degrauwe.pdf

Open economies review 4: 351-379, 1993. @ Kluwer Academic Publishers. Printed in The Netherlands.

A Chaotic Model of the Exchange Rate: The Role of Fundamentalists and Chartists

PAUL DE GRAUWE AND HANS DEWACHTER University of Leuven, National Fund for Scientific Research, Belgium

Key words: exchange rate, chaos, fundamentalism, chartism

Abstract

A monetary model of the exchange rate is constructed in which "fundamentalists" and "chartists" interact. It is shown that the non-linearity of this speculative dynamics leads to chaotic motion of the exchange rate. The model is also capable of generating some of the stylized facts of exchange rate dynamics.

Ever since the empirical breakdown of (linear) structural exchange rate models, the dominant view of exchange rate dynamics has been based on the "news" model. In this model the only sources driving the exchange rate are random events. 1

Recent research has revealed some problems with the "news" model. First, there appears to be more structure in the time series of the exchange rate than the pure stochastic model can account for. This additional structure has been found in most exchange rates. See, for example, Cutler, Poterba and Summers (1990), who report significant autocorrelations in the exchange rates at different lags.

Second, it appears that many, if not most, movements in the exchange rates cannot easily be accounted for by observable "news." In an analysis of high-frequency exchange rate data, Goodhart (1990) documents that very often the exchange rate does not respond to observable news, and that many exchange rate movements cannot be associated with news.

This recent empirical research suggests that in addition to random shocks, there are other driving forces in the exchange market that are important to understanding its dynamics. In this paper we will focus on a (non-linear) speculative dynamics, in which the behavior of "chartists" and "fundamentalists" plays a prominent role. The analysis will be performed in the context of a structural model, the Dornbusch model,

352 DE GRAUWE AND DEWACHTER

which has become the most popular textbook model of the exchange rate. 2 It will be shown that this model, together with a simple non-linear speculative dynamics, is capable of generating a complex behavior of the exchange rate that resembles random shocks. Such behavior has been called chaotic. 3 Very good expositions on chaos theory can be found in Grandmont (1985), Baumol and Benhabib (1989), Brock and Malliaris (1989), Devaney (1989), and Schuster (1985).

At first sight, the difference between chaotic and stochastic motion seems purely academic. One might argue that since these different dynamic processes are not distinguishable in the time domain it does not matter which of them is the true one. However, although these processes look alike, the potential forecastibility of the two systems differs dramatically. While white noise processes are unforecastable, a chaotic process is forecastable in the near future.

This general property of chaotic systems makes the distinction between the two dynamic processes more than academic. It makes practical application possible. If chaos is confirmed, short-term forecasting becomes possible.

The remainder of the paper is organized as follows. In section two we present the model. Sections three and four report the basic properties of the model. It will be shown that the model is able to generate chaotic motion. In section five we replicate some of the stylized facts of exchange rate dynamics. Conclusions close the paper.

1. The Model

I. 1. The Dombusch Model

The version of the Dornbusch model that will be used in this paper consists of the following building blocks'.

1.1.1. The money market equilibrium condition:

Mdt = yta.Pt.( 1 -~- ?'t) -c, Md~ = Mst, (1)

where P~ is the domestic price level in period t, rt is the domestic interest rate, Ms~ is the money supply, and ~ is the (exogenous) level of domestic output. In equilibrium the demand for money equals the supply. Two alternative money supply regimes will be assumed. In one regime the monetary authorities set Ms~ exogenously. In the second regime they follow an interest rate smoothing strategy. We represent this by the equation:

A CHAOTIC MODEL OF THE EXCHANGE RATE 353

Mst = Mst-l((1 + rt)/(1 + rt-1)) o, (2)

where 0 is the interest rate smoothing parameter. Note that when O = 0 the money stock is exogenous (constant), and we are in the first regime.

1.1.2. The open interest parity condition.

Et(St+l)/Et = (1 + rt)/(1 + r#), (3)

where St is the exchange rate in period t (the price of the foreign currency in units of the domestic currency), E~(St+I) is the forecast made in period t of the exchange rate in period t + 1, and r# is the foreign interest rate.

1.1.3. Goods market equilibrium. The long-run equilibrium condition is defined as a situation in which Purchasing Power Parity (PPP) holds, i.e.:

5~t = P;/P}t, (4)

where 5'~ is the equilibrium (PPP) exchange rate, P~t the foreign and Pt* the domestic steady state value for the price level in period t.

The short-term price dynamics is assumed to be determined as follows:

Pt/ Pt-1 = ( St/ S~ ) k, (5)

where k > O. That is, when the exchange rate exceeds its PPP-value, S~, the domestic price level increases. Put differently, when the currency is undervalued this leads to excess demand in the goods market, tending to increase the price level. The opposite occurs when the exchange rate is below its PPP-value (an overvalued domestic currency).

Note that we assume full employment so that adjustment towards equilibrium is realized through price changes. The parameter k measures the speed of adjustment in the goods market. In general the size of this parameter depends on the choice of the units of time. If the unit of time in the model is say, a week, then k will be low compared to a model in which the unit of time is a month or a quarter.

1.2. The Speculative Dynamics

We assume there are two classes of speculators. One class is called "chartists" (noise traders), the other "fundamentalists." (See Frankel and Froot 1986 for a first attempt at formalizing this idea.) A recent microeconomic foundation of this assumption is provided by Cutler,


Poterba, and Summers (1990). Empirical evidence about the impor- tance of these types of speculators is found in Allen and Taylor (1989) and Frankel (1990).

The chartists use the past of the exchange rates to detect patterns which they extrapolate into the future. The fundamentalists compute the equilibrium value of the exchange rate. In our model this will be the (steady state) PPP-value of the exchange rate. If the market rate exceeds this equilibrium value, the fundamentalists expect it to decline in the future (and vice versa). Another way to interpret this dual behavior is as follows. The chartists use past movements of the exchange rates as indicators of market sentiments and extrapolate these into the future. Their behavior adds a "positive feedback" into the model. 4 As will become clear, this is a source of instability. The fundamentalists have regressive expectations, i.e., when the exchange rate deviates from its equilibrium value they expect it to return to the equilibrium. The behavior of the fundamentalists adds a "negative feedback" into the model, and is a source of stability.

A second feature of the speculative dynamics is that we assume that the fundamentalists have heterogenous expectations. As will be shown, this has the effect of making the weights given to chartists and fundamentalists endogenous. Suppose there are N fundamentalists who make a different estimate of the equilibrium value of the exchange rate at time t. Suppose also that these estimates are normally distributed around the true equilibrium value. We then obtain the following picture (see Fig. 1).

When the market exchange rate at time t (S~) is equal to the true equilibrium exchange rate (S;), half of the fundamentalists will find that the market rate is too low, whereas the other half will find that it is too high compared to their own estimates of the equilibrium rate. If we assume that these fundamentalists have the same degree of risk aversion and the same wealth, the amounts of foreign exchange bought by the first half will be sold by the second half. Thus, when the market exchange rate is equal to the equilibrium (fundamental) exchange rate, the fundamentalists do not influence the market. It is as if they are absent from the market. The market's expectations will then be dominated by the chartists' beliefs.

As the market exchange rate starts to deviate from the true equilibrium value, fundamentalists become important again. Take the example of Fig. 1. When the market rate has declined to S~, the number of fundamentalists believing that the market rate is too low, compared to their own estimates of the equilibrium rate, increases so that their expectations become more important in the market. The weight of the fundamentalists on the market's expectation tends to increase. The


F(St)

S't S*t St

Fig. 1. Frequency distribution of estimated equilibrium exchange rate (S~')

same happens when the market exchange rate increases relative to the true equilibrium value. We conclude that, in a world where there is uncertainty about the true fundamental value of the exchange rate, the weight of the fundamentalists' belief in the total market's expectation will increase when the market exchange rate departs from the true equilibrium value.

We now implement these two assumptions about the speculative dynamics as follows. We write the change in the expected future exchange rate as consisting of two components, a forecast made by the chartists and a forecast made by the fundamentalists:

= ' -m`, (6)

where E~(S~+I) is the market forecast made in period t of the exchange rate in period t+ 1; E~(S~+I) and E:t(S~+I) are the forecasts made by the chartists and the fundamentalists, respectively; ~t is the weight given to the chartists; and 1 - ~t is the weight given to the fundamentalists.


We assume that the chartists extrapolate recent observed exchange rate changes into the future, using a moving average procedure, i.e.,

w, f l ~,f2 ~,f3 E~ (St+I)/S~-I = -t_1~t_2~_3. (7)

Most of the popular chartist prediction rules can be formulated as special cases of (7). We will concentrate on one such popular model, in which chartists buy (sell) whenever a short-run moving average is higher (lower) than a long-run moving average. This model can be formalized as:

E~kSt_ I ] = \ LMA ] ' (8)

where 7 is the extrapolation parameter. Equation (8) says that when the short-term moving average of past

exchange rates exceeds the long-run moving average, chartists expect the future exchange rate to increase. We will take a two-period moving average for SMA and a three-period moving average for LMA.

Admittedly this is a very crude assumption, and chartists typically use more sophisticated rules. (in further research we hope to study the implications of using more sophisticated chartists' forecasts.) The use of simple rules, however, is not necessarily a disadvantage if we can show that very complex behavior of the exchange rate is possible even if chartists use these very simple forecasting rules.

The fundamentalists are assumed to calculate the equilibrium exchange rate (i.e., the exchange rate that leads to equilibrium in the model). In our model this is the PPP rate. They will then expect the market rate to return to that fundamental rate (S~) at speed h during the next period, if they observe a deviation today, i.e.,

= h. (9)

As argued earlier, the weights given to chartists and fundamentalists depend on the deviation of the market rate from the fundamental rate. 5 Given the assumptions about the distribution of the estimated equilibrium exchange rates, we postulate a weighting function as follows:

mt = 1/(1 + b(St-1 - $2_1)2), (10)

where rnt is the weight given to the chartists, and b > 0. Graphically we can represent this specification as follows:

From Fig. 2 it can be seen that when the market exchange rate is equal to the fundamental rate, the weight given to the chartists attains its maximum value of one. It is as if there are no fundamentalists in the market. When the market rate deviates from the fundamental rate, the weight of the chartists tends to decline. For very large deviations it


mt 1

0 (S t -S ' t )

Fig. 2. The weighting function of chartists

tends towards zero. The market expectations will then be dominated by the fundamentalists.

Note that parameter b determines the speed with which the weight of the chartists declines. It also measures the degree of divergence of the fundamentalists' estimates of the equilibrium exchange rate. With a high b, the curve in Fig. 2 becomes steeper. This means that the estimates of the true equilibrium rate made by fundamentalists are very precise, i.e., there is little divergence in these estimations. As a result, relatively small deviations of the market rate from the true equilibrium rate lead to a strongly increasing influence of the fundamentalists in the market. The opposite is true when b is low. In that case, there is a lot of uncertainty in the market concerning the true equilibrium rate. As a result, a movement away from the equilibrium rate induces little reaction from the fundamentalists, so that their weight in the market increases little.

From the preceding it will be clear that we have not derived the spec- ifications of the behavior of the speculators from an explicit optimizing


framework. Thus, expectations cannot be called rational. Our defense here is that the accumulated empirical evidence is not very favorable for the rational expectations assumption in the foreign exchange markets. It is therefore worthwhile (although intellectually less appealing) to investigate models that depart from the rational expectations assumption. The bottom line is that we ask the reader to follow us so as to see how far the non-linear specification of the speculative dynamics can go in explaining exchange rate movements. We want to show that the speculative dynamics that we assume here allow us to construct models that come closer to understanding reality than the structural models that have been used up to now. We now proceed toward solving the model.

1.3. Solution of the Model

From the equilibrium condition in the money market and the dynamic of the price level (5) we obtain an expression for (1 + r~) (i.e., we set PF = 1 and rE = 0):

(1 + rt) = r/D(i/l+k)~(k/l+k)~V ~ ;i/fo-1 (1 ] (1 1) LV, t_ l ot . t . . . . t - l k "Jc f't-1)O (1/8+c)

Rewriting the open interest parity condition we obtain the solution of the model:

S,----[Pe-_~iYt-aMs,_l(1 + r,_i)-]~ EtS:+i E,S +I = ( s* / ( , ,sg3 ) TM

=, = i/(i + b(&_, - S*) =) (12) = i/(I + k) = + k/I(1 + k)(o +

= o/(o +

The system has a recursive structure defining a unique equilibrium price (&/g2t-1), where 9t- i is the information set at period t - 1. Given this equilibrium value of St, other contemporaneous variables can be computed. The model is highly non-linear in the speculative price. In order to investigate the time paths of the model we have to iterate the model. Our interest in the model is to know whether it is capable of generating a chaotic dynamic. This will be discussed in the next section.

2. Existence of Chaos

In this section we turn to the question of whether the model is capable of generating chaotic motion in the exchange rates. We will first


define what is meant by chaotic motion. Then, we will look at the different outcomes.

Let us first define chaotic motion, using the definition of Devaney (1989):

A function f : V -+ V, is said to be chaotic on V if

a) it has sensitive dependence on initial conditions; b) it is topologically transitive; o) it has periodic points which are dense in V.

The intuition of this definition can be explained as follows. Accord- ing to (a), a slight change in the initial conditions will (if sufficient time is allowed) lead to a time path of the exchange rate that bears no resemblance whatsoever to the original time path (although the two time paths are bounded by the same attractor). As will become clear, this has far-reaching implications for the predictability of the exchange rate. Item (b) implies that the consecutive exchange rates produced by iteration of equation (13) will eventually move from one arbitrarily small neighborhood to any other. Condition (c) introduces an element of regularity. It ensures that the exchange rate will remain within certain bounds around the steady-state value (a strange attractor). Conditions (b) and (c) together also imply that the exchange rate has infinite periodicity, i.e., no cycles repeat themselves exactly. A chaotic regime can thus be characterized by a seemingly random signal in the time domain (i.e., due to the topological transitivity and the infinite periodicity).

Although the chaotic and random signals cannot be distinguished in the time domain, they can be distinguished if we look into other spaces such as the phase space. Due to Takens' theorem, it is well known that chaotic functions can be "regenerated" in the phase space. According to the theorem, both approaches (the time domain and the phase space representation) are topologically equivalent. This property will be used in the experimental search for chaotic states based on the topological properties in phase space.

Because the model is of higher order, we use experimental methods to find chaotic states. We will look at two characteristics of chaotic motion. First, we will look at the sensitivity to initial conditions of the model. That is, we expect to find large deviations in the time path after some period as a consequence of small changes in the starting values. Second, we will look at the correlation dimension Do. For deterministic chaos the value of the correlation dimension converges to a finite non-integer value as we increase the dimensionality of the


phase space. More specifically:

/9o = lim lim 0 log CMr M-~oo r-~0 0log(r)

1 CM(r) -- T (T - 1) E E ~(Xj, Xi, r) (13)

(Xj, X~, r) is the indicator function taking value:

1 if IXd - X~I < r and 0 otherwise, Xi, Xj E R lxM.

The correlation dimension is explained in more detail in Appendix A. The model presented above has a large parameter space. In order

to reduce this space we will select a particular subspace and scan it for the above-mentioned characteristics. The space we will consider is the (~, ~,) space for various values of the degree of interest rate smoothing (i.e., e = 0, ~ = 1, ~ = 10, 8 = 100). Results of these experiments are presented in Table 1.

Basically we observe from Table 1 that chaotic outcomes become more likely the higher the extrapolation parameter of the chartists and the lower the adjustment speed of the fundamentalists. Fig. 3 presents the correlation dimension estimates for different chaotic regimes. From this picture it becomes clear that the correlation dimension estimates converge to a non-integer finite value, which is an indication of chaos. When 0 = 0, we find a correlation dimension of 2.4; when 0 = 10, the correlation dimension is 3.3. Fig. 4 demonstrates the sensitivity to initial conditions. In the simulations of Fig. 4 we changed the initial exchange rate by 0.1%. The results indicate that this slight change in initial conditions produces a completely different time path of the exchange rate, never reverting to the initial one. Finally in Fig. 5 we show a simulation in the time domain, spanning a relatively long period. This figure shows that the model is also capable of mimicking the relatively long cyclical movements in the exchange rate that have also been observed in reality. These patterns of low frequency cycling have been reported recently by Engle and Hamilton (1990). These authors find that a model of changing trends can outperform the random walk. Similar qualitative behavior is observed in our simulations. In short, the model is capable of generating seemingly random signals, although it is totally deterministic.

3. Predictability of Chaotic Processes

The sensitivity to initial conditions is sometimes interpreted as imply- ing that forecasting on the basis of chaotic models is impossible. If one


e = 0

13

12

11

10

9

B

?

6

5

4

3

2

I

0 I I I I I I I I I I I I 1 2 3 4 ~ 6 "1 B B 10 11 1:2

(a) (continued)

makes a small error in the estimate of the initial state, predictions (i.e., the time path starting from the initial state) will become separated more and more from the real path. As a result, chaotic models are useless. They do not improve predictability and are no better than a random walk as predictive instruments.

This view, however, is not correct. There is a non-trivial difference between chaotic and stochastic regimes (i.e., white noise) with re- spect to forecasting potential. To illustrate this idea we will use the concept of entropy and, more specifically, the Kolmogorov entropy. This measure, K-entropy, is well suited to illustrate the difference in forecasting potential.


0 = 10

t3

11

10

9

8

7

B

5

4

3

2

1

n I I I I I I I I I I t I "1 2 3 4 ~ 7 8 9 10 11 12

~11bedCl I I'~ d I r~ I r~ I on

(b)

Fig. 3. Estimates of the correlation dimension (a) e = 0 (b) e = 10

Consider the set of solutions A = {a e A{a = F(b), Vb E B}. F(.) denotes the transformation of the initial state b into the state one period further (a). For continuous functions, the interval B is transformed into interval A in the next period. The K-entropy is a measure proportional to the number of subintervals, with length 1 necessary to cover A. This number is infinite for stochastic F(.), so that the K-entropy is also infinite. For chaotic systems, however, the K-entropy, is finite and nonnegative. The K-entropy is zero if we can measure the initial state and the parameters of the model exactly. We can then forecast the future without error. If there is some uncertainty about the initial state


1. OB

1, Oe

I , 07

1 .0e

1 .05

1.0~1

1 .03

1 ,02

1 ,01

1

0 .99

O, 98

0 ,87

0 .96

O. 95

O, g4

0.93

O, 92

0 .91

0 .9 I

I

t

m I I I I I 11 21 31 4t 51 81 71

(a) (continued)

(i.e., B is an interval), then due to the sensitivity to initial conditions A = f (B) will be larger than B, i.e., A ~ B. However, unlike a stochastic system, A remains bounded (due to the boundedness of the chaotic attractor) such that the number of subintervals to cover A is finite as well. Since the K-entropy is proportional to this number, this entropy has a finite value for chaotic systems The smaller the K-entropy, the greater the forecasting potential of the model.

One way to measure this K-entropy is to use the correlation dimension estimate for two different embedding dimensions, N1 and N2, and calculate K according to Schuster (1985):

Log cN'(r)- Log eNd(r) I2 = (14) N2- N1

The number of periods one can predict, given that we can locate the


t .14

1 .12

1 .1

1 OB

1 .06

1 .04

1. D,9.

t

O. 98

D. 96

Q.94

0 .92

O.B

O, 66 I I 1 11

I t I I I I 21 31 41 51 B1 71

(b) (continued)

state of the system today with (1 - 6) precision, is given by:

1 1 P = ~ Log

.tt2 ~" (15)

We applied the K-entropy to the exchange rates generated by our model. The results are given in Table 2 for different value of 8. As is clear from Table 2, the K-entropy K is finite. The last column gives the number of periods over which prediction can be done with a certain precision (as measured by A). Setting the precision parameter (1 - 6) = 0.9, the table shows that the exchange rate can be predicted during five periods (if 8 = 0) and during eight periods (if e = 50 ). This example confirms our previous statement that the predictability depends on the actual parameters in a chaotic model. It also confirms that chaos does


1. OQ

I , 08

1.0"~

1 ,05

1 .04

1 .03

1 o;~

1,01

o.ss O. 00

O. g~

0 .96

O, g:~

O. 93

O. 8;~

0 ,91

0 .9

0 .09

0 ,88 I I I I I I I [ ~'1 71

(c)

Fig. 4. Sensitivity to initial conditions (a) 8 = 0 (b) 8 = 50 (c) e = 100

not imply unpredictability. Contrary to random walk models, chaotic models lend themselves to short-term predictions.

4. Explaining the Facts

Up to now the model presented was purely theoretical in the sense that we used an axiomatic approach. In this section we test the model in an indirect way. We will show that it is capable of producing two of the best known empirical phenomena observed in foreign exchange markets.


1,18

1,14

1.1:)

1.1

1.08

"l. 06

1.04

1.02

1

O. 98

0.00

O. 04

n. 02

0.9

0.88

0.86

0.84

O. 82

0.B I I I I I I I I I I 1 51 10t 451 ~01 ~51 304 351 401 451

t l t~

Fig. 5. Time path of simulated exchange rate

4.1. Unit Roots

One of the best-documented empirical phenomena of the behavior of speculative prices (including exchange rates) is the existence of a unit root. We therefore asked whether the simulated exchange rate (using the determistic chaotic model) also exhibits a unit root. For that, we applied the Dickey-Fuller test (ADF) on the simulated exchange rate. Table 3 shows that we cannot reject the unit root hypothesis, although we did not model a unit root process.

These findings can be understood by taking a look at the attractor of the model in the (St, St-l) space. Fig. 6 shows these attractors. The attractor is stretched around the 45 line. The higher the value of 8 the more stretched the attractor becomes. It is precisely this feature that is detected by the ADF test.


1,12

1,1

1.08

1, OB

1. O"t

1.02

t

O. 98

O. 96

O, 84

O. 92

0,8

0

[] 0

% 1 I I I I I

0.9 0,92 0,94 0.96 O,~B 1 I I I i I I

1.0~ 4,04 1.06 1,08 t .1 1.12

St- 1

(a) (continued) Fig. 6. State space representation of the model (a) e = 0 (b) e = 50 (c) e = 100

4.2. The Forward Premium as a Predictor of Future Exchange Rate Movements

Empirical analysis of the exchange markets has led to the conclusion, which is now undisputed, that the forward premium is a biased predictor of the future change in the exchange rate. That is, the forward premium predicts on average a future movement of the exchange rate in the wrong direction. This feature was most dramatic during the period 1980-85, when the dollar was continuously at a discount in the forward market and yet kept increasing in price.

In this section we test for unbiasedness of the forward premium, as generated by our model. If we find that our theoretical model produces


t . 14.

1.12

1.1

1.06

1,06

1.04

1 .O;Z

1

O, 98

0,66

0,94

0.62

0.9

O,BB

0.B6

rt

0

0

D dg

O.'as I 019 I o'~4 I O.'~e I I.'O;Z I t~s I I~t I 1'14 o6e os= o ss 1 1o4 106 11=

St.- 1

(b) (continued)

the same biasedness result, we have additional indirect support for the model.

Before presenting the tests, it is useful first to look at the time series as generated by our model. Fig. 7 shows the simulated forward premium (interest differential) together with the simulated change in the exchange rate for different values of the interest rate smoothing parameter. We have lagged the forward premium by one period, so that each data point has the realized and the predicted change in the exchange rate.

We observe that the realized changes in the exchange rate are a negative multiple of the predicted ones (as measured by the forward premium). This feature of the movements of the exchange rates has also been observed in reality involving both a highly volatile "risk premium" and a bias in market expectation. We illustrate this in Fig. 8, which shows the realized movements of the dollar/yen rate and the


1.2

1 .15

1 .1

1 .OS

1

0 .95

D .9

0 .85

0 .8

n. 75

0 .7 I I I I 1 G.7 0 .8 D.9 1 1 .~ t ,2

(c)

forward premium of the dollar (in the dollar/yen market). Note that in our model this feature becomes more important as the intensity of interest rate smoothing increases.

The bias of the forward premium as a predictor of future exchange rate movements can be found econometrically as follows. One re- gresses the exchange rate change on the forward premium lagged one period:

A In St+l = a + b FPt + Ut+l, (16)

where FP~ is the forward premium; it is also equal to the interest rate differential.

If the forward premium is an unbiased predictor, b should be positive and significantly different from 0. The estimation of the econometric


Table 1. Experimental evidence of chaotic outcomes

~=0

-y 4 CH CH P5 CH P4 P4 P8 CH CH P4

3 CH P125 P5 P23 P4 P4 P4 CH P4 P4

2 Ch LC P123 P89 CH P4 P4 CH P4 P4

1 S S S S S S S S S S

0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

e=l

4 CH CH PIO CH P4 P4 CH CH CH P4

3 CH CH P8 LC LC P4 CH CH P4 P8

3 CH P l l LC LC PIO P4 CH CH CH P4

1 LC LC LC LC LC LC LC LC LC LC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3 '=10

4 Ch CH

3 CH CH

2 CH CH

1 LC LC

CH CH P12 P4 P4 P4 CH CH

P6 P5 CH P4 P4 P4 CH CH

LC LC LC P9 P4 P4 P16 CH

LC LC LC LC LC LC LC LC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3' = 100

4 Ch CH CH CH CH P4 P4 P4 P4 P4

3 CH CH CH P4 P4 P4 P4 CH P4 CH

2 CH CH PIO P9 P4 P4 P4 P4 P4 P4

1 LC LC LC LC LC LC LC LC LC LC

0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0


Table 2. Potential forecastibility of the returns of the speculative price

K2 P

8 =0 0.430 2.33 A* 8 = 10 0.374 2.67 A 8 =20 0.316 3.16 A

8 =50 0.267 3.75 A e =100 0.414 2.43 A

were b = 10000, c= = 0.1, h = 0.1, k = *A denotes the Log 1/5. The other parameters 0.1, c = 1.0, ~ = 4.0

Table 3. Results for the ADF-test on simulated data

8=20

Sample size a 1 r DW R 2 D.E

100 0.998 0.28 1.73 0.63 -0 .37 (0.003) (0.097)

1000 0.999 0.29 1.72 0.69 -0 .59 (o.oo2) (o.o4)

5000 0.999 0.31 1.71 0.69 -1 .10 (o.ooi) (o.oi)

8 = 50

Sample size 1 r DW R 2 D.F.

100 0,998 0.29 1.64 0.57 -0.41 (0.003) (0.10)

1000 0.999 0.27 1.76 0.83 -0 .57 (0.002) (0,04)

5000 0.999 0.30 1.72 0.83 -1 .22

(0.001) (0.01)

e = 100

Sample size al r DW R 2 D.F.

100 0.997 0.25 1.76 0.91 -0.89 (0.003) (0.099)

1000 0.999 0.31 1.70 0.87 -0.59 (0.002) (0.04)

5000 0.999 0.30 1.71 0.89 -1.28 (0.001) (0.01)


12.a :8=0

0.075

0.050

0.025

0.000 i

-0.025

-0.050

-0.075 300 3ZO 340 360 380 400 420 440 460 480 500

(a)

12.b : 0 : 10 0.i0

0.05

0.00

-0.05

-0.10 300 320 340 360 380 400 4ZO 440 .tGO 480 ~00

i oLERI :Rl(-i.~ (b) (continued)


12.c : 0 = 20 0.0?5

0.050

0000 y

-0 .O:/5

-0.050

--0.D75 . . . . . . . . . . . . . . . . . . . . . . . .............................................................................................................. 300 32.0 340 3(50 380 400 420 4.40 460 4~JO 500

DLEXR20 . . . . . IR20( - I~

(c)

Fig. 7. Forward premium and percentage change in exchange rate: simulations from the model

equation (16) on real-life data, in most cases, shows that b is negative and significantly so (see Fama 1984 and De Grauwe 1989).

Our theoretical model is now capable of replicating these empirical results from real-life exchange rate movements. We regressed equation (16) using the simulated numbers of our model. We present the results in Table 4 for different values of the interest rate smoothing parameter.

The striking aspect of these results is that we systematically find the "wrong" negative value for b, indicating that, as in reality, the forward premium (the interest differential) in our model is a biased predictor of future exchange rate movements. This result is all the more striking in that we have assumed that open interest parity holds, i.e., there are no risk premia in the model that could explain these results. Put differently, the fact that the forward premium is a biased predictor of future exchange rate movements can be explained without having to invoke the existence of risk premia (which up to now most researchers have been unable to detect).

How can one explain the result that a model without risk premia generates forward premia that are biased predictors of future exchange


-0

0 n

0.80 - - Percentage change in F the exchange rate

0 .64] - . . . . Forward ) remium

o32048 -o.oo ~ t~ ]~ _,

--0.80 ..... I ........... I ........... h,,; . . . . . . . 1 ........... I .......... ii ........... [ ........... I .......... I ........... I 73 74 75 76 77 78 79 80 81 82

T ime

1.30

1.04

-0 .26 I

-0 .52

- 0.78

-1 .04

-1 .30 ..... I ........... ~ ........... i . . . . . . . . . . . I . . . . . . . . . . . i . . . . . . . . . . . i . . . . . . . . . . . ~ . . . . . . . . . . . ~ ........... I . . . . . . . . . . . 1 73 74 75 76 77 78 79 80 81 82

T ime

F ig . 8 . Forward premium and exchange rate changes for $/$ and $/DM (1973-1982)


Table 4. Estimates of equation (16)

a b R 2 DW

for 8 = O: 0.0 -1.74 0.27 1.5

(0.1) (-19.2)

for 8 = 1: 0.0 -1.91 0.17 1.4

(0.1) (-14.1)

for 8 = 10: -0.0 -2.20 0.04 1.3

(-0.5) (-6.4)

for 8 = 20: 0.0 -1.86 0.02 1.3

(0.0) (-4.23)

rate changes? An increase in the domestic interest rate leads to an instantaneous decline of the spot exchange rate (an appreciation of the domestic currency). To the extent that chartists extrapolate exchange rate movements, the increase in the interest rate can lead to expectations of future appreciations. If the chartists dominate the market, this extrapolative behavior will tend to be self-fulfilling, leading to actual future appreciations. Thus, higher domestic interest rates will on average be associated with future appreciations of the domestic currency.

5. Conclusion

In this paper we have constructed a monetary model of the exchange rate based on the celebrated Dornbusch model. We have added a speculative dynamic in which "chartists" and "fundamentalists" interact and in which the weight given to these two classes of speculators changes depending on market circumstances. The forecasting rules we have assumed for these two groups of speculators are extremely simple, if not crude. The chartists are assumed to extrapolate recent changes in the exchange rate using a simple moving average procedure, whereas the fundamentalists base their expectations on simple PPP calculations. We show that these simple rules implemented in the Dornbusch model are sufficient to produce very complex exchange rate behavior (chaos).

The results of our model allow the development of a more sophisticated view of the role of news in the foreign exchange market. The "news paradigm" that has dominated thinking about the foreign exchange market requires "news" to occur whenever the exchange rate changes. This has led to the situation in which market observers search for news whenever the exchange rate moves. As a rule, these


observers will detect some random event that can be made responsible for the "inexplicable" movement in the exchange rate.

The results of our model lead to a different view. The speculative dynamic produced by the interaction of speculators using different pieces of information is capable of generating complex dynamics that we do not fully understand. Although "news" remains important, we do not need to invoke it to explain every observed movement of the exchange rate. Many such movements are unrelated to the occurrence of news, but follow (as yet) not fully understood dynamics. Our model can therefore be considered to provide a synthetic view of the "news" model, that up to recent times has dominated academic thinking, and the more popular view that exchange rate movements are driven by speculative dynamics. The latter view has recently acquired some academic respectability in the works of, among others, Shiller (1984) and Delong, et al. (1990).

A final implication of the results of our model relates to the difference between stochastic systems, as a description of price dynamics (i.e. random walk), and chaos. Although both dynamic processes appear equivalent, it was shown that these models have a totally different forecasting potential. While stochastic processes are intrinsically not forecastable, chaotic models can be forecasted for some (short) period of time. The length of this period depends on the parameters of the model. Next to the parameters of the model, the other essential parameter for prediction is the precision by which we can locate the systems' state in the actual period. The more precise this information is, the longer the prediction period. Whether or not this might lead to practical applications, however, remains an open question.

We have not attempted to perform rigorous empirical testing of the theoretical model presented in this paper. However, we have shown that the model is capable of reproducing two of the best-known empirical facts concerning exchange rate behavior. One is the unit-root property of exchange rate movements, and the other is the bias of the forward premium as a predictor of future exchange rate changes.

It is obvious that the ability of our model to mimic some important features of exchange rate dynamics is not sufficient to establish its validity. More empirical work will have to be performed to confirm whether models of the kind presented here can fruitfully be used.

Acknowledgments

We are grateful to Casper de Vries and Daniel Gros for useful sug- gestions. We profited from the discussions during seminars at the University of Tilburg (CentER), the University of Amsterdam, and the


University of Siegen. Financial assistance from the Center for Economic Policy Research, the Ford Foundation, and the Alfred Sloan Foundation is gratefully acknowledged.

Appendix A

The experimental way of detecting chaos is partly based on a topological property. This property can be stated as follows.

The Hausdorff dimension has a non-integer finite value for any chaotic signal.

This Hausdorff dimension can be approximated by the correlation dimension. More specifically, the correlation dimension Dc yields a lower bound on the Hausdorff dimension DH; i.e.,

Dc ~

x~ = [x~, x~+l, , X~+M]

x~ = [x j , . . . , xj+M].

The correlation dimension Dc is then given by

Dc = lim lira lim cN(r). r -~O rn---+oo N--*oo

For random systems this value has no finite limit, whereas for chaotic systems this value is finite and non-integer.

Notes

1. See Frenkel and Mussa (1985), Levich (1985), and Mussa (1984). 2. See Dornbusch (1976). 3. in a previous paper, one of the authors used a partial equilibrium model of the

exchange rate. The chaotic results obtained there also depended on the existence


of a J-curve effect (see De Grauwe and Vansanten 1990). Here we discard the assumption of a J-curve, This should make the results stronger,

4. Note that chartists themselves may believe that these movements are unrelated to the fundamentals. They consider these market movements to be important pieces of information reflecting other agents' beliefs about market fundamentals.

5. In De Grauwe and Vansanten (1990) these weights were assumed to be fixed.

References

Allen, H. and M. Taylor (1989) "Charts, Noise and Fundamentals: A Study of the London Foreign Exchange Market," CEPR Discussion Paper No. 341.

Baumol, W. and Benhabib (1989) "Chaos: Significance Mechanism and Economic Ap- plications," Journal of Economic Perspectives 3, 77-105.

Bollerslev. T. (1987) "A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economic and Statistics, 542-547.

Brock, W. and A. Malliaris (1989) Differential Equations, Stability and Chaos in Dynamic Economics. Amsterdam: North Holland.

Cutler, D., J. Poterba, and L. Summers (1990) "Speculative Dynamics," NBER Working Paper No. 3242.

De Grauwe, R (1989) =On the Nature of Risk in the Foreign Exchange Market: Evidence from the Dollar and the EMS Markets," CEPR Discussion Paper No. 352. CEPR, London.

De Grauwe, R and K. Vansanten (1990) "Deterministic Chaos in the Foreign Exchange Market," CEPR Discussion Paper No. 370.

Delong, B., et al. (1990) "Noise Trader Risk in Financial Markets," Journal of Political Economy 98, 703-737.

Devaney, R. (1989) "An Introduction to Chaotic Dynamical Systems", 2nd ed. Menlo Park: Addison Wesley.

Dornbusch, R. (1976) "Expectations and Exchange Rate Dynamics," Journal of Political Economy 84, 1161-1176.

Engle, C. and J. B. Hamilton (1990) "Long Swings in the Dollar: Are They in the Data and Do Markets Know it?" American Economic Review 80, 689-713.

Engle, R. F. and T. Bollerslev (1986) =Modelling the Persistence of Conditional Variances," Econometrics Review 5, 1-50.

Fama, E. (1984) =Forward and Spot Exchange Rates," Journal of Monetary Economics 14, 319-338.

Frankel, J. and K. Froot (1986) "The Dollar as a Speculative Bubble: A Tale of Chartists and Fundamentalists," NBER Working Paper No. 1854.

Frankel, J. (Winter 1989-90) "Chartists, Fundamentalists and Trading in the Foreign Exchange Market," NBER Reporter 9-12.

Frankel, J. and M. Mussa (1985) =Asset Markets, Exchange Rates and the Balance of Payments." In Ronald W. Jones and Peter B. Kenen (eds), Handbook of International Economics, vol. II. Amsterdam: North Holland.

Goodhart, C. (1990) "News and the Foreign Exchange Market," LSE Financial Market Group Discussion Paper No. 71.

Grandmont (1985) "On Endogenous Competitive Business Cycles," Econometrica 53, 995-1045.

Koedijk, K., M. Schafgans, and C. de Vries (1990) "The Tail Index of Exchange Rate Returns," Journal of International Economics 29, 93-108.


Levich, R. (1985) "Empirical Studies of Exchange Rates: Price Behavior, Rate Determina- tion and Market Efficiency," In Ronald W. Jones and Peter B. Kenen (eds), Handbook of International Economics, Vol. II. Amsterdam: North Holland.

Meese, R. and K. Rogoff (1983) =Empirical Exchange Rate Models of the Seventies: Do They Fit Out-of-Sample?" Journal of International Economics 19, 3-24.

Mussa, M. (1984) =The Theory of Exchange Rate Determination." In John F. O. Bilson and Richard C. Marston (eds), Exchange Rate Theory and Practice Chicago: University of Chicago Press, 13-78.

Scheinkman, J and B. Lebaron (1989) "Nonlinear Dynamics and Stock Returns," Journal of Business 62, 311-337.

Schiller, R. (1984) "Stock Prices and Social Dynamics," Brookings Papers on Economic Activity 2, 457-498.

Schulmeister, S. (1987) "An Essay on Exchange Rate Dynamics," Discussion Paper IIM/LMP, Wissenshaftzentrum Berlin fur Sozialforschung, Berlin 87-88.

Schulmeister, S. (1989) "Currency Speculation and Dollar Fluctuations," Banca Nazionale del Lavoro 3, 343-365.

Schuster H. (1985) Deterministic Chaos. Weinheim: Physik-Vedag.

degrauwe.pdf

Documents

Transcript of degrauwe.pdf