A heterogeneous agent exchange rate model with speculators and non-speculators

Journal of Macroeconomics 49 (2016) 203–223

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A heterogeneous agent exchange rate model with speculators and non-speculators Christopher J. Elias Department of Economics, Eastern Michigan University, 703 Pray-Harrold, Ypsilanti, MI 48197, United States

a r t i c l e

i n f o

Article history: Received 17 December 2015 Revised 24 July 2016 Accepted 29 July 2016 Available online 2 August 2016 JEL classification: D83 D84 F31 F37 G12 G15 Keywords: Exchange rate Adaptive learning Expectations formation Heterogeneous expectations Misspecifications Expectation shocks

a b s t r a c t This paper constructs a heterogeneous agent exchange rate model of speculators and nonspeculators from a simple monetary framework. The model replaces rational expectations with an adaptive learning rule that forecasts future exchange rates with an econometric model, and assumes two types of market participants, speculators and non-speculators, that differ by their forecasting model. Speculators employ a correctly specified forecasting model, are relatively short-term oriented, and are subject to momentum and herding effects via an expectation shock; non-speculators utilize a simple forecasting model, have no incentive to be short-term oriented, and are not subject to herding effects. Parameters are calibrated and estimated using the method of simulated moments, and simulation results show that the model is able to replicate foreign exchange market stylized facts better than a model of representative agent rational expectations. Furthermore, the dynamics of the model are shown to derive from both agent heterogeneity and the expectation shock. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In 2013, trading in the foreign exchange (FX) market averaged about $5.3 trillion a day, while the total value of exports and imports amounted to around $92 billion per day.1 These data, consistent with the earlier findings of Mark (2001), suggest that much more currency trading is taking place than is necessary for global commerce. In general, any currency trade can be categorized into one of two types: (1) A trade that has an expectation of profit from anticipated one-way currency fluctuations, and (2) one that does not have any such expectation. The former can be referred to as speculative, and the latter as non-speculative. Put Another way, speculative activities are those that are designed to be subject to FX market risk, while non-speculative activities are not. Examples of speculative activity are going long (or short) a currency with no hedge in place, and arbitrage, while examples of non-speculative activity are trades made for commerce, hedging,

E-mail address: [email protected] The FX trading data come from the Bank for International Settlements 2013 Triennial Central Bank Survey, while the global trade data come from the World Bank. 1

http://dx.doi.org/10.1016/j.jmacro.2016.07.006 0164-0704/© 2016 Elsevier Inc. All rights reserved.

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tourism, etc.2 With this in mind, the goal of this paper is to construct a heterogeneous agent exchange rate model that incorporates the concepts of speculation and non-speculation into a simple monetary framework, and to examine how well the model explains salient facts of the FX market. In the model developed in this paper, the relatively strong assumption of rational expectations is replaced with the more realistic notion of adaptive learning, an expectations modeling technique that assumes agents behave like econometricians who make forecasts using econometric methods.3 Agent heterogeneity results from agents utilizing different forecasting models. The dynamics of the exchange rate result completely from the forecasts of the agent-types; No actual trading takes place. However, the forecasting models employed by the different agent-types reflect their particular trading goals, time horizon, level of knowledge, etc. Specifically, speculators forecast with a “correctly” specified model in the sense that it is of the same form as the rational expectations equilibrium solution. Non-speculators, however, employ a parsimonious model. The intuition is that speculators are attempting to make profits from one-way currency movements and therefore have a profit incentive to be sophisticated market participants. For this reason, they employ fundamental analysis (i.e., conducting research into the FX market, employing computer algorithms to determine discrepancies in exchange rates, studying activities of central banks, etc.) and technical analysis (studying historical price movements to ascertain future price movements). This effort to understand the FX market results in a correctly specified forecasting model. Meanwhile, since non-speculators are unconcerned with making a profit on one-way currency movements, they have no incentive to be sophisticated market participants and are therefore not conducting rigorous analysis of the FX market in the form of fundamental and technical analysis. The result is that they employ a relatively simple forecasting model. This setup is consistent with survey evidence in Cheung and Chinn (2011) that large institutions in the FX market tend to have an informational advantage over other market participants, and in King et al. (2013) who find that non-speculators tend to make no attempt to forecast exchange rates. The two agent-types are also differentiated by their time horizon. Speculators are assumed to be short-term oriented, while non-speculators are assumed to have no specific time-horizon preference. The intuition is that most institutions engaging in speculative activities (banks, hedge funds, proprietary trading firms, etc.) tend to be focused on quarterly trading performance, as measured by profits or investment returns, and can potentially lose assets under management if their quarterly performance significantly lags behind competitor institutions. On the other hand, institutions conducting trades for commerce, hedging, tourism, or other non-speculative activities have no clear incentive to favor any specific time horizon. This time horizon differentiation is captured by the degree to which the two-agent types utilize historical data when estimating the parameters of their forecasting model. In addition to fundamental analysis, survey evidence in Allen and Taylor (1990) and Taylor (1992) suggests that sophisticated FX traders also tend to employ technical analysis, a concept which rests on the idea that historical price action can predict future price movements. These tools measure an exchange rate’s momentum and trend, which practitioners use to make trading decisions.4 One of the results of traders employing these tools is that exchange rate movements can exhibit “herding effects” in that rates that exhibit strong directional movements will attract momentum buyers. To model this idea, I incorporate an expectation shock into the forecasting equation of speculators, which is a relatively simple way of capturing exogenous influences (waves of optimism and pessimism, herding effects, animal spirits, etc.) on agent forecasts. Non-speculators, however, are not subject to expectation shocks because they, due to their lack of incentive towards sophistication, are not conducting technical analysis. Expectation shocks have been employed in a similar manner in Evans and Honkapohja (2003); Milani (2011), and Elias (2016), and are related to the idea of judgemental adjustments to forecasts used in Bullard et al. (2008) and Bullard et al. (2010). The primary contribution of the current paper is the incorporation and estimation of an expectation shock process in a model of flexible exchange rates. The work in this paper is most similar to Kim (2009) and Markiewicz and Lewis (2009), with the main difference from those authors being the inclusion of agent heterogeneity in the adaptive learning mechanism. Agent heterogeneity is a natural extension to the model and empirical evidence in the FX market exists justifying its inclusion. Pesaran and Weale (2006) examine survey evidence at the individual and macro level and find significant evidence of heterogeneity in expectations formation. Ito (1990) looks at panel data of biweekly surveys of the yen/dollar exchange rate and determines that heterogeneity exists in market participant’s expectation formation. Allen and Taylor (1990) find that a significant number of FX dealers use non-fundamental analysis when making trading decisions, suggesting that the market consists of both fundamentalists and non-fundamentalists (i.e., technical analysts). Furthermore, several studies in the heterogenous agent exchange rate literature depict exchange rate markets as the dueling interaction of chartists and fundamentalists, both of which are trading styles that can be categorized as speculative.5 A secondary contribution of the current paper is the broadening of the agent heterogeneity to include both speculative and non-speculative activity. This is consistent with survey evidence of the FX market in King and Rime (2010) who find that a significant amount trading activity can be attributed to algorithmic trading styles, like high frequency trading, which are speculative in nature, and in Bodnar and Hayt (1998) who

2 Hedging would be considered non-speculative in the sense that a hedge is intended to eliminate market risk. In other words, there is no intention to make a profit from a one-way currency movement with a hedge. 3 See Evans and Honkapohja (2001) for more background information on adaptive learning. 4 Examples of momentum-based technical trading tools are moving average convergence divergence and the directional movement indicator. 5 Examples are Frankel and Froot (1987) and Frankel and Froot (1990).

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Table 1 Exchange rate stylized facts I: volatility and autocorrelation structure. Exchange rate returns

UK

GER

JPN

SWZ

Standard deviation Autocorrelation (1) Autocorrelation (4) Autocorrelation (8) Autocorrelation (16)

4.7707 0.1883 0.0493 −0.0096 −0.2839

5.5950 0.1971 0.1247 0.0317 −0.0625

5.6294 0.1297 0.0670 0.0039 −0.1778

6.4948 0.0209 0.0589 0.0070 −0.0166

Deviation from fundamentals Standard deviation Autocorrelation (1) Autocorrelation (4) Autocorrelation (8) Autocorrelation (16)

UK 0.2158 0.9689 0.8230 0.5629 0.1076

GER 0.4159 0.9889 0.9298 0.8121 0.6220

JPN 0.4754 0.9924 0.9550 0.8949 0.8117

SWZ 0.2725 0.9676 0.8449 0.6300 0.2422

The U.S. dollar is the numeraire currency. Exchange rate return moments are in percentage terms. Numbers in parentheses refer to autocorrelation coefficient lag length.

find that the market also consists of corporations (i.e., exporters and importers) that conduct trades that are mostly nonspeculative in nature.6 The monetary model contains a unique rational expectations equilibrium which implies that heterogeneity in this setting necessarily means that some agents forecasting models are misspecified. Misspecification and agent heterogeneity are studied in Branch and Evans (2006a) and applied in a Lucas type monetary model in Branch and Evans (2007). Markiewicz (2012) incorporates these ideas in explaining shifts in volatility in exchange rate returns. That author’s work differs from the current paper in that it employs a more generalized version of the monetary model and allows the degree of agent heterogeneity to change. Markiewicz and Lewis (2009) use misspecification in learning to produce excess volatility in the exchange rate. In those author’s work, however, agents are allowed to switch between forecasting rules and the authors note that both learning and the misspecification rule are necessary to produce their result. The current paper takes a simpler approach by not allowing switching between forecasting rules. Key parameters of the model are estimated with a two stage procedure. In the first stage I calibrate the standard parameters of the monetary model using values consistent with the international macroeconomic literature. In the second stage I estimate the appropriate form of heterogeneous beliefs, as well as additional parameters, with the method of simulated moments. Simulation results imply that the heterogeneous model of speculators and non-speculators replicates a set of stylized facts much better than does a model of representative agent rational expectations. Furthermore, I find that both the heterogeneous agent mechanism and the expectation shock are necessary to generate the dynamics of the model. This paper is organized as follows: Section 2 introduces the monetary model and the stylized facts of the FX market; Section 3 builds the model of speculators and non-speculators; Section 4 discusses the calibration scheme, the method of simulated moments, and parameter estimates; Section 5 presents the numerical results and provides a discussion; Section 6 concludes and appendices cover the data methods utilized and the derivation of the monetary model. 2. The monetary model and stylized facts of the FX market The monetary model of flexible exchange rates employed in this paper, derived in Appendix B, is characterized by two money demand equations, equilibrium in the money market, an uncovered interest parity condition, and a purchasing power parity condition. The reduced form of the model provides an asset pricing-type equation,

st = a1 Et st+1 + b1 ft

(1)

where st is the log exchange rate, ft is the log fundamentals (see Eq. [B.5]), subscripts t on s, f, and E refer to the time period, a1 = λ/(1 + λ ), and b1 = 1/(1 + λ ), where λ is the interest rate semi-elasticity of money demand. Et st+1 is the possibly nonrational, economy-wide expectation of the time t + 1 log exchange rate formed at time t. The deviation from fundamentals (i.e., the error correction term, ξ t ), is defined as the difference between the log exchange rate and the log fundamentals,

ξt ≡ ft − st

(2)

This paper will focus on the stylized facts of Tables 1–3 for the exchange rates between the U.S dollar and four major currencies: the UK pound, the German deutsche mark, the Japanese yen, and the Swiss franc. Table 1 presents volatility and autocorrelation structure characteristics of exchange rate returns and deviation from fundamentals. As pointed out by Mark (2001) and Kim (2009), the currency returns have a standard deviation between four and seven percent, and have low, positive first order autocorrelation that decreases as the lag length increases. At sixteen quarters, all four currencies

6

This paper is related to the large literature on the market microstructure of the FX market. See King et al. (2013) for a recent survey.

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Table 2 Exchange rate stylized facts II: long-horizon insample predictability. k-period ahead change

βk

t

R2

UK pound 1 4 8 16

0.0318 0.1567 0.3632 0.7603

1.36 2.00 3.31 5.81

0.0131 0.0850 0.2012 0.4791

German deutsche mark 1 0.0246 4 0.1125 8 0.2439 16 0.4212

1.83 2.29 3.09 4.96

0.0245 0.1220 0.2454 0.3390

Japanese yen 1 4 8 16

0.0172 0.0742 0.1426 0.2074

1.94 2.22 2.57 3.10

0.0178 0.0790 0.1385 0.1990

Swiss franc 1 4 8 16

0.0531 0.2076 0.3906 0.6228

3.32 4.17 4.97 8.41

0.0463 0.2023 0.3520 0.5600

The U.S. dollar is the numeraire currency. Numbers in column one refer to the k-period ahead change in the dependent variable of the predictability regression in eq. (3).

Table 3 Exchange rate stylized facts III: out-ofsample forecasting. k-period ahead forecast range

U-statistic

UK pound 1 4 8 16

0.9913 0.8679 1.2954 1.4716

German deutsche mark 1 4 8 16

1.1373 1.4424 1.9630 1.9330

Japanese yen 1 4 8 16

1.2052 1.7102 1.6449 1.4877

Swiss franc 1 4 8 16

0.9947 0.9703 0.9646 1.0161

The U-statistic is constructed so that values greater than one imply that the random walk with drift model is forecasting with greater accuracy than the monetary model.

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show negative autocorrelation, indicating possible mean reversion. Additionally, the deviation from fundamentals is much less volatile than exchange rate returns, a fact well known in the literature as the disconnect puzzle of Obstfeld and Rogoff (2001). Lastly, the deviation from fundamentals is highly persistent at short lag lengths, but decreases as lag length increases. Following the formulation of Mark and Sul (2001) and Kim (2009), long-horizon in-sample predictive power is examined by running a series of regressions of the following form,

st+k − st = θk + βk ξt + ζt+k

(3)

where β k is a slope coefficient and ζt+k is a regression error. The economic intuition is such that in periods when the fundamentals are greater than the exchange rate, the future change in the exchange rate will be positive, and vice versa. This implies that β k should be positive. Additionally, if exchange rates exhibit mean reversion, β k , the resultant t-statistic, and the associated adjusted R2 should be positively correlated with k. Table 2 presents regression results of the k = 1, 4, 8, and 16 period-ahead change of the log exchange rate on the current period deviation from fundamentals, where t-statistics are calculated with Newey–West standard errors. Similar to the findings of Kim (2009), the results show that for all four currencies the slope coefficients, t-statistics, and adjusted R2 s increase as time horizon increases. Lastly, out-of-sample fit analysis is conducted in the same manner as Mark (1995); Groen (1999); Faust et al. (2003), and Kim (2009). Specifically, forecasts generated by regressions of the form of equation (3) are compared to those of a random walk with drift model by implementing Theil’s U statistic. Table 3 shows that the random walk with drift model does better at forecasting exchange rate movements except in the case of relatively small values of k for the UK pound and the Swiss franc.7 3. A model of speculators and non-speculators The model’s reduced form consists of the exchange rate equation (Eq. (1)) and the empirical formulation of the fundamentals. Following (Kim, 2009), fundamentals are a trend stationary autoregressive process,

ft = α + δt + ρ f ft−1 + εt

(4)

where ε ∼ i.i.d. N (0, σε2 ), and 0 < ρ f < 1. This paper considers the case of boundedly rational expectations formation in which agents implement an adaptive learning rule that represents a slight deviation from rational expectations. Following Evans and Honkapohja (2001); Berardi (2007), and Berardi (2009), the economy-wide expectation in Eq. (1) is motivated by the idea that the economy, populated by a continuum of agents in the unit interval, consists of different agent-types that are identical in all ways except how they form expectations of the future exchange rate. The economy-wide expectation is then assumed to be a weighted average of the expectations of the different agent-types. Specifically, consider an economy of just two agent-types, speculators and non-speculators. The economy-wide expectation in Eq. (1) is therefore formed by a weighted average of the forecasts of the two agent-types,

Et st+1 = μEtsp st+1 + (1 − μ )Etnsp st+1

(5)

where the superscripts sp and nsp on the expectation operators refer to speculators and non-speculators, respectively, and μ ∈ [0, 1] is the proportion of speculator-type agents in the economy. Rational expectations Begin the analysis by assuming a homogeneous agent (i.e., μ = 1) with rational expectations. The unique, minimum state variable rational expectations equilibrium (REE) solution (as shown in Kim (2009)) found with the method of undetermined coefficients is

st = a + bt + c ft

(6)

where

c=

1 1 + λ − λρ f

b = λcδ a = λ[b + c (α + δ )]

7 Forecasting begins in the first quarter of 1984. The U statistic is constructed so that values of the statistic greater than one imply that the random walk with drift model is forecasting with greater accuracy than does the monetary fundamentals regression model.

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Adaptive learning with speculators and non-speculators Rational expectations assumes agents know: 1. The value of the model’s deep parameters (the interest rate semi-elasticity of money demand [λ], the income elasticity of money demand, [φ ], etc.). 2. The parameters of the fundamentals equation (α , δ , ρ f and σ ε ). 3. The form of the exchange rate REE solution (Eq. (6)). 4. The values of the REE solution coefficients (a, b, and c). The adaptive learning method of expectations modeling makes the more realistic assumption that agents do not know the value of the REE coefficients (i.e., eliminates assumption four above). Because assumptions one through three still apply, the method is only a slight deviation from rational expectations. Moreover, the adaptive learning technique assumes agents behave like econometricians; They employ a model of the exchange rate, use econometric methods to estimate the unknown parameters of their model, and use the model to make forecasts of the future exchange rate. This paper assumes agents use least squares regression when estimating the parameters of their model.8 The form of the REE solution can be thought of as the “correct” structure of the economy. Agents belief about the structure of the economy, known as their perceived law of motion (PLM), is considered correctly specified when it is of the same form as the REE solution. Consider a heterogeneous agent setup where there exists two agent-types, speculators and non-speculators, with each having a different PLM. Assume speculators employ a correctly specified model in that they believe the current period exchange rate is a function of a time trend and current fundamentals. On the other hand, non-speculators employ a parsimonious model by simply believing the exchange rate is a function of the previous period exchange rate. In both cases, constants are added to more closely mimic real-world estimation techniques. The PLMs are written as

stsp = κc1 + κT t + κ f ft + tsp stnsp = κc2 + κ st−1 + tnsp where superscripts on st and  t refer to agent type, κ c1 , κ T , κ f , κ c2 , and κ are parameters to be estimated by agents using least squares regression, and  sp and  nsp are regression errors. The intuition of this setup is that speculators are knowledgeable about the FX market through fundamental analysis, which results in a correctly specified model. Conversely, non-speculators are not conducting fundamental analysis and therefore have less information about the market’s structure. In essence, they are admitting that they don’t know how the market works and simply guess that next period’s exchange rate is a function of the current period’s rate. To avoid timing complications, an assumption must be made about when information is available to agents. This paper makes a standard assumption that the value of the current period exchange rate (the endogenous variable) is not available to agents at the time of expectation formation, but that the current period fundamentals variable is available. Therefore, a timeline of events in the learning process is as follows: 1. st−1 is realized at the end of period t − 1. 2. At the beginning of period t, agents update their model parameter estimates by adding st−1 and ft−1 to their information set and run a least squares regression of st−1 on the variables in their PLM (a constant and ft−1 for speculators, and a constant and st−2 for non-speculators). 3. The value of ft is realized. 4. Agents use ft , st−1 , and their parameter estimates calculated in step two to make their forecast of st+1 . 5. st is realized from the reduced form equation, which, in turn, determines ξ t from Eq. (2), and period t concludes. Iterating the respective PLMs one period forward provides expressions for agents’ forecasts of the future exchange rate (regression errors are omitted for simplicity): sp st+1 = κc1 + κT (t + 1 ) + κ f ft+1 nsp st+1 = κc2 + κ st

Because agents do not know ft+1 or st when they forecast in period t, these variables must be replaced with variables that are known to agents at the time of forecasting. Substitute in expressions for st from the respective PLMs and for ft+1 from the law of motion of the fundamentals (Eq. [4]), take expectations, and collect terms to obtain expressions for agents forecasts:

8

Etsp st+1 = κc1 + κT + κ f (α + δ ) + (κT + κ f δ )t + κ f ρ f ft

(7)

Etnsp st+1 = κc2 (1 + κ ) + κ 2 st−1

(8)

This is the standard assumption in the adaptive learning literature. See Evans and Honkapohja (2001) for more information.

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The way the economy actually evolves over time, known as the actual law of motion (ALM), is determined by the reduced form of the model (Eq. [1]), the economy-wide expectation (Eq. [5]), and the expectation equations for speculators and nonspeculators (Eqs. [7] and [8]). It is helpful to express agents least squares estimation procedure in the form of a stochastic recursive algorithm (SRA), consisting of a set of parameter estimates and a precision matrix, to explicitly demonstrate the updating process. The speculator agent-type’s SRA is written as:

       κc1,t κc1,t−1 1 sp −1 κT,t = κT,t−1 + gsp Rsp,t (t − 1 ) (st−1 − sˆt−1 ) κ f,t κ f,t−1 ft−1    1   (t − 1 ) 1 (t − 1 ) ft−1 − Rsp,t−1 Rsp,t = Rsp,t−1 + gsp ft−1

while the learning algorithm for non-speculators is





  κc2,t κc2,t−1 1 nsp −1 = + gnsp Rnsp,t (st−1 − sˆt−1 ) κt κt−1 st−2

   1 

Rnsp,t = Rnsp,t−1 + gnsp

st−2

1 st−2 − Rnsp,t−1

Time subscripts are added to parameter estimates and precision matrices (Rsp and Rnsp ) to reflect that they change over time, and initial values for the precision matrices and parameter estimates are assumed to be given. The gain parameters, gi , i = sp, nsp, are constants between zero and one and represent the degree to which agents discount historical data. Specifically, agents place a weight of (1 − g)l−1 for a data point l periods in the past, so that as g increases, the degree to which past data is discounted increases. In this manner agents can be thought of as conducting a weighted least squares estimation procedure. The model of speculators and non-speculators assumes speculator-type agents have an incentive to be short-term oriented, while non-speculator-type agents have no such incentive. The economic intuition behind this is that institutions engaging in speculation are focused on short-term trading profits, while institutions not engaging in speculation have no clear incentive to favor any particular time-horizon. The system will become unstable if the parameter on the lag of the exchange rate in the non-speculator’s PLM, κ , becomes larger than one. A projection facility prevents this in that in any period when the condition is violated, the parameter estimates for non-speculators in that period are set to their previous period values. Expectation shocks I incorporate a shock term to speculator-type agents forecasting equation in order to capture momentum and herding effects resulting from the use of technical analysis trading tools, as discussed in Section 1. Eq. (7) becomes:

Etsp st+1 = κc1 + κT + κ f (α + δ ) + (κT + κ f δ )t + κ f ρ f ft + et Assume that the expectation shock et follows a first order autoregressive process,

et = ρe et−1 + νt where νt ∼ i.i.d. N (0, σe2 ) and 0 < ρ e < 1. The intuition for the expectation shock is that the use of technical analysis trading tools causes speculators to make trading decisions based on momentum and trend characteristics of exchange rates. Put another way, these tools can result in speculators becoming either overly optimistic or pessimistic about the future value of the exchange rate for reasons unrelated to fundamentals. This phenomenon is captured by the expectation shock, which causes the forecast in some periods to deviate from the purely model-based value. Interpretation The model formulation presented in this section consists of two agent-types: speculators and non-speculators. The dynamics of the model are such that agent-types only generate forecasts; They don’t actually make trades. However, their forecasting model reflects their trading goals. Speculators, who want to make one-way currency trades for profit, are wellversed in the workings of the foreign exchange market through extensive fundamental research, are relatively short-term oriented because of their focus on quarterly trading profits, and are subject to herding behavior through the use of technical analysis. On the other hand, non-speculators, who do not want to make one-way currency trades for profit, do not perform extensive research, have no incentive to favor any specific time horizon, and are not subject to herding effects. Examples of speculators are market participants going long (or short) a currency with no associated hedge, and arbitragers. Examples of non-speculators are market actors participating in the foreign exchange market for the purposes of commerce, hedging, tourism, etc.

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 4 Monetary fundamentals parameter estimates. ft = α + δt + ρ f ft−1 + εt αˆ δˆ Currency UK GER JPN SWZ

0.0981 (0.1018) 0.0195 (0.0310) −0.1719 (0.0679) 0.0528 (0.0328)

εt ∼ i.i.d. N (0, σε2 ) ρˆf σˆε

−0.0 0 01 (0.0 0 01) −0.0 0 02 (0.0 0 01) 0.0 0 01 (0.0 0 0 0) 0.0 0 01 (0.0 0 01)

0.9823 (0.0183) 0.9893 (0.0170) 0.9648 (0.0147) 0.9709 (0.0174)

Observations

0.0250

140

0.0279

111

0.0251

175

0.0291

172

Note: OLS estimates. Standard errors are in parentheses.

4. Indirect estimation A two-stage indirect estimation process is utilized to obtain values for the model parameters to facilitate simulation. In the first stage, parameters are calibrated using a scheme consistent with the international macroeconomic literature. In the second stage the method of simulated moments is used to determine values for the degree of agent heterogeneity (μ), the expectation shock parameters (σ e and ρ e ), and the adaptive learning gains (gsp and gnsp ). Following Mark and Sul (2001) and Kim (2009), the interest rate semi-elasticity of money demand (λ) is set to 8, and similar to Kim (2009), the income elasticity of money demand is set to 1. Similar to Kim (2009), OLS regressions are run for each currency to obtain estimates of the parameters of the fundamentals specification (Eq. [4]). Fundamentals are constructed using Eq. (B.5) and the methods described in Appendix A. The root mean square error of the regression is used as an estimate of the standard deviation of the innovation of the fundamentals process (σˆ ε ). Table 4 presents the results. Method of simulated moments The degree of agent heterogeneity, the standard deviation of the innovation and autoregressive coefficient of the expectation shock, and the adaptive learning gains are estimated with the method of simulated moments (MSM). The goal of the MSM is to select values of parameters that minimize the numerical distance between specific empirical and simulated moments. The notation used in this section comes from Franke (2009).9 Assume that we are interested in matching nm descriptive statistics in the empirical data with model simulated data. Let  be a np x 1 vector of parameters that we are trying to estimate, mˆ be an nm x 1 vector of moments computed from the empirical data, and uˆ () be an np x 1 vector of moments computed from the simulated data. Define the vector G() as

ˆ G() = uˆ () − m and assume that n p = nm . The MSM selects a parameter vector that minimizes the value of the criterion function

ˆ G() Q () = G()W

(9)

ˆ. with weighting matrix W To ensure consistency of the MSM estimator, the simulated data must be drawn from a stationary distribution. This is accomplished by simulating a data series of length hT, where h is an integer greater than one, removing the first hT/2 data points, and utilizing the second hT/2 data points for moment calculations. Random numbers used to generate shocks must remain unchanged throughout the procedure so that the sole source of randomness derives from parameter changes. Let  = [μ, σe , ρe , gsp , gnsp ] . The moments used in the estimation are the variance, first-order autocovariance, and fourthorder autocovariance of the exchange rate return, as well as the variance and first-order autocovariance of the deviation from fundamentals. These are chosen because the exchange rate return and the deviation from fundamentals are two key characteristics of the model. In general, the moments are

(qt − qemp )(qt−k − qemp ) where qt is the empirical data, qemp is the mean of the empirical data, and the average over time is taken. k = 0 represents the variance, k = 1 represents the first-order autocovariance, and k = 4 is the fourth-order autocovariance. Data on the percentage of foreign exchange turnover by counterparty, as measured by the Bank for International Settlements (BIS) Triennial Survey, are employed to determine a suitable range for the degree of speculation that occurs in the foreign exchange market. As of 2013, the different categories of counterparty include “reporting dealers”, “non-financial customers”, and “other financial institutions”. The majority of the “other financial institutions” category includes non-reporting banks, institutional investors, hedge funds, and proprietary trading firms. I assume that a significant portion of the trading 9

Standard references for the MSM are Lee and Ingram (1991) and Duffie and Singleton (1993).

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Table 5 MSM estimates. Currency

μˆ

σˆ e

ρˆe

gsp

gnsp

UK GER JPN SWZ

0.1786 0.2059 0.2216 0.1616

0.1798 0.2370 0.1862 0.1886

0.8540 0.8503 0.8502 0.8502

0.1549 0.1502 0.1572 0.1504

0.0010 0.0068 0.0060 0.0989

that falls into this category involves speculative activities. The proportion of foreign exchange market turnover represented by this category has increased from 12.5% in 1992 (the first available year of the BIS’s survey) to 53% in 2013. The average over the twenty-one years is 31.8%. I further assume that the amount of speculation occurring in the market is a fraction of this amount. Therefore, I set the constraint on the degree of speculator type-agents in the market for all four currencies to 0.1 < μ < 0.3. An informal grid search is conducted to find ranges of the expectation shock parameters and adaptive learning gains that produce simulated moments that fit the empirical moments well. Furthermore, since the heterogeneous agent model assumes that speculators are short-term oriented and non-speculators have no preference for any specific time-horizon, constraints on the minimization procedure are set so that 0.15 < gsp < 0.25 and 0.001 < gnsp < 0.25.10 The same constraints are used for all four currencies. Matlab’s constrained optimization toolbox using the interior-point algorithm conducts the minimization procedure. The pseudo-random number seed state is set to fourteen which ensures that random number draws used to generate shocks will not vary throughout the process, T is set to the number of observations in the empirical data for each currency series ˆ is an identity matrix (i.e., all moments are weighted equally in the minimization (presented in Appendix A), h is set to 2, W process), and median values of the moments are taken over 20 0 0 replications in each iteration of the procedure. Parameter estimates are displayed in Table 5. 5. Numerical results In this section, five variants of the monetary model for each currency are simulated and results are compared to the stylized facts of Section 1. The first variant, the benchmark specification, is a model with a representative agent with rational expectations (RE1). The second variant extends the benchmark to include the expectation shock (RE2). The third variant is a model of heterogeneous agent adaptive learning with no expectation shock (HL), while the fourth is a model of homogeneous agent adaptive learning with the expectation shock (HML). The fifth is the model of speculators and non-speculators developed in Section 3 (SN). This fragmentation allows the analysis to explicitly show how the key elements of the model of speculators and non-speculators are altering the dynamics of the benchmark formulation. Where applicable, the model variants employ the parameters of Table 5. In each replication the model is simulated for twice the length of the corresponding empirical data set, and then the first half of the observations are discarded, thereby creating a data series of equivalent length to the empirical data with the effects of the initial conditions minimized.11 After each replication, analogous statistics to the stylized facts of Section 1 are calculated, a total of 20 0 0 replications for each model are run, and then median values of the calculated statistics are reported. Shocks are generated with Matlab’s pseudo-random number generator seed state set to fourteen. All calculations are made in accordance with the techniques described in Sections 1 and Appendix A. Volatility Table 6 presents results for the volatility of exchange rate returns and the deviation from fundamentals. In the case of the benchmark under all four currencies, the standard deviations of exchange rate returns are significantly below four to seven percent, and the standard deviations of the deviation from fundamentals are near zero. The SN model produces volatility for all four currencies that is closer to what is seen in the data, in comparison to the benchmark. Also, results of the RE2 model show that adding the expectation shock to the benchmark significantly increases the amount of volatility to both the exchange rate return and the deviation from fundamentals, while the results of the HL model show that the heterogeneous agent adaptive learning mechanism decreases the volatility of the two statistics. On the other hand, the results of the HML model show that homogeneous agent adaptive learning with the expectation shock generates an inordinate amount of volatility in the statistics. Collectively, these results show that the volatility results of the SN model mainly derive from the expectation shock.

10

These are values consistent with other studies in the adaptive learning literature. See Branch and Evans (2006b) and Milani (2007). Carceles-Poveda and Giannitsarou (2007) show that initial conditions have a significant effect on the behavior of adaptive learning algorithms. With this in mind, sensitivity analysis was conducted to ensure that the initial conditions of the learning parameter estimates did not affect the results. 11

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 6 Volatility. Data

Simulated data

Currency

RE1

RE2

HL

HML

SN

Exchange rate returns UK 4.7707 GER 5.5950 JPN 5.6294 SWZ 6.4948

2.1972 2.5566 1.9853 2.3915

17.0355 22.6031 17.7339 18.0996

0.6078 0.4127 0.5205 0.7261

4.18 × 103 107.5673 1.21 × 104 76.1677

4.1650 5.2641 5.0072 4.6857

Deviation from fundamentals UK 0.2158 0.0169 GER 0.4159 0.0153 JPN 0.4754 0.0169 SWZ 0.2725 0.0176

0.2881 0.3699 0.3001 0.3046

0.0471 0.2444 0.0744 0.0524

18.1156 5.4292 18.4565 4.2808

0.2996 0.3064 0.3272 0.3451

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock (benchmark specification), RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. Returns are in percentage terms. Bold values represent the empirical data.

Table 7 Autocorrelation structure: exchange rate returns. Data

Simulated data RE1

RE2

HL

HML

SN

−0.0016 −0.0070 −0.0030 −0.0057

−0.0737 −0.0444 −0.0224 −0.0104

0.7863 0.5161 0.3735 0.1682

0.0081 −0.0020 −0.0017 −0.0082

0.6951 0.2711 0.0634 −0.0780

German deutsche mark AR(1) 0.1971 AR(4) 0.1247 AR(8) 0.0317 AR(16) −0.0625

−0.0073 −0.0082 −0.0 0 06 −0.0012

−0.0707 −0.0511 −0.0192 −0.0063

0.2142 0.1055 0.0941 0.0733

0.1095 0.0878 0.0791 0.0291

0.5200 0.0906 −0.0091 −0.0474

Japanese yen AR(1) 0.1297 AR(4) 0.0670 AR(8) 0.0039 AR(16) −0.1778

−0.0176 −0.0191 −0.0142 −0.0136

−0.0793 −0.0470 −0.0216 −0.0105

0.6553 0.3127 0.1870 0.0294

−0.0019 −0.0040 −0.0054 −0.0065

0.6317 0.1997 0.0185 −0.0881

Swiss franc AR(1) 0.0209 AR(4) 0.0589 AR(8) 0.0070 AR(16) −0.0166

−0.0161 −0.0173 −0.0115 −0.0124

−0.0757 −0.0441 −0.0225 −0.0115

0.8070 0.5021 0.2947 0.0283

0.1245 0.0990 0.0810 0.0431

0.7756 0.3718 0.1014 −0.1325

UK pound AR(1) AR(4) AR(8) AR(16)

0.1883 0.0493 −0.0096 −0.2839

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. AR(k) is the kth order autocorrelation coefficient. Bold values represent the empirical data.

Autocorrelation structure: exchange rate returns Table 7 displays the autocorrelation structure of the exchange rate returns. For all currencies, the RE1 model produces coefficients that are near zero at all lag lengths, while the coefficients of the SN model more closely resemble the pattern seen in the data, although the high autocorrelation seen in the lowest lag lengths are a notable difference from the data and an apparent weakness of the model.

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Table 8 Autocorrelation structure: deviation from fundamentals. Data

Simulated data RE1

RE2

HL

HML

SN

0.9825 0.9312 0.8622 0.7229

0.8330 0.4652 0.1948 −0.0222

0.8888 0.6234 0.3566 0.0147

0.9942 0.9617 0.8857 0.6454

0.9878 0.8869 0.6836 0.2411

German deutsche mark AR(1) 0.9889 0.9894 AR(4) 0.9298 0.9577 AR(8) 0.8121 0.9167 AR(16) 0.6220 0.8441

0.8236 0.4417 0.1613 −0.0438

0.9950 0.9809 0.9626 0.9292

0.9911 0.9479 0.8506 0.5275

0.9819 0.8630 0.6634 0.2813

Japanese yen AR(1) 0.9924 AR(4) 0.9550 AR(8) 0.8949 AR(16) 0.8117

0.9453 0.7940 0.6187 0.3355

0.8357 0.4757 0.2010 −0.0053

0.9549 0.8492 0.7423 0.5932

0.9932 0.9586 0.8779 0.6068

0.9861 0.8737 0.6613 0.2413

Swiss franc AR(1) 0.9676 AR(4) 0.8449 AR(8) 0.6300 AR(16) 0.2422

0.9508 0.8109 0.6446 0.3611

0.8358 0.4789 0.2110 −0.0047

0.8743 0.5732 0.2749 −0.0969

0.9912 0.9497 0.8660 0.6191

0.9880 0.8826 0.6542 0.1566

UK pound AR(1) AR(4) AR(8) AR(16)

0.9689 0.8230 0.5629 0.1076

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. AR(k) is the kth order autorcorrelation coefficient. Bold values represent the empirical data.

The RE2 model produces very similar results to the benchmark. Conversely, the HL model produces coefficients that are positive and become smaller as the lag length increases, although the coefficients don’t become negative at the longest lag lengths, as is seen in the data. The HML model produces mixed results; they are similar to the benchmark for the UK pound and the Japanese yen, but for the other currencies show low, positive autocorrelation at low lag lengths that decreases to near zero as lag length increases. The results indicate that the exchange rate return autocorrelation dynamics of the SN model mainly derive from agent heterogeneity. In addition, it appears that a combination of agent heterogeneity and the expectation shock generates negative autocorrelation coefficients at the largest lag lengths. Autocorrelation structure: deviation from fundamentals Table 8 covers the results of the autocorrelation structure of the deviation from fundamentals. The pattern seen in the data is captured equally well by both the RE1 and SN models. In-sample predictability Tables 9–12 show results of the in-sample predictability regression discussed in Section 1 for the UK pound, the German deutsche mark, the Japanese yen, and the Swiss franc, respectively. The first panel of the tables shows the beta coefficients of the regressions. For all currencies, the RE1 model produces coefficients that have the wrong sign, whereas the other models all produce slope coefficients that follow the pattern seen in the data. The second panel of the three tables shows the median t-statistics from the regressions. Compared to the RE1 model for all currencies, the SN model produces results that are more similar to the data in that the t-statistics increase as the lag length increases. The RE2 and HML models produce results similar to those of the SN model. The HL model’s results are mixed; The t-statistics increase as lag length increases for the deutsche mark, but decrease as lag length increases for the other currencies. The third panel of the four tables show the associated adjusted R2 s of the predictability regression. For all currencies, the RE2, HML, and SN models produce results that match the pattern of the data, while the HL model produces the same

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 9 In-sample predictability: UK pound. k-period

Data

ahead change

βk

Simulated data RE1

RE2

1 4 8 16

0.0318 0.1567 0.3632 0.7603

−0.2108 −0.8352 −1.6106 −2.9879

0.1657 0.5285 0.7999 1.0061

tk 1 4 8 16

1.36 2.00 3.31 5.81

−2.04 −2.49 −3.20 −4.66

3.79 4.84 6.01 7.05

R2 1 4 8 16

0.0131 0.0850 0.2012 0.4791

0.0198 0.0968 0.1922 0.3608

0.0744 0.2519 0.3834 0.4808

HL

0.0966 0.3165 0.5336 0.7996

11.40 7.51 6.03 4.52

0.5594 0.4821 0.4134 0.2987

HML

SN

0.0101 0.0536 0.1385 0.3873

0.0124 0.1134 0.3200 0.7552

0.93 1.35 1.99 3.50

0.68 1.68 2.81 4.95

0.0 0 04 0.0269 0.0764 0.2129

0.0015 0.0554 0.1623 0.3857

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. Numbers in column one represent k-period ahead changes of the dependent variable in the regression. Bold values represent the empirical data.

pattern as the data only for the German deutsche mark; for the other currencies, the adjusted R2 s decrease as the lag length increases. Because the regression results of the HML and SN model are similar, the implication is that the in-sample predictability dynamics of the SN model mainly derive from the expectation shock. Out-of-sample forecasts Table 13 shows for all currencies the median values of U-statistics as calculated by the out-of-sample forecasting exercise described in Section 1.12 If the SN model is true to the data, it should produce a data series in which out-of-sample forecasts based on the monetary model (Eq. [3]) should not significantly outperform those of a random walk model (i.e., should produce U-statistics greater than one). The exceptions are relatively short-period forecasts for the UK pound and the Swiss franc. Both the RE1 and SN models produce data series in which the calculated U-statistics are all greater than one. The Ustatistics of the RE2 model are all less than one. The U-statistics of the HL model are less than one for the UK pound and Swiss franc, but greater than one for the other currencies. They are all greater than one in the HML model. This implies that the expectation shock combined with adaptive learning are driving these results of the SN model. Model comparison based on mean squared error In order to quantitatively assess the performance of all models, the simulation results in Tables 6–13 are used to calculate mean squared errors (MSE) for each major category of stylized fact (volatility, in-sample predictability, and out-of-sample forecasting), and for all categories simultaneously. Mean squared errors are found by squaring the numerical distance between the empirical data and the simulated data and then averaging among the number of dimensions under consideration.13 In the tables that follow, bold numbers represent the lowest MSE for a particular currency. Table 14 shows the MSE’s for the standard deviation of exchange rate returns and the deviations from fundamentals. For exchange rate returns, the SN model performs best (i.e., produces the smallest MSE) for all currencies, while for the

12

Forecasts begin with the same number of periods as was done to calculate the statistics in Table 3. See footnote 7. Specifically, the standard deviations of exchange rate returns and deviation from fundamentals have one dimension each, the autocorrelation structure of exchange rate returns and deviation from fundamentals have four dimensions each (first, fourth, eighth, and sixteenth-order autocorrelation coefficients), the in-sample predictability regressions have twelve dimensions (four beta coefficients along with the associated t-statistics and adjusted R2 ’s that correspond with the first, four, eight, and sixteen-period ahead regressions), and the out-of-sample forecasting exercise has four dimensions (four U statistics). Therefore, the MSE for all categories simultaneously has twenty-six total dimensions. 13

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215

Table 10 In-sample predictability: German deutsche mark. k-period

Data

ahead change

βk

Simulated data RE1

RE2

HL

HML

SN

1 4 8 16

0.0246 0.1125 0.2439 0.4212

−0.2394 −0.9429 −1.8398 −3.3548

0.1750 0.5580 0.8410 1.0332

0.0049 0.0185 0.0355 0.0676

0.0127 0.0667 0.1753 0.4920

0.0164 0.1326 0.3336 0.7186

tk 1 4 8 16

1.83 2.29 3.09 4.96

−1.60 −1.90 −2.42 −3.77

3.48 4.60 5.81 7.14

2.89 2.97 3.32 4.19

0.88 1.29 1.91 3.75

0.74 1.65 2.66 4.68

R2 1 4 8 16

0.0245 0.1220 0.2454 0.3390

0.0118 0.0724 0.1479 0.2839

0.0778 0.2661 0.4084 0.5038

0.0741 0.1836 0.2699 0.3679

0.0 0 0 0 0.0315 0.0896 0.2563

0.0021 0.0643 0.1701 0.3617

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. Numbers in column one represent k-period ahead changes of the dependent variable in the regression. Bold values represent the empirical data.

deviation from fundamentals the SN model is the best fit in the case of the Japanese yen. For the other currencies, the RE2 model is the best fit, but the SN model performs almost as well. Table 15 displays the MSE’s for the autocorrelation structure of exchange rate returns and the deviation from fundamentals. For exchange rate returns, RE1 is the best fit in the case of the Swiss franc, while HML is best for the other currencies. As seen in Table 7 and previously discussed, one of the main weaknesses of the SN model is the relatively high level of exchange rate return autocorrelation generated at low lag lengths. This weakness is reflected in the relatively high MSE’s for the SN model. For the deviations from fundamentals, SN is the best fit in the cases of the UK pound and the Swiss franc, and performs similarly to HML, which provides the best fit in the cases of the other currencies. Table 16 presents the MSE results for the in-sample predictability regressions. SN is the best performer in the case of the UK pound and German deutsche mark, and performs nearly as well as the best performers in the cases of the other currencies. The MSE results for the U statistics of the out-of-sample forecasting exercise are shown in Table 17. SN is the best fit in the case of the Swiss franc. The best fits in the cases of the UK pound, German deutsche mark, and Japanese yen are RE1, HL, and HML, respectively, although all models tend to perform similarly for all currencies. Table 18 shows MSE’s for all categories of stylized facts simultaneously. By this criterion, the SN model provides the best fit of the data for all currencies. In total, the results of Tables 14–18 consist of twenty-eight cases of comparison among the five models. SN represents the best fit in fourteen of those cases, while HML provides the next highest number of “best fits” with seven. These results suggest that among the models compared, SN provides the best fit of the data. Summary and discussion The results of the simulations show that the SN model replicates the data better than the benchmark in several areas: volatility of exchange rate returns, volatility of deviation from fundamentals, and in-sample predictability. Furthermore, if generating exchange rate return autocorrelation that is positive at low lag lengths and negative at high lag lengths is considered the important characteristic of that data, then it can be concluded that SN outperforms RE1 in this area also. On the other hand, both models are equally successful at replicating the autocorrelation structure of the deviation from fundamentals and the out-of-sample forecasting statistics. Moreover, on an MSE basis, the SN model appears to be the best fit of the data. In general, it can be concluded that the SN model replicates the exchange rate stylized facts better than the representative agent rational expectations model, and that agent heterogeneity and the expectation shock are both necessary to produce the dynamics of the SN model. One noticeable weakness of the SN model, however, is the relatively high levels of exchange rate return autocorrelation generated at low lag lengths. The dynamics of the SN model are explained by two key elements. First, the heterogeneous agent adaptive learning mechanism alters the dynamics of the model away from those of the representative agent rational expectations model. Sec-

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 11 In-sample predictability: Japanese yen. k-period

Data

ahead change

βk

Simulated data RE1

RE2

HL

HML

SN

1 4 8 16

0.0172 0.0742 0.1426 0.2074

−0.1944 −0.7311 −1.3509 −2.3735

0.1630 0.5209 0.7969 0.9985

0.0327 0.1002 0.1620 0.2199

0.0157 0.0755 0.1809 0.4623

0.0138 0.1245 0.3349 0.7493

tk 1 4 8 16

1.94 2.22 2.57 3.10

−2.37 −2.86 −3.67 −5.21

4.17 5.36 6.65 7.92

5.28 3.87 3.26 2.40

1.54 2.06 2.84 4.60

0.80 1.96 3.18 5.39

R2 1 4 8 16

0.0178 0.0790 0.1385 0.1990

0.0222 0.0993 0.1892 0.3383

0.0756 0.2522 0.3871 0.4845

0.2205 0.1977 0.1659 0.1124

0.0124 0.0633 0.1338 0.2955

0.0027 0.0608 0.1695 0.3791

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. Numbers in column one represent k-period ahead changes of the dependent variable in the regression. Bold values represent the empirical data.

Table 12 In-sample predictability: Swiss franc. k-period

Data

ahead change

RE1

RE2

HL

HML

SN

0.0531 0.2076 0.3906 0.6228

−0.2110 −0.8139 −1.5295 −2.7520

0.1627 0.5159 0.7798 0.9879

0.0986 0.3244 0.5338 0.7686

0.0104 0.0559 0.1441 0.3969

0.0111 0.1136 0.3387 0.8357

3.32 4.17 4.97 8.41

−2.18 −2.68 −3.47 −4.97

4.11 5.25 6.54 7.65

9.92 7.11 5.81 4.33

0.95 1.42 2.08 3.64

0.65 1.76 3.11 5.71

0.0463 0.2023 0.3520 0.5600

0.0193 0.0899 0.1771 0.3188

0.0742 0.2481 0.3762 0.4719

0.5108 0.4341 0.3603 0.2560

0.0 0 04 0.0256 0.0725 0.2070

0.0012 0.0546 0.1657 0.4114

βk 1 4 8 16 tk 1 4 8 16 R2 1 4 8 16

Simulated data

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and nonspeculators. Numbers in column one represent k-period ahead changes of the dependent variable in the regression. Bold values represent the empirical data.

ond, the expectation shock creates significant variance in the forecasts of speculator-type agents. The combination of these elements makes coordination of beliefs between the two agent-types difficult and prevents the economy from achieving a well-defined equilibrium. This increases the amount of volatility in the exchange rate and, in turn, the deviation from fundamentals. Both elements are necessary to produce the results of the SN model. The increase in the volatility of the exchange rate implies more risk from holding the currency and, qualitatively, should lead to higher returns. Moreover, increasing the volatility of the exchange rate magnifies the mechanism of the exchange

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217

Table 13 Out-of-sample forecasts: U-statistic. k-period ahead

Data

Simulated data

forecast range

RE1

RE2

HL

HML

SN

UK pound 1 4 8 16

0.9913 0.8679 1.2954 1.4716

1.0165 1.0712 1.1320 1.1018

0.9798 0.9272 0.8740 0.8139

0.6545 0.7481 0.8271 0.9483

1.2212 1.4641 1.5570 1.5711

1.0282 1.0615 1.0952 1.0729

German deutsche mark 1 4 8 16

1.1373 1.4424 1.9630 1.9330

1.0247 1.0917 1.1979 1.2520

0.9754 0.9148 0.8566 0.7926

1.0335 1.1336 1.2466 1.3615

1.0938 1.2071 1.2939 1.2565

1.0267 1.0694 1.0761 1.0201

Japanese yen 1 4 8 16

1.2052 1.7102 1.6449 1.4877

1.0299 1.0998 1.1555 1.1037

0.9777 0.9228 0.8706 0.8089

1.0669 1.0294 1.0615 1.1501

1.0475 1.1028 1.1370 1.1387

1.0222 1.0567 1.0953 1.0838

Swiss Franc 1 4 8 16

0.9947 0.9703 0.9646 1.0161

1.0304 1.1013 1.1535 1.1165

0.9785 0.9286 0.8744 0.8127

0.7151 0.7940 0.8715 0.9705

1.0999 1.2522 1.3414 1.3707

1.0200 1.0424 1.0672 1.0373

Median values of 20 0 0 replications. RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The U-statistic is constructed so that values greater than one imply the random walk with drift model is forecasting with greater accuracy than the comparison model. Bold values represent the empirical data. Table 14 MSE for volatility. Model Currency

RE1

RE2

HL

Exchange rate returns UK 6.6230 GER 9.2319 JPN 13.2792 SWZ 16.8369

150.4250 289.2752 146.5199 134.6719

17.3298 26.8563 26.1004 33.2776

Deviation from fundamentals UK 0.0395 GER 0.1605 JPN 0.2102 SWZ 0.0650

0.0052 0.0021 0.0307 0.0010

0.0284 0.0294 0.1608 0.0485

HML

1.74 × 107 1.04 × 104 1.46 × 108 4.85 × 103 320.4041 25.1329 323.3206 16.0663

SN

0.3669 0.1095 0.3871 3.2728

0.0070 0.0120 0.0220 0.0053

RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The numbers reported are the squared differences between the observed data values and the analogous simulated model values in Table 6. Bold numbers represent the minimum MSE for that currency.

rate returning to its fundamental value, explaining the predictability results of Tables 9–12. Adding more volatility to the system should make these movements greater as the time period allowed for the change increases. To understand more fully the dynamics of the SN model, it is helpful to analyze the evolution of agent forecasting model parameter values in the SN model. Figs. 1 and 2 show the evolution of the adaptive learning forecasting model parameter estimates (labeled AL in the figure), along with the associated REE solution values (labeled RE in the figure), for both agenttypes for one replication of the UK pound.14 The figures show that the speculator’s parameter estimates are much more volatile than those of the non-speculator’s, which is an indirect result of the expectation shock. Also, Fig. 2 implies that the non-speculator is using the most recent realized value of the exchange rate as the forecast. This phenomenon may 14

The results are representative for all currencies.

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 15 MSE for autocorrelation structure. Model Currency

RE1

RE2

HL

HML

SN

Exchange Rate Returns UK 0.0292 GER 0.0161 JPN 0.0141 SWZ 0.0019

0.0381 0.0271 0.0213 0.0052

0.2317 0.0057 0.1033 0.2248

0.0278 0.0049 0.0129 0.0053

0.0885 0.0268 0.0695 0.1725

Deviation from Fundamentals UK 0.1200 GER 0.0153 JPN 0.0828 SWZ 0.0039

0.0747 0.2831 0.3508 0.0970

0.0244 0.0299 0.0209 0.0809

0.1033 0.0027 0.0106 0.0523

0.0092 0.0357 0.0967 0.0024

RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The numbers reported are the mean squared differences between the observed data values and the analogous simulated model values of the first, fourth, eighth, and sixteenth order autocorrelation coefficients in Tables 7 and 8. Bold numbers represent the minimum MSE for that currency. Table 16 MSE for in-sample predictability. Model Currency

RE1

RE2

HL

HML

SN

UK GER JPN SWZ

16.8845 12.9768 13.5022 28.6364

1.9360 1.7721 4.6820 0.4370

11.7248 0.2019 1.2396 5.8235

0.6634 0.4012 0.2164 3.7209

0.1305 0.1636 0.6137 1.9846

RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The numbers reported are the mean squared differences between the observed data values and the analogous simulated model values of the in-sample predictability regression slope coefficients, t-statistics, and adjusted R2 s in Tables 9–12. Bold numbers represent the minimum MSE for that currency. Table 17 MSE for U-statistic. Model Currency

RE1

RE2

HL

HML

SN

UK GER JPN SWZ

0.0514 0.2962 0.1976 0.0161

0.1534 0.7073 0.4330 0.0129

0.1552 0.2365 0.2342 0.0300

0.1217 0.2406 0.1934 0.0896

0.0595 0.4428 0.2314 0.0042

RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The numbers reported are the mean squared differences between the observed data values and the analogous simulated model values of the U-statistic of the 1, 4, 8, and 16-period ahead changes in Table 13. Bold numbers represent the minimum MSE for that currency.

help explain why the SN model produces exchange rate returns with relatively large autocorrelation coefficients at low lag lengths; Because the non-speculator represents a significant percentage of the market (approximately 82% based on the result of Table 5), the forecasting model of the non-speculator is generating significant persistence in the exchange rate, which may cause positive returns to follow positive returns, and vice versa. Going further, Fig. 3 depicts the simulated trajectory from a representative replication of the SN and RE1 model of the deviation from fundamentals in levels for the UK pound.15 The SN model produces an exchange rate that can deviate from 15

Again, the results are representative for all currencies.

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219

30 AL RE

Constant

20 10 0 -10

0

50

100

150

Period AL RE

Time

0.05 0 -0.05 0

50

100

150

Fundamentals

Period 6 4 2 0 -2 -4 -6

AL RE

0

50

100

150

Period Fig. 1. Evolution of speculator parameter estimates - UK pound.

1 AL RE

Constant

0.8 0.6 0.4 0.2 0 0

50

100

150

Period

Lagged Exchange Rate

1.4 AL RE

1.2 1 0.8 0.6 0.4 0.2 0 0

50

100

Period Fig. 2. Evolution of non-speculator parameter estimates - UK pound.

150

220

C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 18 Overall MSE. Model Currency UK GER JPN SWZ

RE1

RE2

8.0799 6.4009 6.7959 13.8702

6.7203 12.1005 7.9124 5.3991

HL

HML

SN

6.1424 1.1691 1.6373 4.0212

6.70 × 10 401.1265 5.63 × 106 189.0624 5

0.0988 0.1579 0.3601 1.0696

RE1 is for simulation under homogeneous agent rational expectations with no expectation shock, RE2 is for simulation under homogeneous agent rational expectations with an expectation shock, HL is for simulation under heterogeneous agent adaptive learning with no expectation shock, HML is for simulation under homogeneous agent adaptive learning with the expectation shock, and SN is for simulation under the model of speculators and non-speculators. The numbers reported are the mean squared differences between the observed data values and the simulated model values in Tables 6–13. Bold numbers represent the minimum MSE for that currency.

70 SN RE1

60

Deviation from Fundamentals

50

40

30

20

10

0

-10

-20

-30

0

50

100

150

Period Fig. 3. Trajectory of deviation from fundamentals - UK pound.

the fundamental value by large amounts and for significant periods of time, as compared to the RE1 model. This is most likely a result of the exchange rate persistence generated by the non-speculator’s forecasting equation. Sensitivity analysis It is interesting to understand the sensitivity of the empirical results to the degree of agent heterogeneity. As a simple exercise, a series of simulations of the SN model are run by varying values of μ while holding all other parameters constant. Table 19 shows the resulting criterion function values, based on Eq. (9), for the UK pound. The same moments used in Section 4 are used in this exercise. The criterion function is minimized at the MSM estimate of μ of 0.1786 and grows large as μ approaches 1. The value of the criterion function for the RE1 model is 337, which implies that the SN model outperforms the RE1 model on a criterion function basis only in the approximate range of 0.10 < μ < 0.20. A similar exercise is done for the other currencies, with Table 20 showing the range of values of μ in which the SN model outperforms the RE1 model. From these results it can be ascertained that the empirical results of the SN model are somewhat sensitive to the value of μ.

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Table 19 Sensitivity analysis - UK pound.

μ

Criterion function value

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

512 444 239 148 133 485 1448 3369 6763 12,542 21,640 37,229 62,949 112,370 225,260 541,880 1.9 × 106 11.9 × 106 204.3 × 106 26.0 × 109 3.16 × 101 4

Criterion function values are calculated in the same manner as Eq. (9). The calculated value of the criterion function for the UK pound version of the RE1 model is 337.3.

Table 20 Sensitivity analysis - all currencies. Currency

Range

UK pound German deutsche mark Japanese yen Swiss franc

0.10 0.15 0.10 0.10

< < < <

μ μ μ μ

< < < <

0.20 0.25 0.30 0.20

This shows the approximate range of values of μ in which the SN model outperforms the RE1 model on a criterion function basis.

6. Conclusion This paper employed adaptive learning methods to construct a heterogeneous agent exchange rate model of speculators and non-speculators from a simple monetary framework. Speculators forecasted with a correctly specified model, utilized relatively short periods of historical data when estimating model parameters, and were subject to a persistent expectation shock that captured momentum and herding effects. Non-speculators, on the other hand, implemented a simple forecasting rule, and were not subject to herding effects. The intuition behind this setup is that speculators represent large institutions that conduct sophisticated fundamental and technical analysis of the FX market in an attempt to make a profit from oneway currency movements, while non-speculators represent institutions that make trades for commerce, hedging, tourism, and other purposes not involving attempting to profit from one-way currency movements. The paper utilized a two-stage indirect estimation procedure to facilitate simulation; The first stage calibrated selected parameters using values consistent with the international macroeconomic literature, while the second stage employed the method of simulated moments to estimate parameters unique to the heterogeneous agent model. Simulation results showed that the model was able to replicate several salient facts of the FX market better than does a model of representative agent rational expectations, and that the dynamics of the model derive from both agent heterogeneity and the expectation shock. A noticeable weakness of the model, though, was the relatively high levels of exchange rate return autocorrelation generated at low lag lengths. Areas for future work include incorporating multiple regime shifts in monetary fundamentals, as documented in Kim (2009), analysis of other currency rates (i.e., the US dollar/euro, US dollar/Canadian dollar, etc.), alternative calibration/estimation methods of the expectation shock process (i.e., Bayesian estimation.), and allowing switching between agenttypes based on relative forecasting performance. It would also be beneficial to obtain survey evidence on the degree to which speculative and non-speculative activities characterize the FX market.

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C.J. Elias / Journal of Macroeconomics 49 (2016) 203–223 Table 21 Data ID and sources. Series

Data ID

Source

US/UK spot exchange rate US/Germany spot exchange rate US/Japan spot exchange rate US/Switzerland spot exchange rate US money supply (M2) UK money supply (M0) Germany money supply (M2) Japan money supply (M2) Switzerland money supply (M3) Production of total industry in US Production of total industry in UK Production of total industry in Germany Production of total industry in Japan Production of total industry in Switzerland

EXUSUK EXGEUS EXJPUS DEXSZUS MYAGM2USM052S LPMAVAE MYAGM2DEM189S MYAGM2JPM189S MABMM301CHQ189S USAPROINDMISMEI GBRPROINDQISMEI DEUPROINDQISMEI JPNPROINDQISMEI CHEPROINDQISMEI

FRED FRED FRED FRED FRED BOE FRED FRED FRED FRED FRED FRED FRED FRED

Note: FRED is the Federal Reserve Economic Database. BOE is the Bank of England database.

Acknowledgments I am grateful to William Branch, Fabio Milani, David Brownstone, Steve Hayworth, Jim Saunoris, David Crary, Abdullah Dewan, seminar participants at Eastern Michigan University and the Southern Economics Association 85th Annual Meetings, and three anonymous referees for many useful comments and suggestions. All errors are my own. Appendix A. Data description/methods The data set used in the calculation of the statistics in Tables 1–3 was constructed from quarterly data beginning in the first quarter of 1971 and has different lengths for each country due to data availability. The UK pound data runs through the first quarter of 2006 (141 observations), the German deutsche mark data runs through the fourth quarter of 1998 (112 observations), and the Japanese yen data runs through the fourth quarter of 2014 (176 observations). The Swiss franc data starts in the fourth quarter of 1971 and runs through the fourth quarter of 2014 (173 observations). Quarterly data was constructed from monthly series by using end of period values. The exchange rate is the spot price, in U.S. dollars, of one unit of foreign currency. Exchange rate returns were calculated as:

(exp (y ) − 1 ) × 100

(A.1)

where y is the one-period natural log growth rate of the exchange rate. The money supply is M0 for the UK, M2 for the Germany and Japan, and M3 for Switzerland.16 Fundamentals were calculated using definition (B.5), and the deviation from fundamentals was calculated with definition (2). Following Kim (2009), a country’s total industrial production index was used to proxy for national income. The truncation lag length for Newey-West standard errors was set to floor[4(T/100)2/9 ], where T is the number of observations in the data. Table 21 displays all dataset information. Appendix B. Monetary model under flexible exchange rates The model under consideration is a monetary model of flexible exchange rates that was developed in Frenkel (1976); Mussa (1976), and Bilson (1978). The derivation and notation presented here is from Mark (2001). In this section, all lower case variables represent natural logarithms while upper case variables represent levels, with the only exception being the nominal interest rate, which is always measured in levels. Un-starred variables represent the home country and starred variables represent the foreign country. The model begins with a set of two money demand equations and assumes equilibrium in the domestic and foreign money market, which implies the following,

mt − pt = φ yt − λit

(B.1)

mt∗ − p∗t = φ yt∗ − λit∗

(B.2)

16 The UK pound data ends in 2006 because the M0 money supply series was discontinued in that year. All other measures of the UK money supply are only available from the early 1980s

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where mt is the exogenous money stock, pt is the price level, yt is national income, it is the nominal interest rate, and t is a time subscript. The income elasticity of money demand (0 < φ < 1) and the interest rate semi-elasticity of money demand (λ > 0) are the same for both countries. Uncovered interest parity, the notion that international capital markets are such that arbitrage opportunities (i.e. profiting by borrowing and lending between the home and foreign currencies) don’t exist, implies an international capital market equilibrium,

it − it∗ = Et st+1 − st

(B.3)

where Et st+1 is the expectation of the exchange rate at time t + 1 made at time t. Purchasing power parity is the idea that, in equilibrium, the exchange rate between the two countries should equal the relative price level of the two countries. This condition implies that

st = pt − p∗t

(B.4)

Economic fundamentals are defined as

ft ≡ (mt − mt∗ ) − φ (yt − yt∗ )

(B.5)

Following Kim (2009), assume that the income elasticity of money demand (φ ) is equal to one. To obtain an expression for the current period exchange rate, substitute Eqs. (B.1), (B.2), and (B.3) into (B.4) to obtain

st = a1 Et st+1 + b1 ft

(B.6)

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