Nonlinear phenomena in economics: The exchange rate

Nonlinear

Analysis,

Theory,

Mefhods

& Applicarions, Vol. 30, No. 2, pp. 1043-1049, 1997 Proc. 2nd World Congress ofNonlinear Annly.sts

0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X197 $17.00 + 0.00

PII: SO362-546X(97)00133-8

NONLINEAR

PHENOMENA

IN ECONOMICS:

THE EXCHANGE

RATE

Giuseppe De Arcangelis” and Giancarlo Gandolfo$ *Dipartimento $C.I.D.E.I

di Economia Pubblica and C.I.D.E.I. (Centm Interdipartimentale di Economia Intemazionale), Universita di Roma “La Sapienza”, Via Castro Laurenziano 9, I-00146 Roma, Italy, e-mail:
Keywords

andphrases:

Dynamic

Disequilibrium

Modelling, Out-of-sample Tbeil’s diagram.

Forecasting,

Continuous-Time

Econometrics,

1. INTRODUCTION

Economic models are traditionally presented as linear models or, in a more sophisticated way, as nonlinear models which are then linearized by the usual procedure around some equilibrium solution. But economic phenomena are not necessarily linear and, when they are nonlinear, the tendency to forget that the results obtained by the linear approximation are only locally valid may give rise to serious errors especially when the variables are subject to changes which are not “sufficiently small” but are, on the contrary, rather wide (the exchange rate is a case in point). A few economists, pioneered by Richard M. Goodwin (his 1951 nonlinear trade cycle model, based on relaxation oscillations, and his 1965 growth cycle, based on LotkaVolterra equations, are by now classics) have always advocated the need for nonlinear analysis in economics, but it is only recently that nonlinear analysis has begun to be fairly widely adopted, as many economists have jumped on the bandwagon of the current fashion, which is the application of chaotic dynamics to economic models. For a complete survey of the state of the arts see Gandolfo (1996, Chaps. 24-26). Although we do not jump on this bandwagon (for the reasons detailed in Gandolfo, 1996, Chap. 26, Sect. 26.4), we are firmly convinced that many economic phenomena are inherently nonlinear and have to be analyzed as such. Usually, the linear or nonlinear nature of an economic variable is ascertained empirically, by carrying out one of the various existing tests for nonlinearity. In some cases, however, there are compelling theoretical reasons for a certain economic phenomenon to be intrinsically nonlinear. The exchange rate is one of them.

2. THE EXCHANGE

RATE AS AN INTRINSICALLY

NONLINEAR

PHENOMENON

We shall now show why the exchange rate is theoretically a nonlinear phenomenon, by recalling a few wellknown relations in international economics (see Gandolfo, 1995, Chap. 18). Let CA denote the current account of the balance of payments, NFA the stock of net foreign assets of the private sector, R the stock of international reserves and the operator A the change in the stock (therefore generating a flow). Then the balance-of-payments equation simply states that: CA+

ANFA+

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AR=Q

(1)

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Introduce now the following functional relations: CA = f(E; ) ANFA = g(E,. ) AR = h(E; ) E=cp (. )

(2) (3) (4) (5)

where E is the exchange rate and the dot indicates all the other explanatory variables, that for the present purposes can be considered as exogenous. Given that the system (l)-(5) contains five equations in four unknowns, we can either drop equation (5) and use equation (1) to determine the exchange rate, or keep eq. (5) and drop, say, eq. (3) or eq. (4); then eq. (1) can be used to determine the balance of capital movements (A NFA) or the reserve change (AR) residually. It should be stressed that if the balance-of-payments definition is used to determine the exchange rate, this does not necessarily mean that one is adhering to the traditional (or “flow”) approach to the exchange rate, as was once incorrectly believed. A few words are in order to clarify this point. By following this approach, one is simply using the fact that the exchange rate is determined in the foreign exchange market, which is reflected in the balance-of-payments equation, under the assumption that this market clears instantaneously (as it actually does, if we include the monetary authorities’ demand or supply of foreign exchange as an item in this market).* In fact, no theory of exchange-rate determination can be deemed satisfactory if it does not explain how the variables that it considers crucial (whether they are the stocks of assets or the flows of goods or expectations or whatever) actually translate into supply and demand in the foreign exchange market which, together with supplies and demands coming from other sources, determine the exchange rate. When all these sources - including the monetary authorities through their reaction function on the foreign exchange market, eq. (4) - are present in the balance-of-payments equation, this equation is no longer an identity, but becomes a market-clearing condition. Thus it is perfectly legitimate (and consistent with any theory of exchange-rate determination) to use the balance-of-payments equation to calculate the exchange rate once one has specified behavioral equations for all the items included in the balance of payments. One final observation: the balance-of-payments equation - eq. (1) above - is a nonlinear implicit function in the exchange rate if at least one of the functions A.), g(.) or h(.) is nonlinear, which is very plausible in economic theory (e.g., monetary authorities may maintain a target value for the exchange rate and intervene when there are both positive and negative departures from that value; i.e., their reaction function is quadratic in the exchange rate). As a consequence, also the exchange rate turns out to be a nonlinear function of all the otherendogenous variables(implicit function theorem). Here we havea casein which thereis a strongtheoreticalmotivefor a variable(i.e., the exchangerate) to be nonlinear. 3. NONLINEAR

EXCHANGE-RATE

FORECASTING

Practically all structuraltheoreticalmodelsof the exchangerate are of the type that keepsequation(5) in the previoussection.This equation,moreover,is usuallyspecifiedasa linear(or loglinear)equationin a reduced form. The problemis that, whenone usessuchmodelsfor exchangerate forecasting,one gets resultswhich are strikingly poor,even belowthe ‘benchmark”resultsof a simplerandom-walkmodel(it shouldbe emphasized that this doesnor meanthat one is assumingthe exchangerate to be a random-walkphenomenon, but does simplymeanthat any goodmodelshouldbe ableto do better than the naive forecastingassumptionthat next periods exchangerate is equal to the current exchangerate). This was first shownby Meeseand Rogoff ‘In our approach

this item is given by eq. (4), which defines the monetary

authorities’

reaction function,

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(1983), and confirmed by several later studies; for details see Gandolfo, 1995, Chap. 18, and Gandolfo, Padoan, De Arcangelis, (1993). On the one hand, these results have even led some authors to take the failure of the structural models of exchange-rate determination as a failure of economic theory and as showing the necessity of moving towards pure time-series techniques (prediction without explanation). On the other hand, these results have been interpreted as the most striking proof that the exchange rate is an asset price determined in the “efficient” foreign exchange market. As such, no good prediction of the exchange rate is possible on the ground of the information set of the previous period in order to assure a non-profit condition in the foreign exchange market. This latter assumption, however, cannot be held against the evidence of nonefficiency in the foreign exchange market (see, for instance, Froot and Thaler, 1990).

Actual Differences Figure

I Theil’s

Diagram:

Random

Walk

Among the various reasons that have been set forth to explain this collapse of the theoretical models the most obvious has not come to the mind of the supporters of the traditional models of exchange-rate determination: that the approach of keeping eq. (5) throws away all the intrinsic nonlinearities of the exchange rate and hence is bound to give rise to serious errors, as shown in the forecasting exercises. Since we have been advocating the nonlinearity of the exchange rate (as manifested in dropping eq. (5) and determining the exchange rate through the balance-of-payments equation) for the last twenty years (see Gandolfo, 1976. for the first exposition of this point), we are happy to show in this paper that we are able to beat the random walk as well as the traditional models of exchange-rate determination in out-of%ample forecasting exercises. For this purpose we use the MARK V version of the macrodynamic model of the Italian economy developed by Gandolfo and Padoan (1990). This is a dynamic disequilibrium model specified as a set of 24 nonlinear differential equations and estimated in continuous time by using appropriately developed econometric methods (Gandolfo. 198 1; Wymer, 1993). It would be impossible to prt :ont the model, the estimation procedure and the results here, hence we limit OUT:.;’ .e5
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the model. For an in-depth treatment see Gandolfo and Padoan (1990) and Gandolfo et al (1996). We shall then present the outcome of the forecasting exercise.

3. I. NONLINEARITIES

OF THE

MODEL

Although most of the equations are linear in the logarithms, there are significant nonlinearities in some equations of our model, which require a detailed explanation. In the consumption equation, real private consumption adjusts to its desired level, that is given by the (vacable) average propensity to consume applied to real disposable income, (Y-T/P). Given the logarithmic formulation of the equation, the term log( Y-T/P) is nonlinear in the logarithms. 7.7

w

7.4

5 7.3 7.2 7.1 7

1985: 1

1986:l

1987:l

1988~1

Figure2. Comparisonamong Mark V (*), HML model(0). RandomWalk(x)

1989 and

1

*990:1

actualvalues(+j of theexchangerate.

Another relevant equation is the monetary authorities’ reaction function on international reserves, and is quite complicated because of the change in regime (from fixed to floating exchange rates) which occurred in the sample period. This equation has been specified to reflect the regime change, but also contains ekmei3.s which are independent of the regime in force and which we may call permanent elements in the monetzy authorities’ behaviour. L.et us begin with the permanent elements, which are the leat:l::g-against-tl!e-w:nd policy and the desired reserve ratio. The Italian monetary authorities have generally ;ollowec~ a poiicy of leaning against the wind (even during the Bretton Woods era), to smooth out the path ui‘ Lhe exchange rate and/or to prevent excessive fluctuations. This policy implies that internation?!. re .ervzs mo-je in the opposite direction to the exchange rate. As regards the second permaneilt e!ement, ‘.:. dam ; o:‘It2!: _ k4.. $z;.: ..*J:+iiij a great deal of attention to the months of financial covering of imports, that is hove !ong the ci;;renr flo*a of

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imports can be maintained if the existing stock of reserves are used up for this purpose: this implies the existence of a desired ratio of reserves to the value of imports. Thus also reserves could be modeled via a partial adjustment equation towards the desired level of reserves, which depends on the imports. Let us now come to the elements relating to the exchange-rate regime. During the Bretton Woods era, there was an obligation to maintain the exchange rate within +I% of the official parity, whereas in the flexible period no explicit obligation was present. As a consequence, a switching function according to the exchangerate regime was included in the reaction function of the monetary authorities. The presence of a multiplicative switching variable is an important nonlinear feature of the model. In addition, three equations are definitional equations, linear in the natural values but nonlinear in the logarithms. One equation defines the change in inventories as a residual in the goods market. Another one defines the change in the public sector borrowing requirement. Finally, we have the balance-of-payments definition, which plays a fundamental role in our model: that of determining the exchange rate. This point has already been discussed above in detail.

:.I

-0

L26

-Cl.BB.

-8. e.*

B

8.9.2

8. Bll.

B. L26

Actual Differences Figure3. Theirs Diagram:Mark V. 3.2.EXCHANGE-RATEFORECASTS

The sample period used for estimation was 1960-1984 (quarterly data), and the period 1985:l to 1990: I was used for out-of-sample ex-post forecasts. The same exercise was also carried out for the standard structural models of exchange-rate determination, the best of which turned out to be the Hooper-Morton portfolio model with lags (HML).

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The main results are presented in Figures l-3; all the others are available from the authors upon request. The exchange rate considered is the Italian Lint/US Dollar spot rate; the data are in logarithms, given the logarithmic nature of the models.2 As a first result, the data show that the forecasting performance of the random walk model is rather poor; more precisely, the root mean square error of forecasts (RMSE) is 5.88 while the mean absolute error (MAE) is 5.05. Since the variables are expressed in logarithms, these values practically coincide with percentages. These two indexes, though generally used to assess the forecasting performance of a model, are however not sufficient in this particular case. Given the volatility of the exchange rate, one must also evaluate the models ability to forecast the turning points in the series, namely when the variable changes its behaviour (from decreasing to increasing and vice versa). For this purpose, the Theil (1961, p. 30) diagram is particularly useful. Thus in Figure 1 we plot the forecasted differences against the actual differences. The 45degree line is the locus of perfect forecasts: the nearer the points to it, the better the forecast. But there is more to it tha.n that. The points lying in the first and third quadrant, in fact, represent forecasts that have correctly predicted the direction of change, while all points lying in the second and fourth quadrant represent turning-point errors. Thesecan be of two types: 1) errorsof the first kind, namelya turning point waspredicted,but did not actually take place; 2) errorsof the secondkind, namelyan actualturning point wasnot predicted. Theil’s diagramdoesnot by itself allow the observerto distinguishbetweenthe two types, hencewe have respectivelymarkedthemwith the numbers1 and2. We can further add that, sincetherewere six actual turning pointsin the sampleperiod usedin forecasting, the fact that thereare as many turning pointserrors of the secondkind showsthat the randomwalk model hasneverbeenableto predict a turning point, which is inherentin the definition of randomwalk. Let us now cometo the bestof the standardstructuralmodels(HML). The RMSE and MAE are 5.67 and 5.17 respectivelyfor HML, hencethis modelis unableto outperformthe randomwalk. Theseresultsare consistentwith the Meese-Rogoffresultsand the subsequentliterature. But the resultsfrom the relative Theil’sdiagram(not shownhere,but availableuponrequest)arestill moredevastating:predictionwith HML presentssix type-2 turning-pointerrorsout of six. Sincewhat comesinto considerationin Theil’sdiagramis the qualitative nature of the forecasts(the meredirection of changeindependentlyof its magnitude),and sinceeconomictheory is a setof qualitativepropositions,the positionof thosewho maintainthe uselessness of economictheory might seemto beconfirmed. Fortunately for economictheory there is MARK V. The forecastsof our modelare plotted with the actual valuesin Figure 2 togetherwith the RW andHML forecasts.The superiorityof our forecasts(RMSE:=2.61, MAE=2.09) is confirmedby the strikingly goodresultsof Theil’sdiagram(Figure 3): five turning pointsout of six have beencorrectly predicted,and there are no errors of the first kind. Economictheory, correctly used,is useful. 4. CONCLUSION

Our resultsshowthe essentialimportanceof dealingwith highly nonlinearphenomena by nonlinearmethods - which might seema platitudeto mathematicians but apparentlyis not to economists. *Note that although the models can be specified nonlinearity in the model.

in the logs, eq. (1) must still hold in the levels. This is already

a source of

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REFERENCES FROOT, K.A. and R.H. THALER, Anomalies: Foreign Exchange, Journal of Economic Perspectives 4, pp. 179-192 (1990). GANDOLFO, G., I1 tasso di cambio di equilibria, Milano, F. Angeli (1976). GANDOLFO, G., Qualitative Analysis and Econometric Estimation of Continuous Time Econometric Models, Amsterdam, North-Holland (1981). GANDOLFO, G., International Economics II - International Monetaary Theory and Open-Economy Macroeconomics, Berlin, Heidelberg New York Tokyo, Springer-Verlag (1995). GANDOLFO, G., Economic Dynamics, Berlin, Heidelberg New York Tokyo, Springer-Verlag (1996). GANDOLFO, G. and P.C. PADOAN, The Italian Continuous Time Model: Theory and Empirical Results, Economic Model@ 7, pp. 91-132 (1990). GANDOLFO, G., PC. PADOAN and G. DE ARCANGELIS, The Theory of Exchange Rate Determination, and Exchange Rate Forecasting, in H. Frisch and A. Worgotter (eds.), Open Economy Macroeconomics, London, Macmillan, pp. 332-52 (1993). GANDOLFO, G., P.C. PADOAN, G. DE ARCANGELIS and CR. WYMER. Nonnlinear Estimation of a Nonlinear Continuous Time Model, in W.A.Barnett, G, Gandolfo and C. Hillinger (eds.), Dynamic Disequilibrium Modelling, Cambridge (UK), Cambridge University Press, pp. 127-50 (1996). GOODWIN, R.M., The Nonlinear Accelerator and the Persistence of Business Cycles, Econometrica 19, pp. 1-17 (1951). GOODWIN, A Growth Cycle, paper presented at the first World Congress of the Econometric Society; published in C.H. Feinstein (ed.), 1967, Socialism, Capitalism and Economic Growth, Cambridge University Press (1967). MEESE, R. and K. ROGOFF, Empirical Exchange Rate Models of the Seventies: Do They Fit out-of-sample?, Journal of InternationalEconomics 14, pp. 3-24 (1983). THEIL, H., Economic Forecasts and Policy, 2nd edn., North-Holland, Amsterdam (1961). WYMER, CR., Estimation of Nonlinear Continuous-Time Models from Discrete Data, in P.C.B. Phillips (ed.), Models. Methods and Applications ofEconometrics, Oxford, Blackwell, pp. 91-116 (1993).