Exchange rate target zone models: a Bayesian evaluation

EXCHANGE RATE TARGET ZONE MODELS:A BAYESIAN EVALUATION

KAI LI*

Faculty of Commerce, University of British Columbia, 2053 Main Mall, Vancouver, BC, Canada V6T 1Z2

SUMMARY

This paper develops a Bayesian approach to estimating exchange rate target zone models and rationalexpectations models in general. It also introduces a simultaneous-equation target zone model thatincorporates stochastic realignment risk. Using FF/DM and IL/DM exchange rate data, we ®nd that thesigning of the 1987 Basle±Nyborg Agreement reduces both the magnitude and the likelihood of a centralparity realignment, while the lagged exchange rate deviation from its central parity increases them.Furthermore, the interest rate policies and the monetary conditions of the participating countries signal aforthcoming realignment. In general, we are unable to improve upon a simple random walk model in out-of-sample exchange rate prediction by introducing target zone models. Copyright # 1999 John Wiley &Sons, Ltd.

1. INTRODUCTION

Since the inception of the European Monetary Systems (EMS) in March 1979, eleven govern-ments of European countries have linked their exchange rates through formal participation in theExchange Rate Mechanism (ERM). The essence of the ERM is that each participant country hasan allowed range (target zone) within which its currency may ¯uctuate against the others. The`norm' is+2.25% with respect to the central rate (parity), but Italy and some other countries arepermitted a band of +6%. To keep the exchange rates within these margins, participatingcountries are obliged to intervene in the foreign exchange market, and at times with interest ratepolicies. On the other hand, it is also possible for the monetary authorities to realign the parities,provided that all the members of the EMS agree.

Researchers and practitioners alike are caught up with these institutional arrangements, and anumber of exchange rate target zone models have been introduced. Examples include Krugman(1991), Svensson (1991), Bertola and Caballero (1992), Bertola and Svensson (1993), Mizrach(1995), Bekaert and Gray (1998), Koedijk et al. (1998), and, of particular relevance to this paper,Pesaran and Ruge-Murcia (1999).1

The objective of this paper is twofold: to introduce a simultaneous-equation target zone modelthat incorporates stochastic realignment risk (in terms of both the magnitude and the likelihoodof a realignment) and to develop an encompassing econometric framework to estimate andcompare the existing target zone models. The proposed simultaneous-equation target zone model

CCC 0883-7252/99/050461±30$17.50 Received 22 April 1997Copyright # 1999 John Wiley & Sons, Ltd. Revised 22 October 1998

JOURNAL OF APPLIED ECONOMETRICS

J. Appl. Econ. 14: 461±490 (1999)

* Correspondence to: Professor K. Li, Faculty of Commerce, University of British Columbia, 2053 Main Mall,Vancouver, BC, Canada V6T 1Z2. E-mail: [email protected]

Contract/grant sponsor: Social Sciences and Humanities Research Council of Canada.

1 Pesaran and Ruge-Murcia (1999) develop a discrete-time target zone model in which the bounds can remain ®xed over aperiod of time, but are subject to occasional jumps.

is closely related to Pesaran and Ruge-Murcia's (1999) work, and this paper makes contributionsto the literature along the following directions.

First, we analyse the exchange rate target zone model under a simultaneous-equationmodelling framework, thus allowing for possible interactions in the determination of exchangerate, macro fundamentals and stochastic realignment risk. Furthermore, by explicitly incorpor-ating a correlation structure into the system of equations, we are able to obtain unbiasedestimation results.2

Second, we model both the magnitude and the likelihood of a realignment in central parity as afunction of the participating countries' macroeconomic variables (an independent work byBekaert and Gray, 1998, adopts a similar approach). Also, by including a year dummy variable tothe magnitude and the likelihood of realignment equations, we are able to capture the regimeshifting e�ect of the 1987 Basle±Nyborg Agreement on the exchange rate policies of the EMScountries.

Finally, we develop a Bayesian approach to estimating exchange rate target zone models andrational expectations models in general. Merits of applying the Bayesian inferential proceduresare as follows:

(1) By using the Metropolis-within-Gibbs methodology (Chib and Greenberg, 1995, 1996) withdata augmentation (Tanner and Wong, 1987), we are able to draw from the posteriors ofthese target zone models and avoid direct evaluation of the non-trivial likelihood functions(due to the highly non-linear-in-parameters component and the limited dependent nature ofthe dependent variables within the system).3

(2) Exact ®nite sample results can be obtained in a way that is useful for the treatment of thispaper's small data sets.

(3) Any prior knowledge, such as (non-linear) constraints on the unknown parameters, can beconsistently incorporated via the prior speci®cation during the process of making statisticalinferences.

(4) Full posterior distributions for any quantities of interest can be derived.

In this paper focus will be given to the simultaneous-equation target zone model withstochastic realignment risk (M0) and its four alternatives. They are: Pesaran and Ruge-Murcia's(1999) target zone model with discrete jumps (M1 , with no simultaneity); Svensson's (1991)target zone model with constant realignment risk (M2); Pesaran and Samiei's (1992a,b) targetzone model with known varying bands (M3 , with no realignment risk); and ®nally, Pesaran andSamiei's (1992b) linear rational expectations target zone model (M4 , with no e�ect of the band).

The exact likelihood-based inferential procedures are applied to French Franc/Deutsche Mark(FF/DM) exchange rate data and Italian Lira/Deutsche Mark (IL/DM) exchange rate data. We®nd that the signing of the 1987 Basle±Nyborg Agreement signi®cantly reduces both themagnitude and the likelihood of a realignment in central parity, while the lagged exchange ratedeviation from its central parity increases them. Furthermore, the interest rate policies and the

2Vella (1992) examines the trade-o� between women's weekly hours worked and nonwage labour income. He concludesthat failure to account for the endogeneity of the weekly hours worked variable in the determination of non-wage labourincome produces biased estimates. In this paper the simultaneous-equation target zone model produces signi®cantlydi�erent point estimates for some of the model parameters compared with that of the Pesaran and Ruge-Murcia (1999)model.3 In brief, the Metropolis-within-Gibbs procedure is an extremely versatile Markov chain method to simulate complex,non-standard multivariate distributions. Data augmentation is a scheme to augment the observed data to simplify thelikelihood/posterior.

462 K. LI

Copyright # 1999 John Wiley & Sons, Ltd. J. Appl. Econ. 14: 461±490 (1999)

monetary conditions of the participating countries signal a forthcoming realignment. In general,we are unable to improve upon a simple random walk model in out-of-sample exchange rateprediction by introducing target zone models; this result corroborates those of others usingdi�erent techniques ( for instance, Diebold and Nason, 1990; Meese and Rose, 1991).

The paper is organized as follows. Section 2 introduces the simultaneous-equation target zonemodel and its four alternatives. Section 3 develops the exact likelihood-based inferentialprocedures. The empirical results are presented in Section 4. Section 5 concludes.

2. MODEL FRAMEWORK

2.1. Simultaneous-equation Target Zone Model

Similar to the theoretical target zone models (see Svensson, 1992 for a survey), the structuralexchange rate equation in the system is based on the monetary model of exchange ratedetermination (Pesaran and Ruge-Murcia, 1999 and references therein; our Appendix A providessome derivation details):

e�t � g1E�et j Itÿ1� � g2etÿ1 � b1mt � b2yt � predm

0t � u1t �1�

where e�t is the log of the nominal exchange rate measured as the price of one unit of foreigncurrency in terms of domestic currency units.4 E�et j Itÿ1� is the agents' one-step-ahead forecast ofthe endogenous variable et conditional on their information set at time tÿ 1, which turns out tobe a highly non-linear function of the lagged exchange rate etÿ1 and the macro fundamentals (seeAppendix B). The macro fundamentals of the model are fmt; ytg, where mt is the log of relativemoney supplies, yt the log of relative outputs between the two countries. By construction, thenon-linearity between the exchange rate and the macro fundamentals is captured by a signif-icantly di�erent from zero coe�cient g1 . The agumented set of forcing variables fDmtÿ1;Dmtÿ2;Dytÿ1;Dytÿ2g is contained in predm0

t , which is introduced by Pesaran and Ruge-Murcia (1999,Appendix C) to control for the error involved in estimating the monetary model (1) with thecurrent instead of the future expectations of the exchange rate. u1t is the error term independentand distributed as N�0; s11�.

According to the target zone mechanism of the EMS, the following observation rule on etbecomes applicable:

et �eUt if e�t 5 eUt

e�t if eUt > e�t > eLteLt if e�t 4 eLt

8><>:where eUt; eLt are the corresponding upper and lower bounds on the exchange rate variable attime t, computed by the 2.25% margin above/below the central rates in the FF/DM case and bythe 6% margin in the IL/DM case, et is the observed exchange rate, and e�t is the correspondinglatent exchange rate determined in the monetary model (1). Evidently, (1) is of a two-limit Tobittype.

4Here we take France and Italy as the domestic countries and Germany as the foreign country. We put a superscript � onet to distinguish the latent exchange rate from the actually observed exchange rate. This distinction will becometransparent later.

EXCHANGE RATE TARGET ZONE MODELS 463


To model the forcing variables, fmt; ytg, we ®rst carry out some Bayesian unit root tests to thesemacro fundamentals.5 Since we do not ®nd strong evidence of non-stationarity, the relativemoney supply variable mt is speci®ed as a restricted AR(12) process in level

mt � a1mtÿ1 � a2mtÿ3 � a3mtÿ10 � a4mtÿ12 � a5 � u2t �2�

where u2t is the error term independent and distributed as N�0; s22�. The relative output variableyt is also speci®ed as a restricted AR(12) process in level

yt � f1ytÿ12 � f2 � u3t �3�

where u3t is the error term independent and distributed as N(0, s33). The number of lags in therestricted AR(12) processes fmt; ytg is chosen using the Schwarz Criterion. The twelfth-orderlagged variables are included to account for the seasonal component that may be present in ourmonthly data.

The ®rst innovative component of our simultaneous-equation target zone model is the explicitmodelling of the magnitude and the likelihood of a realignment.6 More speci®cally, followingPesaran and Ruge-Murcia (1999) we assume that the central parity, ect , evolves according to thefollowing stochastic process:

ect � ectÿ1 � st � Zt� ectÿ1 � st � �Jtÿ1 � u4t�

where ectÿ1 is the central parity prevailing at the previous period tÿ 1 (in the pre-target zoneperiod, ec0 is arbitrarily set to the same value as ec1), st is a dummy variable which equals 1 if thereis a realignment at time t and 0 otherwise, and the size of a jump Zt consists of a forecastablecomponent Jtÿ1 and an error term u4t :

Zt � d1Yr87t � d2itÿ1 � d3�etÿ1 ÿ ectÿ1� � d4mtÿ1 � d5mtÿ12 � d6ytÿ1 � d7ytÿ12 � d8 � u4t� Jtÿ1 � u4t

�4�

The justi®cations for including these conditioning variables are given below.7

Yr87t is a dummy variable which equals 1 if the time period t is after the Basle±NyborgAgreement of 1987 and 0 otherwise. In September 1987, the Basle±Nyborg Agreement wasreached on the limited use of EMS credit facilities for intramarginal interventions and greater useof the exchange rate band. According to Dominguez and Kenen (1992), the actual exchange ratebehaviour was signi®cantly di�erent after the Agreement. Our data sets span this transitional

5 Bayesian unit root tests using a ¯at prior have been applied to both the money supply di�erential mt and the real outputdi�erential yt (Koop, 1994, and references therein). The posterior probabilities of mt � I�1�; yt � I�1� are, respectively,0.4105, 0.2185 in the FF/DM case, and 0.0583, 0.1073 in the IL/DM case. Henceforth we will model these macrofundamentals as stationary time series.6 In the spirit of the rational expectations hypothesis, we project both the size and the likelihood of a realignment on aninformation set consisting of lagged macroeconomic variables of the participating countries. In this way, we are able tosolve (numerically) for the rational expectations variable E�et j Itÿ1� in equation (1) (Pesaran and Ruge-Murcia, 1999 andour Appendix B).7 Because of our unprecedented modelling approach and lack of theoretical guidance, we employ the same set ofexplanatory variables in both the size and the likelihood of realignment equations. The set of explanatory variables isrelatively broad and includes many plausible candidates that were used in the previous empirical studies (Bekaert andGray, 1998; Edin and Vredin, 1993; Mizrach, 1995; Pesaran and Ruge-Murcia, 1999; Rose and Svensson, 1994).

464 K. LI


period which provides us an opportunity to investigate aspects of the exchange rate behaviourthat have been changed by the Basle±Nyborg Agreement.

itÿ1 denotes the lagged interest rate di�erential between the home and foreign countries.According to the uncovered interest rate parity hypothesis, we would expect to infer the marketexpectations of a central parity realignment from this interest rate di�erential.�etÿ1 ÿ ectÿ1� is the lagged deviation of the exchange rate from its central parity. Bertola and

Caballero (1992) observe that the realignment probability is related to the position of the EMSexchange rates within their bands, and the likelihood of a realignment increases as the exchangerate approaches the upper limit of its ¯uctuation band. Bekaert and Gray (1998) ®nd that at thelower boundary �etÿ1 ÿ ectÿ1 5 0� , a large deviation can be accommodated within the band (norealignment), but at the upper boundary �etÿ1 ÿ ectÿ1 > 0�, the only possibility is a realignmentwhen the deviation is large. Bertola and Svenssen (1993), Mizrach (1995) and Pesaran and Ruge-Murcia (1999) have all used this lagged deviation of the exchange rate from its central parityvariable to predict the likelihood of a realignment. Our inclusion of �etÿ1 ÿ ectÿ1� captures theasymmetric relationship between the lagged position of the exchange rate in the band and thelikelihood of a realignment empirically.

fmtÿ1;mtÿ12; ytÿ1; ytÿ12g are the lagged macro fundamentals. According to the monetary modelof exchange rate determination, both the money supply and the real output di�erentials betweentwo countries a�ect the exchange rate behaviour. Here we would like to examine whether thesemacro fundamentals also play a role in determining the realignment risk in central parity.

Finally, u4t is the unforecastable component of the jump size (error term) independent anddistributed as N�0; s44�.

The probability of a realignment is formulated as a probit regression

s�t � c1Yr87t � c2itÿ1 � c3�etÿ1 ÿ ectÿ1� � c4mtÿ1 � c5mtÿ12� c6ytÿ1 � c7ytÿ12 � c8 � u5t

st �1 if s�t > 0

0 otherwise

� �5�

where s�t is the underlying latent variable, censored to be either above or below zero, dependingon the value of the dummy variable st . u5t is the error term independent and distributed asN(0, 1).

In the IL/DM case, the set of explanatory variables used in the magnitude and the likelihood ofrealignment equations is fYr87t; ptÿ1; itptÿ1; itspreadtÿ1; itÿ1; �etÿ1 ÿ ectÿ1�;mtÿ1g, where ptÿ1 isthe lagged relative in¯ation between Italy and Germany, itptÿ1, the lagged Italian price level anditspreadtÿ1, the lagged Italian yield curve spread which is obtained by the di�erence between thelong-term government bond rate and the short-term discount rate in Italy. According to Estrellaand Mishkin (1995), itspreadtÿ1 serves as a useful indicator of Italian monetary policy, inparticular, the market expected future path of Italian in¯ation.

The second innovative component of our simultaneous-equation target zone model is that weallow the error structure of the system (1)±(5) to be contemporaneously correlated; that is,Ut � �u1t; u2t; u3t; u4t; u5t�0 �MVN�0;S�, where S is a 5� 5 non-diagonal positive de®nitesymmetric matrix with the ith row jth column element denoted as sij (rij is the correspondingcorrelation coe�cient). The ®fth diagonal element of S is constrained to unity due to the



identi®cation requirement of a probit model (5). This simultaneous-equation target zone modelis, to our knowledge, new to the literature.

2.2. Alternative Target Zone Models

Our simultaneous-equation target zone model (M0) nests the Pesaran and Ruge-Murcia (1999)model (M1) by setting the o�-diagonal terms of the variance±covariance matrix S to zero, andthe Svensson (1991) model (M2) by further setting all the coe�cients in equations (4) and (5) tozero, except for the coe�cients of the intercept terms. As for the Pesaran and Samiei (1992a)model (M3), they assume that the varying band at time t is known to the agents at time tÿ 1,which is a simplifying but unrealistic assumption. In their case, we drop equations (4) and (5)from the system, and modify the rational expectations solution for E�et j Itÿ1� along the lines ofDonald and Maddala (1992) and Lee (1994). Finally, we also examine the benchmark linearrational expectations model (Pesaran and Samiei, 1992b, M4) in which there is no e�ect of theband when agents are forming their rational expectations. As a result the rational expectationssolution for E�et j Itÿ1� is a linear function of the macro fundamentals and the model (M4) onlyincludes equations (1)±(3). By comparing M3 and M4 , we are able to evaluate the importance oftaking into account the band on the exchange rate behaviour when forming expectationsE�et j Itÿ1�.

In summary, the four alternative target zone models can be succinctly characterized as follows:

M1 : includes equations (1)±(5), S is diagonal.M2 : includes equations (1)±(5), S is diagonal and

d1 � d2 � d3 � d4 � d5 � d6 � d7 � 0

c1 � c2 � c3 � c4 � c5 � c6 � c7 � 0

M3 : includes equations (1)±(3), the corresponding variance±covariance matrix is diagonal andthe varying bands are known.

M4 : includes equations (1)±(3), the corresponding variance±covariance matrix is diagonal andthe bands have no e�ect in forming expectations.

Except for M0 , all these other target zone models (M1ÿM4) are under the single-equationmodelling framework.

3. BAYESIAN INFERENCE

3.1. Prior Speci®cation

Our prior speci®cation is motivated by the following considerations:

(1) Priors must be proper to ensure well-behaved estimation results (Poirier, 1996, and referencestherein).

(2) Priors should be relatively non-informative so that Bayesian estimation results mainly re¯ectdata information.

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(3) When some of our models are nested ( for instance, M0 nestsM1 , M1 nestsM2) it is desirablefor the prior to re¯ect the fact that some parameters are common across models (Koop,1991).

In M0 , the unknown parameter Y is composed of two parts: the k-dimensional regressionparameter B which contains all the regression coe�cients �g1; g2; b0s; a0s;f0s; d0s;c0s) in equations(1)±(5) and the (non-diagonal) variance±covariance matrix S. Complication arises due to the factthat the ®fth diagonal element of S is ®xed at unity. Following Li (1998), we reparameterize S to

L� xx0 xx0 1

� �The set of unknown parameter Y then becomes �B; x;Lÿ1�.

We assume the following proper prior distribution:

p�B; x;Lÿ1� / p�B� � p�x� � p�Lÿ1� �6�

where p(B) is a multivariate normal density MVN�B0;Cÿ10 �, p(x) another multivariate normal

density MVN�x0;Aÿ10 �, p�Lÿ1� a Wishart density W�v0;Cÿ10 �.For the coe�cients of the expectations variable and the lagged exchange rate variable g1 , g2 in

equation (1), we consider the following three scenarios for their prior mean speci®cation:

(1) If we believe that the target zone regime induces non-linearity in the exchange rate behaviour,then we adopt E�g1� � 1;E�g2� � 0.

(2) If we believe that the exchange rate follows a random walk with a drift, then we adoptE�g1� � 0;E�g2� � 1.

(3) If we are ignorant of the values that g1 and g2 can possibly take, then we ( freely) adoptE�g1� � 0�5;E�g2� � 0�5, or E�g1� � 0;E�g2� � 0.8

As for the rest of the regression coe�cients in equations (1)±(5), theory does not tell us muchabout the plausible values they can take, and a null vector for the corresponding prior mean isadopted.

To specify the prior variance matrix Cÿ10 for the k-dimensional regression parameter B, wedecompose B as (B01;B

02�0, where B1 contains all the regression coe�cients �g1; g2; b0s; a0s;f0s; d0s�

in equations (1)±(4), k1 � dim(B1), B2 contains only the regression coe�cients c0s in equation(5), k2 � dim(B2). Then Cÿ10 can be accordingly rewritten as

Cÿ11

Cÿ12

� �where Cÿ11 � sc1 � Ik1 , C

ÿ12 � sc2 � Ik2 , I denotes the identity matrix. Both sc1 and sc2 are the

scaling factors to control the dispersion of the prior densities for B1 and B2 , respectively.

The reason that we give separate consideration to the two components of the prior variancematrix Cÿ10 is due to the identi®cation requirement for equation (5). That is, var(u5) � 1 inducesmuch greater variability for the coe�cients (B2) in the probit regression equation (5) than those in

8Note that in order to ensure the existence of a unique rational expectations solution to the system (1)±(5), Pesaran andRuge-Murcia (1999, Proposition 2) require that g1 be in the interval of (ÿ1 , 1]. As a consequence, the priors for g1 willbe truncated normals centred at the di�erent prior means as speci®ed in equations (1)±(3).



the rest of the system (1)±(4) (B1). For instance, in the FF/DM case, the variances of the OLSestimates in equations (1)±(4) are in the range of 4� 10ÿ6 to 4, while the variances of the probitestimates in equation (5) are in the range of 4 to 104; in the IL/DM case, the correspondingvariances are in the ranges of 10ÿ6 to 4, 4 to 2.5� 105, respectively. To ensure the prior densityfor B to be relatively noninformative but not completely di�use, we choose sc1 , sc2 to be largerthan the above variance upper bounds, i.e. sc1 > 4, sc2 > 104 in the FF/DM case; sc1 > 4,sc2 > 2�5 � 105 in the IL/DM case (see Section 4.2 for the actual prior speci®cations).

Choosing the priors for the components of the variance±covariance matrix (x, Lÿ1) can bemore challenging. In general, the larger the values of �A0; v0� the more informative the priors onx, Lÿ1 (S) become. In this paper, we take into account two implementing considerations whenselecting the values for �A0; v0;C0� (Li, 1998). First, given x0 � 0, the prior mean of L� xx0

equals C0=�v0 ÿ 5� � Aÿ10 . We would like the prior on the diagonal elements of L� xx0 to be nottoo far from the corresponding OLS estimates in equations (1)±(4), albeit with a large variance.Second, the key for the choice of prior parameters �A0; v0;C0� is to allocate correctly the variationin L� xx0 to the two components of the sum. That is, we need to have comparable variability forboth L and x. In the end, we come to some relatively ¯at priors on the variance±covariancematrix (to be given in Section 4.2) which enable us to examine the robustness of our posteriorestimates.

3.2. Estimation

Assuming independence among the observations, the likelihood function for the simultaneous-equation target zone model (1)±(5) is as follows:

l�Y� �Y

st�1;Ct�0

Z �10

fm�s�t ; et;wt� ds�tY

st�0;Ct�0

Z 0

ÿ1fm�s�t ; et;wt� ds�t

�Y

st�1;Ct�1

Z Z �10

fm�s�t ; e�t ;wt� ds�t de�t

Yst�0;Ct�1

Z Z 0

ÿ1fm�s�t ; e�t ;wt� ds�t de

�t

�7�

where fm is the m-dimensional multivariate normal p.d.f. �m � 5�;wt � �mt; yt; Zt�, st is theindicator variable de®ned previously, Ct is another indicator variable which equals 1 if theexchange rate is censored (at either the upper or lower target zone boundaries) and 0 otherwise.Since there does not exist a closed-form expression for the posterior density, numerical methodswill be required to conduct the Bayesian inference.9

In this paper, we shall use a Markov chain Monte Carlo method (MCMC), namely theMetropolis-within-Gibbs methodology with data augmentation to overcome the non-linearity(in parameters) and integration involved in evaluating the above likelihood function (Chib andGreenberg, 1995, 1996; Tanner and Wong, 1987).10 In particular, the data-augmentation step is

9 To put our Bayesian estimation of the rational expectations target zone models into perspective, we ®rst summarize theiterative approach used by Donald and Maddala (1992) to estimate the single-equation target zone model. (1) Start withsome initial values of the model parameter Y(0), solve for E�et j Itÿ1� via the Newton±Raphson method as elaborated inAppendix B, for every observation of the sample. (2) Given E�et j Itÿ1�, maximize the likelihood function of the targetzone model to obtain a new set of values for the model parameter Y(1). (3) Use Y(1) to solve iteratively for a new value ofE�et j Itÿ1�. Continue with steps (2) through (3) until at the (s � 1)th iteration Y�s�1� converges to Y�s�.10 I thank John Geweke (the co-editor) and Gary Koop for helpful discussions on the Metropolis-within-Gibbsmethodology adopted in the paper.

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introduced to obtain the continuous (albeit latent) dependent variable s�t in equation (5), and theMetropolis-within-Gibbs step is adopted to generate samples from the joint posteriordistribution by a Markov chain of draws from the proposal densities (see Appendix C fordetails).11

As Appendix B, equation (B4), makes explicit, the rational expectations variable E�et j Itÿ1� inequation (1) is highly non-linear in Y, the parameter vector of the model. Let E�et j Itÿ1� � ~et�Y�we rewrite equation (1) as

e�t � g1 ~et�Y� � g2etÿ1 � b1mt � b2yt � predm

0t � ~u1t �8�

Due to the dependence of ~et�Y� on Y, equation (9) is not an equation from a simultaneous-equation model with limited dependent variables. The fact that Y enters into one of the right-hand-side variables makes the likelihood function (8) quite di�erent from that of a simultaneous-equation model with limited dependent variables.12 The way to overcome this non-linear-in-parameters di�culty is to employ the Metropolis-within-Gibbs procedure. In brief, conditionalon everything else, the natural proposal density for the regression parameter B is a multivariatenormal, while the proposal densities for the components of the variance±covariance matrix Lÿ1,x are a Wishart and a multivariate normal, respectively (see Appendix C). This Bayesianapproach to estimating the simultaneous-equation target zone model (M0) can be applied to thealternative target zone models (M1ÿM4) with some modi®cations.

Since the simultaneous-equation target zone model is based upon Pesaran and Ruge-Murcia's(1999) work, it is desirable to compare these two formulations, as well as the three other modelsintroduced in Section 2.2.

The Bayesian approach to comparing alternative models is often based on the Bayes factor,which is computed as the ratio of marginal likelihoods under alternative model formulations(Kass and Raftery, 1995). In this paper, we adopt the method developed in Gelfand and Dey(1994) and Geweke (1997a, 1997b) to approximate the marginal likelihood of a model using theoutput of a posterior simulator (see Appendix C).

Apart from the within-sample posterior analysis, some predictive analysis of an out-of-samplesubperiod will usually be of interest. Denote y� as any out-of-sample variable we wish to predictand y as the in-sample data. The Bayesian predictive density for y� is formulated as

p�y� j y� �Z

p�y� j y;Y� � p�Y j y� dY �9�

where the parameter Y is treated as a nuisance parameter and has to be integrated out whenmaking predictions. In this paper, we follow Geweke (1994), by combining the MCMC methodwith the importance sampling, to make (point) predictions. More speci®cally, the importancesampling function for each one-step-ahead prediction at time t � 1 is the density function

11The reason that we do not consider the censoring problem of the exchange rate et in our empirical implementation is asfollows. Given the relatively small number of exchange rate observations outside of the band (3 out of 158 in the FF/DMcase; none in the IL/DM case), and their numerically small deviation from the band (the FF/DM exchange ratein October 1980, October 1988, December 1990 was ÿ3.25%, 2.53%, 2.35% away from the respective central parity), itseems unlikely that the results presented in this paper could be signi®cantly a�ected by the way we treat these threeobservations as regular (observable) e0ts. After all, in all these target zone models, it is the e�ect of the exchange rate bandon agents' expectations formation that matters (Donald and Maddala, 1992).12 I thank John Geweke (the co-editor) for pointing this out.



p�yt�1 j yt;Y�. In this way, we can avoid non-trivial reestimation of the model each time when wemake the one-step-ahead prediction.

4. EMPIRICAL RESULTS

4.1. The Data

For illustration, we con®ne the analysis to the three predominant charter ERM currencies: theFrench Franc, Italian Lira and German Deutsche Mark.13 The data on exchange rate and macrofundamentals are obtained from OECD Main Economic Indicators. The interest rate data arefrom IMF International Financial Statistics. The central parity data are extracted from Mizrach(1995) (Table I). 14

The FF/DM exchange rate data are nominal, monthly observations between March 1979and July 1993 (inclusive).15 During this period, the exchange rate was subject to a target-zoneregime, with six (stochastic) jumps taking place in the central parity (Figure 1).16 The IL/DMexchange rate data covers the period March 1979 to September 1992 (inclusive) when the ItalianLira was suspended from the ERM. There were ten jumps taking place in the central parity(Figure 2).17 Summary statistics of the exchange rate data are reported in Table I.

4.2. The Priors

Based on the model speci®cation (1)±(5), we mainly report our ®nal posterior estimates in theFF/DM case under the following four proper priors:

. Prior 1a: E(g1) � 0.5, E(g2) � 0.5, sc1 � 100, sc2 � 100,000, x0 � 0, A0 � 3,000 � Imÿ1,v0 � 8, C0 � 0�002 � Imÿ1.

. Prior 2a: E(g1) � 1, E(g2) � 0, sc1 � 100, sc2 � 100,000, x0 � 0, A0 � 5,000 � Imÿ1, v0 � 10,C0 � 0�004 � Imÿ1.

. Prior 3a: E(g1) � 0, E(g2) � 1, sc1 � 10, sc2 � 20,000, x0 � 0, A0 � 3,000 � Imÿ1, v0 � 8,C0 � 0�002 � Imÿ1.

Table I. Summary statistics of exchange rate data

Mean s.d. Pr(et� I(1) j data) Skewness Excess kurtosis

ln FF/DM 1.1057 0.1403 0.0394 ÿ0.9625 (0.1846) ÿ0.6074 (0.3673)ln IL/DM 6.4512 0.1738 0.0304 ÿ0.6767 (0.1901) ÿ0.8766 (0.3780)

Note: The corresponding standard error is in parentheses.The posterior probability of the exchange rate containing a unit root is computed under a ¯at prior.

13 In this paper, we have focused upon the two bilateral exchange rates FF/DM, IL/DM which were the most visible andmost prone to the central parity realignment. This caveat should be kept in mind throughout the section.14All data are available from the JAE data archive.15We terminate our FF/DM sample period prior to the near breakdown of the ERM in September 1993, because we feelthat our model is not able to incorporate this external shock to the EMS. In fact, after September 1993 the applicabletarget zone for the French Franc and other major currencies is widened to+15% around their respective central parities.16 The realignment on the FF/DM parity took place in September 1979, October 1981, June 1982, March 1983, April1986 and January 1987.17 The realignment on the IL/CM parity took place in September 1979, March 1981, October 1981, June 1982, March1983, July 1985, April 1986, January 1987, January 1990 and September 1992.

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Figure 1. FF/DM exchange rate and EMS target zone

Figure 2. IL/DM exchange rate and EMS target zone



. Prior 4a: E(g1) � 0, E(g2) � 0, sc1 � 10, sc2 � 20,000, x0 � 0, A0 � 5,000 � Imÿ1, v0 � 10,

C0 � 0�004 � Imÿ1.

m � 5 is the number of the equations in the simultaneous-equation target zone model (M0), Idenotes the identity matrix.

Prior 1a is the base prior, and we assess the sensitivity of our posterior results by making

changes to the base prior.18 For instance, in Prior 2a, the prior means for the coe�cients of the

expectations variable and the lagged exchange rate variable g1 , g2 are changed to 1 and 0 ( from

0.5 and 0.5), respectively; the prior on x ��MVN�x0;Aÿ10 �� becomes tighter by changing the A0

matrix to 5000 � Imÿ1 ( from 3000 � Imÿ1�; the prior on Lÿ1 ��W�v0;Cÿ10 �� also becomes tighter

by changing both the degrees of freedom parameter (v0) and the location parameter (C0) to 10

and 0�004 � Imÿ1 ( from 8 and 0�002 � Imÿ1�, respectively; the prior mean for L � xx0 stays as

0�001 � Imÿ1 throughout. In Prior 3a, the prior means for g1 , g2 are changed to 0 and 1 ( from 0.5

and 0.5), respectively; the prior on B ��MVN�B0;Cÿ10 �� becomes tighter by changing the scalars

sc1 , sc2 , which control the dispersion of the prior density for B, to 10 and 20,000 ( from 100 and

100,000), respectively. In Prior 4a, the prior means for g1 , g2 are changed to 0 and 0 ( from 0.5 and

0.5), respectively; the prior on B ��MVN�B0;Cÿ10 �� becomes tighter by changing the scalars sc1 ,

sc2 to 10 and 20,000 ( from 100 and 100,000), respectively; the prior on x ��MVN�x0;Aÿ10 ��becomes tighter by changing the A0 matrix to 5000 � Imÿ1 ( from 3000 � Imÿ1�; the prior on

Lÿ1 ��W�v0;Cÿ10 �� also becomes tighter by changing v0 and C0 to 10 and 0�004 � Imÿ1 ( from 8

and 0�002 � Imÿ1�, respectively. In brief, compared with the base prior (Prior 1a), Prior 2a is

di�erent in terms of the prior speci®cation on the variance±covariance matrix S�x;Lÿ1�; Prior 3ais di�erent in terms of the prior on the regression parameter B; Prior 4a is totally di�erent.

In the IL/DM case, the following four proper priors are considered:

. Prior 1b: E(g1) � 0.5, E(g2) � 0.5, sc1 � 1,000, sc2 � 1,000,000, x0 � 0, A0 � 3,000 � Imÿ1,v0 � 8, C0 � 0�002 � Imÿ1.

. Prior 2b: E(g1) � 1, E(g2) � 0, sc1 � 1,000, sc2 � 1,000,000, x0 � 0, A0 � 5,000 � Imÿ1,v0 � 10, C0 � 0�004 � Imÿ1.

. Prior 3b: E(g1) � 0, E(g2) � 1, sc1 � 100, sc2 � 250,000, x0 � 0, A0 � 3,000 � Imÿ1, v0 � 8,

C0 � 0�002 � Imÿ1.. Prior 4b: E(g1) � 0, E(g2) � 0, sc1 � 100, sc2 � 250,000, x0 � 0, A0 � 5,000 � Imÿ1, v0 � 10,

C0 � 0�004 � Imÿ1.

Prior 1b is the base prior in the IL/DM case, and we assess the sensitivity of our posterior

results by making similar changes as we did in the FF/DM case, to the base prior.

In summary, relatively non-informative priors have been chosen for the regression parameter B

and they are centred over values that are not implausible in light of theory. The prior variance

Aÿ10 for x and the prior hyperparameters (v0 , C0) for Lÿ1 are also selected in a way to re¯ect

the great uncertainty over location and dispersion of the error structure S (Li, 1998 and our

Section 3.1).

18 I thank John Geweke (the co-editor) and an anonymous referee for suggesting this approach to implementing the priorsensitivity analysis.

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4.3. Estimation

In the sampling process we ignore the ®rst 1,000 draws (the `burn-in'), and collect the next 20,000

draws. These are used to approximate the posterior distributions of �B; x;Lÿ1�. Using a 266MHz

PentiumII laptop (64 RAM), the most complicated model (M1) requires less than 2 hours'

execution time. All programming was done in Gauss-386i VM version 3.2.16.

Given that the estimation algorithm involves simulating the latent variable s� and the high-

dimensional parameter vector Y � �B; x;Lÿ1�, we evaluate the convergence of the Metropolis-

within-Gibbs procedure by carefully following Geweke (1994). In particular, we compute

functions of interests, such as the ®tted probabilities of realignment, by using di�erent seeds of

the random number generator, di�erent initial values for the parameter Y, and di�erent prior

speci®cations. In general, the results are quite robust.19

Estimation of the structural exchange rate equation (1) is reported in Table II for the FF/DM

exchange rate case (under Prior 1a) and in Table III for the IL/DM case (under Prior 1b). Under

M3 where the target zone band is assumed to be fully credible and exogenously determined, we

®nd strong non-linearity between the FF/DM exchange rate variable and the macro

fundamentals (i.e. g1 is signi®cantly di�erent from zero according to the 95% highest posterior

Table II. Estimates of the FF/DM exchange rate determination equation (1), Prior 1a

M0 M1 M3

E�et j Itÿ1� 0.1149 (0.0988) 0.1418 (0.1157) 0.4351 (0.1763)mt 0.0107 (0.0068) 0.0105 (0.0067) 0.0103 (0.0417)yt ÿ0.0060 (0.0086) ÿ0.0053 (0.0083) ÿ0.0130 (0.0532)etÿ1 0.8769 (0.0978) 0.8484 (0.1139) 0.5388 (0.1742)Dmtÿ1 0.0015 (0.0150) 0.0062 (0.0193) ÿ0.0204 (0.1274)Dmtÿ2 0.0181 (0.0148) 0.0145 (0.0192) ÿ0.0131 (0.1229)Dytÿ1 0.0022 (0.0064) 0.0025 (0.0076) 0.0061 (0.0488)Dytÿ2 0.0068 (0.0064) 0.0051 (0.0078) 0.0079 (0.0510)Constant ÿ0.0024 (0.0085) ÿ0.0006 (0.0086) 0.0173 (0.0534)

Note: The corresponding posterior standard deviation is in parentheses.M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.M3 is the target zone model with no realignment risk.

Table III. Estimates of the IL/DM exchange rate determination equation (1), Prior 1b

M0 M1 M3

E�et j Itÿ1� 0.0963 (0.0901) 0.1129 (0.0927) 0.2607 (0.1816)mt 0.0453 (0.0148) 0.0425 (0.0131) 0.0341 (0.0124)yt ÿ0.0045 (0.0032) ÿ0.0034 (0.0037) ÿ0.0023 (0.0032)etÿ1 0.8559 (0.0870) 0.8414 (0.0898) 0.7032 (0.1725)Dmtÿ1 ÿ0.0058 (0.0173) 0.0006 (0.0195) 0.0008 (0.0154)Dmtÿ2 0.0354 (0.0179) 0.0244 (0.0208) 0.0207 (0.0180)Dytÿ1 0.0003 (0.0025) 0.0007 (0.0027) 0.0010 (0.0024)Dytÿ2 0.0008 (0.0027) 0.0010 (0.0029) 0.0008 (0.0027)Constant 0.0072 (0.0342) 0.0127 (0.0315) 0.0063 (0.0329)

19All convergence results are available from the author upon request.



density interval criterion) and relatively weak non-linearity between the IL/DM rate and the

macro fundamentals. In summary, under M3 agents take into account the band on the exchange

rate when forming expectations E�et j Itÿ1�, which subsequently induces non-linearity in the

exchange rate (while in the IL/DM case, the relatively frequent realignments reduce the e�ect of

the band on expectations). On the other hand, under M0 and M1 where stochastic realignment

risk in central parity is introduced, it is hard to detect any non-linear relationship between the

exchange rate and the macro fundamentals in either the FF/DM or the IL/DM case. This result

in fact is quite intuitive because once allowing for stochastic adjustment in central parity, the

pressure of the exchange rate band is not strong enough to generate signi®cant non-linearity in

the exchange rate any more. This paper provides empirical evidence against non-linearity in the

EMS exchange rates, particularly, the FF/DM rate and the IL/DM rate. Our conclusion on non-

linearity is later con®rmed in our model comparison results (Section 4.4). The data does not

support the non-linear model M3 against models M0 ÿM2 and the linear model M4 .

Two important equations of the system are for the size and the propensity of a realignment,

and our estimation results under M0 and M1 are given respectively in Tables IV and VI in the

FF/DM case (under Prior 1a), and in Tables V and VII in the IL/DM case (under Prior 1b). In

the realignment size equation (Tables IV and V), the signing of the 1987 Basle±Nyborg

Agreement signi®cantly reduces the size of a realignment, while the lagged exchange rate

deviation from its central parity increases it. Furthermore, the lagged interest rate di�erential

between France and Germany (Table IV) and the lagged Italian yield curve spread (Table V)

increase their respective size of a central parity realignment.

Table IV. Estimates of the size of realignment (FF/DM) equation (4), Prior 1a

M0 M1

Yr87t ÿ0.0052 (0.0026) ÿ0.0031 (0.0033)itÿ1 0.9431 (0.3574) 1.9215 (0.4607)etÿ1 ÿ ectÿ1 0.3758 (0.0802) 0.3517 (0.1053)mtÿ1 0.0041 (0.0123) ÿ0.0045 (0.0148)mtÿ12 ÿ0.0053 (0.0112) ÿ0.0019 (0.0142)ytÿ1 0.0021 (0.0101) ÿ0.0009 (0.0110)ytÿ12 ÿ0.0006 (0.0100) ÿ0.0029 (0.0106)Constant 0.0019 (0.0108) 0.0042 (0.0111)

Note: M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.

Table V. Estimates of the size of realignment (IL/DM) equation (4), Prior 1b

M0 M1

Yr87t ÿ0.0118 (0.0043) ÿ0.0125 (0.0052)ptÿ1 ÿ0.0002 (0.2378) 0.1849 (0.2807)itptÿ1 0.0002 (0.0002) 0.0002 (0.0002)itspreadtÿ1 3.7127 (1.6687) 6.4009 (1.9662)itÿ1 0.5508 (0.8864) 0.0614 (1.0324)etÿ1 ÿ ectÿ1 0.3122 (0.0577) 0.3533 (0.0743)mtÿ1 ÿ0.0032 (0.0217) 0.0109 (0.0256)Constant 0.0108 (0.1310) ÿ0.0815 (0.1540)

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In the realignment propensity equation (Tables VI and VII), the Basle±Nyborg Agreementdummy reduces the likelihood of a realignment after 1987, while the lagged exchange ratedeviation from its central parity increases it. The ®rst result is consistent with the general beliefthat the EMS target zones became more credible after 1987 by exhibiting few realignments in thecentral parity. The second result is consistent with the observation made by Bertola andCaballero (1992) and Bekaert and Gray (1998) that the likelihood of a realignment increases asthe exchange rate approaches the upper (not the lower) limit of its ¯uctuation band. Theasymmetric relationship between the deviation variable �etÿ1 ÿ ectÿ1� and the realignment risk isalso in agreement with the fact that both the French and Italian monetary authorities have beenless tolerant of positive deviations of the FF/DM and IL/DM rates from their respective centralparities in the earlier period of the EMS. In the IL/DM case (Table VII), the expected futurein¯ation in Italy (itspreadtÿ1) also increases the propensity of an IL/DM realignment. Aseemingly puzzling result in Table VII is that the interest rate di�erential between Italy andGermany has a negative e�ect on the propensity of an IL/DM realignment, which is incontradiction to the prediction of the uncovered interest rate parity hypothesis. This result is,however, consistent with the traditional `competitiveness' argument. That is, a large value of itÿ1is associated with large returns on an investment in Italian Lira, which consequently inducesgreat demand for the Italian Lira in the foreign exchange market and reduces its depreciationprobability with respect to the Deutsche Mark.

Figures 3 and 4 plot the computed probability of realignment using both the simultaneous-equation target zone model (M0 , under Priors 1a±4a, Priors 1b-4b) and the Pesaran and Ruge-Murcia variation (M1 , under Priors 1a, 1b). Tables VIII and IX give the ®tted probability in the

Table VI. Estimates of the probability of realignment (FF/DM) equation (5),Prior 1a

M0 M1

Yr87t ÿ8.3311 (2.4741) ÿ23.0890 (12.7856)itÿ1 106.5013 (117.2890) 276.5908 (132.4135)etÿ1 ÿ ectÿ1 124.0662 (40.4062) 165.6086 (50.5074)mtÿ1 ÿ17.6778 (9.1305) ÿ17.5261 (10.0406)mtÿ12 14.0686 (7.7158) 16.9777 (8.5556)ytÿ1 ÿ2.4019 (3.0174) ÿ2.1314 (3.1146)ytÿ12 4.8849 (6.0368) 1.3208 (6.0415)Constant 1.7436 (4.0913) ÿ3.2328 (4.7494)

Table VII. Estimates of the probability of realignment (IL/DM) equation (5),Prior 1b

M0 M1

Yr87t ÿ4.0854 (3.2061) ÿ2.9103 (1.4203)ptÿ1 43.9509 (79.9360) 83.9957 (63.1353)itptÿ1 ÿ0.1433 (0.0840) ÿ0.0069 (0.0579)itspreadtÿ1 2216.7924 (570.3637) 1385.0851 (423.1104)itÿ1 ÿ1469.4127 (342.8871) ÿ562.8197 (244.8476)etÿ1 ÿ ectÿ1 84.7097 (45.3473) 69.4859 (17.6300)mtÿ1 37.3331 (10.3352) 11.7447 (8.5552)Constant ÿ228.3116 (61.3670) ÿ75.5652 (51.5357)



Figure 3. Fitted time-varying probability of FF/DM realignment

Table VIII. Fitted probability of FF/DM realignment (1979 :3±1992 :4)

M0 M1

Prior 1a Prior 2a Prior 3a Prior 4a Prior 1a

September 1979 0.4500 0.3004 0.3916 0.2649 0.2646(0.2540) (0.2368) (0.2429) (0.2145) (0.2291)

October 1981 0.0832 0.1551 0.1145 0.0980 0.2545(0.1097) (0.1744) (0.1520) (0.1131) (0.1981)

June 1982 0.3469 0.5087 0.4275 0.3932 0.7157(0.2422) (0.2612) (0.2713) (0.2433) (0.1840)

March 1983 0.1038 0.0727 0.0693 0.0849 0.2533(0.1537) (0.1324) (0.1035) (0.1206) (0.1925)

April 1986 0.0649 0.0861 0.0640 0.0840 0.1565(0.0845) (0.0932) (0.0772) (0.0928) (0.1308)

January 1987 0.7145 0.7484 0.6219 0.7897 0.8163(0.2691) (0.2293) (0.3243) (0.2311) (0.2172)

Note: The corresponding posterior standard deviation is in parentheses.M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.

476 K. LI


month of the actual realignment. The probability of realignment varies considerably during the

sample periods (1979 :3±1992 :4 for FF/DM, 1979 :3±1991 :6 for IL/DM), especially prior to the

Basle-Nyborg Agreement (1987 :9), demonstrating the importance of allowing for time-varying

realignment propensity in the target zone models. Furthermore, Figure 4 also depicts the

enduring problems of Italian monetary and exchange rate policies by displaying large realign-

ment probabilities across the sample period.20

Finally, the computed 5� 5 (non-diagonal) variance±covariance structure in Table X reveals

that there exist strong correlations among the exchange rate determination equation (1) and the

size and the propensity of realignment equations (4) and 5. Ignoring these important correlations

could induce biased estimation results, such as the case in Table IV under M1 , where we do not

detect a signi®cant relationship between the year dummy variable (Yr87t) and the magnitude of a

realignment variable (Zt), while we ®nd the opposite result under M0 .

Figure 4. Fitted time-varying probability of IL/DM realignment

20 In comparison to the realignment probabilities computed by other researchers (Bekaert and Gray, 1998; Koedijk et al.,1998), we make the following observations. Both Bekaert and Gray (1998) and Koedijk et al. (1998) used the weekly dataat time t to compute the next period's realignment probability; while we used the monthly data (as the macrofundamentals are only available at the monthly frequency). In Bekaert and Gray (1998), their ®tted FF/DM realignmentprobabilities are mostly in the range of 0.10, and their highest realignment probability of 0.34 occurred some timein March 1983. All the rest realignment probabilities on the actual event dates are in the range of 0.14. In this paper, wehave the ®tted probabilities to be as high as 0.79 in the case of the January 1987 realignment, and the rest of realignmentprobabilities in the actual event months are between 0.002 ± 0.51. In Koedijk et al. (1998), they computed their IL/DMrevaluation/devaluation probabilities separately, and they used the +2.25% band (rather than the +6% band) for Italyin their computation.



Table X. Estimates of the variance±covariance matrix S

FF/DM (Prior 1a) IL/DM (Prior 1b)

s11 1.2092E-4 (1.3842E-5) 1.0795E-4 (1.3010E-5)s22 1.1321E-3 (1.2866E-4) 1.0267E-3 (1.2030E-4)s33 1.0593E-3 (1.2021E-4) 3.8924E-3 (4.5043E-4)s44 1.6117E-4 (1.8930E-5) 1.8684E-4 (2.2603E±5)r12 ÿ0.0014 (0.0849) ÿ0.1198 (0.0987)r13 0.0640 (0.0833) ÿ0.0471 (0.0853)r14 0.6940 (0.0426) 0.5483 (0.0579)r15 0.4983 (0.1249) 0.4821 (0.1094)r23 0.0537 (0.0806) 0.1149 (0.0799)r24 0.0005 (0.0845) 0.0067 (0.0852)r25 ÿ0.1378 (0.3461) ÿ0.0155 (0.1888)r34 0.0144 (0.0828) ÿ0.1015 (0.0845)r35 0.0703 (0.3249) ÿ0.0228 (0.2133)r45 0.5600 (0.1190) 0.6790 (0.0802)

Note: The corresponding posterior standard deviation is in parentheses.

Table IX. Fitted probability of IL/DM realignment (1979 :3±1991 :6)

M0 M1

Prior 1b Prior 2b Prior 3b Prior 4b Prior 1b

September 1979 0.0687 0.0645 0.0602 0.0893 0.0570(0.0979) (0.0887) (0.0896) (0.1008) (0.0695)

March 1981 0.4449 0.4493 0.4252 0.2250 0.4762(0.3142) (0.2810) (0.2695) (0.2222) (0.1896)

October 1981 0.3662 0.4893 0.3688 0.3630 0.5812(0.2206) (0.2430) (0.2442) (0.1980) (0.1775)

June 1982 0.3557 0.5001 0.4463 0.4727 0.6063(0.2753) (0.2715) (0.3113) (0.2689) (0.2109)

March 1983 0.0021 0.0340 0.0159 0.0392 0.0915(0.0048) (0.0624) (0.0296) (0.0532) (0.0798)

July 1985 0.0493 0.0676 0.1015 0.0766 0.1827(0.0517) (0.0670) (0.0784) (0.0723) (0.1006)

April 1986 0.2325 0.2373 0.2035 0.2382 0.3509(0.1245) (0.1308) (0.1295) (0.1357) (0.1497)

January 1987 0.1861 0.0747 0.1147 0.0939 0.1021(0.1383) (0.0708) (0.1037) (0.0857) (0.0810)

January 1990 0.1642 0.2000 0.2868 0.1600 0.3604(0.1892) (0.1859) (0.1936) (0.1446) (0.2181)

Note: The corresponding posterior standard deviation is in parentheses.M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.

478 K. LI


4.4. Model Comparison21

Other than estimating the (encompassing) simultaneous-equation target zone model (M0), we

also compare this extended target zone model to some existing alternatives using the logarithmic

Bayes factor �BFij�.22 Tables XI and XII show that the target zone model with constant

realignment risk (M2) is overwhelmingly supported by both data sets under our di�erent prior

speci®cations. Intuitively, in our FF/DM (IL/DM) sample of 158 (148) observations, there were

Table XI. Logarithm of the Bayes factor (FF/DM)

Prior 1aHypothesis BF Hypothesis BF Hypothesis BF

M0 vs M1 2.88 M1 vs M2 ÿ51.61 M2 vs M3 397.43M0 vs M2 ÿ48.73 M1 vs M3 345.82 M2 vs M4 179.00M0 vs M3 348.70 M1 vs M4 127.39 M3 vs M4 ÿ218.43M0 vs M4 130.27


M0 vs M1 ÿ13.85 M1 vs M2 ÿ53.86 M2 vs M3 450.59M0 vs M2 ÿ67.71 M1 vs M3 396.73 M2 vs M4 178.56M0 vs M3 382.88 M1 vs M4 124.70 M3 vs M4 ÿ272.03M0 vs M4 110.85





Note: BF denotes the logarithmic Bayes factor.M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.M2 is the target zone model with constant realignment risk.M3 is the target zone model with no realignment risk.M4 is the linear rational expectations target zone model.

21We know, from Lindley's paradox, that if Model B (such as M1) restricts certain parameters in Model A (such as M0)to be zero (in our case, the variance±covariance matrix S in M1 becomes diagonal), and if the prior for these parametersin Model A is su�ciently di�use (as our priors 1a±4a, 1b±4b), then the Bayes factor must favour the restricted Model B.This caveat should be kept in mind when interpreting our model comparison results reported in this section. I thank JohnGeweke (the co-editor) and an anonymous referee for pointing this out.22According to Kass and Raftery (1995), the logarithmic Bayes factor of Mi versus Mj (BFij) with a value greater than 5provides decisive evidence against Mj .



only six (nine) jumps. Therefore, it is di�cult to draw strong conclusions on the unknownparameters in (4) and (5) by utilizing data with only six (nine) realignments. As a result, the targetzone model with constant realignment risk (M2) ®ts the data best. On the other hand, with itsunrealistic assumption of known varying bands, the Pesaran and Samiei model (M3) ®ts the datapoorest. Given our ®nding in Section 4.3 (Tables II and III) thatM3 is the only target zone modelexhibiting non-linearity between the exchange rate and the macro fundamentals, this paperprovides empirical evidence against non-linearity in the EMS exchange rates.

4.5. Prediction

In this paper, we have used 158 monthly observations on the FF/DM exchange rate forestimation and leave 15 observations for one-step-ahead exchange rate prediction (1992 :5±1993 :7). In the IL/DM exchange rate case, we have used 148 monthly observations for estimationand 15 observations (1991 :7±1992 :9) for prediction. Tables XIII and XIV report the meansquared errors of the out-of-sample one-step-ahead prediction of the exchange rate for the ®vetarget zone models and the random walk model. The predictive performance of various target

Table XII. Logarithm of the Bayes factor (IL/DM)

Prior 1bHypothesis BF Hypothesis BF Hypothesis BF








480 K. LI


zone models is mixed. Overall, the random walk model produces the smallest mean squared

prediction error.

In summary, we are generally unable to improve upon a simple random walk model in out-of-

sample ®rst moment prediction of the exchange rate by introducing target zone models, this

result comes as no surprise given similar ®ndings by Diebold and Nason (1990) and Meese and

Rose (1991).23

5. CONCLUSION

By employing the Metropolis-within-Gibbs methodology with data augmentation, this paper

develops a Bayesian approach to estimating exchange rate target zone models and rational

expectations models in general. It also introduces a simultaneous-equation target zone model

that incorporates stochastic realignment risk. Under the system of equations framework, we are

able to examine both the magnitude and the likelihood of an adjustment in the central parity as

well as the correlated error structure of the system. The probabilistic structure we introduce is

¯exible enough to generate realistic relationships among observables, i.e. exchange rate, macro

fundamentals and central parity realignment.

Table XIII. Mean squared prediction error under alternative models FF/DM exchange rate, 1992 :5±1993 :7

M0 M1 M2 M3 M4 RW

Prior 1a 5.712E-5 5.584E-5 5.692E-5 8.869E-5 5.895E-5 4.857E-5Prior 2a 6.106E-5 5.623E-5 5.682E-5 1.144E-4 5.933E-5 4.854E-5Prior 3a 5.644E-5 5.685E-5 5.721E-5 9.037E-5 5.919E-5 4.852E-5Prior 4a 5.417E-5 5.600E-5 5.727E-5 9.168E-5 5.874E-5 4.848E-5

Note: M0 is the simultaneous-equation target zone model.M1 is the target zone model with no simultaneity.M2 is the target zone model with constant realignment risk.M3 is the target zone model with no realignment risk.M4 is the linear rational expectations target zone model.RW denotes the random walk model of exchange rate.

Table XIV. Mean squared prediction error under alternative models IL/DM exchange rate, 1991 :7±1992 :9

M0 M1 M2 M3 M4 RW

Prior 1b 1.330E-3 1.326E-3 1.323E-3 1.339E-3 1.327E-3 1.231E-3Prior 2b 1.314E-3 1.330E-3 1.325E-3 1.312E-3 1.326E-3 1.231E-3Prior 3b 1.306E-3 1.328E-3 1.323E-3 1.289E-3 1.328E-3 1.231E-3Prior 4b 1.306E-3 1.334E-3 1.336E-3 1.297E-3 1.326E-3 1.230E-3

23Diebold and Nason (1990) employ univariate non-parametric time series methods to forecast the conditional mean ofspot exchange rate, but ®nd little improvement in predictive accuracy over a simple random walk. Meese and Rose (1991)consider ®ve structural exchange rate models and ®nd all of them display a uniform lack of ability to out-predict arandom walk alternative signi®cantly. Although the target zone model is useful in many ways, such as making inferencesabout the size and the likelihood of a forthcoming realignment, examining the validity of the monetary model ofexchange rate determination, etc. it is simply not the right model for prediction. Taken together, this paper provides fairlystrong evidence against exploiting asset price non-linearities for enhanced point prediction.



The simultaneous-equation target zone model is compared with alternative target zonemodels: (1) with no simultaneity; (2) with constant realignment risk; (3) with no realignment risk(known varying bands); (4) with no e�ect of target zone (linear rational expectations). Using FF/DM and IL/DM exchange rate data, the model comparison results are overwhelmingly in favourof the target zone model with constant realignment risk. The predictive performance of varioustarget zone models is mixed. In general, the target zone models are not able to outperform therandom walk model of exchange rate in terms of ®rst moment prediction, which is in agreementwith the general ®ndings in the literature.

On examining the exchange rate determination in the target zone, we conclude that onceallowing for stochastic realignment risk in the target zone models, neither the FF/DM rate northe IL/DM rate exhibits non-linearity any longer.

On examining the stochastic realignment risk of the target zone, we ®nd that the signing of the1987 Basle±Nyborg Agreement signi®cantly reduces both the magnitude and the likelihood of acentral parity realignment, while the lagged exchange rate deviation from its central parityincreases them. Furthermore, the interest rate policies and the monetary conditions of theparticipating countries signal a forthcoming realignment.

APPENDIX A

This appendix derives the structural exchange rate equation (1) adopted in the paper. FollowingPesaran and Ruge-Murcia (1999) and many others (see Svensson, 1992 for a review), themonetary model of exchange rate determination consists of the following equations:

E�et�1 ÿ et j It� � it �A1�et ÿ etÿ1 � f� �et ÿ etÿ1� �A2�

�et � p0 � p1it ÿ p2yt � p3mt � ut p1; p2; p3 > 0 �A3�

where equation (A1) is (approximately) a risk-neutral arbitrage condition (the uncovered interestrate parity condition), et is the logarithm of the nominal exchange rate, E�� j It� is the agents' one-step-ahead forecast of the endogenous variable, conditional on their information set at time t,and it denotes the interest rate di�erential between the home and foreign countries. Equation (A2)describes the adjustment process of the exchange rate towards its equilibrium level �et, which inturn is given in equation (A3). yt and mt are as de®ned in the main text.

Solving for et in terms of the macro fundamentals we obtain

e�t � g1E�et�1 j It� � l1etÿ1 � wt �A4�

where

g1 �fp1

1� fp105 g1 5 1

l1 �1ÿ f1� fp1

05 l1 5 1

482 K. LI


and

wt �1

1� fp1

� ��p0fÿ p2fy2 � p3fmt � fut�

As proved by Pesaran and Ruge-Murcia (1999, Appendix C), the saddle path solution of a linearrational expectations model with future expectations (A4) is mathematically equivalent to arational expectations model with current expectations, given the model's forcing variables areaugmented with their lagged di�erences. Accordingly, (A4) can be approximated by the follow-ing non-linear current expectations model:

e�t � g1E�et j Itÿ1� � g2etÿ1 � b1mt � b2yt � predm

0t � u1t �A5�

where g2=�1ÿ g1� ��d� is the root of the quadratic equation l1dÿ1 � g1d � 1 that lies inside the

unit circle.24 The augmented set of forcing variables fDmtÿ1;Dmtÿ2;Dytÿ1;Dytÿ2g is contained inpredm0

t , u1t is the error term.

APPENDIX B

Following Pesaran and Ruge-Murcia (1999), as well as work by Donald andMaddala (1992) andLee (1994), this appendix sketches the rational expectations solution for E�et j Itÿ1� in thesimultaneous-equation target zone model (1)±(5). First we decompose this conditionalexpectation of et into two parts:

E�et j Itÿ1� � E�et j Itÿ1; st � 0� � Pr�st � 0� � E�et j Itÿ1; st � 1� � Pr�st � 1� �B1�

where the time t band ect experiences a discrete jump Zt with probability Pr(st � 1) as given inequation (5).

Let

"t � u1t � �b1 b2� �u2tu3t

� �s2" � s11 � b21s22 � b22s33 � 2b1s12 � 2b2s13 � 2b1b2s23xt � "t ÿ u4t

s2x � s11 � b21s22 � b22s33 � s44 � 2b1s12 � 2b2s13 ÿ 2s14 � 2b1b2s23 ÿ 2b1s24 ÿ 2b2s34

We standardize these two random variables as

tt � "t=s"vt � xt=sx

24Here it is the lagged observed exchange rate variable etÿ1 that appears as a r.h.s. explanatory variable. This formulationcaptures the dynamics of the exchange rate variable, in the meantime, it is analytically simpler. Refer to Maddala (1983,pp. 186±187) for the alternative dynamic Tobit model.



De®ne

c0it � �eitÿ1 ÿ g1E�et j Itÿ1� ÿ b1m

et ÿ b2y

et ÿ predmt�=s" i � L;U

c1it � �eitÿ1 � Jtÿ1 ÿ g1E�et j Itÿ1� ÿ b1m

et ÿ b2y

et ÿ predmt�=sx i � L;U

where predmt � fg2etÿ1; predm0t g, and the superscript e represents E�� j Itÿ1�.

Under the no realignment case (st � 0), the observation rule applicable to et is

et �eUtÿ1 if tt 5 c0Ut

g1E�et j Itÿ1� � b1met � b2y

et � predmt � "t if c0Ut > tt > c0Lt

eLtÿ1 if tt 4 c0Lt

8><>:we come to

E�et j Itÿ1; st � 0� � E�et j Itÿ1; tt 5 c0Ut� � Pr�tt 5 c

0Ut�

� E�et j Itÿ1; c0Ut > tt > c0Lt� � Pr�c0Ut > tt > c

0Lt�

� E�et j Itÿ1; tt 4 c0Lt� � Pr�tt 4 c

0Lt�

� eUtÿ1 � �1ÿ F�c0Ut��

� �g1E�et j Itÿ1� � b1met � b2y

et � predmt� � s"

f�c0Lt� ÿ f�c0Ut�F�c0Ut� ÿ F�c0Lt�

� �� F�c0Ut� ÿ F�c0Lt�� eLtÿ1 � F�c0Lt� �B2�

Under the realignment case (st � 1), the observation rule applicable to et is

et �eUt if vt 5 c1Ut

g1E�et j Itÿ1� � b1met � b2y

et � predmt � "t if c1Ut > vt > c1Lt

eLt if vt 4 c1Lt

8><>:we come to

E�et j Itÿ1; st � 1� � E�et j Itÿ1; vt 5 c1Ut� � Pr�vt 5 c

1Ut�

� E�et j Itÿ1; c1Ut > vt > c1Lt� � Pr�c1Ut > vt > c

1Lt�

� E�et j Itÿ1; vt 4 c1Lt� � Pr�vt 4 c

1Lt�

� �eUtÿ1 � Jtÿ1� � �1ÿ F�c1Ut��


et � predmt� � sx

f�c1Lt� ÿ f�c1Ut�F�c1Ut� ÿ F�c1Lt�

� �� F�c1Ut� ÿ F�c1Lt�� eLtÿ1 � Jtÿ1� � F�c1Lt� �B3�

484 K. LI


Substituting equations (B2) and (B3) into (B1), we arrive at the following implicit equation forE�et j Itÿ1�:

E�et j Itÿ1� �neUtÿ1 � �1ÿ F�c0Ut�� eLtÿ1 � F�c0Lt�� g1E�et j Itÿ1� � b1m

et � b2y

et � predmt� � �F�c0Ut� ÿ F�c0Lt��

� s" � �f�c0Lt� ÿ f�c0Ut��o� Pr�st � 0�

�n�eUtÿ1 � Jtÿ1� � �1ÿ F�c1Ut�� eLtÿ1 � Jtÿ1� � F�c1Lt�


et � predmt� � �F�c1Ut� ÿ F�c1Lt��

� sx � �f�c1Lt� ÿ f�c1Ut��o� Pr�st � 1�

� F �E�et j Itÿ1�� B4�

where F and f are the standard normal c.d.f. and p.d.f., respectively. Since both c0it and c1it(i � L, U) are functions of E�et j Itÿ1�, the dependence of rational expectations on the parametersand variables of the model is implicit and highly non-linear, evidently there does not exist aclosed-form solution for E�et j Itÿ1�. Let q � E�et j Itÿ1�, we rewrite equation (B4) asG�q� � qÿ F�q�, then

@G�q�@q� 1ÿ g1�F�c0Ut� ÿ F�c0Lt�� Pr�st � 0� ÿ g1�F�c1Ut� ÿ F�c1Lt�� Pr�st � 1�

Following Pesaran and Ruge-Murcia (1999), we can prove that as long as g14 1, there exists aunique solution for E�et j Itÿ1�. The Newton±Raphson iterative method is employed in the paperto solve numerically for the rational expectations variable E�et j Itÿ1� in equation (1).

APPENDIX C

This appendix ®rst summarizes the Metropolis-within-Gibbs methodology (Chib and Greenberg,1995, 1996) with data augmentation (Tanner and Wong, 1987) used in estimating thesimultaneous-equation target zone model (M0), then brie¯y explains the method developed inGelfand and Dey (1994) and Geweke (1997a,b) for approximating the marginal likelihood.

Given the rational expectations variable E�et j Itÿ1� in equation (1) is highly non-linear in themodel's parameter vector Y � �B; x;Lÿ1�, we denote E�et j Itÿ1� � ~et�Y� � ~et�B; x;Lÿ1�. Theaugmented likelihood function for model M0 is given as

l�B; x;Lÿ1; ~et�B; x;Lÿ1�; s�� / jSÿ1jn=2 exp�ÿ12�Y� ÿ XB�0Sÿ1 In�Y� ÿ XB�� C1�

where Y� is the dependent variable (e;m; y; Z; s�) in equations (1)±(5), X, which includes thehighly non-linear-in-parameters component ~et�B; x;Lÿ1�, is the regressor in equations (1)±(5), Bis the regression parameter (g1; g2; b

0s; a0s;f0s; d0s;c0s) in equations (1)±(5), s� in equation (5) is



the latent data to be generated in the estimation process:

S � L� xx0 x

x0 1

!

n is the sample size.25

To implement the Metropolis-within-Gibbs procedure, the target density is obtained as theproduct of the prior (6) and the augmented likelihood function (C1), and it is intractable. In thispaper, to simplify the search for suitable proposal densities we apply the Metropolis-within-Gibbs method in turns to subblocks �B; x;Lÿ1) of the parameter vector Y, rather thansimultaneously to all elements of the vector.

What are the (e�cient) proposal densities for parameters B; x;Lÿ1, respectively? The answerlies in recognizing the following fact. Were there not a rational expectations variable ~et�B; x;Lÿ1�in the model, the product of the prior (6) and the augmented likelihood function (C1) would haveled to the following full conditional densities for B; x;Lÿ1:

(1) Conditional on Y�; x;Lÿ1,

S � L� xx0 xx0 1

� �is given, and with a prior on B, p�B� �MVN�B0;C

ÿ10 �,

B �MVN� ~B; ~Cÿ1� �C2�

where ~B � ~Cÿ1�X0�Sÿ1 In�Y� �C0B0�, ~C � X0�Sÿ1 In�X�C0. Y� is the augmenteddependent variable (e;m; y; Z; s�) in equations (1)±(5), X is the regressor in equations (1)±(5),B is the k-dimensional regression parameter (g1; g2; b

0s; a0s;f0s; d0s;c0s) in equations (1)±(5),n is the sample size.

(2) Conditional on Y�;Lÿ1;B, then u5 and u: � �u1; u2; u3; u4� are given, and with a prior on x,p�x� �MVN�x0;Aÿ10 �,

x �MVN�~x; ~Vÿ1� �C3�

where ~x � ~Vÿ1�Lÿ1u0:u5 � A0x0�; ~V � u05u5Lÿ1 � A0. u5 is the error term in equation (5), and

u: � �u1; u2; u3; u4� is the error term in equations (1)±(4).(3) Conditional on Y�; x;B, then u5 and u: � �u1; u2; u3; u4� are given, and with a prior on

Lÿ1; p�Lÿ1� �W�v0;Cÿ10 �,

Lÿ1 �W� ~v; ~Cÿ1� �C4�

where ~v � n� v0;~C � �u:ÿ u5x

0�0�u:ÿ u5x0� � C0, n is the sample size.

25During the sample estimation period the FF/DM exchange rate falls out of the band only three times, and the IL/DMrate never falls out of the band. Hence, we do not consider the censoring problem of the exchange rate et in our empiricalimplementation (see footnote 11).

486 K. LI


Given the highly non-linear-in-parameters variable ~et�B; x;Lÿ1� is one of the regressors in thesystem (1)±(5), the above full conditionals (C2)±(C4) become the natural choices for the proposaldensities of B; x;Lÿ1, respectively. Our Bayesian estimation of the simultaneous-equation targetzone model (M0) proceeds as follows.

At the (s � 1)th iteration (given the values B�s�; x�s�;Lÿ1�s��, we ®rst simulate the latent datas��s�1� from a censored normal density

s� � CN�m5j:; s5j:� �C5�

where m5j: � X5c�s� � x�s�0�L�s� � x�s�x�s�0�ÿ1�Y:ÿ X:�s�B:�s��; s5j: � 1ÿ x�s�0�L�s� � x�s�x�s�0�ÿ1x�s�.

CN denotes the censored univariate normal distribution, and s� is censored to be eitherabove or below zero, depending on the value of the binary variable s.26 X5 is the regressor inequation (5), c is the regression parameter in (5). . denotes the corresponding component inequations (1)±(4). In particular, X. is the regressor in (1)±(4), B. is the regression parameter(g1; g2; b

0s; a0s;f0s; d0s) in (1)±(4), Y. is the dependent variable (e;m; y; Z) in (1)±(4).Second, conditional on s��s�1�; x�s�;Lÿ1�s�, we generate a candidate B� from the multivariate

normal proposal density as given in equation (C2) with mean ~B�s��s�1�; x�s�;Lÿ1�s�� and variance±covariance matrix ~Cÿ1�s��s�1�; x�s�;Lÿ1�s��. We ensure that the constraint imposed on g1 ofequation (1) (see Appendix B) is satis®ed by only retaining those draws that meet the condition.We then accept B�s�1� � B� with probability

minp�B��l�B�; x�s�;Lÿ1�s�; ~et�B�; x�s�;Lÿ1�s��; s��s�1��=qB�B�s�;B� j s��s�1�; x�s�;Lÿ1�s��

p�B�s��l�B�s�; x�s�;Lÿ1�s�; ~et�B�s�; x�s�;Lÿ1�s��; s��s�1��=qB�B�;B�s� j s��s�1�; x�s�;Lÿ1�s��

; 1

( )

and otherwise set B�s�1� � B�s�, where p(.) is the prior, l(.) the likelihood, qB(.) the proposal

density as given in equation (C2). More speci®cally, the last expression in the numerator of theabove equation gives the proposal density of (C2) evaluated at B�, where B(s) is used to calculate eÄ;similarly, the last expression in the denominator gives the proposal density of (C2) evaluated atB�s�, where B� is used to calculate eÄ.

Third, conditional on s��s�1�;B�s�1�;Lÿ1�s�, we generate a candidate x� from the multivariatenormal proposal density as given in equation (C3) with mean x�s��s�1�;B�s�1�;Lÿ1�s�� andvariance±covariance matrix ~Vÿ1�s��s�1�;B�s�1�;Lÿ1�s��. We accept x�s�1� � x� with probability

minp�x��l�B�s�1�; x�;Lÿ1�s�; ~et�B�s�1�; x�;Lÿ1�s��; s��s�1��=qx�x�s�; x� j ��

p�x�s��l�B�s�1�; x�s�;Lÿ1�s�; ~et�B�s�1�; x�s�;Lÿ1�s��; s��s�1��=qx�x�; x�s� j ��; 1

( )

and otherwise set x�s�1� � x�s�, where p(.) is the prior, l(.) the likelihood, qx(.) the proposal densityas given in (C3). . in the proposal density denotes �s��s�1�;B�s�1�;Lÿ1�s��. The last expression inthe numerator of the above equation gives the proposal density of (C3) evaluated at x�, where x(s)

is used to calculate eÄ; similarly, the last expression in the denominator gives the proposal densityof (C3) evaluated at x(s), where x� is used to calculate eÄ.

26 In this paper, the e�cient algorithm developed by Geweke (1991) is adopted, and I thank Gary Koop for providing theGauss code to generate these censored univariate normals.



Finally, conditional on s��s�1�;B�s�1�; x�s�1�, we generate a candidate Lÿ1� from the Wishartproposal density as given in (C4) with vÄ degrees of freedom and location matrix~Cÿ1�s��s�1�;B�s�1�; x�s�1��. We accept Lÿ1�s�1� � Lÿ1� with probability

minp�Lÿ1��l�B�s�1�; x�s�1�;Lÿ1�; ~et�B�s�1�; x�s�1�;Lÿ1

� �; s��s�1��=qL�Lÿ1�s�;Lÿ1� j ��p�Lÿ1�s��l�B�s�1�; x�s�1�;Lÿ1�s�; ~et�B�s�1�; x�s�1�;Lÿ1�s��; s��s�1��=qL�Lÿ1�;Lÿ1�s� j ��

; 1

( )

and otherwise set Lÿ1�s�1� � Lÿ1�s�, where p(.) is the prior, l(.) the likelihood, qL(.) the proposaldensity as given in (C5). . in the proposal density denotes �s��s�1�;B�s�1�; x�s�1��. The lastexpression in the numerator of the above equation gives the proposal density of (C4) evaluated atLÿ1�, whereLÿ1�s� is used to calculate eÄ; similarly, the last expression in the denominator gives theproposal density of (C4) evaluated at Lÿ1�s�, where Lÿ1� is used to calculate eÄ.27

The Bayes factor is the basis of model comparison, and it is computed as the ratio of marginallikelihoods under alternative model formulations. In this paper, we follow Gelfand and Dey(1994) and Geweke (1997a,b) to compute the marginal density of the sample data m(y) givenparameter draws from the above MCMC posterior simulator.

More speci®cally, Gelfand and Dey (1994) observe that for any p.d.f. f(g) whose support iscontained in G, G � Rk,

Ef �g�

p�g�l�g��

� m�y�ÿ1 �C6�

where p(g) is the properly normalized prior density, l(g) the properly normalized data density (thelikelihood). We can approximate (C6) from the output of any posterior simulator, but for thisapproximation to have a practical rate of convergence, f �g�=p�g�l�g� should be uniformlybounded.

From the output of the posterior simulator with I iterations, de®ne

gI �1

I

XIi�1

g�i� SI �1

I

XIi�1�g�i� ÿ gI��g�i� ÿ gI�0

Then for some c 2 �0; 1�, de®ne

GI � fg : �gÿ gI�0Sÿ1I �gÿ gI�4w21ÿc�k�g

and take

f �g� � cÿ1�2p�ÿk=2jSIjÿ1=2 exp�ÿ1

2�gÿ gI�0Sÿ1I �gÿ gI��wGI�g�

where w(.) is the characteristic function wGI�g� � 1 if g 2 GI; wGI

�g� � 0 if g =2 GI. According toGeweke (1997b), smaller values of c will result in better behaviour of f �g�=p�g�l�g� over the

27 In our actual estimation of the simultaneous-equation target zone model (M0), the acceptance rates for B; x;Lÿ1 arearound 0.20, 0.99 and 0.99, respectively.

488 K. LI


domain GI, but greater simulation error. In this paper, as a compromise we report our marginallikelihood results by setting c � 0.5 (Geweke, 1997a).28

ACKNOWLEDGEMENTS

I would like to thank Dale J. Poirier and Gary Koop for all their guidance. I am very grateful toJohn Geweke (the co-editor) and two anonymous referees for their constructive and helpfulcomments which have signi®cantly improved the paper. I also wish to thank J. Brander,M. Campolieti, G. Donaldson, R. Heinkel, N. Joseph, M. Levi, B. McCabe, K. Pasula, P. Pauly,M. Puterman, F. Ruge-Murcia (especially), G. Tauchen, E. Tsionas, R. Uppal, and theparticipants of the seminar at Simon Fraser University, the 1996 Canadian EconomicsAssociation meetings (St Catherines), the 1997 European Finance Association meetings (Vienna)for helpful comments. Financial support from the Social Sciences and Humanities ResearchCouncil of Canada and use of the BACC software jointly developed by S. Chib and J. Geweke aregratefully acknowledged. The usual disclaimer applies.

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Exchange rate target zone models: a Bayesian evaluation

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