Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New Analytical Insights and Agent-Based Explorations

Mignot, Sarah; Westerhoff, Frank Article — Published Version Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New Analytical Insights and Agent-Based Explorations Open Economies Review Provided in Cooperation with: Springer Nature Suggested Citation: Mignot, Sarah; Westerhoff, Frank (2022) : Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New Analytical Insights and Agent-Based Explorations, Open Economies Review, ISSN 1573-708X, Springer US, New York, NY, Vol. 34, Iss. 1, pp. 155-169, https://doi.org/10.1007/s11079-022-09667-5 This Version is available at: https://hdl.handle.net/10419/309896 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. 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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/ Vol.:(0123456789) Open Economies Review (2023) 34:155–169 https://doi.org/10.1007/s11079-022-09667-5 1 3 RESEARCH ARTICLE Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New Analytical Insights andAgent‑Based Explorations SarahMignot1· FrankWesterhoff1 Accepted: 28 March 2022 / Published online: 27 April 2022 © The Author(s) 2022 Abstract We apply recent stability and bifurcation results to provide an analytical characterization of Paul de Grauwe’s chaotic exchange rate model. We prove that the model’s fundamental steady state becomes unstable due to a Neimark-Sacker bifurcation when chartists extrapolate past exchange rate trends too strongly, a phenomenon that gives rise to cyclical exchange rate dynamics. In contrast, fundamentalists’ mean reversion strength only has a stabilizing effect on the model’s dynamics when the exchange rate is out of equilibrium. We also show that agent-based versions of Paul de Grauwe’s exchange rate model may produce endogenous exchange rate dynamics, too. Keywords Foreign exchange markets· Exchange rate dynamics· Chartists and fundamentalists· Stability and bifurcation analysis· Agent-based modeling JEL Classification D84· F31· G14 1 Introduction In the early 1990s, a research group led by Paul de Grauwe proposed an inspiring theoretical model of the foreign exchange market that is capable of producing chaotic exchange rate dynamics. See, for instance, De Grauwe and Dewachter (1992, 1993) and, in particular, De Grauwe etal. (1993). For ease of exposition, we call their work “Paul de Grauwe’s chaotic exchange rate model” in our paper. In a nutshell, its basic functioning may be explained as follows. Endogenous exchange rate dynamics arise due to nonlinear interactions between chartists and fundamentalists. Chartists have destabilizing extrapolative expectations. They dominate the foreign exchange market near its fundamental value and regularly set bubble dynamics in motion. In contrast, fundamentalists have stabilizing regressive expectations. They * Frank Westerhoff frank.w[email protected] 1 Department ofEconomics, University ofBamberg, Bamberg, Germany 156 S.Mignot, F.Westerhoff 1 3 become increasingly relevant for the determination of the exchange rate as its mispricing worsens, driving back the exchange rate to more moderate levels. Since this development automatically leads to a revival of chartism in the foreign exchange market, the exchange rate remains in constant motion. Paul de Grauwe’s chaotic exchange rate model paved the way for numerous financial market models with heterogeneous interacting agents, studying the functioning of stock, commodity and foreign exchange markets, as documented by De Grauwe and Grimaldi (2006), Westerhoff (2009), De Grauwe and Rovira Kaltwasser (2012a), Hommes (2013) and Dieci and He (2018). The goal of our paper is twofold. First, we apply recent stability and bifurcation results obtained by Lines etal. (2020) and Gardini etal. (2021) to study Paul de Grauwe’s chaotic exchange rate model. Our analytical results reveal that the exchange rate becomes unstable due to a Neimark-Sacker bifurcation, giving rise to cyclical exchange rate dynamics, when chartists’ extrapolation strength is strong. While the local stability of the exchange rate also depends on speculators’ discount factor, it is independent of fundamentalists’ expected mean reversion speed and of the uncertainty surrounding the fundamental value of the foreign exchange market. However, these model elements may affect the out-of-equilibrium behavior of the exchange rate. Our analytical results aim at complementing the current available numerical simulations for this approach. Second, Paul de Grauwe’s chaotic exchange rate model rests on a few simplifying assumptions, typical for this line of research. To address some of these issues, we propose two agent-based versions of Paul de Grauwe’s chaotic exchange rate model that relax these assumptions and show that these setups may produce endogenous exchange rate dynamics, too. Our numerical explorations underline the robustness of the key insights offered by Paul de Grauwe’s chaotic exchange rate model. We proceed as follows. In Sect.2, we recall Paul de Grauwe’s chaotic exchange rate model, as presented in Chapter3 of the influential book by De Grauwe etal. (1993). In Sect.3, we derive our main results. In Sect.4, we develop two agentbased versions of Paul de Grauwe’s chaotic exchange rate model and study them numerically. In Sect.5, we offer some concluding remarks. 2 Paul de Grauwe’s Chaotic Exchange Rate Model The starting point of Paul de Grauwe’s foreign exchange market model is that the exchange rate in period t is determined by where Xt denotes a reduced form equation capturing structural forces and exogenous influence factors that drive the exchange rate in period t ; Et( S t+1) stands for the aggregate expectations held by speculators in period t about the exchange rate in period t+1 ; and 0<b<1 is a discount factor. In line with empirical evidence, Paul de Grauwe’s chaotic exchange rate model considers two types of speculators: chartists and fundamentalists. Chartists use past (1) St =X t E t( S t+1)b, 157 1 3 Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New… movements of the exchange rate as an indicator of market sentiment and extrapolate them into the future. Fundamentalists have regressive expectations and believe that the exchange rate will return to its fundamental value. See Menkhoff and Taylor (2007) for empirical evidence of these two archetypical trader types. Speculators’ aggregate exchange rate expectations are defined as where EC t( S t+1) is the forecast made by chartists, mt is the weight of chartists, EF t( S t+1) is the forecast made by fundamentalists, and 1−mt is the weight of fundamentalists. Note that speculators predict the exchange rate for period t+1 at the beginning of period t on the basis of information available to them up to the end of period t−1 . Chartists employ a simple moving average rule to forecast the evolution of the exchange rate. In particular, chartists expect an increase (decrease) in the exchange rate when a short-term moving average of the exchange rate crosses a long-term moving average of the exchange rate from below (above). Their predictions are represented by E C t ( St+1 ) =St−1 ( SMA(St−1) LMA ( S t−1))2𝛾 , where SMA( St−1 ) = S t−1 St−2 is a short-term moving average of the exchange rate and LMA( St−1 ) = ( St−1 St−2)0.5( St−2 St−3)0.5 = ( St−1 St−3)0.5 is a long-term moving average of the exchange rate. Chartists’ extrapolation strength is captured by parameter 𝛾>0 . Accordingly, chartists’ expectations are equivalent to and depend on the exchange rate in periods t−1 , t−2 and t−3 . As we will see, the exchange rate is thus driven by a three-dimensional dynamical system. Fundamentalists calculate the fundamental value of the exchange rate S∗ t via (1) by setting S ∗ t =S t =E t( S t+ 1 ) , yielding S ∗ t = ( X t)1 1− b . Since fundamentalists expect the exchange rate to return to its fundamental value, their forecasts read Parameter 0<𝛼<1 denotes fundamentalists’ expected mean reversion speed. For simplicity, De Grauwe etal. (1993) set Xt=1 . Hence, the fundamental value of the exchange rate is equal to S∗ t=1 . The key building block of Paul de Grauwe’s chaotic exchange rate model concerns the setup of the weights of chartists and fundamentalists. Quite realistically, its creators take into account that fundamentalists have heterogeneous views with respect to the fundamental value of the exchange rate. For simplicity, they assume that these views are normally distributed around the (true) fundamental value of the exchange rate and develop the following train of thought. (2) E t ( St +1) =EC t( St +1)m tEF t( St +1)1 − m t , (3) E C t ( St+1 ) =St−1 ( St−1St−3 S t− 2 2 )𝛾 (4) E F t ( St+1 ) =St−1 (S∗ t St−1)𝛼 . 158 S.Mignot, F.Westerhoff 1 3 • When the exchange rate is equal to its fundamental value, half of the fundamentalists will find that the exchange rate is too low, whereas the other half will find that it is too high. In such a situation, fundamentalists do not influence the foreign exchange market – their foreign exchange market transactions neutralize each other – and the exchange rate depends solely on the expectations of chartists. • As the exchange rate starts to deviate from its fundamental value, however, fundamentalists’ expectations become relevant. When the exchange rate is overvalued, for instance, a majority of fundamentalists believe that the exchange rate is too high. Accordingly, the impact of fundamentalists’ expectations on speculators’ aggregated exchange rate expectations will increase when the exchange rate departs from its fundamental value. Based on these considerations, the weight of chartists in Paul de Grauwe’s chaotic exchange rate model is approximated by the weighting function where parameter 𝛽>0 controls how rapidly the weight of chartists declines as the mispricing of the foreign exchange market increases. Its creators regard parameter 𝛽 also as a measure of the degree of divergence of fundamentalists’ estimates of the fundamental value of the exchange rate. For instance, a low value of parameter 𝛽 reflects a lot of uncertainty in the foreign exchange market concerning the fundamental value of the exchange rate. A given mispricing then has a relatively low impact on their weight.1 3 Main Analytical Results Combining (1)-(5) and recalling that S∗ t=1 enables us to express Paul de Grauwe’s chaotic exchange rate model in the form of a three-dimensional nonlinear map, given by (5) m t= 1 1+𝛽 ( S t −1−S∗ t−1) 2 , (6) ⎧ ⎪ ⎨ ⎪ ⎩ St=Sb(1+𝛾mt−𝛼(1−mt)) t−1X−2b𝛾mt t−1Yb𝛾mt t−1 Xt=St−1 , Yt=Xt−1 1 The bell-shaped weighting scheme (5) may also be reconciled with a setup in which speculators switch between heterogeneous expectation rules. The main argument goes as follows. If the exchange rate is near its fundamental value, the majority of speculators opt for an extrapolative expectation rule, hoping to benefit from the current momentum of the exchange rate. As the mispricing of the foreign exchange market increases, however, more and more speculators conclude that a fundamental price correction is about to set in and, consequently, favor a regressive expectation rule. See Westerhoff (2003) for an example. 159 1 3 Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New… where m t= 1 1+𝛽 ( S t− 1−1 ) 2 and Xt and Yt are auxiliary variables. De Grauwe etal. (1993) state that dynamical system (6) precludes an analytical treatment due to its multiple nonlinearities and high dimension, which is why they resort to the messy world of numerical experiments. Due to recent results by Lines etal. (2020) and Gardini etal. (2021), however, we are able to conduct a rigorous local stability and bifurcation analysis of the dynamical system (6). By applying the set of equilibrium conditions S =S t =S t−1 =X t−1 =Y t , straightforward computations reveal that map (6) possesses only one economically meaningful steady state, namely Since the exchange rate corresponds to its fundamental value when the dynamics is at rest, we may regard this steady state as a fundamental steady state. Obviously, this steady state implies that the weight of chartists is equal to m=1 , i.e. fundamentalists’ market impact is zero at the model’s fundamental steady state. This observation allows us understand our main analytical results. To study the local stability properties of the model’s steady state, we have to evaluate the Jacobian matrix of (6) at the FSS , yielding from which we get the characteristic polynomial where a1=−b(𝛾+1) , a2=2b𝛾 and a3=−b𝛾 . Using the stability results derived in Lines etal. (2020), we can conclude that the model’s steady state loses its local stability if one of the following three stability conditions becomes broken: (i) 1+a1+a2+a3>0 , (ii) 1−a1+a2−a3>0 and (iii) 1 − a2 + a1a3 − a3 2 > 0 . Moreover, Gardini etal. (2021) demonstrate that a separate violation of stability condition (i), (ii) or (iii), while the other two stability conditions hold, is associated with the emergence of a fold, a flip and a NeimarkSacker bifurcation, respectively. Since stability conditions (i) and (ii) require that 1−b>0 and 1+b (4 𝛾+1)>0 , respectively, they are always satisfied. Stability condition (iii) necessitates that 1−b𝛾(2−b)>0 . Solving this stability condition for parameter 𝛾 results in If this inequality becomes violated, the model’s steady state loses its stability via a Neimark-Sacker bifurcation, giving rise to cyclical exchange rate dynamics. Accordingly, only two model parameters affect the local stability of the model’s (7) FSS = ( S,X,Y ) =(1,1,1 ) . (8) J (FSS)= ⎡ ⎢ ⎢⎣ b(1+𝛾)−2b𝛾b𝛾 1 00 0 10 ⎤ ⎥ ⎥⎦ , (9) P (𝜆)=𝜆 3 +a 1 𝜆 2 +a 2 𝜆+a 3, (10) 𝛾< 1 b( 2 −b). 160 S.Mignot, F.Westerhoff 1 3 fundamental steady state: chartists’ extrapolation strength 𝛾 and speculators’ discount factor b . Put differently, the local stability of the model’s fundamental steady state does not depend on fundamentalists’ expected mean reversion speed 𝛼 or on the weighting parameter 𝛽 . The following proposition summarizes our main analytical results. Proposition 1: The fundamental steady state of Paul de Grauwe’s chaotic exchange rate model loses its local stability due to a Neimark-Sacker bifurcation if the inequality 𝛾<𝛾 NS crit ∶= 1 b(2−b) becomes violated, an outcome that is associated with the emergence of endogenous cyclical exchange rate dynamics. Let us briefly illustrate our main analytical results. Figure1 displays a bifurcation diagram for parameter 𝛾 , assuming that b=0.95 , 𝛼=0.65 and 𝛽= 10,000 . As predicted by Proposition 1, we observe a Neimark-Sacker bifurcation at 𝛾NS crit ≈ 1.0025 . For 0 <𝛾<𝛾 NS crit , the exchange rate converges towards its fundamental value. At 𝛾 =𝛾 NS crit , a Neimark-Sacker bifurcation occurs, triggering endogenous cyclical exchange rate dynamics. As parameter 𝛾 increases further, additional (global) bifurcations emerge, during which the amplitude of the exchange rate fluctuations tends to increase, underlining the destabilizing role played by chartists’ extrapolation behavior. These outcomes are also visible in Fig.2, which portrays the dynamics of Paul de Grauwe’s chaotic exchange rate model in the time domain for specific values of Fig. 1 Bifurcation diagram. The panel depicts the behavior of the exchange rate as a function of parameter 𝛾 , assuming that b=0.95 , 𝛼=0.65 and 𝛽= 10,000 161 1 3 Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New… parameter 𝛾 , again using b=0.95 , 𝛼=0.65 and 𝛽= 10,000 . We depict the dynamics for 160 periods, after a single one-percent exogenous shock has hit the foreign exchange market in period t=1 . The left panels of Fig. 2 show the evolution of the exchange rate and the right panels the corresponding weights of chartists. For 𝛾=0.95 (top panels), the exchange rate converges towards its steady-state value S=1 , implying that the weight of chartists eventually approaches its maximum value m=1 . For 𝛾=1.05 (middle panels), the exchange rate is subject to endogenous cyclical dynamics, an outcome that we also find for the weights of chartists. Since mispricing of the exchange rate remains rather small for 𝛾=1.05 , the weight of chartists remains relatively high. For 𝛾=3 (bottom panels), the exchange rate displays chaotic motion, and the weight of chartists fluctuates wildly, too. The latter parameter setting, i.e. b=0.95 , 𝛼=0.65 , 𝛽= 10,000 and 𝛾=3 , is the lead parameter setting in De Grauwe etal. (1993). Fig. 2 Time series plots. The left (right) panel shows the exchange rate (weight of chartists) in the time domain, assuming that 𝛾=0.95 (top), 𝛾=1.05 (middle) and 𝛾=3 (bottom). Remaining parameters: b=0.95 , 𝛼=0.65 and 𝛽= 10,000 162 S.Mignot, F.Westerhoff 1 3 We may summarize the functioning of Paul de Grauwe’s chaotic exchange rate model as follows. When the exchange rate is near its fundamental value, fundamentalists disagree about the question whether the foreign exchange market is overvalued or undervalued. Consequently, chartists dominate the dynamics of the foreign exchange market. As is clear from Proposition 1, chartists’ extrapolation behavior pushes the exchange rate away from its fundamental value, provided that stability condition (10) is violated. Far away from the fundamental value, however, fundamentalists rule the dynamics of the foreign exchange market. When the exchange rate is relatively high (low), a majority of fundamentalists conclude that the exchange rate is overvalued (undervalued). Fundamentalism now matters and exercises a stabilizing effect on the dynamics of the foreign exchange market, driving the exchange rate closer towards its fundamental value. Unfortunately, this (desirable) movement of the exchange rate is responsible for a comeback of the destabilizing behavior of chartists, and the above pattern repeats itself, either in a cyclical or in a chaotic manner, depending on chartists’ extrapolation strength. 4 Agent‑Based Simulations Recall that there are two types of speculators in Paul de Grauwe’s chaotic exchange rate model: heterogeneous fundamentalists and homogeneous chartists. Fundamentalists are heterogeneous because their estimates of the fundamental value of the exchange rate are normally distributed around the true fundamental value of the exchange rate. This natural assumption has been used to motivate (the powerful) switching function (6), which, in turn, enters the expression that defines speculators’ aggregate exchange rate expectations, i.e. (2). In this section, we propose agent-based versions of Paul de Grauwe’s chaotic exchange rate model that do not rely on this approximation. In fact, 30years after its invention, modern computers and software tools allow us to state and explore the behavior of heterogeneous fundamentalists exactly as envisioned by its creators. Chartists, following expectation rule (3), are homogenous because they have identical expectations. Since we do not have any clear guidance on how to turn homogeneous chartists into heterogeneous chartists, we explore two different agentbased versions of Paul de Grauwe’s chaotic exchange rate model. What both versions have in common is that the market impact of heterogeneous chartists decreases when the exchange rate becomes more turbulent, the latter being captured by either stronger mispricing or stronger volatility, which has an additional stabilizing effect on the dynamics of the foreign exchange market.2 2 Two comments are in order. First, after conducting numerous additional simulations, it appeared to us that either the exchange rate approaches its fundamental value or it explodes when we abstain from taming chartists’ behavior. Note that a similar taming device is present in the original model – chartists’ impact on the aggregate exchange rate expectations of the foreign exchange market diminishes with the current mispricing. Second, as stated in footnote 1,weighting function (5) is consistent with a setup in which speculators switch between extrapolative and regressive expectation rules. It seems worthwhile to us to develop agent-based versions for that scenario, too. Preliminary simulations suggest that they may be able to replicate key statistical properties of the foreign exchange market quite well. 169 1 3 Revisiting Paul de Grauwe’s Chaotic Exchange Rate Model: New… Declarations Ethical Statement The authors declare that they have no relevant or material financial interests that relate to the research described in this paper. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. References Day R, Huang W (1990) Bulls, bears and market sheep. J Econ Behav Organ 14:299–329 De Grauwe P, Dewachter H (1992) Chaos in the Dornbusch model of the ex-change rate. Kredit Und Kapital 25:26–54 De Grauwe P, Dewachter H (1993) A chaotic model of the exchange rate: the role of fundamentalists and chartists. Open Econ Rev 4:351–379 De Grauwe P, Dewachter H, Embrechts M (1993) Exchange rate theory - chaotic models of foreign exchange markets. Blackwell, Oxford De Grauwe P, Grimaldi M (2006) The exchange rate in a behavioral finance framework. Princeton University Press, Princeton De Grauwe P (2011) Animal spirits and monetary policy. 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