross-moments for actual and simulated exchange rates. Actual Cov ( D M / N G , D M / F F )
0.011
Simulated 0.033
(0.0o4) [1.000] Cov ( D M / N G , F F / N G )
0.032
0.033*
(0.004) [0.497] Cov ( D M / F F , F F / N G )
Note: See Table 2.
130
- 0.719
- 0.027 (0.004) [1.000]
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
TABLE 4. Statistics for actual and simulated first differences of exchange rates. DM/NG Actual
Mean
- 3 . 1 0 -s
DM/FF
Simulated 4.10 -6 (2.10 -4) [0.587]
FF/NG
Simulated
Actual
Simulated
-0.003
- 3 . 1 0 -4 (1.10 -4) [1.000]
0.003
3.10 .4 (1.10 -4) [0.000]
0.015" (5" 10 -4) [0.6751
0.015
0.015" (0.001) [0.3531
Actual
Variance
0.005
0.016 (0.001) [1.000]
0.015
Skewness
- 0.054
- 0.004 (0.057) [0.820]
- 0.327
- 0.127 (0.058) [1.000]
0.223
0.130 (0.056) [0.056]
6.724
3.022 (0.113)
7.944
3.094 (0.127)
7.990
3.094 (0.130)
Kurtosis
[o.ooo] Jarque-Bera
ARCH(l)
ARCH(5)
431.9
85.18
96.93
2.185 (2.270)
[o.ooo] 774.1
8.549 (6.732)
[o.ooo] 781.3
8.836 (7.069)
[o.ooo1
[o.ooo1
[o.ooo1
{0.073}
{0.560}
{0.596}
1.144 (1.691)
78.41
1.489 (2.158)
44.59
1.335 (1.980)
[o.ooo1
[o.ooo1
[o.ooo1
{0.060}
{0.101}
{0.086}
5.531 (4.003)
131.8
6.465 (4.430)
87.84
6.277 (4.263)
[o.ooo]
[o.ooo]
[o.ooo]
{0.082}
{0.123}
{0.124}
See Table 2. Jarque-Bera is a test for normality and ARCH(z) is the LM test for no z order A R C H effects. Curly brackets denote the proportion of simulated test statistics exceeding the 5% significance level from the appropriate g 2 distribution.
V. Conclusion Using no-arbitrage arguments, this paper proposed an arbitrage free trilateral model of a credible target zone regime with bands on each bilateral exchange rate. The triangular no-arbitrage condition that characterizes highly competitive foreign exchange markets reduces the system from three to two dimensions. In particular, the third redundant exchange rate becomes the difference between the two independent rates. Therefore, in addition to their own boundaries around the central parity rate, the movements are restricted by the requirement that the redundant rate obeys its band. This implies that traditional target zone models defined in a bilateral setting do not directly apply to a 131
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen r,D
~4
I
I
0
I
L
I
I
I
i
I
I
I
I
I
I
i
i
t
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
Simulated observation I R o u ~ 5. Simulated series corresponding to Figure 4. K E Y : _
D M / N G
DM/FF
_
F F / N G ................
Margins
multilateral setting (as in the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS)), where a large number of currencies are linked together in a cobweb of bands on bilateral exchange rates. Technically, the exchange rates are postulated to follow a mean reverting process, with a local covariance matrix that depends on the positions of all three exchange rates relative to their respective bands. This assumption is motivated on two grounds. First, given perfect credibility of the target zone regime all deviations from long-run levels should be transitory. Second, mean reversion prevents any exchange rate from being absorbed by the boundaries and thus rules out an obvious arbitrage opportunity. When either exchange rate approaches one of its boundaries, the local variance, which is the unpredictable component of the exchange rate movement, must approach zero in order to prevent potential violation of the band. The inherent mean reversion accomplishes this by forcing the exchange rate away from the boundaries and towards the parity. Also, as the redundant rate approaches its boundaries, its local variance vanishes, and the other rates must approach perfect correlation. Although somewhat arbitrary, the variance structure imposed in this paper is sufficiently flexible to mimic a variety of policy scenarios, such as intramarginal intervention. 132
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
Since the model has no known analytical solution, it is estimated by the method of simulated moments on three currencies within the ERM in the EMS. By limiting the focus to a subset of currencies participating in the ERM, the empirical results presented in the paper are exposed to the criticism raised that the model leaves out important information from neglected currencies. Although parameter estimates take on plausible values, evidence from simulations of the estimated model is mixed. While the parameter estimates appear quite robust with respect to the moments used in estimation other moments are badly matched. The estimated mean reversion parameter appears larger than expected and, therefore, the model is unable to generate ARCH effects to the extent found in real data. However, the model succeeds in generating the stylized empirical fact of non-normal returns. This paper has several potential extensions. Most importantly, the model may currently be too symmetric to characterize real world exchange rate behavior. Second, alternative specifications of mean reversion that depend on the position of current exchange rates could imply restrictions between the expected return and volatility of exchange rates that determine risk premia. Finally, placing the model in an equilibrium framework may provide better guidance in choosing the appropriate stochastic processes. No~s 1. Credibility and related problems have been addressed in, e.g. Bertola and Caballero (1992), Bertola and Svensson (1991), and Svensson (1992). 2. An extension of Krugman's model to three countries would involve finding investor's expectations towards exchange rate movements by solving a three dimensional stochastic partial differential equation under piece-wise linear boundary conditions. Since we could not guess the existing solution, an arbitrage approach was chosen. 3. If the target zone is less than perfectly credible, the probability of a realignment will affect the results even if the realignment does not occur in sample, see Krasker (1980). 4. See Ingersoll (1987), p. 384. 5. For simple mean reverting processes, this could be achieved by imposing parameter restrictions between the drift and variance, see Cox et al. (1985), p. 391. 6. Though the current trilateral target zone model has a Markov property for the exchange rates, it is not appropriate to pool data in order to increase the sample size. The exchange rates are functions of their relative positions within the bands and a realignment typically causes a given exchange rate to alter this. In particular, an exchange rate which is near the upper limit of the band will often continue much closer to the lower edge of the new band after a realignment, see Rose and Svensson (1991). Therefore, the relative positions within the bands will unavoidably display discontinuities, although it is possible to obtain smooth curves for the exchange rates. 7. In reality, the band is not just + 2.25 percent, but rather the nominal central parities of one are multiplied (divided) by the factor 1.022753 to give upper (lower) limits, see e.g. Grabbe (1991) pp. 39-41 for an explanation of this factor. In the present context the upper and lower limits are then 100 ln(1.022753) and 100 ln(1/1.022753) respectively and the weights in the variance formulas are restated accordingly. 8. For distributional results, properties, and applications of this approach, see, inter alia, Duffle and Singleton (1993), Gregory and Smith (1990), and Smith and Spencer (1992). 133
Trilateraltargetzone model: B N Jorgensen and H 0 A~ Mikkelsen
References
BERTOLA,GIUSEPPEAND RICARDOJ. CABALLERO,'Target zones and realignments,' American Economic Review, June 1992, 82: 520-536. BERTOLA, GIUSEPPE AND EARS E. O. SVENSSON, 'Stochastic devaluation risk and the empirical fit of target zone models,' NBER Working Paper No. 3576, 1991. COX, JOHNC., JONATHANE. INGERSOLL,Jr., AND STEPHENA. ROSS, 'A theory of the term structure of interest rates,' Econometrica, March 1985, 53: 385-407. DU~aE, DARRELLAND KE~CETH J. SINGLETON,'Simulated moments estimation of Markov models of asset prices,' Econometrica, July 1993, 61: 929-952. FROOT, KENNETH A. AND MAURICE OBSTFELD, 'Exchange rate dynamics under stochastic regime shifts,' Journal of International Economics, January 1991, 31:203-229 (1991a). FROOT, KENNETH A. AND MAURICE OBSTFELD, 'Stochastic process switching: some simple solutions,' Econometrica, January 1991, 59:241-250 (1991b). GRABBE, J. ORHN, International Financial Markets, New York, NY: Elsevier, 1991. GREGORY, ALLAN W. AND GREGOR W. SMITH, 'Calibration as estimation,' Econometric Reviews, 1990, 9: 57-89. HANSEN, LARS P. AND Josi~ A. SCHE~KMAN, 'Back to the future: generating moment implications for continuous time Markov processes,' Working Paper, University of Chicago, January 1992. INGERSOLL,Jr., JONATHANE., Theory of Financial Decision Making, Ottowa, NJ: Rowman and Littlefield, 1987. KRASKER, WILLIAM S., 'The Peso problem' in testing the efficiency of forward exchange markets,' Journal of Monetary Economics, 1980, 6: 269-276. KRUGMAN, PAUL R., 'Target zones and exchange rate dynamics,' Quarterly Journal of Economics, August 1991, 1116: 669-682. LINDBERO,HANS AND PAUL SODERLIND,'Testing the basic target zone model on Swedish data,' IIES Seminar Paper No. 488, 1991. LINDBERG,HANSAND PAUL SODERLIND,'Target zone models and the intervention policy: the Swedish case,' IIES Seminar Paper No. 496, 1992. ROSE, ANDREW K. AND LARS E. O. SVENSSON, 'Expected and predicted realignments: the F F / D M exchange rate during the EMS,' NBER Working Paper No. 3685, 1991. SMITH, GREGOR W. AND MICHAEL G. SPENCER, 'Estimation and testing in models of exchange rate target zones and process switching,' in P. Krugman and M. Miller, eds, Exchange Rate Targets and Currency Bands, Cambridge: Cambridge University Press, 1992. SVENSSON, I.ARS E. O., 'Target zones and interest rate variability,' Journal of International Economics, 1991, 31: 27-54. SVENSSON,LARS E. O., 'The foreign exchange risk premium in a target zone with devaluation risk,' Journal of International Economics, 1992, 33: 21-40.
134
Journal oflnternationalMoney and Finance, Vol. 15, No. 1 pp. 117-134, 1996 Copyright @ 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved
0261-5606(95)00050-X
0261-5606/96 s15.00+ 0.00
An arbitrage free trilateral target zone model B J O R N N JORGENSEN*
J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL 60208-2002, USA AND HANS O L E hE MIKKELSEN
School of Business Administration, Universityof Southern California, Los Angeles, CA 90089-1421, USA This paper proposes an arbitrage free trilateral model of a credible target zone regime with bands on each bilateral exchange rate. The no arbitrage condition reduces the system to only two dimensions. Any two rates must obey their own boundaries but, in addition, the free rein for movements is restricted by the band of the redundant rate. Therefore, target zone models defined in bilateral settings do not apply to general systems with a cobweb of bilateral bands, such as the European Monetary System. Since the model has no known analytical solution it is estimated by the method of simulated moments. (JEL F31).
Following the pioneering work of Krugman (1991), there has been a considerable amount of research directed towards the behavior of bilateral foreign exchange rates in a target zone regime. In the presence of publicly announced credible restrictions on the fluctuations of an exchange rate around a central parity rate, several non-trivial problems arise which are absent under freely floating rates. Basically, Krugman's model is a classical monetary model with imposed bands, giving rise to a less than proportional S-shaped relation between the logarithmic exchange rate and some fundamental determinants rather than the proportional relation arising in a floating regime. Hence a credible target zone has a stabilizing effect on the exchange rate relative to the situation under free float. The exchange rate dynamics in Krugman's model are * Most of this paper was written while the second author was affiliated with Aarhus University and visiting Northwestern University. We thank Robert Hodrick for helpful discussions and suggestions. Comments from participants at the 1993 Econometric Society European Meeting, the 1993 meeting of the European Finance Association, seminar participants at the University of Aarhus, and an anonymous referee are acknowledged. Any remaining errors are the responsibility of the authors. 117
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
driven by fundamentals that are regulated Brownian motions. Froot and Obstfeld (1991a) extend the analysis to allow for a drift in the fundamentals and Froot and Obstfeld (1991b) consider mean reverting fundamentals. Due to technical obstacles outlined below, this paper takes an approach quite different from these models and postulates a consistent, but admittedly ad hoc, stochastic behavior for three exchange rates in a target zone. Besides the behavior in a credible target zone, the case of imperfect credibility is of interest 1. The success of a publicly announced target zone system depends critically on the credibility of governments' and central banks' commitment to enforce the regime. If credibility is low, some of the currencies involved are likely to be subject to speculative attacks, forcing them to realign or eventually pushing them out of the regime. Credibility issues, of course, are complex since they involve numerous factors such as the number of countries participating in the zone, differences in economic development, and whether the bilateral exchange rate is defended by both countries. However, it could be quantified by realignment expectations as in Rose and Svensson (1991), but empirical results are sensitive to the classification into expected movements within the band and expected realignments. The exchange rate mechanism (ERM) in the European Monetary System (EMS) is an example of a system with a relatively large number of participating countries, where each bilateral exchange rate is defended by both countries. The non-EMS Scandinavian countries recently imposed unilaterally enforced exchange rate bands on their currencies against the European currency unit (ECU). The fact that their exchange rate bands were unilateral and therefore not enforced by the EMS central banks may have caused a credibility problem contributing to their collapse. To the authors' knowledge, all previous target zone models are defined in the context of a two-country target zone, where only one band for the single bilateral exchange rate matters. In a more general target zone system such as the ERM in the EMS several exchange rates are linked together in a cobweb of bands on bilateral exchange rates. Here, the behavior of exchange rates located well inside their own bands may be restricted by the fact that other exchange rates are close to their bands. To see this, consider a target zone on the three bilateral exchange rates existing between three countries. Since only two rates are independent due to triangular arbitrage, one may wish to model these two by, say, the Krugrnan model. However, since the band on the third exchange rate must also be maintained, the movements of the two exchange rates must be further constrained. As an example, the third exchange rate can be close to its band while the other two lie well inside their respective bands, and hence the two rates must move in a way that keeps the third rate inside its band although none of their own bands are binding. In a general N-country target zone system there are ( N - 1)5//2 bilateral exchange rates and bands, of which at most N - 1 are independent. Therefore, exchange rates have to move in accordance with a large number of restrictions in addition to their own bands. 2 Intuitively, the free rein for the movements of the independent exchange rates will be inversely proportional to the number of other exchange rates being close to their bands. 118
Trilateral target zone model: B N Jorgensen and H 0 AS Mikkelsen
If a target zone system starts out in a situation where all exchange rates are at their parities, differences in monetary policy or general economic development between the participating countries may over time trigger a situation in which several exchange rates are close to their bands. Such a situation will severely reduce the flexibility of all other exchange rates because of the above mentioned triangular no-arbitrage requirements. Since the target zone is viewed as a system, these other rates can still fluctuate but the movement of individual exchange rates is restricted conditional on values for the other rates in the system. The higher the number of countries, the more flexibility is lost. Besides drastic and possibly unpleasant policy changes through, say, increased interest rate differentials, this problem could be resolved by occasional realignments of parities. The inverse variability of the exchange rates in a target zone may then be a good predictor of future realignments. Using a three-country economy, this paper attempts to account for the above mentioned restrictions and interdependencies between exchange rates in a fully credible target zone. The model is then estimated on the exchange rates between three large integrated EMS economies: Germany, France, and the Netherlands. The sample period is restricted to a period with no realignments; therefore the results are open to the criticism of the so-called Peso problem. 3 Since analytical expressions for the asymptotic distributions of the exchange rate processes and their moments cannot be derived, standard estimation procedures do not apply. However, the method of simulated moments (MSM) seems well suited to handle this problem, see Smith and Spencer (1992) and the references cited therein. The idea is that, even though theoretical moments cannot be derived, the process can be simulated and the simulated moments used instead. Lindberg and S6derlind (1991) use MSM to estimate the basic two-country target zone model with infinitesimal interventions at the margins, and Lindberg and S6derlind (1992) consider the case of mean-reverting fundamentals. The paper proceeds as follows. Section I proposes a continuous time model with mean reversion and demonstrates the well known implication of no arbitrage of reducing the number of linearly independent dimensions. Based on constraints on the model consistent with the regulatory environment in the ERM imposed in Section II, Sections III and IV estimate the model using MSM and compare the results of simulations with the estimated model to the actual data. Finally, Section V concludes the paper. I. A trilateral model It is well known from continuous time finance that not all stochastic processes are permissible descriptions of prices. 4 This was a recurring problem in the development of the theory of the term structure of interest rates. Two general problems would arise: first the price process itself might allow for intertemporal arbitrage and, second the internal consistency requirements among the available assets could overly strain the model. The model of the term structure of interest rates due to Cox et al. (1985) is internally consistent, in that it does not permit such arbitrage opportunities. The goal of this paper is similar, 119
Trilateraltargetzone model:B N Jorgensenand H 0 AF,Mikkelsen namely to identify a system of stochastic differential equations (SDE) that characterizes arbitrage free time-series behavior of three exchange rates in a target zone regime. Assume that investors in all countries are allowed to hold foreign currency that does not yield interest. The first problem alluded to above could then occur if one exchange rate hits a time-invariant upper or lower bound of its band since the direction of the exchange rate process is then predictable. There are many potential solutions to this problem. One approach is to assume a stochastic band or a non-stochastic, widening band. Another approach is a Markov-switching model, where the exchange rate is affected by stochastic discretionary shocks that force the country to either change the band or leave the target zone regime. Since these approaches are not consistent with the credible regime considered in this paper, the next section assumes that any exchange rate can get arbitrarily close to its band without ever touching it. 5 This property is consistent with the Krugrnan model. There, however, the exchange rate is a monotone function of a regulated fundamental and nothing prevents this fundamental from hitting its time-invariant upper or lower band, where the exchange rate will be perfectly predictable. The remainder of this section performs a basic inspection of the consequences of the no arbitrage assumption without specific reference to target zone models. In a fully competitive market without transactions costs, no arbitrage opportunities can exist. The high liquidity, implying low bid-ask spreads in the foreign exchange markets, makes this a reasonable assumption. One fundamental property of the three spot exchange rates existing between three countries is the triangular no arbitrage condition that must hold for any time t:
(l)
S12(t) -- S13(t) = S32(t),
where lower case letters denote the logarithm of the nominal amounts, i.e., sjj(t) denotes the log of the spot exchange rate between country i and j prevailing at time t expressed in units of currency i per unit of currency j. For notational purposes define s 1 = $12, s 2 S13, and s 3 $32 and suppress the time index. Written in SDE form, a general error correction model of the exchange rates would be =
[ dS1 (2)
Ids2 Lds3
=
K1 (Jt/~l--S1) ] X2 ( lU,2- S2) d t + K3 ( ~[Z3-- S3 )
=
[ dzl 1
vldz /' [dg3J
where /xi is the long-run level of exchange rate i, r i is the speed of adjustment of deflations from the long-run level, Z i are three independent standard Brownian motions, and V is a 3 x 3 matrix that can vary with the current levels of the exchange rates. When the terms standard deflation, variance and covariance are applied to dsi they refer to the stochastic component of ds i. Thus, V is the Cholesky decomposition of the local covariance matrix, VV'. The model would be the three-dimensional Ornstein-Uhlenbeck process in the special case where V = trL but in this paper V will invariably depend on the current exchange rates.
120
Trilateral targetzone model: B N Jorgensen and H 0 A~ Mikkelsen
From (2} the drift of an exchange rate process is a constant proportion of its deviation from a natural level, so that the expected future rate is linear in the current rate. In the Krugman model this drift equals the interest rate differential between two countries and is approximately linearly decreasing in the fundamental determinants of the exchange rate for narrow target zones (Svensson, 1991). Furthermore, since the exchange rate reacts less than proportionally (S-shaped) to these fundamentals the absolute drift of the exchange rate is an increasing function of the current rate. For the Krugman model extended to mean-reverting fundamentals (intra-marginal interventions), the nonlinear relationship between the exchange rate and the fundamentals is much less pronounced because of a smaller probability of getting close to the band. Therefore, the drift of the exchange rate is approximately linear in the current level, see Lindberg and S6derlind (1992). This suggests that the drift in (2) is an appropriate description for the situation of intra-marginal interventions. The three dimensional system in (2) can be reduced using the no-arbitrage condition from (1). First, the no-arbitrage condition must hold for the long-run levels (3)
11/'1 -- ~[/'2 = ~U'3"
Since the no-arbitrage condition holds for all realizations of the underlying stochastic processes, the condition applies to the drift term, i.e., ~1 (/Xl - s l ) (4)
[1 - 1 -11
K 2 (~t/,2--S 2)
=0.
K 3 (ill, 3 - $ 3 )
Using (1) and (3) in (4) and rearranging terms yields <5)
( K 3 -- K1) (S 1 -- ~/q) -t- ( K 2 -- /I;3) (S 2 -- IA~2) = 0
that must hold for any possible pair of values, (s 1, s2). This is only possible if (6)
K= K1= K2 = K3.
Having analyzed the drift term of the three exchange rate processes, we apply the no-arbitrage arguments to the stochastic term (7)
dZ1 [1 - 1 - 1 1 V [dZ 2 =0, [dZ3
which also must hold independent of the realizations of dZ i and therefore reduces to the three equations (8)
[1-1
- l l Z = [O 0 0].
Obviously, V is not of full rank, so the third column is arbitrarily chosen to consist of zeros. Therefore, only two independent stochastic processes are needed to describe the dynamics of three exchange rates. This is not to say that there cannot be three independent, country-specific stochastic processes, rather, 121
Trilateral target zone model: B N Jorgensen and H 0 dE Mikkelsen
it implies that at most two components can be identified from three exchange rate series. Further reduction is possible by applying the no-arbitrage condition on the elements of the covariance matrix (9)
VAR[ds3] = VAR[ds 1 - ds2] = V i R [ d s l ] + V i R [ d s 2 ] - 2COV[dSl, ds 2 ]
or, alternatively stated with o-i as the local standard deviation of exchange rate i, the correlation between exchange rate 1 and 2 is o-? +
(10)
P12 =
-
2O-lO,2
The model can therefore be rewritten as the no-arbitrage condition in (1) and (11)
Lds2 ]
r
/z2 - s2
where (12)
,
o ]
g =
012 o-2 o - 2 ~ is the Cholesky decomposition of the covariance matrix, (13)
EE, = [0-~ [ P12 O"10"2
/9120"10"2] 0.2 J"
Note, that the volatility of the third exchange rate does affect the two others through the correlation ,o12 from equation (10). This section repeatedly applied the basic principle of no arbitrage (equation (1)) to a trilateral exchange rate model with mean reversion and reduced the rank of the SDE by one. The next section will impose a target zone regime on the above model.
II. Target zone regime We assume that the three countries participate in a fully credible target zone, where each bilateral exchange rate is allowed to fluctuate within a symmetric band around some parity rates. These parities are set in accordance with the no arbitrage condition and, without loss of generality, are normalized to zero. Henceforth, s i is interpreted as the log deviation from the central parity. Any target zone with these properties involves the constraints (14)
Isil <
~7, Vi,
where 100r/ approximately equals the maximal allowed percentage deviation from the parity rate. Obviously, the drift terms must tend to relax any restrictions arising from one or two exchange rates approaching their bands since otherwise the exchange rates would either leave the band or be absorbed
122
Trilateral target zone model." B N Jorgensen and H 0 A~ Mikkelsen
by the barriers. This translates into the following two conditions on the drift term (15)
I il < 71, ¥ i
(16)
r > O.
From the no-arbitrage condition in (1), at most two exchange rates can be linearly independent, in the sense that the third is given by the other two. Off hand, one might model the exchange rates between three countries by considering only two of the exchange rates explicitly such that the possible exchange rate combinations seem bounded within the full square (17)
{[sa $2 $3 ] ~.,~3: [Sl[ ~ 77 A 1S2[ ~ '/7 AS 3 =-S 1 --$2}
depicted on Figure 1. However, since the fluctuations in the two exchange rates should maintain the band of the third exchange rate, (18>
[s3l=ls 1 - s 2 1 < ~/
follows from (1) and (14). Thus, in the case of a trilateral target zone regime, possible exchange rates are really confined to (19> {[Sl s2 s3 ] ~ 3 :
iSll < r/Als2l< n A _ 7 1 + s 2 < s l < Tl+s 2 As 3 = s l _ s 2 } .
Consider first the hypotenuses of triangles A and B in Figure 1. On these lines, the third exchange rate is always on its band. This implies that exchange rate combinations within the square but outside the shaded area do not have the third exchange rate adhering to its boundaries. When the third exchange rate is on its limit, the local variance of this process must be zero. If this were not the Ca
11
A
SI
B
FIGURE1. Restrictions in a trilateral target zone. 123
Trilateral target zone model: B N Jorgensen and H 0 AE Mikkelsen
case, that exchange rate could move out of its band and out of the shaded region with a strictly positive probability and, therefore, the two other exchange rates must be perfectly correlated. In this case, equation (10) reduces to (20) 2o'lo"2 = o'1z + o"2 implying that o'1 0"2" Intuitively, the identical variances make sense because there is no asymmetry in our formulation of a target zone. Of course, this argument holds for all exchange rates. For any of the six vertices of the shaded hexagon, two exchange rates are on their band simultaneously. By the same no arbitrage argument, the last exchange rate cannot have any variability component even though it is on its parity. From the above discussion two properties must characterize any model of a credible trilateral target zone. First, the local variance of any exchange rate must approach zero as the exchange rate approaches its band. This is ensured by forcing the two others to approach perfect correlation. Second, when two exchange rates approach their bands, the local variance of the third should approach zero. Therefore, a natural but indeed arbitrary local standard deviation of exchange rate i is chosen as follows =
Ear (21)
O"1 = g a p
(22)
a i =
Vj--/=i
2 ' where p is a non-negative parameter and the normalized weights, a i,
,77-s rli2
,
ensure that the exchange rates possess all properties of a target zone system as discussed above. Since in certain cases 0"1 = o'2, the variance scaling parameter, 8, has to be common across exchange rates. To analyze the above standard deviation, consider the ith exchange rate and fix the other two rates at zero corresponding to a bilateral target zone. Figure 2 depicts the standard deviation as a function of the deviation from the central parity rate for three different values of p and with the scaling parameter, 8, equal to unity. When s --0, the exchange rate volatility is maximized at unity and o. is locally insensitive to s. Furthermore, as s ---> + ~/(~/= 2.25 percent in the figure) the standard deviation tends to zero. The manner in which o. approaches zero is controlled by the parameter p. For p = 1 the reaction of the standard deviation to s is moderate with a rate of decay that increases linearly with the absolute value of s. As p decreases, e . g . to 0.1, the standard deviation gets less sensitive for small departures from the parity but declines rapidly as the exchange rate gets close to the bands. When p is increased to, say, 10, the standard deviation displays a rapid decay to zero. It follows that the chosen functional form is flexible enough to mirror a variety of patterns for the standard deviation between the extremes of a collapse to zero only at the bands (as p ~ 0) and a collapse immediately as the exchange rate tends to leave the 124
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
central parity (for p ~ ~). The corresponding standard deviation in the Krugman model with interventions only at the margin is given by 8(~s//af), where f is a regulated Brownian motion process with diffusion coefficient 8, see Svensson (1991, 1992). The derivative of s with respect to f is a function which is qualitatively similar to the a P function defined above giving rise to an inverted U-shaped relationship similar to Figure 2 for p approximately equal to unity, as in Figure 1 of Svensson (1992). For the Krugman model with mean-reverting fundamentals c~s/af is almost linear except perhaps for values of s close to the band. If the long-run level of the exchange rate was zero, the standard deviation would again be maximized (cTs/~f= 1) for s = 0 and be zero (c~s/c~f=0) at s = + ~7. Therefore the relationship between an exchange rate and its standard deviation would be shaped like an inverted V if drawn in Figure 2, resulting in a less volatile exchange rate since shocks are partly neutralized within the band. Such a shape can easily be mimicked by (21) with p above unity, say, 1.25. It follows that the selected parameterization is sufficiently flexible to capture a variety of policy scenarios including marginal as well as intra-marginal intervention. Thus, p is labelled the policy parameter. The summation in (21) ensures that the standard deviation of an exchange rate approaches zero as the remaining two rates both get close to their bands. In Krugman's model, the exchange rate had an S-curved relation to the O
SS
/
o
:
i
~%
:
i
/ o
,
:
\
: 0
\
: ,: •"
0 0
".
,.,,"
-3
-2
%,...,
-1
0
1
2
3
s F]GURE 2. ~ as a ~ n c t J o n o f p=O.1
w i t h other e x c h ~ g e rates ~ ¢ d at zero. K E Y : ~ p = l . O ................ p = l O . O
si
125
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
underlying fundamental. Here, since there are two independent forcing processes, the reaction function is S-curved in both dimensions. In Figure 3, the x- and y-axes represent forcing processes and the z-axis is the corresponding exchange rate levels giving rise to a snake-shaped relation. If exchange rates were freely floating the reaction function would be depicted as a hyperplane. The main empirical implication of this section is an issue of model (mis)specification. A test of e.g. Krugman's two country model to a target zone that has more than two exchange rates must take into consideration restrictions imposed by no arbitrage from non-modelled exchange rates. The subsequent application of the above model to the EMS is subject to such critique since more than three countries participate in the target zone. IlL MSM estimation The data used in this section are the three bilateral exchange rates between Deutsche-mark (DM), Dutch guilder (NG), and French franc (FF) from Citibase. In particular D M / N G is chosen for s 1, D M / F F for s 2, and F F / N G for s 3. Data points are sampled daily as the average of spot buy and sell
eq
Q
FIGURE 3. Reaction of exchange rates to underlying processes: The snake effect. Note: This figure was generated from an approximate numerical integration of the trilateral model with a given parameterization. 126
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
quotations during the period from January 13, 1987 until January 4, 1990. This period was chosen because it is the longest interval available without any realignments during the ERM, yielding a total of 748 observations on each exchange rate. 6 In order to make the data consistent with the model, units are chosen so that logarithmic exchange rates have a central parity of zero. The resulting exchange rates are then multiplied by 100 to give percentage deviations from the parity rate. Figure 4 shows the time series behavior of the data. 7 The absence of analytical asymptotic distributional results for the exchange rate processes renders the use of traditional estimation techniques infeasible. The recently developed method of simulated moments (MSM) provides a convenient tool to overcome this problem. 8 The philosophy underlying this approach is that even though true moments are unknown, the model can always be simulated and simulation moments calculated. Given an N x 2 matrix, dZ, of simulated standard normal random variables, the parameter estimates can be chosen as those minimizing some distance between the resulting simulated moments and the actual moments of the data. This paper follows Lindberg and SiAderlind (1991) and Smith and Spencer (1992) in estimating an exactly identified system in which the number of moments to be matched equals the number of parameters to be estimated. The estimation procedure in these papers is restricted MSM where one parameter
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FIGURE 4. Actual series from January 13, 1987 through January 4, 1990. KEY: F F / N G ................ D M / N G _ _ DM/FF Margins 127
Trilateral target zone model." B N Jorgensen and H 0 AE Mikkelsen
is estimated conditional on arbitrary values for the others, while Lindberg and S6derlind (1992) uses a multidimensional grid. In the present analysis a suitable multidimensional minimization algorithm is used to simultaneously chose the parameters that minimize the following quadratic loss function: 5
(23)
Loss= ~
(Msi-MAi) 2,
i=1
where Msi is the ith simulated sample moment, and MAi is the ith actual sample moment. Note that in this exactly identified system parameters may be chosen that bring the loss function arbitrarily close to zero. Also note that the credible target zone structure ensures the existence of all moments. The above procedure is then repeated for R = 100 times (i.e. for R different random draws of forcing processes, dZ). The final parameter estimates are given as the arithmetic average from these R replications. Monte Carlo standard deviations are reported to allow evaluation of the statistical significance of the estimates. Table 1 reports results from MSM estimation of the trilateral target zone model with N = 2000. The moments used are the mean of s I and s 2, the covariance between s 1 and s 3 and the variance of the first difference of s 2 and s 3. All parameters appear to be estimated with sufficient precision, i.e. standard errors are very small compared to parameter estimates, except for/.t 1 which is not significantly different from zero. The loss function was essentially always equal to zero, with a mean value of 3.4.10 -7. The values o f / z 1 and /z2 reveal the difference between the positions of D M / N G and D M / F F within the band, with the latter being relatively weak during the sample period. By the no arbitrage argument, the estimated value o f / z 3 = / z 1 -/~2 = 0.6869 corresponds to the relative position of F F / N G in its band. The estimate for r indicates that 12.5 percent of a given deviation from the long-run level is discounted away in each step which seems somewhat higher than would be expected from inspection of the data. Finally, the estimate of p is close to but significantly above 1. Overall, the estimated model appears to catch some of the characteristics of the given exchange rates. This study encounters the aliasing problem that numerous continuous time models can match the discrete data. Also, an approximation error arises from the discrete time simulation of the underlying model. Under usual regularity conditions convergence of the traditional GMM estimator is assured, see Hansen and Scheinkman (1992), and with additional regularity conditions related to the approximation error there is no reason to suspect problems in TABLE 1. M S M estimates of the trilateral target zone model.
Estimate Std. Dev.
r
/x 1
/~2
8
p
0.1250 (0.0189)
0.0034 (0.0225)
- 0.6835 (0.0216)
0.1413 (0.0019)
1.0824 (0.0003)
Note: T h e table reports the m e a n and standard deviation across 100 replications.
128
Trilateral target zone model: B N Jorgensen and H 0 AE Mikkelsen
the convergence of the MSM estimator. Since the data have been time-aggregated on a daily basis, the quantitative estimates presented above may not correspond exactly to the parameters in the true, underlying continuous time process. In particular, since a discrete time interval contains (infinitely) many continuous time differentials, the true r and 8 will be smaller than the estimates.
IV. Properties of the estimated model The previous section estimated the model by exact matching of sample and simulation moments, and the parameter estimates appear to be fairly precise in terms of standard deviation relative to the mean. In order to further examine the precision of the estimates, it is of interest to investigate how well the estimated model performs on a large number of new random forcing processes with fixed parameters. The results from this exercise will reveal the stability of the model and show how well moments that are not used in estimation are matched. In particular, 2000 observations from the estimated model, as it appears in Table 1, are simulated 1000 times and various moments and statistics are calculated as well as for the actual data. Table 2 compares the first four moments of the actual data to the simulated data in logarithmic levels. It is evident that the first order moments are well matched, consistent with what one would expect, given that they are matched
TABLE 2. Moments for actual and simulated exchange rates. DM/NG Actual
Simulated
DM/FF Actual
FF/NG
Simulated
Actual
Simulated
Mean
0.005
0.003* (0.022) [0.457]
- 0.680
- 0.680* (0.021) [0.468]
0.685
0.685 (0.021) [0.501]
Variance
0.043
0.065 (0.006) [1.000]
0.730
0.060 (0.005) [0.0001
0.752
0.060 (0.005) [0.0001
Skewness
0.877
- 0.004 (0.111) [0.000]
0.490
0.328 (0.120) [0.090]
Kurtosis
3.172
2.866 (0.176) [0.047]
1.977
3.037 (0.261) [1.000]
-
0.609
2.058
- 0.330 (0.119) [0.986] 3.039 (0.272) [1.000]
Note: Simulated moments are calculated as the average across 1000 replications of the
estimated model with 2000 observations, parentheses denote Monte Carlo standard deviations and square brackets the fraction of simulated models with a larger moment than the actual data. An * indicates a moment used in estimation.
129
Trilateral target zone model: B N Jorgensen and H 0 t E Mikkelsen
in estimation. For both variance and kurtosis, only D M / N G is within reasonable distance to the actual values. In contrast, the skewness for D M / N G is far away from the true value. Table 3 reports the actual and simulated covariances of the level processes. Again, since the covariance between D M / N G and F F / N G is used in estimation, it is well matched in the simulations. The covariance between D M / N G and D M / F F seems reasonable while the covariance between D M / F F and F F / N G is at odds with the actual data. Table 4 reports the first four moments, the Jarque-Bera test for normality and ARCH tests for the first difference of the actual and simulated series. Similar to previous results, the moments that are used in estimation are well matched while the results for the others are mixed. Interestingly, the Jarque-Bera test rejects the null of normality in more than 50 percent of the simulations for D M / F F and F F / N G at the 5 percent significance level. Though the statistic never reaches the same absolute value as for the actual series, this is interpreted as evidence in favor of the model's ability to generate non-normality as observed in actual data. The ARCH statistics are somewhat disappointing in that the rejection rate under the null of no ARCH effects is only 10 percent (at the 5 percent significance level). On the other hand, rejection occurs more frequently than allowed by the nominal size of the test, but whether this can be attributed to finite sample size distortions of the test or simply low power is beyond the scope of this paper. Another feature of the simulations is that the model is not capable of generating sufficient asymmetry across the three exchange rates. As an example, the three simulated covariances in Table 3 are all of equal magnitude but the covariances for the actual data are very different. Figure 5 shows simulations of the estimated model of equal length as the actual time series and starting at the same initial values. Casual inspection of the graphs seems to indicate that there is a more pronounced mean reversion in the simulated series than in the actual data. This might indicate why some moments in Tables 2, 3 and 4 are not well matched.
TABLE 3. Cross-moments for actual and simulated exchange rates. Actual Cov ( D M / N G , D M / F F )
0.011
Simulated 0.033
(0.0o4) [1.000] Cov ( D M / N G , F F / N G )
0.032
0.033*
(0.004) [0.497] Cov ( D M / F F , F F / N G )
Note: See Table 2.
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- 0.719
- 0.027 (0.004) [1.000]
Trilateral target zone model: B N Jorgensen and H 0 A~ Mikkelsen
TABLE 4. Statistics for actual and simulated first differences of exchange rates. DM/NG Actual
Mean
- 3 . 1 0 -s
DM/FF
Simulated 4.10 -6 (2.10 -4) [0.587]
FF/NG
Simulated
Actual
Simulated
-0.003
- 3 . 1 0 -4 (1.10 -4) [1.000]
0.003
3.10 .4 (1.10 -4) [0.000]
0.015" (5" 10 -4) [0.6751
0.015
0.015" (0.001) [0.3531
Actual
Variance
0.005
0.016 (0.001) [1.000]
0.015
Skewness
- 0.054
- 0.004 (0.057) [0.820]
- 0.327
- 0.127 (0.058) [1.000]
0.223
0.130 (0.056) [0.056]
6.724
3.022 (0.113)
7.944
3.094 (0.127)
7.990
3.094 (0.130)
Kurtosis
[o.ooo] Jarque-Bera
ARCH(l)
ARCH(5)
431.9
85.18
96.93
2.185 (2.270)
[o.ooo] 774.1
8.549 (6.732)
[o.ooo] 781.3
8.836 (7.069)
[o.ooo1
[o.ooo1
[o.ooo1
{0.073}
{0.560}
{0.596}
1.144 (1.691)
78.41
1.489 (2.158)
44.59
1.335 (1.980)
[o.ooo1
[o.ooo1
[o.ooo1
{0.060}
{0.101}
{0.086}
5.531 (4.003)
131.8
6.465 (4.430)
87.84
6.277 (4.263)
[o.ooo]
[o.ooo]
[o.ooo]
{0.082}
{0.123}
{0.124}
See Table 2. Jarque-Bera is a test for normality and ARCH(z) is the LM test for no z order A R C H effects. Curly brackets denote the proportion of simulated test statistics exceeding the 5% significance level from the appropriate g 2 distribution.
V. Conclusion Using no-arbitrage arguments, this paper proposed an arbitrage free trilateral model of a credible target zone regime with bands on each bilateral exchange rate. The triangular no-arbitrage condition that characterizes highly competitive foreign exchange markets reduces the system from three to two dimensions. In particular, the third redundant exchange rate becomes the difference between the two independent rates. Therefore, in addition to their own boundaries around the central parity rate, the movements are restricted by the requirement that the redundant rate obeys its band. This implies that traditional target zone models defined in a bilateral setting do not directly apply to a 131
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen r,D
~4
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Simulated observation I R o u ~ 5. Simulated series corresponding to Figure 4. K E Y : _
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_
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Margins
multilateral setting (as in the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS)), where a large number of currencies are linked together in a cobweb of bands on bilateral exchange rates. Technically, the exchange rates are postulated to follow a mean reverting process, with a local covariance matrix that depends on the positions of all three exchange rates relative to their respective bands. This assumption is motivated on two grounds. First, given perfect credibility of the target zone regime all deviations from long-run levels should be transitory. Second, mean reversion prevents any exchange rate from being absorbed by the boundaries and thus rules out an obvious arbitrage opportunity. When either exchange rate approaches one of its boundaries, the local variance, which is the unpredictable component of the exchange rate movement, must approach zero in order to prevent potential violation of the band. The inherent mean reversion accomplishes this by forcing the exchange rate away from the boundaries and towards the parity. Also, as the redundant rate approaches its boundaries, its local variance vanishes, and the other rates must approach perfect correlation. Although somewhat arbitrary, the variance structure imposed in this paper is sufficiently flexible to mimic a variety of policy scenarios, such as intramarginal intervention. 132
Trilateral target zone model: B N Jorgensen and H 0 / E Mikkelsen
Since the model has no known analytical solution, it is estimated by the method of simulated moments on three currencies within the ERM in the EMS. By limiting the focus to a subset of currencies participating in the ERM, the empirical results presented in the paper are exposed to the criticism raised that the model leaves out important information from neglected currencies. Although parameter estimates take on plausible values, evidence from simulations of the estimated model is mixed. While the parameter estimates appear quite robust with respect to the moments used in estimation other moments are badly matched. The estimated mean reversion parameter appears larger than expected and, therefore, the model is unable to generate ARCH effects to the extent found in real data. However, the model succeeds in generating the stylized empirical fact of non-normal returns. This paper has several potential extensions. Most importantly, the model may currently be too symmetric to characterize real world exchange rate behavior. Second, alternative specifications of mean reversion that depend on the position of current exchange rates could imply restrictions between the expected return and volatility of exchange rates that determine risk premia. Finally, placing the model in an equilibrium framework may provide better guidance in choosing the appropriate stochastic processes. No~s 1. Credibility and related problems have been addressed in, e.g. Bertola and Caballero (1992), Bertola and Svensson (1991), and Svensson (1992). 2. An extension of Krugman's model to three countries would involve finding investor's expectations towards exchange rate movements by solving a three dimensional stochastic partial differential equation under piece-wise linear boundary conditions. Since we could not guess the existing solution, an arbitrage approach was chosen. 3. If the target zone is less than perfectly credible, the probability of a realignment will affect the results even if the realignment does not occur in sample, see Krasker (1980). 4. See Ingersoll (1987), p. 384. 5. For simple mean reverting processes, this could be achieved by imposing parameter restrictions between the drift and variance, see Cox et al. (1985), p. 391. 6. Though the current trilateral target zone model has a Markov property for the exchange rates, it is not appropriate to pool data in order to increase the sample size. The exchange rates are functions of their relative positions within the bands and a realignment typically causes a given exchange rate to alter this. In particular, an exchange rate which is near the upper limit of the band will often continue much closer to the lower edge of the new band after a realignment, see Rose and Svensson (1991). Therefore, the relative positions within the bands will unavoidably display discontinuities, although it is possible to obtain smooth curves for the exchange rates. 7. In reality, the band is not just + 2.25 percent, but rather the nominal central parities of one are multiplied (divided) by the factor 1.022753 to give upper (lower) limits, see e.g. Grabbe (1991) pp. 39-41 for an explanation of this factor. In the present context the upper and lower limits are then 100 ln(1.022753) and 100 ln(1/1.022753) respectively and the weights in the variance formulas are restated accordingly. 8. For distributional results, properties, and applications of this approach, see, inter alia, Duffle and Singleton (1993), Gregory and Smith (1990), and Smith and Spencer (1992). 133
Trilateraltargetzone model: B N Jorgensen and H 0 A~ Mikkelsen
References
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