Economic Modelling 19 Ž2002. 509᎐529 Toward an econometric target zone model with endogenous devaluation risk for a small open economy 夽 Michel A. Klaster 1, Klaas H.W. KnotU De Nederlandsche Bank (DNB), Amsterdam, The Netherlands Accepted 24 October 2000 Abstract A number of econometric target zone models is estimated for the Belgian franc and the Dutch guilder vis-a-vis the deutsche mark, with a particular focus on the modeling of ` endogenous devaluation risk. Both currencies can be characterized by mean reversion, whereas the theoretical S-effect is observed only for the Belgian franc. Exchange rate volatility can be adequately modeled by means of a GARCHŽ1,1. process. For the Belgian franc, exchange rate tensions have been induced by movements in the inflation differential vis-a-vis ` Germany and the level of foreign exchange reserves, whereas for the Dutch guilder the interest rate differential vis-a-vis Germany and the level of foreign exchange reserves ` have been particularly important. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Exchange rate target zones; Endogenous devaluation risk JEL classifications: E44; F31 1. Introduction This article focuses on the movements in exchange rates within a system of target zones such as the Exchange Rate Mechanism ŽERM. of the European 夽 Any views expressed in the paper are the authors’ only, and do not necessarily represent the position of DNB, andror the KLM Pension Fund. U Corresponding author. Tel.: q31-20-524-22-45; fax: q31-20-524-36-69. E-mail address: k.h.w.knot@dnb.nl ŽK.H.W. Knot.. 1 Michel Klaster is currently at the KLM Pension Fund. 0264-9993r02r$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 0 0 . 0 0 0 6 9 - 9 510 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 Monetary System ŽEMS.. In particular, it will focus on the modeling of endogenous devaluation risk. The models developed in the paper will be applied to the experience of the Belgian franc and the Dutch guilder within the ERM. While both currencies have been irrevocably fixed with the coming into being of the EMU and the adoption of the euro, the experience from their ERM participation might contain valuable lessons for the countries currently participating in ERM-II ŽDenmark and Greece., as well as those countries ᎏ mainly Central and Eastern European ᎏ for which future participation in ERM-II will be a prerequisite for joining the euro area. Belgium and the Netherlands have been participating in the ERM since its establishment in 1979. Both countries may be characterized as small open economies that attach great value to stable exchange rates. To a large extent, monetary policy in EMS countries has always been aimed at a stable exchange rate against the deutsche mark. Underlying this choice is the importance of Germany as a partner in foreign trade and, above all, the pronounced anti-inflation reputation of the Bundesbank. Before monetary policy was completely subordinated to the exchange rate target, both currencies experienced a number of devaluations ŽUngerer et al., 1990; Knot and De Haan, 1995.. The Dutch guilder, for instance, was devalued twice in the early years of the ERM Ž1979᎐1983. before being anchored definitively in a narrow band around central parity. Before 1987, the Belgian authorities even devalued as many as seven times, as the country’s deteriorating fundamentals frequently caused speculative pressure on the franc. From March 1990 on, the National Bank of Belgium has adhered to the so-called franc fort policy, under which the franc is virtually pegged at central parity. In the theoretical target zone models developed in the late 1980s, the probability of a realignment is often assumed to be exogenous or is sometimes not even modeled at all ŽKrugman, 1991; Svensson, 1991; Lindberg and Soderlind, 1994.. To ¨ circumvent these limitations and to investigate the empirical behavior of exchange rates and devaluation risk in a target zone, various authors have constructed econometric target zone models. After an extensive study of different model specifications, Nieuwland et al. Ž1991. conclude that an ARŽ1. ᎐GARCHŽ1,1. jump model best describes exchange rate developments within the ERM. They model the probability of a jump by means of a Poisson distribution, while the observed clustering of extreme values necessitates a GARCH specification. In a MAŽ1. ᎐GARCHŽ1,1. jump model presented by Vlaar Ž1992., the probability of a jump is conditioned on economic fundamentals such as the inflation differential vis-a-vis Germany and the trade surplus. The study shows that the Dutch and ` French probabilities of a jump are affected by developments in the inflation differential and that the Danish probability of a jump is related to developments in the trade surplus. For Belgium, Ireland, and Italy, no significant relationships are reported. Ball and Roma Ž1993. adopt a different approach to model exchange rate dynamics within the ERM. Their decomposition of the exchange rate into the central parity Ž c t . and the exchange rate within the band Ž x t . is more in line with the theoretical target zone models. Assuming that x t follows an Ohrnstein ᎐ M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 511 Uhlenbeck process owing to intramarginal interventions and that the likelihood of a realignment depends on the position within the band, they show that both the jump element and the mean reversion element are important aspects in the modeling of ERM exchange rates. Engel and Hakkio Ž1994. emphasize the fact that within the ERM extreme exchange rate changes have a tendency to cluster. Their model is characterized by a ‘quiet’ distribution and a jump distribution in which the probability of a sampling from one of the two distributions depends on the position within the band and the type of distribution of the previous sampling. They find that the probability of a jump increases as the exchange rate approaches the upper band andror as the previous observation also involved a jump. Finally, one of the most advanced econometric target zone models at present is that of Bekaert and Gray Ž1996.. Their model, which is estimated on the basis of data for the FFrDM exchange rate, distinguishes itself from other econometric target zone models by the large number of explanatory variables with which devaluation risk is endogenized. In the present study, three of these models will be described and estimated for the Belgian franc and the Dutch guilder. Various economic fundamentals will be identified that influence the probability of a realignment. Apart from being used to endogenize the probability of realignment, a number of practical applications will be considered. Models in international finance are often based on specific assumptions regarding the exchange rate distribution ŽBoothe and Glassman, 1987., for instance, the assumption of normally distributed exchange rate changes in the construction of an efficient asset portfolio or valuation methods of currency options. By charting the stochastic processes underlying the exchange rate movements within target zones, the legitimacy of such assumptions may be assessed. The paper is organized as follows. Section 2 first analyzes the exchange rate movements of the Belgian franc and the Dutch guilder from the foundation of the ERM in March 1979 up to just before the widening of the fluctuation margins in July 1993. The results of this analysis serve as a guideline for the estimation of an elementary jump model in Section 3. To allow for endogenous devaluation risk, the model presented by Bekaert and Gray Ž1996. is subsequently described and estimated in Section 4. On the basis of the outcomes for both Belgium and the Netherlands, Section 5 then presents a new specification whose predictive performance is tested in Section 6. Section 7 offers some concluding remarks. 2. An analysis of the Belgian and Dutch exchange rate data The results of a thorough statistical analysis of exchange rate data may help to find a suitable model for exchange rate changes. For the analysis of the Belgian franc and the Dutch guilder, we used weekly observations on the price of the deutsche mark expressed in units of the domestic currency, provided by Datastream ŽWednesday’s closing rates. over the period 13 March 1979 through 30 July 1993. Weekly data were chosen to avoid the problems of daily effects with respect to exchange rate volatility.2 For both Belgium and the Netherlands, the set of data 512 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 comprised 749 observations, in which the original exchange rates were converted into percentage logarithmic exchange rate changes. Table 1 reveals that average exchange rate changes Ž ⌬ St . over the period under review were small but positive. This is a logical consequence of the fact that the currencies involved were devalued repeatedly against the deutsche mark but were never revalued. The t-values show that only for the Belgian franc has the depreciation tendency vis-a-vis ` the deutsche mark been significant. In addition, the sizeable values for skewness and kurtosis indicate that normal distribution is out of the question, as it would be characterized by a near-zero skewness and a kurtosis of approximately three. The extremely significant values in the Bera᎐Jarque and Kiefer᎐Salmon tests of normality reaffirm this notion. Another striking feature is that the rejection of the hypothesis of normality is considerably stronger for Belgium than for the Netherlands. This may be attributed mainly to the 8.5% devaluation that took place in February 1982. Without this outlier, the skewness and kurtosis values would be similar. The results of the autocorrelation tests in the Table 1 Basic statistics of weekly exchange rate changes: ⌬ St s 100 = lnŽ StrSty1 . a Mean S.D. t-Test Skewness Kurtosis B-J test KS-1 test KS-2 test ␳ Ž1. QŽ10. QŽ50. Belgium The Netherlands 0.0358 0.3531 2.77UU 12.03 238.41 UU 1 750 373 UU 18 025 UU 1 732 347 1.13 42.55UU 134.54UU 0.0057 0.1537 1.01 2.35 21.72 UU 11 596 UU 685 UU 10 911 3.71UU 18.59U 94.11UU a NB: Sample: 13 March 1979 until 30 July 1993. The t-test measures the significance of the mean wt s ␮ ˆ sr␴ˆsr6N, where N s 749 observationsx. The Bera᎐Jarque normality test combines skewness and Žexcess. kurtosis; under the null hypothesis of normally distributed exchange rate changes it is asymptotically distributed as ␹ 2 Ž2.. The normality tests of Kiefer and Salmon are asymptotically distributed as ␹ 2 Ž1.. Under the null they assume zero skewness and a kurtosis of three. ␳ Ž1. denotes n times the first autocorrelation coefficient. Under the null of no autocorrelation this statistic has a standard normal distribution ŽTaylor, 1986, p. 136.; In finite samples QŽ10. and QŽ50. are modified Portmanteau tests, that use the first 10 or 50 autocorrelations. Since autocorrelations are independent under the null, QŽp. is asymptotically distributed as ␹ 2 Žp. ŽHarvey, 1990, p. 45.. Rejection at the U 5% and UU 1% significance level is indicated. 2 On Mondays, for instance, exchange rate changes have a greater variance than on other days because the weekend is a longer time period during which new information may come in ŽHsieh, 1989.. Usually, significant differences also exist in exchange rate gains on the different days Žsee for instance Taylor, 1986.. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 513 Table 2 Heteroscedasticity tests Ž749 weekly exchange rate changes. a LMŽ1. LMŽ5. QU Ž10. QU Ž30. QU Ž50. Belgium The Netherlands 10.29UU 18.49UU 90.15UU 157.31UU 172.89UU 7.25UU 8.99 22.92UU 63.50UU 111.53UU a NB: Sample: 13 March 1979 until 30 July 1993. To avoid an extremely large kurtosis, the Belgian 8.5% devaluation of February 1982 is omitted. LMŽp. is equal to the number of observations times the R 2 of a regression of the squared exchange rate changes on an intercept and p lags. Under the null of homoscedasticity the LM test is asymptotically distributed as ␹ 2 Žp.. QU Žp. is a Portmanteau test on squared exchange rate changes, which is also asymptotically distributed as ␹ 2 Žp.. Rejection of the null at the U 5% and UU 1% significance level is indicated. bottom three rows of Table 1 indicate that for both currencies under investigation the null hypothesis of no autocorrelation should be rejected.3 The significantly positive values of the skewness could argue for a model that contains two distribution functions: a distribution for ‘normal or quiet’ times and a jump distribution for ‘crisis’ periods, with a much higher average and variance. This could also offer a partial explanation for the extraordinarily high values of the kurtosis found. The high kurtosis could be described by a distribution in which the variance within the model is conditioned on actual values for the variance, a so-called GARCH model.4 This would also provide a potential explanation of the clustering of extreme values noticed by Engel and Hakkio Ž1994.. To detect such an autoregressive component in the variance, several direct tests can be used. Table 2 shows a number of direct tests for ŽG.ARCH effects, such as two Lagrange Multiplier tests Žsee Breusch and Pagan, 1979. and three Portmanteau tests based on squared exchange rate changes ŽHarvey, 1990.. For all tests, the null hypothesis is homoscedasticity; the alternative hypothesis for the various tests is a first, fifth, 10th, 30th and 50th order ARCH specification for the variance, respectively. As the GARCH model may be seen as an infinite ARCH model, the higher-order tests are particularly important in the testing for GARCH. The results show that the null hypothesis of homoscedasticity can safely be rejected for Belgium and the Netherlands by means of both types of tests.5 It is fair to 3 If we apply these tests to the absolute values Ž< ⌬ St <. as well, the rejection turns out to be even stronger Žnot shown.. 4 The ARCHŽq. model was introduced by Engle Ž1982. and is essentially a MAŽq. process for the variance of the innovations, while the GARCHŽp,q. model may be interpreted as an ARMAŽp,q. process in terms of this variance ŽBollerslev, 1986; Greene, 1993.. 5 Meaningful interpretation of these tests requires finite kurtosis, which is not the case for the Belgian data ŽTable 1.. We therefore decided to omit the 8.5% devaluation in February 1982 from the tests presented in Table 2. 514 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 Fig. 1. The Belgian franc in the ERM: 1979᎐1993. conclude, therefore, that exchange rate movements in both countries might be adequately modeled by means of a GARCH model. 3. A first step towards an econometric target zone model The analyses presented in the previous section suggest three elements that a successful model of exchange rate movements in the ERM should incorporate. First, the excessive skewness in the distribution points to the necessity of adding a jump distribution in order to adequately explain the positive outliers. For the modeling of such stochastic jump processes, the economic literature usually applies Poisson distributions, although Bernoulli distributions would also be suitable. As the number of realignments is too small to arrive at sound estimates, the jumps will have to relate to both realignments and exchange rate ‘jumps’ within the band that are comparable in size to the actual realignments. This can be seen in Figs. 1 and 2, Fig. 2. The Dutch guilder in the ERM: 1979᎐1993. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 515 which show the exchange rate movements together with the relevant target zones of Belgium and the Netherlands, respectively. Second, significant first-order autocorrelation coefficients ᎏ especially for the Netherlands ᎏ point to the incorporation of an MAŽ1. or an ARŽ1. component. Curiously enough, however, the autocorrelations appear positive. This contrasts with various studies that have reported negative first-order autocorrelations Žsee, for instance, Vlaar and Palm, 1993., which are attributed to intramarginal interventions. Third, the results of the autocorrelation and heteroscedasticity tests suggest that the developments in the variance can be described by means of a GARCH specification. Given the fact that the spike in the first lag of the autocorrelation function is more pronounced than the corresponding one of the partial autocorrelation function, an MAŽ1. specification is preferred, although the difference from an ARŽ1. specification seems marginal. The selection of the type of jump distribution can be made on purely pragmatic grounds. This hardly influences the estimation results, while a model with a Bernoulli jump distribution is considerably easier to estimate ŽVlaar and Palm, 1993.. In addition, a choice was made for a GARCHŽ1,1. specification because this is usually sufficient to adequately describe conditional changes in the variance. The first and second components together may be represented as follows: Ž1. ⌬ St s ␮ q ␭␯ q ␺␧ ty1 q ␧ t , where ␮ is the mean of a ‘quiet’ distribution, ␭ the jump intensity, ␮ q ␯ is the mean of the jump distribution, ␺ the MAŽ1. parameter and ␧ t the disturbance term. Since h2t represents the variance of the quiet distribution and h 2t q ␦ 2 the variance of the jump distribution, the total distribution for ␧ t is given by N Ž0, h2t q ␭␦ 2 .. In addition, the GARCH Ž1,1. specification is given by: 2 h2t s ␣ 0 q ␣ 1 h 2ty1 q ␣ 2 ␧ ty1 . Ž2. Table 3 A MAŽ1. ᎐GARCHŽ1,1. ᎐Bernoulli jump modela ␮ ␺ ␣0 ␣1 ␣2 ␭ ␯ ␦2 Belgium The Netherlands 0.0053 Ž1.39. y0.0420 Ž0.86. 0.0023U Ž2.49. 0.4562UU Ž4.41. 0.3520UU Ž3.53. 0.0412UU Ž2.91. 0.2041 Ž1.14. 2.1011 Ž1.06. y0.0024 Ž0.91. 0.0265 Ž0.52. 0.0001 Ž1.07. 0.8005UU Ž6.72. 0.1516 Ž1.40. 0.0450U Ž2.36. 0.2658U Ž1.98. 0.1506U Ž1.96. a NB: Sample: 13 March 1979 until 30 July 1993. Heteroscedasticity consistent t-values are within brackets. Significance at the U 5% and UU 1% level is indicated. The number of observations is 749 for both countries. 516 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 The variance at time t is, therefore, conditioned in a GARCHŽ1,1. model on the variance and the squared innovation of time t y 1. In order to arrive at estimation results for this model, the log-likelihood function had to be maximized over eight parameters. As with the other models in the present study, we employed the GAUSS Maxlik module. By choosing suitable initial values, this model could be estimated using the algorithm of Broyden, Fletcher, Goldfarb and Shanno Žsee Broyden, 1965.. Table 3 shows the results for Belgium and the Netherlands. For both countries the MAŽ1. component is not significant. For the Netherlands the sign is even positive, which is generally associated with mean aversion rather than mean reversion. This would seem remarkable as various studies do report mean reverting behavior by the guilder᎐deutsche mark rate ŽSvensson, 1993; Knot and De Haan, 1995..6 Mean reversion in a credible target zone could arise from Žexpectations concerning. two types of central bank interventions: ‘marginal’ interventions, taking place whenever the exchange rate approaches the upper or lower limit of its target zone, leading to the well-known S-shaped exchange rate function; and ‘intramarginal’ ᎏ or ‘leaning-against-the-wind’ ᎏ interventions. However, the link between mean reversion and MArAR parameters, which is often drawn in the literature, is not cast-iron. Mean reversion, as tested in the above-mentioned studies, applies to the relation between the current movement and the current position of the exchange rate, whereas the MArAR parameters link the current movement to the preceding movement. Therefore, to correctly test for mean reversion, a specification is required in which exchange rate movements are linked directly to the position within the band. Another conclusion from Table 3 is that for Belgium the variances of the quiet w ␣ 0rŽ1 y ␣ 1 y ␣ 2 .x as well as the jump distributions Ž ␦ 2 . are considerably larger than for the Netherlands. This might be expected because, as shown in Section 2, the franc has experienced larger fluctuations than the guilder. Additionally, the mean of the quiet distribution for the Netherlands is negative. Apparently, Dutch exchange rate movements have been characterized by a slight appreciation tendency vis-a-vis the deutsche mark, which has sometimes been corrected by a jump ` that has nullified or even exceeded the earlier appreciation. Finally, it can be observed that the two GARCH parameters for Belgium are both significant, while for the Netherlands only ␣ 1 is significant.7 4. The econometric target zone model of Bekaert and Gray In this section, the model presented by Bekaert and Gray Ž1996. will be 6 The positive sign of the MA parameter does correspond to the autocorrelation coefficient found Žsee Table 1.. In addition, financial series almost always exhibit positive first-order MA parameters ŽTaylor, 1986.. 7 Positive GARCH parameters are guaranteed by estimating 6␣ i rather than ␣ i . Allowance should be made, however, for the fact that t-values generated in this way are not entirely consistent. In order to obtain consistent t-values, these have been calculated without the non-linear transformations. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 517 introduced. This model also contains a quiet distribution and a jump distribution. Further, following Nieuwland et al. Ž1991. and Vlaar Ž1992., the variance is modeled by means of a GARCHŽ1,1. specification. Finally, the position of the exchange rate within the band, just as with Ball and Roma Ž1993. and Engel and Hakkio Ž1994., plays an important role in explaining exchange rate movements. Apart from these similarities, various innovative elements can also be found in the model that allow for a direct test of some of the theoretical implications of the Krugman Ž1991. model. For instance, the Krugman model stipulates an S-type link between economic fundamentals and the exchange rate, stemming from central bank interventions taking place whenever the exchange rate approaches the upper or lower limit of its target zone. A testable implication of this notion is that exchange rate movements should display a lower variance near the edges of the target zone. By the same token, expectations about these marginal interventions should induce an observable process of mean reversion in the exchange rate. Aside from testing the characteristics of the Krugman model, the model allows for an assessment of the probability of a realignment because both the probability of a jump and the specification of the jump distribution ᎏ as a function of a specific set of fundamentals ᎏ are known at any given moment. The probability of a jump can be interpreted as an indicator for the lack of target zone credibility. Three exogenous variables are introduced to explain the conditional mean of the jump distribution: the relative reserve position of the central bank Ž LR ., the slope of the yield curve Ž SYC . and the interest rate differential vis-a-vis Germany Ž ID .. ` This way, both the probability of a jump and its conditional mean have been endogenized. The model will first be expressed in terms of the distribution of exchange rate changes, conditioned on the available information: f Ž ⌬ St < Ity1 .. Ity1 represents the available set of information and f Ž. <.. indicates the conditional density function. The bivariate structure of the model is reflected by the fact that this function is subdivided into a part in which there is no jump and a part in which there is a jump, with a conditional probability of 1 y ␭ ty1 and ␭ ty1 , respectively: 2 f Ž ⌬ St < Ity1 . s TN Ž ␮ ty1 , ␴ty1 ,⌬ L ty 1 ,⌬ Uty 1 . Ž 1 y ␭ ty1 . 2 q N Ž ␳ ty1 , ␳ ty1 ␦ 2 . ␭ ty1 , Ž3. with ␭ ty1 s ⌽ Ž ␤ 1 q ␤ 2 SYCty1 . ␳ ty1 s ␤ 3 q ␤4 LR ty1 q ␤5 < PBty1 < q ␤6 IDty1 q ␤ 7 CIDty1 ␮ ty1 s ␤ 8 q ␤ 9 PBty1 2 2 2 ␴ty1 s ␤ 10 q ␤ 11Ž 1 y RDty1 . ␧ ty1 q ␤ 12 ␴ty2 q ␤ 13 < PBty1 < . Ž4. 518 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 Appendix A lists the definitions of all variables and reports on the expected signs of their parameters. The first part of Eq. Ž3. represents a truncated normal ŽTN . distribution of exchange rate changes, given that there is no jump. This distribution is truncated at the boundaries of the target zone because realignments are by definition impossible. In this case, therefore, the exchange rate behaves in accordance with the principles of a completely credible target zone. The largest possible positive Ž ⌬U . and negative Ž ⌬ L. changes cause the exchange rate to end up exactly at either the upper or lower limit of the target zone. The probability density function of a truncated normal distribution is therefore zero outside the truncation points, and inside them the probability is heightened proportionally. The probability density function inside the truncation points can be represented as follows: ␾ f Ž ⌬ S t < Ity1 , ␭ ty1 s 0 . s ⌽ ž ž /' ' / ž ' ⌬ St y ␮ ty1 2 ␴ty1 ⌬Uty1 y ␮ ty1 '␴ 2 ty1 y⌽ 1 2 ␴ty1 ⌬ L ty1 y ␮ ty1 2 ␴ty1 / , Ž5. where ␾ Ž.. is the standard normal density function and ⌽ Ž.. the cumulatively normal density function Žsee Greene, 1993, p. 683, for details on the truncated normal distribution.. The conditional mean Ž ␮ ty1 . depends on the position within the band Ž PBty1 ., so that the mean reverting behavior of exchange rates within credible target zones 2 . can be tested directly. The conditional variance Ž ␴ty1 of the truncated distribution follows a GARCHŽ1,1. process, which, by means of a dummy variable Ž RDty1 ., also allows for whether or not a realignment took place recently. Without the dummy the innovation w ␧ ty1 s ⌬ Sty1 y Ety1Ž ⌬ S ty1 .x would wrongly indicate that the predicted variance is large, whereas, in fact, immediately after realignments volatility is typically below average. The variance also depends on the deviation from central parity: < PBty1 <, the principal determinant of the variance in the Krugman model. In the second part of the conditional density function wEq. Ž3.x, the possibility of a realignment is taken into account. It is distributed normally Ž N . and has a considerably higher mean variance than the truncated distribution. The probability of a jump Ž ␭ ty1 . is specified as a function of the slope of the yield curve Ž SYCty1 .. As in the probit model, the cumulatively normal distribution has been used, so that ␭ can only assume values between 0 and 1. A negative slope ᎏ when short-term interest rates exceed long-term interest rates ᎏ is taken as a harbinger of heightened exchange rate tensions and, hence, an increased probability of a jump.8 8 An inverse yield curve could also result from a temporary monetary tightening in Germany that was being copied by the pegging country. The fact that such a situation need not, in principle, be related to exchange rate tensions could diminish the significance of the parameter in question. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 519 Since macro-economic data such as industrial output and unemployment can only be obtained on a monthly basis and with a 6- to 8-week lag, such influences on the probability of a jump were neglected. The conditional mean of the jump distribution Ž ␳ ty1 . depends on four factors: the reserve position of the central bank Ž LR ty1 ., the deviation from central parity Ž< PBty1 <., the 1-week interest rate differential vis-a-vis Germany Ž IDty1 . and the cumulative inflation differential ` vis-a-vis Germany Ž CIDty1 ..9,10 To avoid identification problems, the conditional ` variance of the jump distribution is deemed to increase proportionally with the conditional mean. High jump means are therefore accompanied by larger variances. To estimate this model, the conditional log-likelihood function had to be optimized over 14 parameters.11 For the estimates, we used the optimization algorithms provided by Berndt et al. Ž1974. and by Broyden, Fletcher, Goldfarb and Shanno ŽBroyden, 1965.. A thorny problem in estimating this model was finding the correct initial values. This was resolved by initially equating all parameters to zero and subsequently removing these restrictions one by one, employing the resulting estimates as initial values for the next round. While the final estimates generally appeared to be independent from the exact sequence in which the restrictions were lifted, sizeable differences in the speed of convergence could be noticed. This method nonetheless yielded the best ‘guarantee’ that the global optimum could indeed be approached. As noted in the introduction, in March 1990 a significant trend break occurred in Belgian monetary policy. For this country we therefore included observations on the period preceding March 1990 only.12 In the Netherlands, the transition toward a more credible monetary policy has taken place much more gradually, so that all observations could indeed be included here. The number of significant parameters for the Netherlands is considerably higher than that for Belgium Ž10 vs. five; Table 4.. However, for Belgium only ␤5 has the wrong sign, while this phenomenon occurs three times for the Netherlands: ␤6 , ␤ 7 and ␤ 13 .13 Also, ␤ 9 appears to be negative for both countries and for Belgium almost significantly so. Hence, some form of mean reversion can be observed by 9 The deviation from central parity has been included in absolute terms, because jumps could also relate to sizeable jumps within the band. While the direction of a realignment is typically in line with the deviation from central parity, the potential size of a jump within the band is larger when the jump originates from the opposite edge of the band. 10 The lagged availability could have been a consideration not to include the inflation differential in the model. For reasons of comparability with Bekaert and Gray Ž1996., the variable was included nevertheless. It is, incidentally, not clear whether Bekaert and Gray made allowance for the lagged availability of the figures. Our estimations have, however. 11 The statistical motivation for the use of this method and the exact specification of the conditional log likelihood function are given in Bekaert and Gray Ž1996, p. 13.. 12 The period 1979᎐1990 was chosen because the 1990᎐1993 period provided too few observations for this model. Additionally, exchange rate changes are so small that attempts to generate estimates for the later period would inevitably face identification problems. 13 For the sake of comparison: for France, six out of 10 parameters were significant, and two were incorrectly signed, the latter being ␤5 and ␤ 13 ŽBekaert and Gray, 1996.. 520 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 Table 4 The model of Bekaert and Gray a ␤1 Ž ␭. ␤2 ŽSYC. ␤3 Ž ␳ . ␤4 ŽLR. ␤5 Ž<PB <. ␤6 ŽID. ␤7 ŽCID. ␤8 Ž ␮ . ␤9 ŽPB. ␤10 Ž ␴ . ␤11 Ž ␧y1 . ␤12 Ž ␴y2 . ␤13 Ž<PB <. ␦ Log-likelihood Belgium Ž03r1979᎐03r1990. The Netherlands Ž03r1979᎐07r1993. y1.6590UU Ž5.508. y0.1799 Ž1.046. y0.0372 Ž1.930. y0.2750 Ž1.928. y0.1908 Ž1.741. 0.0176 Ž1.573. 0.2660 Ž1.897. 0.0386UU Ž2.802. y0.0399 Ž1.736. 0.0021 Ž1.573. 0.2396UU Ž3.467. 0.7199UU Ž8.383. y0.0024 Ž1.512. 4.9992U Ž1.982. 0.309438 y2.2459UU Ž5.982. y0.0992 Ž0.628. 3.2321U Ž1.986. y2.1632UU Ž2.782. 0.4389UU Ž2.875. y0.2558UU Ž5.101. y0.2455UU Ž4.740. y0.0018 Ž0.709. y0.0302 Ž1.234. 0.0007 Ž0.746. 0.2135UU Ž3.535. 0.6980UU Ž10.46. 0.0070UU Ž2.706. 0.1054UU Ž3.656. 0.921054 a NB: Heteroscedasticity consistent t-values are within brackets. Significance at the U 5% and level is indicated. The number of observations is 583 for Belgium and 749 for the Netherlands. UU 1% means of this model. Another prediction by the Krugman Ž1991. model is the S-type relation between the fundamental and the exchange rate. This implies that exchange rates move more slowly if they are close to the bands. A negative value of ␤ 13 would be in line with this. This is found for Belgium, but for the Netherlands this parameter is significantly positive. A possible explanation may be the fact that, from 1983 on, the Nederlandsche Bank has adopted an implicit bandwidth much smaller than 4.5%. Thus, well before the boundaries of the target zone were reached, exchange rate movements were tempered by intramarginal interventions Žpredominantly interest rate changes., which impedes ‘clean’ testing of the hypothesis involved for the Netherlands. The GARCH parameters Ž ␤ 11 and ␤ 12 . are highly significant for both countries. This is not in line with the Krugman model either, as that model would imply a variance that is independent from previous exchange rate observations. The jump parameters Ž ␤ 2 , ␤4 , ␤5 , ␤6 and ␤ 7 . for Belgium are all insignificant. It should be noted, however, that relative to the 10% level three of the five jump parameters are significant. For the Netherlands, it is not so much the significance that is disappointing, but rather the fact that the two most significant parameters Ž ␤6 and ␤ 7 . both seem to have the wrong sign. For the cumulative inflation differential, the negative sign is indeed hard to explain, as it would suggest that a higher inflation rate in the Netherlands vis-a-vis Germany would strengthen rather than weaken ` the guilder. The negative sign in the interest rate differential can, on second thought, be justified by the high credibility of the guilder peg and the preventative nature of Dutch interest rate policy. The underlying idea of the expected positive sign was M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 521 that exchange rate tensions are usually attended by considerable interest rate increases. These marginal interventions have often been used by countries such as France and, to a lesser extent, also Belgium. Dutch authorities have always resorted to intramarginal interventions to withstand any emerging pressure on the guilder. This way, they communicated to markets that the exchange rate target would prevail upon domestic targets. Hence, such increases were implemented in a situation of relatively low speculative pressure, and often resulted in exchange rate appreciation. This explanation is confirmed by the fact that during the ‘franc fort’ policy of the 1990s Belgian exchange rate changes have also been negatively related to the interest rate differential Žnot shown.. 5. Development of an alternative model Estimating the model of Bekaert and Gray Ž1996. for Belgium and the Netherlands has revealed a number of shortcomings. First, there is an overparametrization of the mean of the jump distribution. For Belgium, this results in a large number of insignificant parameters and for the Netherlands, in an incorrectly signed coefficient on the inflation differential. Further analysis showed that the parameter ␤ 7 of the inflation differential is strongly correlated with the parameter ␤6 of the interest rate differential. Bekaert and Gray’s model, therefore, is characterized by multicollinearity, which makes identification of the parameters considerably more difficult. A second problem is provided by the variable that determines the jump probability, the slope of the yield curve. As in Bekaert and Gray’s original estimates for France, this variable is significant for neither country. Finally, the variance of the jump distribution is affected by the low predictive value of the variables that determine its mean. The misspecifications and the large number of parameters to be estimated yield unsatisfactory convergence properties for both countries. In order to resolve these problems, a number of changes have been implemented. First, to avoid multicollinearity, the number of predictors of the mean of the jump distribution has been reduced to one. Initially we chose the inflation differential for both countries, given the strong theoretical link between inflation differentials, competitiveness and devaluation. Additional analysis demonstrated, however, that the interest rate differential would be a better predictor for the Netherlands, in terms of both its significance and the overall likelihood of the model. We will therefore present both models for the Netherlands. Second, to improve the predictive value of the jump probability, the slope of the yield curve has been replaced by the central bank’s level of reserves. Given the fact that speculative pressure on the currency will typically also be reflected in the reserve position, this variable can be justified on intuitive grounds. Third, the truncated normal distribution has been replaced by an ordinary normal distribution because this simplifies the interpretation of the mean reversion parameter. A property of the truncated distribution is that the mean of the underlying normal distribution to 522 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 be estimated has a tendency toward mean aversion.14 Correction for this would imply that the mean reversion parameter and the corresponding confidence interval must be calculated on the basis of non-linear functions of the initial position of the exchange rate and a number of other parameters from the model, which appears to be impossible ŽBekaert and Gray, 1996.. As a result of these changes, the model is as follows: 2 .Ž 2 f Ž ⌬ St < Ity1 . s N Ž ␮ ty1 , ␴ty1 1 y ␭ ty1 . q N Ž ␳ ty1 , ␴ty1 q ␦ 2 . ␭ ty1 , Ž6. with ␭ ty1 s ⌽ Ž ␥ 1 q ␥ 2 LR ty1 . Ž7. ␳ ty1 s ␥ 3 q ␥4 JX ty1 ␮ ty1 s ␥ 5 q ␥6 PBty1 2 2 2 ␴ty1 s ␥ 7 q ␥ 8 Ž 1 y RDty1 . ␧ ty1 q ␥ 9 ␴ty2 q ␥ 10 < PBty1 < . The specification of the quiet distribution is identical to that of the model presented in Section 4, apart from the truncation. As noted above, the exogenous variable that determines the mean of the jump distribution Ž JX ty1 . varies across countries: for Belgium we used only the cumulative inflation differential Ž CIDty1 . whereas for the Netherlands we will report on estimates for the interest rate differential Ž IDty1 . also. Generating estimates appeared more satisfactory for all three models, as was evidenced by a more rapid convergence to a Žlog. likelihood level which, despite the smaller number of parameters, turned out to be higher also for the Netherlands. In particular, the results for the Belgian franc have profited markedly from the implementation of the various changes ŽTable 5.. The number of significant variables has increased from five to eight and all parameters now have the expected sign.15 For both currencies, the level of reserves may well function as a jump indicator. Another striking result is that the mean reversion parameter ␥6 has now become significant. Apparently, eliminating the truncation has been essential to allow for mean reversion in this model, while the model presented in Section 3 was incorrectly specified to detect mean reversion. The number of significant explanatory variables for the Netherlands cannot lead us to conclude that this model provides better results than the model presented in Section 4, as this number has decreased from 10 to six in both cases. Still, the 14 This is because the actual mean of the truncated distribution is by definition closer to central parity than the underlying mean, as the truncated distribution always has the largest tail on the side of central parity. 15 The significance of intercept parameters is less relevant as the significance of these parameters was influenced by the rescaling of the explanatory parameters. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 523 Table 5 An alternative modela ␥1 Ž ␭. ␥2 ŽLR. ␥3 Ž ␳ . ␥4 ŽJX. ␥5 Ž ␮ . ␥6 ŽPB. ␥7 Ž ␴ . ␥8 Ž ␧y1 . ␥9 Ž ␴y2 . ␥10 Ž<PB <. ␦ Log-likelihood Belgium JX s CID Ž03r1979᎐03r1990. The Netherlands JX s CID Ž03r1979᎐07r1993. The Netherlands JX s ID Ž03r1979᎐07r1993. 0.8267 Ž1.800. y3.5620UU Ž4.870. y0.0136 Ž0.152. 0.6090U Ž2.060. 0.0458UU Ž4.075. y0.0609UU Ž3.553. 0.0063UU Ž3.657. 0.2918UU Ž4.166. 0.5522UU Ž7.530. y0.0061UU Ž3.561. 1.6757 Ž1.181. 0.291084 1.0963 Ž1.714. y3.1073UU Ž4.182. 0.1801 Ž1.549. 0.0919 Ž0.875. y0.0014 Ž0.628. y0.0375UU Ž2.887. 0.0001 Ž1.669. 0.1048U Ž2.181. 0.8319UU Ž12.90. 0.0014UU Ž3.265. 0.3717UU Ž3.130. 0.932634 1.4377 Ž1.755. y3.4810UU Ž3.606. 0.3620 Ž1.947. y0.1297U Ž2.179. y0.0016 Ž0.693. y0.0368U Ž2.208. 0.0004 Ž0.746. 0.0953U Ž1.986. 0.8423UU Ž13.09. 0.0013 Ž1.947. 0.3687UU Ž3.171. 0.935212 a NB: The specification of the JX variable is given in the column headings. Heteroscedasticity consistent t-values are within brackets. Significance at the U 5% and UU 1% level is indicated. changes may be considered an improvement for the guilder as well. The modified model can be interpreted more easily on economic grounds and its likelihood is higher, despite the reduced number of parameters. As with Belgium, the exchange rate’s mean reverting behavior now clearly comes to the fore; ␥6 is negative and now also significant. This is a plausible result in light of the high credibility of the Dutch exchange rate policy in the period under review, although the parameter measuring the S-effect Ž␥ 10 . remains positive. To test the hypothesis that this result may be attributed to the smaller implicit bandwidth supposedly adhered to by the Dutch central bank since 1983, the model was re-estimated with a bandwidth of "1% from March 1983 onward Žnot shown.. This also provided a positive, albeit now clearly insignificant, parameter, so that no S-effect can be established for the guilder. Finally, the interest rate differential turns out to be a better predictor of the mean of the jump distribution for the guilder than the cumulative inflation differential. It should be noted, though, that the parameter capturing the link between the interest rate differential and the average jump Ž␥4 . has lost some of its significance. One explanation may be the fact that its significance level in Bekaert and Gray’s model had been boosted by multicollinearity with other variables in the specification. As was correctly pointed out by the referee, the evidence presented in Tables 3᎐5 can only provide tentative indications as to which model is the ‘best’. Factors such as speed and ease of convergence need not always be relevant ways of comparing models, nor is the number of significant coefficients. The log likelihood function certainly has some content but it is not an easy tool for comparison in the case of non-nested models. A theoretically superior way would have been to 524 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 include the various models in an out-of-sample forecast comparison exercise. This avenue proved to be filled with insurmountable obstacles, however. Owing to the overparametrization and multicollinearity identified above, the original Bekaert and Gray model ŽTable 4. did not lend itself to several re-runs over shorter subsamples with even fewer jumps. Without having achieved satisfactory convergence properties, the resulting out-of-sample forecasts would have been hard to interpret, let alone to compare with those from the other models. Below, we will therefore restrict ourselves to the ‘most reasonable’ or ‘preferred’ one, i.e. the alternative model of Table 5. 6. The model’s predictive power Owing to the extensive specification of the jump distribution, models with an endogenous devaluation risk may be used to chart and, if necessary, predict exchange rate tensions ex post. In particular, the model streamlined in the previous section could have assisted investors in deciding on the optimum allocation across various currencies because it includes sound explanatory variables and is marked by a relatively satisfactory convergence of the estimation process. The probability of a jump and the complete jump specification are known at all times. That is why the probability of an exchange rate movement outside the target zone can be determined by means of the model. Both statistically and politically, the probability of a realignment in the form of a revaluation of another ERM currency vis-a-vis ` the deutsche mark has always been negligible. Within the ERM, therefore, only the likelihood of de valuation is relevant. Figs. 3 and 4 show the probability of devaluation for Belgium and Netherlands, respectively, over the entire ERM period up to the widening of the bands in August 1993. For the Belgian franc, six of the seven devaluations, indicated by a vertical line, are preceded by an increase in the devaluation probability to at least 3%. Before Fig. 3. Probability of a devaluation of the BFrDM parity. M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 525 Fig. 4. Probability of a devaluation of the NGrDM parity. the most recent realignment, the devaluation probability had increased to 2%. While the probabilities are relatively limited in absolute size, they are sufficiently pronounced in order to be distinguished from those in tranquil periods.16 Our model therefore quite adequately signals mounting exchange rate tensions. The exact timing of the devaluations, however, cannot always be predicted by our model. Occasionally, such as with the fifth devaluation in March 1983 and the sixth in July 1986, the devaluation is preceded by an increase in the devaluation probability to over 3%, which already set in many months earlier. In addition, the devaluation probability rose substantially twice without being followed immediately by a realignment ŽApril 1980 and June 1984.. For both months, Fig. 1 suggests that the franc had reached the Žweak. upper limit of the target zone. A comparison of Figs. 1 and 3, however, shows that the franc also weakened quite regularly without automatically triggering noticeable devaluation expectations. If an exchange rate within a target zone is characterized by negligible devaluation probability, the target zone may be considered completely credible. Similarly, a relatively high devaluation probability may be associated with a low-credibility target zone. Various studies of the ERM have concluded that its credibility has increased significantly since 1987, and hence, that the crisis of 1992r1993 came as a complete surprise ŽRose and Svensson, 1994; Frankel and Phillips, 1992.. Looking at the Belgian devaluation probability in this context, it can be concluded that the credibility of Belgian exchange rate policy has indeed increased significantly since 1987. The announcement in 1990 of a monetary policy exclusively aimed at 16 The devaluation probabilities are also comparable to what is usually found in the literature ŽVlaar, 1992; Bekaert and Gray, 1996.. In addition, it should be recalled that our estimates represent a lower limit for the actual probability of a realignment. 526 M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 exchange rate stability has undoubtedly supported this process. Finally, it can be noted that the tensions of 1992r1993 could not have been predicted by our model either. In our model for the Dutch guilder, the first devaluation is preceded by a few periods with a discernible devaluation probability Ž4% maximum., while the second devaluation can be considered a complete surprise. This devaluation was the result of political rather than economic developments, and the former developments are not included in the model. Credibility has been consistently high since the early 1980s; the probability of devaluation has remained negligible throughout the latter period. In order to assess the model’s real predictive performance only out-of-sample results should be utilized. Given the complexity of generating estimates with a very limited number of observations, however, this would have been a rather fruitless effort for the initial years of the ERM, when, unfortunately, the exchange rate tensions were most pronounced. Our second test therefore supplements the within-sample results from Figs. 3 and 4 by providing a comparison of our preferred model’s out-of-sample forecast errors with the forecast errors of the random walk model and those implied by uncovered interest rate parity ŽUIP.. Table 6 lists the results for predictions from 1 to 4 weeks ahead. The table shows that for both countries our preferred model’s predictive power is larger than that of the two alternative models. This is striking, as virtually no empirical model can beat the random walk ŽMeese and Rogoff, 1983., as is confirmed once more by a comparison of the latter with the UIP-model. Another striking feature is that the model’s forecasts even improve if the forecast period is extended to 3 weeks, even though the model has been estimated on the basis of 1-week observations. All in all, it is fair to conclude that the model has a certain degree of predictive power. As far as both currencies are concerned, however, this is primarily a matter of historical relevance, as exchange rate movements between the deutsche mark, the Belgian franc and the Dutch guilder have come to an end since the recent adoption of the euro. Table 6 Relative improvements in 1- to 4-week forecast errors a Belgium Ž% against RW. Belgium Ž% against UIP. The Netherlands Ž% against RW. The Netherlands Ž% against UIP. a 1 week 2 weeks 3 weeks 4 weeks 0.54 2.79 3.77 6.68 4.10 10.72 6.49 13.43 7.15 16.60 6.71 17.00 4.59 16.81 1.71 13.83 NB: The forecasted period is April 1991 until August 1993. The parameters have been estimated over 3 preceding years and have been updated each half year Žso that the maximum forecast horizon is 26 weeks ahead.. RW denotes the ‘random walk’ model: Ew Stq 1 < It x s St , and UIP denotes the forecast obtained by uncovered interest parity: Ew Stq 1 < It x s St q i t y i tDM . M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 527 7. Summary and conclusion We have reviewed the movements in exchange rates within target zone exchange rate systems such as the ERM, focussing on the experiences of the Belgian franc and the Dutch guilder in particular. In the theoretical target zone literature, which mainly emerged during the late 1980s, the modeling of realignments stemming from speculative attacks is rather limited. Target zone models are, however, suitable for analyzing the movements in exchange rates within a credible target zone. These models predict, for instance, that exchange rates have a tendency to return to the central parity Žmean reversion. and that their variability diminishes near the boundaries of the target zone Žthe S-effect.. These implications can be tested by means of econometric target zone models. Our estimations have shown that an S-effect can only be observed for the Belgian franc, while mean reversion can be asserted for both currencies. Additionally, our results suggest that exchange rate volatility can be adequately modeled by means of a GARCHŽ1,1. process. This implies that the volatility for both currencies depends on previous observations and that there is a clustering of extreme values. Exchange rate tensions in Belgium may be foreshadowed by movements in the inflation differential and the level of reserves, whereas for the Netherlands, sizeable exchange rate movements have been preceded by changes in the level of reserves and changes in interest rate differentials. The latter jumps were, however, rarely high enough to force a realignment. Finally, our model displayed a reasonably adequate degree of predictive power with respect to both currencies. Acknowledgements We would like to thank Peter van Bergeijk, Lex Hoogduin, Pieter Otter, Job Swank and an anonymous referee for useful comments on an earlier version. Janet Bungay provided expert editorial assistance. Appendix A. Definition of the model variables This appendix lists in greater detail the variables used in the model presented in Section 4 and will also briefly dwell upon the expected signs of the accompanying parameters. 䢇 䢇 For the exchange rate changes Ž ⌬ S t . logarithmic differentials were taken: lnŽ S trSty1 ., where S t denotes the price of the deutsche mark expressed in units of the domestic currency. The relative position within the band Ž PBt . of the ERM was defined as Ž S t y Ct .rw1r2ŽUt y L t .x. In this formula, Ct is the log of central parity, Ut the log of the upper limit and L t the log of the lower limit of the target zone. In M.A. Klaster, K.H.W. Knot r Economic Modelling 19 (2002) 509᎐529 528 䢇 䢇 䢇 䢇 䢇 addition, y1 - PBt - 1, with PBt ) 0 if the francrguilder is relatively weak against the deutsche mark. Owing to the mean-reverting behavior that results from intramarginal interventions, ␤ 9 is assumed to be negative. The distance from the central parity Ž< PBt <. is given by the modules of the PBt defined above. If the exchange rate is closer to the boundaries of the target zone, it could make relatively larger jumps within the band and the probability of a realignment Žwhich is often accompanied by a jump. is also enhanced. The expected sign of ␤5 is therefore is positive. However, ␤ 13 is likely to be negative if it is assumed that the Krugman model applies to the quiet distribution Žin which exchange rate movements near the edges of the band are tempered by expectations concerning imminent policy interventions .. The slope of the yield curve Ž SYCt . may be approximated by subtracting the 1 1-year euro interest rate from the 1-month euro interest rate: i 12 t y i t . Euromarket interest rates were chosen because the euro currency markets are the most efficient. As noted in Section 4, an inverting yield curve is associated with exchange rate tensions; the expected sign of ␤ 2 is therefore negative. The central bank’s relative level of reserves Ž LR t . for this model was defined as the present stock divided by a 4-week moving average: R tr 14 Ý R t y i . If LR t - 1, this indicates a relative deterioration of the reserve position. Such a deterioration is assumed to trigger an increase in the mean of the jump, implying that the accompanying coefficient Ž ␤4 . would be positive. The 1-week interest rate differential vis-a-vis ` Germany Ž IDt . may be defined as i t y i tD M and is based on Euromarket interest rates too. Speculative tensions are typically neutralized by monetary authorities through an increase in shortterm interest rates. Hence, such an increase points to rising exchange rate tensions and, consequently, to a larger jump; ␤6 is expected to be positive. 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