explaining and forecasting the euro/dollar exchange rate through a non-linear threshold model

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Explaining and forecastingthe euro/dollar exchange ratethrough a non-linear thresholdmodelAsmara JamalehPublished online: 15 Oct 2010.

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Explaining and forecasting the euro/dollarexchange rate through a non-linear thresholdmodel*ASMARA JAMALEHIntesaBci – Research Department, Via Borgonuovo, 2 – 20121 Milan, ItalyEmail address: [email protected]

A linear econometric error correction model (ECM) model is built, based on shortinterest rates, gross domestic product (GDP) growth expectations and in� ation differ-entials, in order to explain the euro/dollar exchange rate dynamics and provide reliableforecasts. This speci� cation performs well. However, the introduction of non-linearthreshold dynamics provides a better understanding of ‘abnormal’ features other thandeviations from long-run equilibrium levels, allowing for the possibility of asymmetricbehaviour. Empirical evidence of this is found in the actual dynamics of the euro. Thenon-linear speci� cation performs better than the linear model in both in-sample � ttingand out-of-sample forecasting, showing that fundamentals hold, working also throughsome non-linear mechanism, in explaining the euro/dollar dynamics.

Keywords: euro/dollar exchange rate, economic fundamentals , long-run equilibrium,outliers, non-linearity, threshold models

1. INTRODUCTION

When the euro was introduced on 1 January 1999 few observers expected itwould have experienced such a prolonged depreciation against the US dollar; infact, the euro dropped by about 27% from its introduction to September 2000.Both its status as a young currency and its somehow peculiar behaviour requirea deeper understanding of the factors driving its dynamics. In this respect, the� rst natural question that arises is whether the euro is not driven byfundamentals and to what extent its developments were predictable. Ex post theanswer is yes, in the sense that the euro/dollar exchange rate evolved in linewith GDP growth and short-term interest rate differentials between the EuroArea and the USA. Hence, in the presence of correct expectations on funda-mentals, the weakening of the euro would have not been a surprise. Howeverthe magnitude of the depreciation could be debated, raising the doubt of anundervaluation of the euro with respect to the dollar.

* To save journal pages, a number of � gures and tables are in a separate appendix. This is availablefrom the author on request. When such � gures and tables are referred to in the text they aredenoted Fig. A1, Table A1 and so on.

The European Journal of FinanceISSN 1351-847X print/ ISSN 1466-4364 online © 2002 Taylor & Francis Ltd

http:/ /www.tandf.co.uk/ journalsDOI: 10.1080/13518470210167301

The European Journal of Finance 8, 422–448 (2002)

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A behavioural linear ECM model is hence speci� ed, which is aimed at both� nding the main determinants of the euro/dollar exchange rate and obtainingreliable forecasts. The hypothesis is tested that short interest rate differentials,expected GDP growth differentials and in� ation rate differentials possibly drivethe long term dynamics of the euro/dollar exchange rate, on the ground ofempirical evidence. The adopted speci� cation can be judged as satisfactory, asit beats the random walk and provides the correct positioning with respect toone-month forward rates in more than 50% of cases.1 However, the linear modelis not able to capture outliers and some possibly ‘abnormal’ features of theeuro/dollar exchange rate (if compared with the base model assumptions) , asthese are not simply related to a problem of temporary deviation fromequilibrium levels.

The presence of non-linearity is hence assumed to be the main cause of thesepeculiarities , and a set of alternative threshold regression models is built,maintaining, apart from the introduction of non-linear dynamics, the samestructure of the linear model. The non-linear speci� cation proposed allows thepossibility of taking into account asymmetric responses of the euro/dollarexchange rate to similar impulses, depending on some ‘state’ condition being inplace. The better in-sample � tting and out-of-sample forecast performanceexhibited relative to the linear model seems to con� rm this hypothesis, showingthat, for instance: (i) monetary policy interventions may make sense only whena signi� cant degree of undervaluation of the euro, which puts at risk thein� ation stability condition, is underway, while the same consideration does notnecessarily hold in the opposite case; (ii) the euro seems to be more vulnerablewhen GDP growth differentials are unfavourable while, in the opposite situation,positive factors may amplify their upward in� uence by reinforcing their cross-effects; (iii) extraordinarily positive stock market performances may temporar-ily decouple exchange rate dynamics from macroeconomic fundamentals .Evidence of these general � ndings is present in the actual behaviour of the euro/dollar exchange rate already during its � rst one and a half years of life.

2. THE ‘YOUNG CURRENCY PROBLEM’ AND THE DATA

The choice of the data used for the models described has required someassumptions as far as the period preceding 1 January 1999 is concerned, sincethe euro was introduced on that date.

1 From an exquisitely market-oriented perspective, it is important for a model to provide forecastswhich make possible � nancial gains with respect to the forward (exchange rate) prediction. If, forinstance, the exchange rate appreciates, and the model forecasts a larger appreciation than theforward exchange rate does, it is convenient to enter a long position at the forward price, and closethe position with a sale at the � nal spot market price. In this case, the indications provided by themodel are good within a speculative context. The ‘good performance’ condition with respect to theforward exchange rate prediction can be summarised as follows:

(Mt 2 Ft) 3 (At 2 Ft) . 0,

where Mt is the model forecast for time t, Ft is the forward exchange rate and At is the actual valueof the exchange rate.

423Explaining and forecasting the euro/ dollar exchange rate

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The basic assumption is that before the introduction of the single currencythe ‘natural’ substitute for the Euro Area as the counterpart of the USA wasGermany. Another alternative could have been that of building an appropriateweighted average of data taken by each of the countries which now belong tothe Euro Area, but from an exquisitely economic perspective this would nothave made much sense. On the contrary our choice is also consistent with thefact that German interest rates played a major role in the convergence processwhich took place in the run-up to the European Economic and Monetary Union.As a consequence, the DEM/USD exchange rate corrected for the EUR/DEMparity is used before 1999. Short interest rates are 3-month interbank rates forboth Germany and the USA. Both exchange and interest rate data are monthlyaverages of daily data. GDP data are GDP annual growth rate expectations, takenfrom Consensus Economics Inc. (Consensus in the following), which collectsevery month the predictions of ‘over 200 prominent � nancial and economicforecasters’ on a range of macroeconomic and � nancial variables (available inthe publication Consensus Forecasts). These data are a valid proxy for aggregateexpectations of GDP growth. Monthly data taken from Consensus Forecasts arethen used to calculate weighted monthly averages of expected growth for thecurrent and the following year, where the weights for the current year decreaseas year-end approaches, while the weights for the following year correspond-ingly increase. German GDP growth data are used until December 1997, while for1998 a weighted average of German and aggregate Euro Area data is employed,where the weights for German growth decrease as year-end approaches, whilethe weights for the Euro Area growth correspondingly increase. Finally, in� ationdata are annual in� ation rates of CPI, both for Germany, the Euro Area and theUS. The same procedure as for the GDP series is applied here to German andEuro Area in� ation data in 1998, in order to smooth the transition path.

Monthly data are used, from January 1992 to August 2000. All data, bar for theConsensus’ data, are taken from the data provider Datastream. In-sampleestimation results cover the full period while, when the models are used toobtain forecasts, estimation is conducted on a reduced sample, ending inDecember 1998, and the remaining data form the forecast set.

3. A LINEAR MODEL BASED ON ECONOMIC FUNDAMENTALS

The model proposed here is aimed at seeking the key determinants of the euro/dollar exchange rate, with the main purpose of producing forecasts for asuf� ciently large forecasting horizon even in a more than one-step aheadprediction framework. It is based on economic fundamentals , in order tocapture the trend developments of the exchange rate in line with a basemacroeconomic scenario, and exploits the idea of a long-run ‘equilibrium’relation between the exchange rate and its determinants.

As it is variously shown in the literature, the key assumption which is testedhere is that the main driving forces of the exchange rate are the differentials ofinterest rates, GDP growth and in� ation rate between the two areas involved.However, both economic growth and in� ation enter in this model in a slightlydifferent way with respect to the traditional approach. In fact, following a

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relatively new research line already in use in � nancial markets, the hypothesisconsidered here is that the exchange rate is driven by the differentials of theexpectations of GDP growth, in place of the actual differentials, thus re� ectingthe fact that inevitably market participants take their decisions using a forward-looking approach. The other point is quite similar since, turning to the in� ationrate differential, this enters the model one-period lagged, thus taking intoaccount a sort of ‘announcement’ effect, since in� ation � gures are usuallyavailable with a one-month lag.

Consequently, the long-run relation is as indicated in equation (1) :

EURDOLt 5 a 0 1 a 1 SRt 1 a 2 GDPt 1 a 3 CPIt 2 1 1 ut (1)

where EURDOLt is the euro/dollar exchange rate, SRt is the short interest ratedifferential, GDPt is the expected GDP growth differential and CPIt 2 1 is thein� ation rate differential one-time lagged, and ut is a disturbance term. Alldifferentials are between the Euro Area variable and the correspondent USvariable, hence the short rate differential is the difference between the EuroArea interest rates and the US interest rates, and so on.

According to economic theory, the coef� cients a 1 and a 2 should have apositive sign, while a 3 should be negative. In fact: (i) higher short rates, asopposed to long rates, which incorporate an in� ation premium, should favour acurrency, since they make � nancial investments attractive in relative terms,through a liquidity effect (MacDonald, 1994); (ii) higher GDP growth, as far as ithints at solid economic conditions of the business cycle, should bene� t acurrency by attracting � ows of business investments ; while (iii) higher in� ationshould negatively affect a currency, by discouraging both � nancial and businessinvestments , due to a lack of con� dence with respect to the healthy state ofeconomic fundamentals .

For our purposes, a cointegration analysis is conducted following theJohansen methodology (Johansen, 1991, 1995). In fact, if EURDOL, SR, GDP andCPI are linked through a cointegration relation, the so-called long-run equation(which enters the ECM term in the � nal equation) – which includes thesevariables – can be estimated.

First of all the hypothesis of integrated variables has to be tested in order toperform a cointegration analysis. If it can be accepted that GDP growth rates orin� ation rates are non-stationary, it might seem puzzling that the differencebetween European and US short rates, GDP growth rates and in� ation rates arenon-stationary as well. But important changes in the considered variablesthroughout the sample period examined might in part explain this � nding.German short rates, for instance, experienced a pronounced fall from 10% to 4%from 1992 to 1995, which in turn favoured a marked acceleration in GDP growthstarting from 1993. Also German in� ation enjoyed a sharp decrease from 6% to2% during the same period. But this hypothetical arguments � nd a morerigorous support in the results of some nonstationarity tests. In fact, using theAugmented Dickey–Fuller test, it has been found that all series individuallytaken (i.e., EURDOL, SR, GDP and CPI) are I(1) processes (see Table A1).


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At this point, the Johansen cointegration rank tests can be conducted. TableA2 summarises the results and shows that the cointegration condition is met,and that the tests indicate the presence of one cointegrating relation.

For the purposes of this paper, the assumption is made of weak exogeneity forall the explanatory variables with respect to the exchange rate, since the mainfocus here is on the exchange rate dynamics conditional on a given macro-economic scenario. A normalization with respect to the EURDOL variable istherefore made here in order to solve the problem of the non-identi� cation ofthe cointegrating vector. Letting the coef� cient of EURDOL be equal to 1, thenormalized cointegrating coef� cients of the other variables obtained from theJohansen procedure are contained in Table 1. As can be seen from Table 1 thecoef� cients have the correct signs and are statistically signi� cant according tothe ï 2 ï t-value criterion.

At this point, using the aforementioned hypothesis of weak exogeneity, aunivariate Error Correction Mechanism (ECM) including the above-estimatedlong-run relation and some dynamic terms is speci� ed for the euro/dollarexchange rate as shown by equation (2):

D EURDOLt 5 b 1D EURDOLt 2 1 1 b 2D EURDOLt 2 2 1 b 3ECMt 2 1 1 b 4 D SRt1 b 5 D GDPt 1 b 6 D CPIt 2 1 1 b 7D SRt 2 1 1 . . . 1 b i D GDPt 2 11 . . . b k D CPIt 2 2 1 . . . 1 . . . 1 e t (2)

where the ECM term contains the above long-run relation. All the othervariables are differenced, hence D xt 5 xt 2 xt 2 1. Using the general-to-speci� cmethodology, only three dynamic terms for the exogenous variables resulted tobe signi� cant. Estimation results are reported in Table 2.

Also here, estimation results show that all coef� cients have the correct signsand are signi� cant. In particular, the autoregressive component is relevant,which is not a surprise for exchange rates but, above all, the ECM term issigni� cant, thus indicating that both economic fundamentals and any deviationfrom the long-run equilibrium level play a key role in the determination of theexchange rate dynamics. Among the exogenous dynamic terms, only thewidening or the narrowing of the interest rate and of the expected GDP growthdifferential were signi� cant throughout the variables-selection cycle.

Preliminary output statistics are in favour of a goodness-of-� t judgement, as itcan be easily seen even from Table 2 and from Fig. A1, which plots the actualand � tted values of the euro/dollar exchange rate changes together. As Fig. A1shows, the model is able to reproduce quite closely the developments of the

Table 1. Estimation results of the long-run equation

SR coeff.: a 1 GDP coeff.: a 2 CPI coeff.: a 3 Constant termcoeff.: a 0

Coef�cient 13.69 32.05 –15.46 1.41t-value 3.52 3.30 –2.38 18.94

Note: these are the normalised cointegrating coef�cients for the cointegrating equation.

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exchange rate, even capturing the direction of changes, which is very importantin this contest, also from an operative point of view. However, it does notperform so well in dimensional terms, since larger variations generally remainunexplained, which is not a surprise within the linear framework, an issue thatwill be analysed more carefully in the following. This is not a surprise evenlooking at the R2 value, which is relatively good if compared with empiricalevidence in the applied literature, but is not so high in absolute terms. Finally,estimated residuals are checked. As Table A5 shows, autocorrelations andpartial autocorrelations con� rm the outcome of the h-Durbin test (in Table 2) ofuncorrelated residuals. Even the hypotheses of serial correlation and hetero-skedasticity are strongly rejected, while normality is accepted (Tables A6and A7).

The Chow tests (Table A8) do not detect any structural change at the timewhen the euro was introduced. However, both the shorter dimension of thesecond period used for testing and the procedure employed in order to smooththe transition from German to euro data may have contributed to this outcome.In any case, even bringing the break point backwards in time, no structuralchanges are individuated. Also the Ramsey tests (Table A9) show that theestimated relation is stable. But it is worth mentioning that in some sub-samplessometimes this test fails, thus indicating that the linear speci� cation may needsome improvement . At this point, also considering the aforementioned fact thatthis model is unable to capture larger variations, this hint at the possibility thattaking into account the presence of non-linearity may make sense.

But before passing to non-linear analysis, forecasts from the model areobtained and an interpretation of it is provided.

Table 2. Estimation results for the linear ECM model

Variable Coef�cient t-value p-value

D EURDOLt–1 0.24 2.60 0.01ECMt–1 –0.03 –2.25 0.03D SRt–2 –2.72 –1.97 0.05D GDPt 4.73 3.48 0.00D GDPt–1 –2.77 –2.30 0.02

R2 0.21 Mean dependentvariable

–0.003

Adjusted R2 0.18 Std. deviation of dep.variable

0.03

Standard error ofregression

0.03 Akaike informationcriterion

–4.39

Sum of squaredresiduals

0.07 Schwarz criterion –4.27

h-Durbin 0.59 F-statistic (probability) 6.50 (0.00)Durbin-Watson statistic 1.95 Log likelihood 233.45

Note: this set of diagnostics shows that the adopted speci�cation is good.


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3.1 INTERPRETATION OF THE LINEAR MODEL AND FORECASTSOverall, the linear model can be considered good, both from an econometricpoint of view and for the applied purposes of market operators.

It shows that fundamentals work in describing the exchange rate dynamics,both in static terms, when the possibility of temporary deviations from the long-run equilibrium is allowed, and in dynamic terms, especially when it isdemonstrated that changes in the relative GDP growth expectations are highlysigni� cant in determining the direction of the exchange rate. This is what thevalues and the statistics of the ECM and D GDP coef� cients say.

But this can also be clear looking at the out-of-sample properties of the model,i.e. by examining its forecasting performance . One-step and multi-step aheadpredictions are obtained, taking parameters estimated up to December 1998 inboth cases. As far as the exogenous variables are concerned, both actual valuesand Consensus’ forecast values have been employed. The former choice isconsistent with exact expectations on the underlying macroeconomic scenario,and has the double advantage of (i) avoiding any distortion related to not-correct forecasts on the exogenous variables, and (ii) showing whether theeuro/dollar exchange rate movements were really unpredictable even wheneconomists and analysts were able to predict macroeconomic fundamentals .The latter alternative can be useful for operational purposes, since it gives agauge of how much of the forecast error is due to an incorrect prediction of theunderlying macroeconomic scenario.

Considering the predictions obtained using the actual values for the exoge-nous variables, as Fig. 1 and Table A10 show, the good forecasting performanceof the model is evident, especially in terms of precision and direction of change,when looking at the one-step forecasts (as also quantitative methods offorecasting performance evaluation will show a few sections below). But it isperhaps even more relevant that the full path of the euro/dollar exchange rate,

0.90

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Note: model values have been obtained asforecasts from the model with actualexogenous data, while consensus values havebeen obtained as forecasts from the modelwith Consensus’ exogenous data.

Note: the same considerations as for Fig. 1apply to Fig. 2.

Fig. 1. One-step forecasts from thelinear model

Fig. 2. Multi-step forecasts from thelinear model

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with a � rst downward phase, a second stage of up-trending and a � nal phase offurther depreciation, is well captured when the model is employed to providemulti-step ahead forecasts (see Fig. 2, Fig. A4 and Table A11) . As for the smallerprecision of multi-step predictions – especially when the time horizon is broughtforward – this might also be interpreted as a gauge of the under-valuation of theeuro (widely reckoned by analysts and institutional players, such as theEuropean Central Bank itself) with respect to the dollar. In fact this model hasthe double merit of taking account of dynamics (which is consistent with adepreciating/appreciating euro when GDP growth differential widened/nar-rowed) and the long-run equilibrium level, which, apart from the direction ofchanges, measures the sign and magnitude of the deviation from equilibrium. A� nal consideration is worth mentioning. In fact, the computation of the dynamicforecasts requires an hypothesis of strong exogeneity. In this case, theConsensus’ forecasts of the GDP growth rates likely take into account the pastperformance of the system. It would therefore be reasonable to deem that thepossible lack of strong exogeneity could explain the poor performance of themodel in providing the multi-step predictions. However, the Granger causalitytest accepted the null hypothesis that D EURDOL does not Granger cause D GDP(with a 32.5% probability) while it rejected the hypothesis that D GDP does notGranger cause D EURDOL (with a 1.3% probability). Given the assumption ofweak exogeneity made earlier, this shows that there should not be a problem oflack of strong exogeneity. In addition, when the t1 1, t1 2, t1 3, . . . multi-stepahead predictions are computed, also the Consensus’ GDP growth rate forecastsemploy the information available up to time t. It is therefore likely that theprediction error is also due to the erroneous GDP growth scenario provided byConsensus at time t.

This is even more evident when one looks at the same � gures and tablesabove, but considering the predictions obtained using the Consensus’ forecastvalues for all the exogenous variables. In fact, in this case the importance ofdesigning a good underlying macroeconomic scenario is evident. This is not soevident in the case of the one-step forecasts, since the adjustment takes placestep by step, but it is apparent in the multi-step case. Consensus’ forecasts hada scenario much more favourable to the euro than it actually was, since it hadpredicted that both GDP growth and interest rates would have evolved in favourof the euro. This was responsible for both the distance and the divergenceobtained for the dynamic (Consensus’) forecasts with respect to both actual andmodel’s forecast (consistent with actual values for the exogenous variable)values of the euro/dollar (EUR/USD) exchange rate. The conclusion is that ifeconomists and analysts had been good enough at forecasting the macro-economic environment, they would have been able to partly predict thedepreciation of the euro against the dollar.

But is this effectively true, and did both the actual and forecast euro-dollarexchange rate go in line with fundamentals? Figure A5 hints to a positive answer,since it shows that in most cases the actual euro/dollar exchange rate is in linewith the economic fundamentals , especially with the GDP growth differentialsbut, albeit in less cases, also with the interest rate and in� ation differentialsbetween the Euro Area and the USA.


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Finally, as far as the forecasting performance of the model with respect to theforward rates is concerned, this is acceptable (see Fig. A6) , since the percentageof successful indications with respect to the forward predictions is 53%.

4. NON-LINEARITIES IN THE EURO/DOLLAR EXCHANGE RATE

Even though the above-built linear model provides a satisfactory speci� cationfor the euro/dollar exchange rates, some already-mentioned elements hint at thepossibility that some form of non-linearity is present in the euro–dollardynamics. These elements are (i) the incapability of capturing larger variations,(ii) the negative outcome of the RESET test in some sub-samples, (iii) the � ndingthat the ECM term in the dynamic equation is not equally signi� cant acrosstime, even though these latest results might be distorted by the fact thatestimating sub-sets of the full database reduces the dimension of the samples.However this evidence allows nothing more than conjectures, hence someformal linearity tests have to be performed in order to derive more reliableconclusions.

The form of non linearity which is taken into account here is the threshold-type non-linearity (Tong and Lim, TL80 in the following; Guegan, 1994). A simplebut general two-regimes threshold regression model is as equation [3] shows:

a0 1 a1yt2 1 1 . . . 1 aj0xt 1 aj1xt2 1

1 . . . 1 am0zt 1 am1zt2 1 1 . . . 1 s 1e t if wt-d # s1yt 5 (3)

b0 1 b1yt2 1 1 . . . 1 bj0xt 1 bj1xt2 1

1 . . . 1 bm0zt 1 bmlzt 1 1 1 . . . 1 s 2e t if wt-d . s15

where wt-d is the threshold variable (which could be a lagged endogenousvariable or an exogenous variable) , d is the lag parameter, s1 is the thresholdvalue. In this model [yt] is an observable output and [xt], [zt], . . . are observableinputs. [ s j e t, j 5 1, 2] are sequences of heterogeneous strict white noises, with 0mean and 1 variance, each independent from the input sequences [xt] , [zt], . . . .These two error sequences are also independent from each other. This model isnon-linear across time, but is locally linear in the dimensional space of thethreshold variable. The threshold value individuates , time by time, the break-point between the two regimes, thus causing, deterministically, the transitionacross regimes. Of course, more regimes can be individuated. More precisely,this model is a Closed-loop Threshold Autoregressive System or TARSO (TL80,Tong, 1983, 1995), where both an exogenous variable and a lagged endogenousvariable may be chosen as the threshold variable. In this case, yt is D EURDOL,while the input variables are ECMt 2 1, D SRt, D GDPt, D CPIt 2 1. Examples in theliterature of applications of threshold models to � nancial variables andexchange rates are provided, among others, by Cao and Tsay, 1993) (CT93 in thefollowing) , Chappell et al. (1996) , Lundbergh and Terasvirta (1998) and Zakoian(1994).

Given the assumption of weak exogeneity, illustrated earlier, the non-linearityanalysis may be conducted with reference only to the single equation forD EURDOL (equation (3)). In this way, there should not be any problem with

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conducting inference in the starting linear model in the usual way, since thepresence of threshold non-linearity is a successive alternative hypothesis,which gives rise to a competitive model. As for the choice of keeping thecointegration vector constant across regimes and leaving the cointegratingrelationship to be linear, this � nds support in both an economic rationale andempirical evidence in the currency market. In fact, the ECM model (equation(2)) allows for both the presence of a long-run relationship and temporarydeviations from this equilibrium. Given also that estimates of the long-runequation have been found to be highly signi� cant, the focus here is on thedynamic form only, where the problem of the deviation from the equilibriumbasically arises. This is also in line with empirical evidence, which shows that agiven impulse does not always cause the expected response in the exchangerate movements, or because this is not large enough or because otherconditions are in place that play a more important role than the impulse itself.This does not necessarily mean that the long-run relationship has changed, butsimply that temporary conditions may occur that alter the dynamical adjust-ment towards the equilibrium. As a consequence , non-linearity focuses only onthe dynamical part of the model.

The linearity tests applied here (Table A12) are the threshold test and thegeneral non-linearity test, which are speci� cally designed to detect threshold-type non-linearity. These are F-tests, based on the so-called arranged autoregres-sion. In the case of the threshold test, for a prespeci� ed AR order p an arrangedautoregression (i.e. an autoregression where data are arranged according to theincreasing dimension of the threshold variable) of order p is � tted recursively tothe considered series ( D EURDOL, in this case). The standardized predictiveresiduals from this autoregression are calculated and then regressed in turn onthe lagged terms (from lag 1 to lag p) of D EURDOL and a second series ofresiduals is obtained. An F-test statistic is � nally provided, stemming from thecomparison between the summation of the standardized predictive residualsand the second group of residuals, adjusted for the degrees of freedom (foredetails see CT93) . The general non-linearity test differs from the threshold testmainly for the introduction of further terms in the regression from which thesecond series of residuals is calculated (for details, see Tsay, 1989).

As Table A12 shows, the null of linearity is rejected if autoregressive dynamicsof at least order 2 is considered. However, in line with the speci� cation of thelinear model (see Table 2) and with the results obtained from the autocorrela-tion analysis (see Table A5) of the euro/dollar changes, only the � rst twoautoregressive terms are considered, since higher-order terms were not sig-ni� cant. These tests are speci� cally designed for the case in which thethreshold variable is a lagged variable, hence their implementation depends onthe delay parameter which is speci� ed. Consequently the table above refersonly to the case where the threshold variable is D EURDOLt 2 1, since differentalternatives seem to be less straightforward from the point of view of theeconomic theory. In fact, as it will be seen in the following, simultaneous (notlagged) exogenous threshold variables are entertained here (with the obviousexception of the ECM term) . In fact, the focus here is on the static relationbetween the euro/dollar exchange rate and its key determinants .


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However this is not a problem, since another way of detecting threshold-typenon-linearities which are tied to speci� c threshold variables is provided by therecursive local � tting approach (RLF) . This is a recursive estimation of theregression coef� cients which is conducted in the space of the thresholdvariable. Results can be shown through scatterplot graphs (see Fig. 3 or Fig. A14and Figs A7–A16) , which plot recursive estimates of the coef� cients against theordered threshold variable. If non-linearity is present, estimates should appearinitially stable, then should show some unstable behaviour in the form of ajump, and � nally might become stable again. In this case two regimes areindividuated , but the extension to more regimes is straightforward . The RLFregression involves the same six regressors of the linear case, that is:D EURDOLt 2 1, D EURDOLt 2 2, ECMt2 1, D SRt, D GDPt, D CPIt 2 1. Five regressions are thenperformed, which use D EURDOLt 2 1, ECMt 2 1, D SRt, D GDPt, D CPIt 2 1 respectively asthe threshold variable. In each case � ve graphs are plotted, one for everycoef� cient.

As Figs A7–A16 show (not all the graphs are reported here, but only somewhich are among the most signi� cant), a break is evident almost in everyscatterplot, and is usually located around the zero-value, hence individuatingtwo regimes for each regression. When the threshold variable is (i) D EURDOLt 2 1,the � rst (second) regime describes the case of a weakening (strengthening)exchange rate, (ii) ECMt 2 1, the � rst (second) regime describes the case of anundervalued (overvalued) exchange rate, (iii) D SRt, the � rst (second) regimedescribes the case of falling (increasing) interest rate differentials, (iv) D GDPt,the � rst (second) regime describes the case of falling (increasing) GDP growthdifferentials, (v) D CPIt 2 1, the � rst (second) regime describes the case of falling(increasing) in� ation differentials.

Fig. 3 Fig. 4

-1

-0.5

0

0.5

1

1.5

-0.025 0.025 0.075

Note: on the Y-axis there is the coef�cient ofECMt–1, and on the X-axis the thresholdvariable D GDPt in increasing order.

Note: on the Y-axis there is the coef�cient ofD EURt–1, and on the X-axis the thresholdvariable D NASt in increasing order.

Figs 3–4. RLF coef�cients’ estimates of the threshold models

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This means that the coef� cients of the same variables may be differentdepending on what � nancial and/or macroeconomic conditions are in place, i.e.depending on which regime is active, using the model’s syntax. This introducesthe possibility of an asymmetric reaction of the exchange rate to identicalqualitative/quantitative impulses.

4.1 How to take the role of the stock market into account

Up to now the analysis has considered only macro-variables. However, thequestion sometimes arises whether international stock exchange developmentscould exert some in� uence on the exchange rate. Given the leading role of theUS equity market within the international environment, Nasdaq monthlypercentage variations are employed here, since they exhibit a larger correlationwith the euro/dollar dynamics than the S&P and the Dow Jones indexes. Thisissue became especially pressing beginning with November 1999, when the eurostarted an uninterrupted depreciating path to hit its historical lows in May 2000.The extraordinarily large gains posted by Nasdaq stocks in November, attracting� ows of capitals towards the US market, cast the doubt that this could add evenmore to the euro weakness.

In order to take into account the role of the equity prices, an experiment isconducted here using stock yields as the threshold variable, without introduc-ing them as an explanatory variable in the regression. If it comes out to besigni� cant, this solution has the advantage of not requiring detailed expecta-tions as far as the bourse’s performance is concerned, but simply to expresswhether it is expected to fall below or go above a given threshold value.Scatterplots are shown as indicated.

All scatterplots (Fig. 4 and Figs A17–A21) show a major break, often with aclear parameter shift, around 3%, thus discriminating two regimes, i.e. where the� rst one includes losses and ‘normal’ gains, and the second one onlyconsiderable gains.

4.2 Non-linear estimation results

Six non-linear threshold models are speci� ed here, which use the sameexplanatory variables of the linear model, as equation (4) shows:

a1 D EURDOLt 2 1 1 a2 D EURDOLt 2 2 1 a3ECMt 2 1 1 a4 D SRt

1 a5 D GDPt 1 a6D CPIt 2 1 1 s 1 e t if wt 2 d# s1D EURDOLt 5 (4)

b1 D EURDOLt 2 1 1 b2 D EURDOLt 2 2 1 b3ECMt 2 1 1 b4D SRt

1 b5 D GDPt 1 b6 D CPIt 2 1 1 s 2 e t if wt 2 d . s15

The threshold variable wt 2 d is respectively (i) in model (1) D EURDOLt 2 1, (ii) inmodel (2) ECMt 2 1, (iii) in model (3) D SRt, (iv) in model (4) D GDPt, (v) in model (5)D CPIt, and (vi) in model (6) D NASt. For each model, after having individuatedfrom the scatterplots a plausible range for the threshold value, the correctspeci� cation is chosen according to the Akaike or the Schwarz criteria and the� nal estimation is hence conducted (Tsay, 1989) . Estimation results are shownin Tables 3–8.


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Estimates clearly show that, when different macroeconomic conditions occur,i.e. when one regime is activated in the place of the other, the same explanatoryvariables may play a different role. Speci� cally, a given determinant of theexchange rate may be signi� cant in one regime and not signi� cant in the other,or may have a different weight. It is therefore important to be able to recognizewhich regime is active, since the same event could have to be read in a differentway. An inspection of the outcome of every model proposed follows here:

Model [1] – Table 3: if at the previous time the exchange rate posted negativeor small positive changes, the short rate differential has a relevant weight,which may re� ect the potentially active role of monetary policy in defending acurrency or, which is more consistent with the euro case, it re� ects the majorrole of policy rates when euro weakness poses some threat to price stability.Instead, when some currency strengthening occurred at t2 1, the autoregressivecomponent and GDP growth differentials become relevant. However, given thesame magnitude of the AR(1) and AR(2) coef� cients, in the presence of twosimilar positive variations in a row, the prevailing role is played by GDPdifferentials. If, at time t2 2, the exchange rate depreciated, this should insteadstrengthen at time t, unless GDP spreads are unfavourable.

Model [2] – Table 4: the outcome of model (2) is even more relevant. In fact,it shows that if the euro is highly undervalued with respect to its long-runfundamental equilibrium level (i.e., the ECM term is negative), only the short-rate differential may play a signi� cant role. This con� rms the importance of anactive monetary policy when the euro is weaker with respect to somefundamental equilibria, therefore threatening price stability, not when it is weakper se. This is consistent with the way the ECB conducted monetary policy in

Table 3. Estimation results for threshold model (1)

Threshold variable: D EURDOLt–1

Regime 1 Regime 2

Variable Coeff. t-value Coeff. t-value

D EURDOLt–1 0.29 1.40 0.28 2.19D EURDOLt–2 0.08 0.35 –0.28 –2.65ECMt–1 –0.02 –0.97 –0.01 –1.56D SRt 7.72 2.46 –1.23 –0.72D GDPt 1.35 0.27 3.83 2.96D CPIt–1 1.23 0.88 –0.91 –0.78

eq. std. err. 0.03 0.02R2 0.42 0.231AIC crit. –6.83 –7.41Schwarz crit. –6.51 –7.17

Threshold Overall AIC Overall Schwarz

0.004157 –14.25 –13.68

Notes: eq. std. err. = equation standard error; AIC crit. = Akaike information criterion;Schwarz crit. = Schwarz criterion.

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the � rst months of 2000, and even with central bank of� cials’ statements, whowere prudent in linking a rate hike to a euro slide, unless this re� ected somemacroeconomic unbalance. When the undervaluation degree is small, or theeuro is correctly priced or overvalued, again a potentially re-equilibrating role isplayed by the autoregressive components and by the relative business cycleconditions.

Model [3] – Table 5: unfavourable interest rate developments are not relevant,unless the euro has a GDP growth disadvantage, because the only signi� cantmacro-variable is the GDP spread in this regime. When rates are favourable,these are highly important, hence potentially making a ‘strength position’ of thecurrency to persist if this was in place at t2 1.

Model [4] – Table 6: when economic growth expectations worsen, the onlysigni� cant variable is the AR(1) component, thus describing the possibility that,if some currency weakness is present, this might persist, unless expectationsimprove. In this case, this favourable factor plays a leading role. In addition, thesigni� cance of the autoregressive and the ECM terms means that if, in theprevious time, a depreciation occurred which made the euro undervalued , thisdisequilibrium may be corrected. But it is worth noting that in this case justGDP relative growth expectations may give a key contribution to mark a turningpoint for the euro. This could re� ect quite well the reversal in the second half ofMay, when the euro began a fast appreciation after having hit its historical lows,on some news pointing at the possibility of a gradual cooling-off of the USeconomy.

Model [5] – Table 7: when in� ation conditions are favourable, the relevance ofthe GDP term means that if this goes together with healthy business cycle


Threshold variable: ECMt–1

Regime 1 Regime 2


D EURDOLt–1 0.14 0.291 0.23 2.25D EURDOLt–2 0.17 0.473 –0.27 –2.76ECMt–1 –0.04 –0.96 –0.01 –1.42D SRt 7.96 2.09 –0.66 –0.42D GDPt 3.35 0.37 3.96 3.20D CPIt–1 2.19 1.26 –1.36 –1.31



–0.1008 –13.97 –13.39



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Threshold variable: D SRt

Regime 1 Regime 2


D EURDOLt–1 0.30 1.99 0.24 1.74D EURDOLt–2 –0.13 –0.84 –0.27 –2.02ECMt–1 –0.004 –0.42 –0.01 –1.26D SRt 0.33 0.17 6.66 2.39D GDPt 4.57 2.16 2.13 1.18D CPIt–1 0.98 0.70 –1.45 –1.24



–0.00035 –14.36 –13.80



Threshold variable: D GDPt

Regime 1 Regime 2


D EURDOLt–1 0.28 2.05 0.34 2.00D EURDOLt–2 –0.07 –0.49 –0.37 –2.44ECMt–1 0.005 0.49 –0.02 –2.02D SRt 2.13 0.93 –0.86 –0.39D GDPt 2.98 1.66 5.88 2.15D CPIt–1 1.01 0.94 –1.74 –1.12



–0.00017 –14.31 –13.75


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conditions then the euro can appreciate; if, on the contrary, lower in� ation is anindication of worse GDP expectations , the euro could instead weaken.

Model [6] – Table 8: � nally, when the US bourse has a negative or a ‘normal’positive performance, again GDP expectations play a relevant role. On the otherside, when Nasdaq stocks post extraordinarily large gains, this should notnecessarily damage the single currency, since the ECM term is signi� cant, thuspotentially playing a defensive role. However, the fact that the other variablesare not signi� cant, with the only exception of in� ation (which has, however, anincorrect sign), might indicate that neither the autoregressive dynamics norfundamentals have some relevant weight, therefore eventually causing someinstability to the currency.

Overall, empirical evidence shows that non-linear dynamics might explainsome exchange rate behaviour which can be judged to be unusual if one has inmind that a given variable should exert a given in� uence independently of any‘static situation’ being in place, that is of the active regime. A further analysismay consider also links between regimes based on different threshold variables.Of course these regimes overlap, but it is interesting to analyse to what extentthey coincide or differ.

The most signi� cant case is when the threshold variable is the AR(1) term ofD EURDOL. In fact we found here that the short interest rate component isrelevant (hinting to the importance of the role of monetary policy) if previouslythe exchange rate suffered a depreciation. This is perfectly consistent with thecase when the depreciation degree was so large that the exchange rate becamemore than 10% undervalued , which is the regime 1 case of the model where the


Threshold variable: D CPIt–1

Regime 1 Regime 2


D EURDOLt–1 0.39 2.74 0.17 1.10D EURDOLt–2 –0.21 –1.41 –0.26 –1.80ECMt–1 –0.01 –1.07 0.001 0.11D SRt 0.04 0.02 3.31 1.49D GDPt 4.69 2.30 2.18 1.15D CPIt–1 0.73 0.63 –2.94 –1.78



0.00019 –14.25 –13.70



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ECM term is used as the threshold variable. Furthermore , in both models theautoregressive term and the GDP growth variable are signi� cant in regime 2.

These two models have a signi� cant relation also with model (3) , where it isshown that if the interest rate spread turns out in favour of the euro this has alarge weight in driving the dynamics of the EUR/USD exchange rate. It is thenplausible that short rates may play a role when the exchange rate depreciatedor became undervalued (in regime 1 of both model (1) and (2) only the SRvariable is signi� cant). It is also interesting that when the euro is largelyundervalued , only the short interest rate spread is signi� cant, hinting at thepossible consideration that in this case monetary policy may be more effectivethan any natural adjustment in the macroeconomic conditions.

Another important consideration is that the GDP component is alwayssigni� cant in the euro-favourable regime of all the models, with the onlyexception of the interest rate-led model or model (3). This seems to hint at thefact that an eventual relative advantage of the Euro Area in terms of GDP growthmight not be suf� cient or might not matter at all for the euro to appreciate ifother conditions (high in� ation, for instance) are not favourable, whichdescribes quite well what happened to the euro in its � rst two years of life. Thisis also reasonable because the interest rate spread may give rise to speculative� nancial opportunities (carry trades for instance, when investors borrow thecurrency with lower interest rates and invest in the currency with higher yields,which in the end should drive the former currency lower) which make aneventual disadvantage in terms of GDP growth or in� ation less important in thatcase. On the other hand, the evidence of the third model might be consistent


Threshold variable: D NASt

Regime 1 Regime 2


D EURDOLt–1 0.41 2.81 0.03 0.22D EURDOLt–2 –0.29 –1.94 –0.21 –1.35ECMt–1 –0.01 –0.77 –0.05 –1.96D SRt 0.57 0.34 3.55 0.78D GDPt 3.77 1.99 3.94 1.46D CPIt–1 –1.05 –0.78 2.55 1.83



0.03764 –14.34 –13.75


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with the case when, for instance, an expansionary monetary policy by the ECB(regime 1) could favour the euro through better GDP growth prospects.

The � nal consideration is that in all models the CPI variable is neversigni� cant in dynamical term, which may be perhaps interpreted with the factthat in� ation conditions are being kept under control through the conduct ofmonetary policy, which is the case of Europe, or these are more thancounterbalanced by the GDP growth picture, which is more in focus at the Fedin the USA.

Two � nal considerations apply to the relations between a model where avariable is signi� cant under a given regime and the model where that variable isused as the threshold.

Comparing, for instance, model (2) with model (4) it can be noted that whenthe euro is overvalued (regime 2 of model (2)) better GDP growth prospectsmight accentuate this positive disequilibrium. On the contrary, when the GDPcomponent becomes favourable to the euro (regime 2 of model (4)) , the ECMterm always plays a corrective role.

Another relation is between model (1) and model (4). In fact, if the euro hasappreciated (regime 2 of model (1)), unfavourable GDP developments may playa corrective role, since the AR(1) and the AR(2) elements might compensateeach other while the GDP variable exerts a determinant role. Symmetrically,when GDP conditions have become euro-unfavourable (regime 1 of model (4)),the leading role is played by the AR(1) variable, with a potentially re-balancingeffect.

4.3 An inspection of non-linear � tting properties

Fitted versus actual values graphs are another way of looking at the greatercapability of non-linear threshold models to capture some exchange ratefeatures which a linear model is not able to explain. Some improvement withrespect to the linear speci� cation is already evident from traditional � ttingplots. However, it becomes even more evident if an alternative approach is used,as Figs A22–A27 and 5–6 (or A28, A29) show.

Graphs in Figs A22–A29 plot � tted values of the euro/dollar changes togetherwith actual values, but data are ordered according to the increasing magnitudeof actual values. In this way it is possible to see whether outliers, or simplylarger positive or negative changes, can be captured by the model. It is evidentthat almost all the non-linear models overperform the linear one as far as thecapability of capturing outliers is concerned. This is a typical theoretical featureof threshold models, which gains support even here, where it is relevant given‘extreme’ dynamics exhibited by the euro/dollar exchange rate during its � rstone and a half years of life. Especially interesting is the case of the model usingthe ECM term as the threshold variable. In fact, simply choosing as thethreshold a largely negative value (which is sub-optimal according to a rigoroususe of the Akaike information criterion (AIC) or the Schwarz criterion, butremains globally acceptable) , the model performs even better with respect toboth negative and positive extreme values (see Fig. A27). This indicates thatdiscriminating between a situation of large undervaluation of the exchange rate


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and a ‘normal’ situation might prove to be extremely important. This considera-tion should apply to the opposite case of a large overvaluation, but it is lessrelevant here, given the marked downward trend experienced by the euro fromits introduction in January 1999 to May 2000.

4.4 Non-linear versus linear forecasting performanceNon-linear out-of-sample forecasts from the six threshold models previouslyspeci� ed are obtained here, in order to judge whether the capability of thethreshold speci� cation in capturing both extreme values and the asymmetricbehaviour of the euro/dollar exchange rate results in an even better forecastingperformance.

As far as one-step ahead forecasts are concerned, graphs are not shown here,since they are quite similar to the linear case, even though predictions generallyprove to be more precise, even in terms of direction-of-change forecasting . Table9 (or A19) con� rms this � nding, since it contains some quantitative indicators offorecasting performance , that is the mean absolute error, the mean square errorand the percentage of correct signs. Larger MAE and MSE indicate less preciseforecasts, while higher percentages of correct signs indicate a better perform-ance in terms of direction-of-change forecast. It is evident that the non-linearspeci� cations considered perform better than the linear one, both in terms ofprecision and of direction-of-change, with the only exception of the model usinginterest rate differentials as the threshold variable.

A comparison is also conducted with the random walk predictor, which istraditionally considered to be a benchmark in one-step forecasting of exchangerates. The good result is that both the linear model and all the non-linearspeci� cations entertained beat the random walk, thus contributing furtherevidence to that part of the applied literature according to which exchange ratemovements may be approximated by, but are not, a random walk.

Fig. 5 Fig. 6

-0.1

-0.05

0

0.05

0.1

-0.1

-0.05

0

0.05

0.1

Note: �tted values are taken from the linearmodel. Both actual and �tted data are inincreasing order.

Note: �tted values are taken from Model (2),where ECMt–1 is the threshold variable, but thethreshold value is a sub-optimal one. Bothactual and �tted data are in increasing order.

Figs 5–6. Estimated models – �tted versus actual values

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Finally, the ‘forward’ model provides the worst indications , both in terms ofpredictive accuracy and in terms of direction-of-change predictability.

In order to assess whether the forecasting performance of the non-linearspeci� cation is signi� cantly better than the linear one, two encompassing tests,i.e. the so-called ENC-T and the ENC-REG test (Clark and McCracken, 1999; CM99in the following), are conducted against the null that the linear modelencompasses the non-linear ones and vice-versa. The same tests are appliedeven to the random walk predictor versus both the linear and the non-linearmodels. The ENC-T test derives this denomination from the fact that it uses at-statistic for the covariance between the predictive errors of the � rst modeland the difference between the predictive errors of the � rst and the secondmodel. The ENC-REG test is a regression-based variant of the ENC-T test, wherethe t-statistic is associated with the coef� cient from the ordinary least squares(OLS) regression between the predictive residuals of the � rst model and thedifference between the predictive residuals of the � rst and the second model(for details, see CM99) .

These tests con� rm (see Table 10) the better performance of the non-linearformulation (with the only exception of model (3)), which is preferable to boththe linear ECM model and the random walk speci� cation. A stronger proof isprovided here since the null that the non-linear model encompasses thealternative speci� cations is accepted while the null that the alternative modelsencompass the non-linear models is rejected. On the other hand, the testsindicate that it is dif� cult to assess whether the linear ECM model is better thanthe random walk or not.

A � nal test was conducted, which judges the goodness of a model not in termsof point forecasting performance, but in terms of density (distribution)

forecasting ability (Berkowitz, 1998) , which required some Monte Carlo simula-tions. In fact, through a Monte Carlo design it is possible to obtain the forecastdensities of the actual realizations of the variables under examination. When the

Table 9. One-step predictive performance of different models

MAE (%) MSE (%) Correct signs %

Linear model 1.74 0.05 62

D EURDOLt–1 1.62 0.04 80ECMt–1 1.61 0.04 67D SRt 1.82 0.05 67D GDPt 1.61 0.04 73D CPIt–1 1.65 0.04 67D NASt 1.49 0.03 73

Random walk 1.98 0.05 –

Forward exchange rate 2.23 0.06 24

Notes: MAE = mean absolute error; MSE = mean square error; correct sign % =percentage of correct predictions of the direction of change.


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model forecast density corresponds to the true predictive density, the probabil-ity integral transforms of the actual realizations of the variables with respect tothe forecast densities of the considered model are independently and identicallydistributed. So, a likelihood-ratio test of independence across observations ofthe aforementioned transforms and a likelihood-ratio test also for the hypoth-esis of equal distribution can be formulated (Clements et al., 1999). Also thiskind of analysis is conducted here, because non-linear models might notforecast better than linear ones in the case of point forecasts, but they might bepreferable if precision is measured with forecast densities. The results of thetwo above-mentioned tests are shown in Table 11. This shows that only twothreshold models pass the test of good density forecast performance, i.e. model(2) or the model using the ECM term as the threshold variable, and model (6),which uses Nasdaq stock yields as the threshold variable. This provides furtherevidence in favour of the above-mentioned � nding, especially con� rming theworse performance of the linear speci� cation. As far as the other non-linearformulations are concerned, results might bring to the conclusion that the ECM-based threshold model outperforms the alternative models, as also previousresults suggest. This means that the degree of disequilibrium of the exchange

Table 10. One-step predictive performance of different models (encompassing tests)

ENC-TEncompassing

test

ENC-REGEncompassing

test

ENC-TEncompassing

test

ENC-REGEncompassing

test

H0: the linear model encompassesthe non-linear model

H0: the non-linear modelencompasses the linear model

D EURDOLt–1 0.02 0.02 0.92 0.93ECMt–1 0.01 0.06 0.37 0.41D SRt 0.72 0.75 0.10 0.05D GDPt 0.03 0.08 0.58 0.56D CPIt–1 0.04 0.06 0.66 0.64D NASt 0.02 0.02 0.77 0.70

H0: the random walk modelencompasses the non-linear model

H0: the non-linear model encom-passes the random walk model

D EURDOLt–1 0.03 0.03 0.37 0.39ECMt–1 0.03 0.02 0.08 0.17D SRt 0.18 0.13 0.04 0.06D GDPt 0.03 0.04 0.31 0.34D CPIt–1 0.04 0.03 0.16 0.24D NASt 0.03 0.01 0.38 0.42

H0: the random walk modelencompasses the linear model

H0: the linear model encompassesthe random walk model

0.13 0.09 0.15 0.18

Note: results are expressed as rejection frequencies. Both test statistics are standard normal sincethese are applied to non-nested forecasts.

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rate with respect to macroeconomic fundamentals played a relevant role,conditioning the dynamics of the euro/dollar exchange rate.

As Tables A22–A23 show, the encouraging results obtained for one-step pre-dictions � nd further support in the multi-step ahead forecasts performance.

The most evident and signi� cant element is that, apart from capturing quitewell the direction-of-change dynamics, threshold models are even more precisethan the linear one, even when forecasts are brought forward (i.e. from the one-step to the 16-step-ahead forecast case), with the usual exception of model (3).The second-best performer is the ECM-led model, thus offering further evidencein favour of the key role of fundamentals, especially in the sense of the existenceof a long-run equilibrium relation, which leads exchange rate dynamics over alonger time horizon than the one-step frame allows. Also Fig. 7 (or A30) and FigsA31–A32 show the good predictive performance of the ECM-based model asopposed to the interest rate-based model, whose dynamics diverge from actualvalues.

Table 11. One-step predictive performance of different models (density forecastevaluation)

Model LR test (1 dof) LR test (3 dof)

Linear model 0.02 0.01Threshold model (1) 0.008 0.01Threshold model (2) 0.05 0.12Threshold model (3) 0.01 0.05Threshold model (4) 0.008 0.02Threshold model (5) 0.06 0.03Threshold model (6) 0.27 0.07

Note: results are expressed as rejection frequencies. Under the null, these likelihood-ratiotests are distributed c 2 with one and three degrees of freedom (dof) respectively.

1.09

1.11

1.13

1.15

1.17

1.19

01/99 03/99 05/99 07/99 09/99 11/99 01/00 03/00

0.94

0.99

1.04

1.09

1.14

1.19

ECMt-1 D EURDOLt-1 actual (rhs)

Note: actual values are on the right-hand scale, while forecasts are on the left-handscale.

Fig. 7. Multi-step-ahead forecasts from non-linear models


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5. ECONOMIC INTERPRETATION OF NON-LINEAR RESULTS APPLIED TOTHE EUROHISTORY

Among the major features of threshold models there are the capability ofreproducing asymmetrical behaviours and of capturing outliers, which oftencharacterize exchange rate movements. This a priori ability � nds empiricalevidence in the results obtained here.

Asymmetry takes the form of a different contribution and signi� cance of thesame variable, depending on which regime is active. For instance, it is shownhere that when the undervaluation degree of a currency is relevant, thus puttingthe objective of price stability at risk, then monetary policy may play a role,through higher interest rates. This simply emerges from the signi� cance of theshort interest rate component when the exchange rate is below its long-runfundamental equilibrium level, while this same component turns out to beinsigni� cant in the case of fair- or over-valuation. However, there is anotherimportant element. In fact, it is not suf� cient that the exchange rate isundervalued to trigger a monetary policy action, but the undervaluation degreehas to be suf� ciently large in order to be considered an element of risk for theprice stability objective. This concept is simply contained in the thresholdvalue, which in fact in this case is estimated to be not zero, but around 2 10%in terms of deviation of the exchange rate from its equilibrium level. This isconsistent with the conduct of monetary policy by the ECB in the � rst monthsof 2000, when it was made clear that it was not a weak euro per se whichtriggered any rate hike, but precisely a euro whose weakness could threatenprice stability by contributing to macroeconomic unbalances.

As far as the GDP growth differential is concerned, this is of relevance in twoways. In fact, this is the variable which appears to be more frequently signi� cantacross all the non-linear speci� cations entertained . Moreover, it is relevant asthe conditioning variable of the possible asymmetric reaction of the exchangerate to similar impulses depending on the relative state of business cycleexpectations. In fact, regression estimates indicate that, when economic growthexpectations are not in favour of the euro, the most signi� cant variable is theAR(1) term, which may exert further pressure on the euro if this is alreadyweaker, as it happened in the � rst half of 1999, when the euro fell much morerapidly than the worsening of relative GDP growth expectations indicated. Thismeans that in the presence of unfavourable economic growth conditions, theeuro is even more vulnerable than it would otherwise be. On the contrary, whenthe regime of favourable business cycle conditions is active, the ECM term alsoturns out to be relevant, with the possibility that upward in� uences could bereinforced, even through the channel of the deviation-from-equilibrium correc-tion. This happened from July to October 1999 and might have occurred in thesame way in the second half of 2000, given the expected path of relative GDPgrowth (see the last graph of par. 3.1, also for the following consideration).

As far as the autoregressive dynamics are concerned, the asymmetricbehaviour of the exchange rate is evident. In fact, if a downward trend prevails,only short rates are relevant, while GDP conditions are not important. Theopposite occurs when the exchange rate is heading upwards. This was evident

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at the end of the summer of 1999, when, between unfavourable interest ratesconditions and favourable cyclical conditions, these latter prevailed, pushingthe euro up.

Interesting is also the interpretation of the CPI-led model, where thesigni� cance of the GDP term when in� ation conditions are favourable to thesingle currency makes it possible to distinguish whether lower in� ation goestogether with healthy business cycle conditions or is an indication of a weakcyclical evolution. The former case occurred in October 1999 and the latter casefrom February to April 2000.

Less relevant is the role of short rates as far as the non-linear behaviour isconcerned, as forecast results show. But this might also be due to the ‘softer’dynamics of this variable during the period under examination.

As far as the role of the stock market is concerned, the � nding that, when theUS bourse has a negative or a ‘normal’ positive performance, GDP expectationsare signi� cant, might signal the possibility that even the relative weakness of thestock market in May 2000, together with some incipient signs of cooling-off ofthe US economy, may have given a contribution to a path-reversal of the euro(see Fig. A33).

On the contrary, when Nasdaq stocks perform extraordinarily well, the singlecurrency should not necessarily be affected, thanks to the potentially balancingrole of the ECM term. However, the minor role of autoregressive terms andfundamentals may potentially cause some instability to the currency, thustemporarily decoupling exchange rate dynamics from fundamentals, as ithappened in November 1999, when both GDP growth expectations and shortrate spreads were in favour of an appreciation of the euro. This instead starteda sharp and uninterrupted fall which ended only in May 2000.

6. THE EURO: WHAT’S IN THE FUTURE?

We report the forecasts of the euro/dollar exchange rate for the remainingmonths of 2000 using both the linear ECM model and the ECM-based non-linearthreshold speci� cation (which is considered perhaps the most representativemodel among the non-linear speci� cations previously entertained , both for itsgood performance and for its interpretability from an economic perspective).The underlying macroeconomic scenario is provided by Banca CommercialeItaliana (2000). As Fig. A34 shows, this mainly involves a narrowing of both the(negative) GDP growth differentials and the short rate spreads throughout thesecond half of 2000 and the � rst quarter of 2001.

Consequently, the expected result should be that of an appreciation of theeuro, as both models show, with the main difference that the non-linearformulation provides higher predictions (Figs A38, A39 and Tables A24, A25).

Previous results are obtained with data available up to August 2000. Ifestimations are instead stopped at July 2000, and the August euro/dollarexchange rate is considered an outlier (consequently, the September forecast isproduced using the August prediction), there is the possibility that the euro/dollar exchange rate � nally returns above parity with the linear model (see Fig.A40 and Table A26).


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On the contrary, according to the non-linear formulation , the euro/dollarexchange rate at the end of the forecasting horizon would remain below parity(Fig. A41 and Table A27) . In addition, its � nal value would even be lower than inthe previous (non-linear) case. This stems from the fact that the non-linearmodel allows for regime switches. In fact, using historical data up to August(which is the ‘outlier month’), the starting point for the following forecasts islower. As a consequence , the euro/dollar predicted value remains lower (or, theother way round, ‘more undervalued’, given the new historical lows reached inAugust) , thus triggering a shift into the � rst regime some time (two months inthis case) before the same happens using data up to July. The correction withrespect to the higher undervaluation degree (given by the value of the ECM termwith respect to the threshold) and consequently the return to a level which isnearer to the equilibrium is quicker, thus allowing a higher � nal level at the endof the forecasting horizon.

7. CONCLUDING REMARKS

Empirical evidence obtained in this work supports the hypothesis that eco-nomic fundamentals effectively drive the dynamics of the euro/dollar exchangerate, thus rejecting the random walk hypothesis. This is straightforward bothlooking at (i) the euro/dollar exchange rate path exhibited after its introduction ,which appears to be consistent with underlying fundamentals (hence contrast-ing the general disappointment of economists and analysts who were not able toforecast the depreciation of the euro), and (ii) the good performance of thelinear model, which is not only based on fundamentals , but also takes intoaccount a long-run equilibrium relation.

The even better � tting and out-of-sample forecasting performance of the non-linear threshold models entertained here provides further support to thisconclusion, adding evidence that not only fundamentals play a major role in theeuro/dollar exchange rate dynamics, but also that their in� uence works througha non-linear mechanism. In fact, the same variable may exert a differentin� uence on the exchange rate in dependence on which relevant ‘state’ (orregime) of the macroeconomic scenario is in place, which allows the possibilityof asymmetrical behaviours. Furthermore, and in contrast to linear models,threshold models are capable of capturing outliers, which are not extraneous toexchange rate dynamics.

Important is also the role of the threshold values, which describe quite wellthe fact that some effects may be produced on the exchange rate only whentriggering factors are suf� ciently signi� cant. The main example, in this sense, isthat it is not suf� cient that the euro is weak to make the ECB raise rates, sinceit also has to be suf� ciently undervalued as to threaten the objective of stablein� ation.

As for the possibility of an interaction between international stock marketperformances and the exchange rate, this can be taken into account through anon-linear structure, simply working as a sort of ‘disturbance’ factor, which maycause, in extraordinary situations, some decoupling of exchange rate dynamics

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from fundamentals . This seems to produce some improvements in the under-standing of the euro/dollar exchange rate, but without introducing equity yieldsas an explanatory variable.

Finally, the choice of a sub-optimal threshold (biased towards smaller orlarger values with respect to the optimal one according to a rigorous use ofselection criteria) , may improve the performances of a model, if this isconsistent with the prevailing regime during the relevant period. This canbecome useful when scenario analysis is conducted, since a bias towards whatis judged to be the most likely scenario is allowed.

ACKNOWLEDGEMENTS

The author acknowleges useful imput from Jeremy Berkowitz, Todd E. Clark,Rodolfo Dozio, Michael W. McCracken, Luca Mezzomo, Domenico Sartore andTimo Terasvirta. Any remaining errors and inaccuracies are the author’s. Theopinions expressed do not necessarily represent those of IntesaBci.

REFERENCES

Banca Commerciale Italiana (2000) Economic and � nancial perspectives – quarterlymacroeconomic outlook , Research Division, Milan.

Berkowitz, J. (1999) Evaluating the forecasts of risk models.Cao, C.Q. and Tsay, R.S. (1999) Nonlinear time-series analysis of stock volatilities, in

Pesaran, M.H. and Potter, S.M. (eds) Nonlinear Dynamics, Chaos and Econometrics,New York: John Wiley & Sons, pp. 157–77.

Chappell, D., Padmore, J., Mistry, P. and Ellis, C. (1996) A threshold model for the Frenchfranc/Deutschmark exchange rate, Journal of Forecasting, 15, 155–64.

Clark, T.E. and McCracken, M.W. (1999) Tests of equal forecast accuracy and encompass-ing for nested models.

Clements, M.P., Franses, P.H. and Smith, J. (1999) On SETAR non-linearity and forecast-ing.

Consensus Economics Inc. Consensus Forecasts – A Digest of International EconomicForecasts.

Guegan, D. (1994) Series Chronologiques Non Lineaires a Temps Discret, Paris: Econom-ica.

Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors inGaussian vector autoregressive models, Econometrica, 59, 1551–1580.

Johansen, S. (1995) Likelihood-based Inference in Cointegrated Vector AutoregressiveModels, Oxford University Press.

Lundbergh, S. and Terasvirta, T. (1998) Modelling economic high-frequency time serieswith STAR-STGARCH models.

MacDonald, R. and Taylor, M.P. (1994) The monetary model of the exchange rate: long-run relationships, short-run dynamics and how to beat a random walk, Journal ofInternational Money and Finance, 13(3), 276–290.

Tong, H. (1983) Threshold Models in Non-linear Time Series Analysis, New York: Springer-Verlag.

Tong, H. (1995) Non-linear Time Series. A Dynamical System Approach, Oxford/New York:Clarendon Press.


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Tong, H. and Lim, K.S. (1980) Threshold autoregression, limit cycles and cyclical data,Journal of the Royal Statistical Society, B-42(3) , 245–92.

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explaining and forecasting the euro/dollar exchange rate through a non-linear threshold model

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