The Role of Foreign Exchange Intervention in a Chaotic Dornbusch Model

Da Silva, Sergio Article The Role of Foreign Exchange Intervention in a Chaotic Dornbusch Model Kredit und Kapital Provided in Cooperation with: Duncker & Humblot, Berlin Suggested Citation: Da Silva, Sergio (2000) : The Role of Foreign Exchange Intervention in a Chaotic Dornbusch Model, Kredit und Kapital, ISSN 0023-4591, Duncker & Humblot, Berlin, Vol. 33, Iss. 3, pp. 309-345, https://doi.org/10.3790/ccm.33.3.309 This Version is available at: https://hdl.handle.net/10419/293419 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Kredit und Kapital, Heft 3/2000 Seiten 309-345 The Role of Foreign Exchange Intervention in a Chaotic Dornbusch Model By Sergio Da Silva, Birmingham and Brasilia* I. Introduction A possible explanation for exchange rate movements away from the level consistent with macro fundamentals is given by destabilizing expectations of speculators who use supposedly recurring patterns in graphs to make forecasts. Such 'technical' or 'chart' analyses may be a source of nonlinearity leading to chaos in the Dornbusch (1976) model, as shown by De Grauwe and Dewachter (1992) and De Grauwe, Dewachter, and Embrechts ((1993), Chapter 5). These chaotic models are able to mimic the alleged random walk pattern of actual exchange rates despite the fact that the 'stochastic' behavior is produced by a deterministic solution. The model presented here belongs to the same line of research. It is able to conciliate the two apparently divergent pieces of evidence that the nominal exchange rate appears to follow a random walk although it also seems to be explained by fundamentals. A martingale process (i. e. a random walk with heteroskedasticity) may be a solution of a chaotic version of the Dornbusch model in which fundamentals still matter. In De Grauwe and Vansanten (1990) intervention could stabilize a chaotic exchange rate within a framework that was the forerunner of the models of De Grauwe and Dewachter (1992) and De Grauwe, Dewachter, and Embrechts (1993). However, nonlinearities were introduced in the De Grauwe-Vansanten model by assuming the existence of a J-curve. Such a restrictive assumption was dropped by De Grauwe and Dewachter, and * This paper draws on chapter III of my PhD thesis at the University of Birmingham (Da Silva (1999a)). I am grateful to John Fender, Paul De Grauwe, Peter Sinclair, Nick Horsewood, David Kelsey, Somnath Sen, and an anonymous referee for comments on previous drafts. Any errors and shortcomings are my responsibility. Financial support from CAPES (Brazilian Post-Graduate Federal Agency) under grant BEX-118995-8 is acknowledged. Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 310 Sergio Da Silva De Grauwe, Dewachter, and Embrechts, henceforth DD (1992) and DDE (1993) respectively. This paper generalizes further the extension of the Dornbusch model accomplished by the DD (1992) and DDE (1993) models by rescuing the point made in the De Grauwe and Vansanten paper. A major novelty presented here refers to the introduction of a policy rule linking the nominal exchange rate to the nominal money supply The aim is to show that massive interventions can remove chaos from the foreign exchange market in the DD (1992) and DDE (1993) models. Section II. sets up the model, whose simulated solutions are presented in Section III.; the results are then contrasted with selected stylized facts and previous work (Section IV.); and Section V. concludes. Formal tests for chaos are relegated to an appendix. II. The Model 1. The Building Blocks The model is made up of equations (l)-(8) displayed in Table 1. Variables and parameter x are defined as follows. Variable St stands for the nominal exchange rate (the price of the foreign currency in units of the domestic currency) at time period t; S*t is the equilibrium nominal exchange rate at t\ Pt represents the domestic price level at t; Pt~i is the domestic price level at t— 1; P*t stands for the steady-state value of the domestic price level at t; and Pft* is the steady-state value of the foreign price level at t. Parameter x ^ (0? measures the actual speed of adjustment in the goods market and may be seen as a proxy for the degree of domestic price level flexibility, as explained below. Equation (1) gives the long-run equilibrium condition, which is defined as a situation in which purchasing power parity (PPP) holds. Since PPP is one of the long-run properties of the Dornbusch model, equation (1) states that explicitly. Equation (1) is employed by both the DD (1992, p. 28) and DDE (1993, p. 129) models. Equation (2) is a substitute for the Phillips curve in describing the short-run price dynamics by linking domestic price level changes and nominal exchange rate deviations from equilibrium. It states that since X > 0 whenever the nominal exchange rate St exceeds its PPP-value S*t the domestic price level increases, i.e. Pt >Pt_i. So whenever the currency is undervalued an excess demand in the goods market follows, Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 311 Table 1 The Extended Nonlinear Dornbusch Model with Speculative Dynamics and Foreign Exchange Intervention Pt (SA* Pt-i \s;J (MA (stY_ y? \Pj\sJ (1 + i, Sj+1 _(cSt+1\ct(Fsuiy-Ct St-1 \St-i) KSt^J FSUI _ (SUAX St-1 Ut-J causing the domestic price level to increase, and vice versa. Equation (2) is the same as the one employed in the DD ((1992), p. 28) model. Since parameter x measures the speed of adjustment in the goods market, the value of x may be interpreted as a proxy for the degree of domestic price level flexibility in the context of this model. Price rigidity occurs at the borderline case where x —• whereas full price flexibility is represented by x oo. The money market equilibrium is given by equation (3). Variable Y^ stands for the domestic real income at time period t, which equals the Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 312 Sergio Da Silva exogenous level of domestic output by assumption; it represents the domestic nominal interest rate at t\ Mt gives the central bank target to the domestic nominal money supply at and St is the nominal exchange-rate target at t. Parameters 6 G (0, oo) and 6 G (0, oo) are proxies for the income elasticity of money demand and the absolute value of the interest elasticity of money demand respectively. As will be seen, parameter S will not appear in the solution to this model (equation (14)). So the results presented in Section III should hold regardless of the value of the income elasticity of money demand. The central bank parameter 0 captures the degree of official intervention in the foreign exchange market. A major novelty in the model in Table 1 lies in equation (3); thus, it deserves a more detailed rationale. That equation is a standard LM such as Mt/Pt = Yf 6/(1 + it)0, where Mt stands for the domestic nominal money supply at time period t, to which more structure is given by the introduction of the following policy rule: where parameter 0 is zero under free float and approaches either plus or minus infinity to a fixed exchange rate; leaning-against-the-wind intervention is represented by 0 G (-oo, 0), whereas leaning into the wind is given by 0 G (0, oo). Policy rule (9) can be found, for instance, in Marston (1985, p. 910), who use its natural-logarithm version in other contexts. The economy is under free float when 0 = 0 because in such a situation the central bank focuses exclusively on the target to the domestic nominal money supply abstaining from any intervention in the foreign exchange market (0 = 0 produces Mt = Mt in (9)). When Mt = Mt and 0 = 0, equation (3) collapses to the standard LM referred to above, which makes up most versions of the Dornbusch model, including the DD (1992, p. 27) and DDE (1993, p. 128) models. Accordingly, it may be argued that the Dornbusch model implicitly presupposes free float. Thus, the DD (1992) and DDE (1993) models turn out to be particular cases of the model in Table 1 in that these implicitly assume 0 = 0. The fixed exchange-rate regime holds when 0 —• ± oo because in such a situation the authorities focus exclusively on the nominal exchange-rate target without concern for the domestic nominal money supply (0 —• ± oo yields St = St in (9)). It might also be noted that, since credibility issues are not Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 313 taken into account, 0 —> ± oo can be thought of as a fixed exchange-rate regime with perfect credibility. Leaning against the wind is the intervention operation that attempts to move the exchange rate in the opposite direction from its current trend. Leaning into the wind is the operation that is motivated by the central bank's desire to support the current exchange-rate trend. In the context of this model, both leaning-against-the-wind and leaning-into-the-wind interventions are carried out by changes in Mt. Whether such changes are sterilized is not discussed here. If St > St (St < St) for any reason, the aim of leaning against the wind is thus to reduce (increase) the current nominal exchange rate St\ in this model, that can be achieved by reducing (increasing) Mt in (9) when 4> < 0. By contrast, since leaning into the wind signifies supporting the current exchange-rate trend, if St > St (St < St) such an intervention operation means increasing (reducing) Mt in (9) when 0 > 0. We will see in Section III that the degree of such intervention also matters in this model for a successful stabilization of a chaotic nominal exchange rate. Equation (4) is the uncovered interest rate parity (UIP) hypothesis. Variable S*+1 stands for the forecast made at time period t for the nominal exchange rate at t + 1; and i[ represents the foreign nominal interest rate at t. Both it and i{ are the nominal interest rates available on similar domestic and foreign securities respectively, with the same periods to maturity. UIP states that the expected foreign-exchange gain from holding one currency rather than another (the expected nominal exchangerate change) must be just offset by the opportunity cost of holding funds in this currency rather than the other (the nominal interest rate differential). UIP is a basic ingredient in even the simplest versions of the Dornbusch model. Equation (4) is also used in both the DD (1992, p. 27) and DDE (1993, p. 128) models. Equations (5)-(8) describe the speculative dynamics of the model by introducing chartist behavior among speculators. In equations (5)-(8), speculators are assumed to take positions in the market at time period t based on the forecasts they have made for t + 1, and these forecasts were made by them using information available at t— 1. That is the reason why St-i appears rather than St in these equations. Since St is the solution obtained when speculators have taken their market positions, St is not observable by these agents at the moment they make their forecasts. This point is observed by DDE (1993, p. 74). Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 314 Sergio Da Silva Equation (5) splits expectations between two components - the expectations based on charts CSt+i* anc* the expectations based on the fundamentals of the model fS*+1. Variable St~i is the nominal exchange rate at time period t1, whereas Ct e (0, 1) stands for the weight given to charting at t. As far as the weight of chartist activity is concerned, if Ct G (0, 0.5) then there is less charting than forecasts based on fundamentals. If Ct = 0.5 then half of the speculators are involved in charting and the other half are making forecasts based on fundamentals. If Ct e (0.5, 1) then expectations are dominated by chartists. Equation (5) is the same as the one employed in the DDE (1993, p. 131) model. The expectation rule for the forecasts based on charts is given by equation (6). Variables St-2 and St-3 are the nominal exchange rates at time periods t - 2 and t — 3 respectively; and parameter v E (0, oo) stands for the degree of past extrapolation used in technical analysis. Since v > 0, the greater v, the more the past will be extrapolated into the future in exchange rate forecasts, and chartists will expect the nominal exchange rate at time period t + 1 to be less than the nominal exchange rate prevailing at t— 1. Rule (6) is employed by DDE ((1993), Chapter 3, p. 80) in a simple chaotic model without money. Here it is used in the context of the Dornbusch monetary model. Doing so, no principle is apparently violated. LeBaron (1996) demonstrates empirically significant forecastability from a simple moving average trading rule (similar to (6)) for series of the US dollar against the mark and the yen that uses both weekly and daily data. Equation (6) may seem a little odd at first sight, but the further discussion presented below will help to clarify it. The rationale to (6) is provided following DDE (1993, pp. 78-80). Speculators expect an increase in the nominal exchange rate whenever a short-run moving average of past exchange rates Sf crosses a long-run moving average of past exchange rates S\ from below (Figure 1). In such an event a buy order of the foreign currency is given by them. In turn, they expect a decline of the nominal exchange rate whenever Sf crosses St from above. In the latter case speculators order a selling of the foreign currency. This can be postulated as Equation (10) states that since v > 0, whenever Sf > S\ (Sf < chartists expect an increase (fall) of the nominal exchange rate relative to the Kredit und Kapital 3/2000 (10) OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 315 most recently observed value St_ By assumption, the short-run moving average Sf is based on a one-period change, i. e. and the long-run moving average S\ is based on a two-period change, i. e. Rule (6) can be obtained by plugging (11) and (12) into (10). De Grauwe (1996, pp. 181-185) presents a microeconomic foundation for this chartist behavior. While making forecasts based on the fundamentals of the model, speculators are assumed to use the rule given by equation (7). Variable S*t_x represents the equilibrium nominal exchange rate at time period t— 1, whereas parameter A e (0, oo) gives the expected speed of return of the current nominal exchange rate toward its equilibrium value. According to (7), whenever fundamentalists observe a market rate above (below) the PPP-value, they will expect it to decline (increase) in the future. Since A > 0, the greater A, the higher the expected speed of return toward the fundamental rate. The greater A, the faster fundamentalists will expect the nominal exchange rate to increase (fall) toward its equilibrium value if St-i < SI _1(St-1 > SI _1). Values of A greater than one mean that fundamentalists expect some sort of overshooting (DDE (1993), p. 111). Equation (7) is also employed in the DDE (1993, p. 131) model. The weight of charting is endogenized by equation (8). The amount of technical analysis used by speculators is made dependent on the size of the deviation of the current nominal exchange rate from its equilibrium (fundamental) value. Equation (8) states that if (St_i — J2 —> oo then Ct —> 0, i.e. whenever deviations from PPP increase, the expectations based on charts will be reduced. If (St-1 - S*t_x)2 —• 0 then Ct —• 1, which means that whenever deviations from PPP tend to be eliminated, charting will grow in importance among speculators. Parameter t e (0, oo) stands for the speed at which forecasts based on charts switch to forecasts based on fundamentals. The higher t, the faster chartist activity will decrease, and vice versa. The same weighting function (8) is found in the DDE (1993, pp. 75-78) model. In accordance with (8), LeBaron (1994, p. 400) points out that predictability appears to be higher during periods (11) (12) Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 316 Sergio Da Silva of lower volatility, a phenomenon used by chartists to achieve some small out-of-sample improvements in forecasts. This completes the description of the model. We can proceed toward the solution to the model. The eight endogenous variables are: St, Pt, iu S*t, Set + 1, cSet+i, and Ct. The model is recursive in that the block made up of equations (5)—(8) runs first. An additional assumption beforehand helps to simplify matters. The rate at which speculators expect the nominal exchange rate to return toward its fundamental value is assumed to be the same as the speed at which prices in the goods market actually adjust, i. e. One known property of the Dornbusch model is that after a possible overshooting of the nominal exchange rate in the impact period, it asymptotically moves back toward its equilibrium value at the same pace as the domestic price level movement (Dornbusch (1976), p. 1165). Therefore it is not unreasonable to think that such piece of information can be taken into account by fundamentalists. Assumption (13) is also employed in both the DD (1992, p. 34) and DDE (1993, p. 131) models. Substituting (13) in (7) and then plugging the resulting equation along with (6) and (8) into (5) yields an expression for Set + 1. Next, inserting (1) into (2) obtains an expression for Pt\ and inserting the expression for Pt into (3) gives an expression for 1 + if. Then, substituting the latter expression into (4) produces an expression for St. Without loss of generality we may assume additionally every appropriate exogenous (fundamental) variable to be constant and normalized to unity (and i{ = 0). Considering this assumption in the expression obtained earlier for St it becomes apparent that it depends only on a term for Set+1. That assumption also implies S*t = S*t_± = 1 so that PPP holds in equilibrium. After inserting this result into the expression for Set + 1 and substituting it in the expression for St we obtain the solution to the model for the nominal exchange rate given by the following weighted geometric moving average: 2. Solution (13) A = X- (14) St = St-1fl St.2h St.3/3, Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 323 with recent attempts to revive explanations based on fundamentals to beat the simple random-walk model (e.g. Mark (1995)). Here fundamentals matter because the interaction between speculative private behavior and foreign exchange intervention can give rise to chaos and thus mimic a random walk. Fundalmentals also matter because massive foreign exchange interventions are able to stabilize these chaotic motions and thus influence the nominal exchange rate. There is a piece of indirect evidence addressing the implication that intervention may stabilize a chaotic nominal exchange rate. In accordance with that, Dominguez (1993) presents evidence that foreign exchange intervention actually reduces the volatility of the nominal exchange rate. There is a puzzling piece of evidence, too. Using both weekly and daily data of foreign exchange intervention and foreign exchange series of the US dollar against the mark and the yen, LeBaron (1996) shows that after removing periods in which the Federal Reserve is active, the ability to predict future exchange rates coming from technical trading rules is dramatically reduced. This suggests that central bank intervention may introduce noticeable trends into the evolution of the nominal exchange rate and thus create profit opportunities coming from speculation against the central bank. In this connection, Taylor (1982) and Leahy (1995) find evidence that central banks make money on their foreign exchange intervention operations; and Silber (1994) presents evidence in a cross sectional context that technical rules have value whenever governments are present as major players. More strikingly, Szpiro (1994) argues that an intervening central bank may even induce chaos in the nominal exchange rate. Thus, these results suggest that the more central banks intervene, the more they give incentives to chartists to enter the market, thereby increasing the chance of chaos. Chaos is also shown to be possible in a microfounded sticky-price model by Da Silva (1999b), where the model developed recently by Obstfeld and Rogoff (1995; 1996) is extended to encompass the speculative dynamics and the modeling of foreign exchange intervention discussed in this paper. In the Da Silva (1999b) model chaos is possible under a non-zero amount of foreign exchange intervention, which is in line with the 'puzzling' findings referred to in the paragraph above. By contrast, another effect of intervention is analyzed in the present paper: the more central banks intervene, the less the likelihood of chaos. Most precisely, massive intervention can remove chaos from the foreign exchange market. However, it might be noted that the 'puzzling' effect of Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 324 Sergio Da Silva intervention generating chaos also appears in the results displayed in Tables 2-5, where low volumes of foreign exchange intervention can also lead to chaos. Here low amount of intervention can be interpreted as either intervention toward profitability on the part of the central bank or non-credible attempts at stabilization of the nominal exchange rate. V. Conclusion This paper examines whether a result obtained in De Grauwe and Dewachter (1992) and De Grauwe, Dewachter, and Embrechts (1993, Chapter 5), that chaos can be generated in a Dornbusch style model if some traders are chartists, will continue to hold when central banks engage in foreign exchange intervention. The answer seems to be a qualified no, that is, generally massive interventions are able to stabilize the chaos. Central bank intervention is introduced to represent anti-chartist behavior through a rule connecting the nominal exchange rate to the nominal money supply. The analysis of both leaning-against-the-wind and leaning-into-the-wind interventions becomes possible and the Dornbusch model is reduced to the particular case of free float. Chaotic, cyclical, and unstable nominal exchange-rate series under free float and low amount of intervention are shown to collapse to stable ones as long as massive central bank interventions are carried out. The results in this paper show that past extrapolation into the future in chartists' forecasts can produce very complex dynamics in the Dornbusch model. The possible solutions to the model range from stability and cycles of different periodicities to chaos (with or without crashes) and instability. Under both free float and low volumes of intervention, the more chartists extrapolate the past into the future, the larger the variance of the chaotic nominal exchange rate, and the greater the currency crashes. The higher the past extrapolation by chartists, the larger must be intervention to stabilize the nominal exchange rate. Also, the emergence of chaos does not show dependence on the speed at which forecasts based on charts switch to forecasts based on fundamentals. Massive interventions can also remove chaotic, cyclical, and destabilizing movements even if the assumption that prices are sticky is relaxed. Chaos, crashes, cycles, and instability emerge with both free float and low volumes of intervention, when the interest elasticity of money demand assumes sensible values. Massive interventions can again proKredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 325 duce stability in such a scenario. However, intervention plays no role in the context of the liquidity trap. This study shows consistency with a number of stylized facts and previous results in the literature on exchange rates and foreign exchange intervention. Most remarkably, the model presented here is able to conciliate the two apparently divergent pieces of evidence that the nominal exchange rate appears to follow a martingale process although it also seems to be explained by fundamentals. The random-like behavior of the nominal exchange rate in which crashes crop up is generated by deterministic solutions to the model, and since massive foreign exchange interventions are able to stabilize these chaotic motions, fundamentals - most precisely, exchange rate policy - also influence the nominal exchange rate. Appendix The decision that a given solution to the model is chaotic is made in Tables 2-5 on the grounds that no datapoint repeats itself in the range of 10000 periods. A problem with simulations is that there is no formal guarantee that a reached conclusion still applies for the simulation range plus one. For that reason, formal tests for chaos are carried out in this Appendix for the obtained chaotic solution with parameters v = 15, <f> = 0, ¿=104, x — ^ = 0.45, 0 = 0.95, and initial values 1.00000000, 0.99000000, and 1.02000000. A data record of 15000 points is here taken, and the first 100 values are skipped to allow for the nominal exchange rate to settle into its final behavior. The program employed for data analysis was Chaos Data Analyzer: The Professional Version 2.1® by J. C. Sprott, copyright © 1995 by the American Institute of Physics. The pictures in Figure 5 were obtained using such a software. Information regarding description of statistics as well as suggestions for analysis strategy were taken from the PC user's manual of that software by Sprott and Rowlands (SR) (1995). Panel a in Figure 5 suggests that the nominal exchange rate exhibits a randomlike behavior similar to the graph of data of a white (uncorrelated) noise (e.g. SR (1995), p. 51). No obvious structure is seen. However, even in well-known chaotic systems - such as the logistic and Henon maps - this sort of graphing of data usually reveals no structure (e.g. SR (1995), pp. 50-51). But a discernible structure emerges in a three-dimensional embedding plotting (panel b in Figure 5), i.e. there is a 'strange attractor'. Strange attractors are suggestive pictures that can be plotted from chaotic series showing some order in fake randomness. The shape of the attractor in panel b in Figure 5 is very similar to the one displayed in DD (1992, p. 39), which is generated by a plotting of 6000 observations in a twodimensional diagram. It also retains a certain resemblance to the two-dimensional phase diagram shown in DDE (1993, p. 136). A blow-up showing that no datapoint repeats itself in this attractor is given by DD (1992, p. 39). If the data were random, a formless cloud of dots ('random dust') would have emerged, whereas a Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 326 Sergio Da Silva cycle and a nearly periodic system would have displayed a closed and a fuzzy loop respectively (SR (1995), p. 17). Thus, since the data are aperiodic but not random, they are chaotic. Indeed, chaos is defined as apparently stochastic behavior occurring in deterministic systems (Stewart (1997), p. 12). Panel c in Figure 5 shows a logarithmic plot of the probability distribution of the data record, using 32 bins into which the datapoints are sorted according to their value. The bins all have the same width and are spread uniformly between the lowest and highest data value. Looking at the left-hand side in panel c in Figure 5, the distribution might look Gaussian like the one emerging in genuinely random data (e.g. SR (1995), pp. 51-52). However, as the right-hand side is also taken into account, a 'fractal' shape associated with chaos can be recognized. If the data were periodic, a simple histogram with sharp edges would have appeared (SR (1995), p. 18). Table 6 provides a summary of the standard statistics obtained from the data set as well as the other statistics presented below. A calculated Hurst exponent of about 0.04 indicates that the data are not actually random. The Hurst exponent gives a measure of the extent to which the data can be represented by a random walk (or a fractional Brownian motion). White (uncorrelated) noise has a Hurst exponent of 0.5 (SR (1995), pp. 22, 37). Also, the IFS dumpiness test (panel d in Figure 5) shows that the data cannot be represented by a random walk. In a picture displaying the IFS dumpiness test, white noise fills it uniformly whereas chaos or colored (correlated) noise generates localized clumps (SR (1995), p. 18). Relative LZ complexity gives a measure of the algorithmic complexity of a time series. Maximal complexity (randomness) has a value of 1.0, whereas perfect predictability (cycles) has a value of 0 (SR (1995), p. 36). The data are not periodic either, because the calculated LZ complexity is around 0.6. It might be noted that this is the same value as the one the program calculates for the Henon map. The fact that the data are not periodic or quasi-periodic is confirmed, too, as their power spectrum is visualized. The power spectrum is reasonably broad (panel e in Figure 5) if calculated by a fast-Fourier transform on the data record (SR (1995), pp. 8-9 provide a technical discussion of this method). Panel e in Figure 5 displays the log of the power spectrum versus frequency, using 32 frequency intervals. Chaos and random data give rise to broad spectra, while cycles and quasi-cycles generate a few dominant peaks in the spectrum (SR (1995), p. 20). The dominant frequency refers to the frequency at which the power spectrum has its maximum value (SR (1995), p. 37). Calculated by the fast-Fourier transform method, the dominant frequency is about 0.02. This suggests that the data are more likely to be chaotic or random, not periodic or quasi-periodic. Using the maximum-entropy method, there is virtually no dominant frequency (panel / in Figure 5) (SR (1995), p. 21 give a technical account for this method). Panel / in Figure 5 is a log-linear view of dominant frequencies, using 2 complex poles of discrete frequency. It suggests that the data are neither period nor quasi-periodic. An attractor can be quantified by measures of its dimension and its Lyapunov exponents. The dimension evaluates the complexity of the attractor, whereas the Lyapunov exponent measures the sensitivity to initial conditions, i.e. the famous 'butterfly effect' of chaotic series. Capacity dimension and correlation dimension are major measures of dimension of a chaotic attractor (a description of these staKredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 327 tistics is presented by SR (1995), pp. 8, 24, 25). Values greater than about 5 for these measures give an indication of randomness, whereas values less than 5 provide further evidence of chaos (SR (1995), p. 25). Extreme sensitivity to tiny changes in initial conditions and therefore evidence of chaos is obtained as long as the largest Lyapunov exponent is positive. A zero exponent occurs near a bifurcation; periodicity is associated with a negative Lyapunov exponent; and white (uncorrelated) noise is related to an exponent approaching infinity (SR (1995), pp. 23, 37). The correlation dimension calculated from the data is about 1.7. This gives evidence of chaos. Panel g in Figure 5 shows a saturation in the calculated correlation dimension as the embedding dimension is increased. Such a well-defined plateau indicates an appropriate embedding dimension in which to reconstruct the attractor. Another indication of proper embedding is the minimum dimension for which the number of false nearest neighbors falls to zero (panel h in Figure 5). Such a picture suggests the proper embedding as given by 3. Thus, panels g and h in Figure 5 provide an indication of low-dimensional chaos, albeit some quasi-periodic data (such as the one generated from the so-called two incommensurate sine waves) may also exhibit the properties shown in these pictures (SR (1995), p. 50). The capacity dimension calculated from the data is about 1.9. Similar to the correlation dimension, a capacity dimension greater than about 5 would have implied random data. So the calculated capacity dimension is compatible with the presence of chaos in the data. The largest Lyapunov exponent calculated from the data is about 0.3, whereas the largest Lyapunov exponent to the base e is about 0.2. These calculations considered the proper embedding dimension as given by 3, using 3 time steps and accuracy of 10~4. Such positive values for the Lyapunov exponents give further evidence of chaos. Entropy is a measure of disorder in the data. It is given by the sum of the positive Lyapunov exponents to the base e (SR (1995), p. 37). Its reciprocal gives roughly the time over which meaningful prediction is possible (SR (1995), p. 25). The approximated entropy calculated from the data set is about 0.3. It might be noted that this is exactly the same value as the one the software calculates for the Lorenz attractor. Also, the calculated entropies of the Henon and logistic maps are about 0.4 and 0.5 respectively. Thus, the entropy of this chaotic solution to the Dornbusch model is very similar to those expected for well-known chaotic systems. The BDS statistic is a measure of deviation of the data from pure randomness (SR (1995), p. 37). The BDS statistic calculated from the data record is around 1. It is worth pointing out that the program calculates exactly the same BDS statistic to the logistic map. To test whether the evidence of hidden determinism in the data is robust, it is prudent to repeat the calculations of the quantitative measures of the attractor using surrogate data that resemble the original data but with the determinism removed. Robustness implies that analysis of these surrogate data should provide values that are statistically distinct from those calculated from the original data (SR (1995), pp. 14-15). As observed, "this test is a very important one and is rarely Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 328 Sergio Da Silva included in papers claiming observation of low-dimensional chaos in experimental data" (SR (1995), p. 15). The most useful method is to Fourier-transform the data, randomize the phases, and then inverse Fourier-transform the result to get a new time series with the same spectral properties as the original but lacking determinism (SR (1995), p. 14); this implies a different probability distribution. Panel i in Figure 5 shows a bell-shaped distribution (indicating lack of determinism) which differs from the distribution of the original chaotic data in panel c. The major quantitative measures of the surrogate data are indeed different from those of the original data. The largest Lyapunov exponent and the largest Lyapunov exponent to the base e are 0.847 ±0.011 and 0.587 ± 0.008 respectively. The correlation dimension is 4.505 ± 0.089, and the calculated capacity dimension is 2.238 ±0.128. To know whether this difference is statistically significant, ideally one should generate many surrogate-data sets and see whether the results from the original data lie within the range of values corresponding to the surrogates. If they do, then the difference is not statistically significant and the original data are indistinguishable from colored noise (SR (1995), p. 15). By repeating the above test twice, the calculated Lyapunov exponents were found to be greater than the values shown above. Another surrogate-data set was generated simply by shuffling the original data values, as one shuffles a deck of cards. The calculated Lyapunov exponents were still greater than the ones obtained from the original data. Thus, the conclusion that the data obtained from the Dornbusch model are chaotic and distinguishable from colored noise seems to be robust. Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 329 IN O ¡D EH CO EH co EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO O CM U CSI U EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO CJ O CO U CSI U CSI O CSI CJ CSI O EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO CM O ID CSI o CSJ o CSI o CSI U CSJ U CSI U CSI U CSJ U CSI U CSI U CSI O CSI U EH CO EH CO EH CO EH CO EH CO o |D CSI U CSJ U CSJ u CSI u CSI U CSJ o CM U CSI U (M O CSJ O CSI U CSJ U CSJ U CSI O CSI O CSI U EH CO -|D ¡D ID ÍD ¡D ID ÍD ID ID ¡D E ID ¡D ¡D ¡D ¡D ID LO o ¡D ¡D E U E O E U E u E U E U E U E O E U E u W u W u E u E u E u EH CO o ID |D E o E u E u E o E o E O E O W U E U E U FFI U K U E o E o E u EH CO lO o 1 ¡D ¡D ¡D E U E o E u E o E o E o E u E u E u a o K u E o E u ^ o EH CO 1 E U ¡D ¡D E u E u E u E u E u E u E u E u E u co U W u E o E u u EH CO o 13 U ID P E u E u E u E u E u E o E u ffi O ^ o ^ o ^f u EH CO EH CO CM o 1 co U u U E o u ^ u u u ^ u ü ^ o ^ o EH CO EH CO EH CO EH CO EH CO EH CO co O 1 E o ^ u co O EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO o 1 ^ o ^ o EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO m O 1 ^ u EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO EH CO w O o 'o o o m o IO CSJ IO CSI CSI o o CSJ IO r-o IO LO CSJ es O LO C-o LO LO CSI LO o LO o -e- <v rC 2 uoS Ì'CS ö 'S II 2 cu y £ * .2 ^ o o 11 fi X C O ^ " O) li s ° o O w o CO o _ <N 2 " La I ! g* í .2 o ) -P o ' fi O 3 <Ü O : t g L fi ® u fi <2 o tí ° S § <z> 2 EH CO Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 330 Sergio Da Silva fi « S CO T5 fi fa fi o CQ M 2 o fa C/3 Vi tÌ « s .2 « g § K W Ci « fi C/5 ** CO V « g. 42 § Cfl J3 co u « X & W £ a II •O 0) V a co a cn « J3 » O pfi O IH IO O H CO Eh C/3 Eh co Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO o Eh CO Eh C/3 Eh C/3 Eh C/3 H C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO PO o CSI U CSI O CSI O CSI U CSI U CSI U CSI O CSI U (M U CSI U CSI U CS] U CSI O CSI U CSI U CSI U CSI U ¡3 o CSI U CS] U CSI u CSI O CSI o CSI U CSI U CSI U CSI u CSI U CSI U CSI U CSI U CSI U ¡3 ¡3 ¡3 W U o CSI o CM U CSI u CSI U O co O CSI U CSI U CSI U P D P P ¡3 ¡3 W u -P ¡3 P ¡3 £ D P P P Ì3 D P £ ¡3 ¡3 P P io © « u K u e U E o E o E u e u e u e u w o W U w u ¡3 ffi U K u W u P P o w u W u E u E u p E u E o E u P W u ffi u W o ffi U W u ffi u K u ffi u S LO © 1 E o E u !D p t> & D D ¡3 13 D P ^ W u W u p 1 W o K O p D D D W o w U W u W u D D o 1 £ £ u ^ u TjH U u £ u P W u ffi u W u P <o 1 U E o ^ u ^ o u o o ^ u ^ u ^ u C16 ^ u u ^ u P ^ W u w u o 1 Eh C/3 Eh CO Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Tt< o 1 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO IO o 1 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO -e-in O o PO o o o io o o co IO tCSI o LO CSI LO <M CSI o o CSI o LO c-o LO LO CSI o CSI o LO o II Ö , CU O CTj SÌ E u O V, rO X o o~ o ai io a> n > Il ì:- 2 o £ ° o o m o : CO O 3 csi > m o »•fi09 .2 ©~ c ® u ö • .2 M ! o : i a> co H CO Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 331 in O H CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO EH CO H CO Eh CO Eh CO Eh CO Eh CO Eh CO O Eh co Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO H CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO « O Eh CO Eh CO Eh CO E ¡3 E E E U <M U CM U CM O CM O CM U CM u CM U CM O CM U CM U IM O Eh CO Eh CO co U CD U CD O E E E u E E u E o E o E u E u CM O CM U CM U o Eh CO Eh CO Eh CO CO U CD U C12 E u co U E u E u ¡3 £ E E u CM U 00 u CM U CM U -H C0 Eh CO Eh CO co U CO O C12 E u C12 E u E u E u E u E o E o CO U E u E E o LO o Eh co Eh CO Eh CO co O co O C12 E u C24 E u E u E u E u E u CO U E o E u E u E o Eh CO Eh CO Eh CO co U co U C12 E u C24 co U E o E u E u E u co U E u E u E u Eh CO io o 1 H CO Eh CO Eh CO co U co U C12 E u E u co O E o E o E u E u E u E u E o E u E U 1 Eh CO Eh CO Eh CO co O co O C12 E u E o co O E o E u E u E u E u E O E o E o *E u o 1 Eh CO Eh CO Eh CO co U co U C12 E u E o co U C12 co U E u E o E u ^ u E o E E u N O 1 Eh CO Eh CO Eh CO Eh CO CIO U C12 E o E u E u E u E u u ^ u ^ u E u E u C16 n O Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO H CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO o 1 Eh CO Eh CO Eh CO Eh CO Eh CO H CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO lO O 1 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO -e-m O o co O o o os o o 00 o o c-o o co o o ID o IO CM o o CM es O o IO IO CM o IO IO o -e- <u X! 1: £ O rO O I E O % > 11 I S§ « a§ £ 8 II ÖX) •rH < (-1 ( a ñ : £ O O) o u o o o „ II II l x -eco Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 332 Sergio Da Silva IfS O CSI u CS] u e u CSI u Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO EH CO Eh CO Eh CO Eh CO Eh CO O E u p CSI U CSI O CM u E U Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO EH CO Eh CO Eh CO Eh CO Eh CO Eh CO co O E u e U D CSI U CSI U CSI U CSI U X U CSI U CSI U CSI U CSI U CSJ U Eh CO Eh CO Eh CO Eh CO Eh CO IM o E u E u e u ¡3 CSI U (M u CS] u CSJ U CSI U CSI u CSI U CSI U CSI u CSI U CSI O CSI U CSJ U Eh CO o E u E u E u e u E u ¡3 CSI U CSJ u CSI U CSJ U CSJ u CSI O CSI o CSI U CSI u CSI U CSI U Eh CO -E u E u E u E u E u E u w u ¡3 ¡3 p P K u K u CSI o CSI u CSI o Eh CO IO o E u E u E u E u E u E u W u w u W U W U w o w u w o W u w U w u p Eh CO o E u E u E o E u E u E u W u W u W u W o ^ u K u W u W u W u CD o p Eh CO LO o 1 E o E u E u E u E u E ü u W u W o W u w u W u co U p p Eh CO 1 E u E u E u E u E u e u W u W u W u W u W u co U C12 & D D Eh CO o 1 E u E u E u E u E u w u D ¡3 D ID W U w u w u w u Eh CO CS] O 1 E u a u E u E u £ W u W u K u C68 u ^ U ^ u ^t1 u o ^ u Eh CO Eh CO 00 O 1 E u E u E u ¡3 ¡3 e u ^ u W u Eh CO Eh CO H co Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO O 1 E u *E u P C24 u w u Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO m O 1 E Ü £ e o a u Eh co Eh C/3 Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO Eh CO -oin o o co O o o IO o IO IO CTS co CD LO co CSI ^H o co g 3 er £ J tuo ì ! 'S : ^ * o +-> « o o ^ o CO IO u c Si o > •• rt OJ o O s O £ ° O <X> O N e« O fi W g flj (1) V cu II +-> o fi o 0) o ! ö : 1 0) co a s o> o tH O a : (11 1 w 1 fi QJ CO ' Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 339 a) Graph of Data 1 Tine 15600 This graph of data looks like white (uncorrelated) noise and shows no obvious structure. b) Strange Attractor This plot in a three-dimensional embedding reveals a clear structure in the data: a strange attractor can be recognized. Figure 5: Chaotic Solution with Parameters v = 15, <j> = 0, i = 104, x = ^ = 0.45, (9 = 0.95, and Initial Values St_3 = 1.00000000, St_2 = 0.99000000, and St_! = 1.02000000. Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 340 Sergio Da Silva c) Probability Distribution Logarithmic plot of the probability using 32 bins into which the datapoints are sorted according to their value. This distribution looks Gaussian (as in random data) if only the left-hand side is considered; however, a 'fractal' shape associated with chaotic data emerges when the right-hand side is also taken into account. d) IFS dumpiness : :. . • - . ' ' - - 1_______J Test This picture shows localized clumps indicating chaos or colored (correlated) noise. Figure 5 (continued) Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 341 e) Power Spectrum (Fast-Fourier) Log of the power spectrum versus frequency, using 32 frequency intervals. Since this picture shows a reasonably broad power spectrum, the data are more likely to be chaotic or random, and not periodic or quasi-periodic. f) Power Spectrum (Entropy) Log-linear view of dominant frequencies in the power spectrum, using 2 complex poles of discrete frequency. This picture shows no dominant frequency, which indicates that the data are neither periodic nor quasi-periodic. Figure 5 (continued) Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 342 Sergio Da Silva | | ! ! 1 \ \ g) Correlation Dimension: Saturation Embedding DiMETIS ion This picture shows a clear saturation in the calculated correlation dimension as the embedding dimension is increased; the well-defined plateau gives an indication of the proper embedding associated with chaos, although some quasi-periodic data may also exhibit such a property. Enbedding Dimension h) Correlation Dimension: Neighbors This picture gives another indication of the proper embedding (associated with chaos) as the minimum dimension for which the number of nearest false neighbors falls to zero. Figure 5 (continued) Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 343 o i) Surrogate Data o X z Logarithmic plot of the probability distribution of the surrogate data using 32 bins into which the datapoints are sorted according to their value. This picture shows a bell-shaped distribution (indicating lack of determinism) which differs from the distribution of the original chaotic data in panel c. Da Silva, S.: Exchange Rate Models, Chaos, and Foreign Exchange Intervention, PhD Thesis, Birmingham University Department of Economics, unpublished, 1999a. - Da Silva, S.: Exchange Rate Dynamics Redux and Chaos, Birmingham University Department of Economics Discussion Paper, No. 99-08, 1999b. - De Grauwe, P.: International Money: Postwar Trends and Theories, Oxford: Oxford University Press, Second Edition, 1996. - De Grauwe, P and Dewachter, H.: Chaos in the Dornbusch Model of the Exchange Rate, Kredit und Kapital 25(1), pp. 2654, 1992. - De Grauwe, P., Dewachter, H. and Embrechts, M.: Exchange Rate Theory: Chaotic Models of Foreign Exchange Markets, Oxford and Cambridge, MA: Blackwell, 1993. - De Grauwe, P. and Vansanten, K.: Deterministic Chaos in the Foreign Exchange Market, CEPR Discussion Paper, No. 370, January 1990. - Domínguez, K. M.: Does Central Bank Intervention Increase the Volatility of Foreign Exchange Rates?, NBER Working Paper, No. 4532, November 1993. - Dornbusch, R.: Expectations and Exchange Rate Dynamics, Journal of Political Economy 84(6), pp. 1161-1176, 1976. - Gärtner, M.: Macroeconomics under Flexible Exchange Rates, New York: Harvester Wheatsheaf, 1993. - Goodhart, C. A. E.: 'News' and the Foreign Exchange Market, LSE Financial Markets Group Discussion Paper, No. 71, January 1990. - Jeanne, O. and Masson, P.: Currency Crises, Sunspots and Markov-Switching Regimes, CEPR Discussion Paper, No. 1990, October 1998. - Leahy, M. P.: The Profitability of US Intervention in the Foreign Exchange Markets, Journal of International Money and Finance 14(6), pp. 823844, 1995. - LeBaron, B.: Chaos and Nonlinear Forecastability in Economics and Finance, Philosophical Transactions of the Royal Society of London A 348 (1686), References Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 344 Sergio Da Silva pp. 397-404, 1994. - LeBaron, B.: Technical Trading Rule Profitability and Foreign Exchange Intervention, NBER Working Paper, No. 5505, March 1996. - Mark, N. C.: Exchange Rates and Fundamentals: Evidence on Long-Horizon Predictability, The American Economic Review 85(1), pp. 201-218, 1995. - Marston, R.: Stabilization Policies in Open Economies, in Jones, R. W. and Kenen, P. B., eds., Handbook of International Economics, Amsterdam: North-Holland, Volume 2, pp. 859916, 1985. - Obstfeld, M. and Rogoff, K.: Exchange Rate Dynamics Redux, Journal of Political Economy 103(3), pp. 624-660, 1995. - Obstfeld, M. and Rogoff, K.: Foundations of International Macroeconomics, Cambridge, MA and London: The MIT Press, 1996. - Silber, W. L.: Technical Trading: When It Works and When It Doesn't, The Chartered Financial Analyst Digest 24(3), pp. 66-68, 1994. - Sprott, J. C.: Chaos Data Analyzer: The Professional Version 2.1, University of Wisconsin, Madison, WI 53706, 1995. - Sprott, J. C. and Rowlands, G.: PC User's Manual of Chaos Data Analyzer: The Professional Version, New York: American Institute of Physics, 1995. - Stewart, I.: Does God Play Dice? The New Mathematics of Chaos, London: Penguin Books, Second Edition, 1997. - Szpiro, G. G.: Exchange Rate Speculation and Chaos Inducing Intervention, Journal of Economic Behavior and Organization 24(3), pp. 363-368, 1994. - Taylor, D.: Official Intervention in the Foreign Exchange Market, or, Bet Against the Central Bank, Journal of Political Economy 90(2), pp. 356-368, 1982. Summary The Role of Foreign Exchange Intervention in a Chaotic Dornbusch Model Massive foreign exchange interventions are shown to remove chaos, cycles, and instability in the models of De Grauwe and Dewachter (1992) and De Grauwe, Dewachter, and Embrechts (1993), where the exchange rate can behave chaotically in the framework of the Dornbusch model. (JEL F31, F41, F47) Zusammenfassung Die Rolle der Devisenmarktintervention im Chaotischen Dornbusch-Modell Modelle von De Grauwe und Dewachter (1992) und De Grauwe, Dewachter und Embrechts (1993), in denen der Wechselkurs im Rahmen des Dornbusch-Modells chaotischen Tendenzen unterliegen kann, zeigen, daß starke Devisenmarktinterventionen Chaos, Zyklen und Instabilität eliminieren. Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23 The Role of Foreign Exchange Intervention 345 Résumé Le Rôle de l'Intervention dans les Marchés du Cours du Change dans la Version Chaotique de la Modèle du Dornbusch L'intervention massif dans les marchés du cours du change a prové d'emmener des chaos, des cercles et de l'instabilité dans la modèle du De Grauwe et Dewachter (1992) et dans la modèle du De Grauwe, Dewachter et Embrechts (1993). Dans ces modèles le cours du change peux s'amener chaotiquement en la version de la modèle du Dornbusch. Kredit und Kapital 3/2000 OPEN ACCESS | Licensed under CC BY 4.0 | https://creativecommons.org/about/cclicenses/ DOI https://doi.org/10.3790/ccm.33.3.309 | Generated on 2023-01-16 13:16:23