Abstract
In this paper, we present a stochastic agentbased model and extend an artificial financial market by considering a stochastic process of market fundamentals. The model predicts that groups of noise traders are busy communicating when market uncertainty is high. In particular, we examine the effects of social interactions on price movements, based on parameter estimation of the group behavior. As traders’ reactions to new information act much like an endogenous shock on return volatility, however, we cannot easily find an exact solution for the model with social interactions. Thus, simulationbased inference is used for the model validation; we investigate whether our artificial economy can match the empirical moments observed in five major foreign exchange data sets (as closely as possible). The results indicate that the return volatility under scrutiny can be robustly decomposed into news (45–55 %) and social interaction effects (45–55 %).
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Notes
Nonlinear structural equations are often used to represent and explain economic behavior and expectation formation process in ABMs. In addition, most behavioral equations are subject to a high degree of stochastic noise. Given this, we do not use gradientbased methods to examine model validation. Instead, the evaluation of the model is to be based on a direct search method or heuristic optimization technique; on this point, see also Winker and Maringer (2007), as well as Gilli and Schumann (2010). The author is currently preparing a manuscript which explains detailed issues regarding the optimization in this paper.
The variation in the equilibrium distribution is expressed in a logarithmic form. For instance, \(L= \ln \frac{1.99}{0.01}\simeq 5.29 \) for \(N = 100\), whereas the normalization constant \(L\) for \(N=1000\) amounts to 7.60.
According to Alfarano et al. (2007), analytical moment conditions are based on tractable opinion dynamics from a simpler version of this model. In our case, however, the mathematical derivation is much more complicated because of the interactions between news and the group behavior. See Appendix 1 for details.
If a small step size (i.e., \(\varDelta t=0.01\)) is used for approximating the price dynamics, then the computation will be expensive. For instance, in order to generate 10,000 price series, we need to compute 10,000 \(*\, \frac{1}{0.01}\) prices.
See Carrasco and Florens (2002) for a detailed survey on the SMM.
Here, the word “robustness” means that timedependent structures in the datagenerating process do not change when the observation size shrinks or expands.
I am grateful to an anonymous referee for this point. An important topic for future research is to find the relation between the selection of moments and the power of the model validation test as models might not be rejected because sufficient moments have not been included in the study.
The term in the square brackets is derived from an approximation of the binominal distributions of the optimists and pessimists; the Gaussian distribution from the Langevin equation is used for its approximation; see Appendix 1.2 in Alfarano and Lux (2007) for its derivation.
I am indebted to an anonymous referee for this point.
In our study, ‘good’ approximations of the true optimum in the objective function can be evaluated by the simulated moments. Then, we can directly compare the simulated and empirical moments, and see whether a small change in parameter points provides different objective function values (where this indicates lack of robustness). This will be discussed later.
The data were retrieved from the Sauder School of Business, University of British Columbia, http://fx.sauder.ubc.ca/data.html.
To save space, we report the case of DM/US. The results of this experiment are almost the same across the other FX data.
As the price dynamics in Eq. (9) are driven by the volatility of the fundamental price and opinion dynamics, the prices will fluctuate between the relative ratio of trading volumes from the fundamentalists and the chartists (\(\frac{N_{C}T_{C}}{N_{F}T_{F}} = \frac{1}{10}\)); that is, \(1 \pm 0.1\).
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Acknowledgments
A preliminary version of this paper was presented in seminars at the University of Kiel (CAU) and the Kiel Institute for the World Economy (IfW), at the 12th ZEW Summer Workshop “Expectations and Expectation Formation on Financial Markets” in Mannheim, and at the 5th Computational Statistics & Data Analysis (CSDA) International Conference on Computational and Financial Econometrics (CFE’11) in London, and in seminars at the Bank of Korea (BoK), the Korea Institute for Industrial Economics and Trade (KIET), and the Korea Institute for International Economic Policy (KIEP). The author expresses his gratitude to all the participants for their active involvement in the conference. The author benefited from discussions with Reiner Franke, Jaba Ghonghadze, Simone Knief, Roman Liesenfeld, Thomas Lux, Veronika Penner, Stephen Sacht, AkiHiro Sato, Peter Winker, and three anonymous referees. The support from the German Academic Exchange Service (DAAD) and the Möller fund from CAU is gratefully acknowledged.
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Appendices
Appendix 1: Conditional Probability for Market Price and Herd Behavior
The stochastic differential equation for the price can be discretized with the Euler method, as follows:
where \(\varDelta t\) denotes the step size for a short discretetime interval.
To approximate the above equation over a small time step, we take one step back in time:
By subtracting the above from Eq. (19), we arrive at:
where the fundamental price follows a Brownian motion in Eq. (10).
As the stochastic process in the fundamentals is affected by the switching behavior at the micro level, we cannot easily derive the analytical expression for the system. However, the dynamic transitions in the opinion index can be simplified using a discretetime Langevin equation:
where \(\eta \) is a random variable drawn from a standard normal distribution.
To see the effects of news and opinion dynamics on the price movements, we derive the transition density for the market price from Eqs. (21) and (22). Thus, the conditional probability for the price can be approximated by the following Gaussian distribution:
where the price fluctuations can be decomposed into two parts; namely, news and social interaction effects.
However, a workable solution for the transition density is based on a strong assumption regarding the opinion index; that is, that the trajectory for the opinion dynamics is based on an approximated Gaussian distribution. For example, the Langevin equation can provide a poor approximation of highly dynamical transitions in the opinion formation process, especially when different time steps are used. Because of the approximation errors, we use SMM for the model validation instead of classical likelihood methods.
2: The Behavior of the Opinion Index at the Boundaries
The model predicts that strong social interactions will occur when the optimistic and pessimistic traders are clearly divided. At the point of \(N_{O}=N_{P}\), the conditional probability for the changes in the noise traders’ configuration is to be the highest value of \(\nu \varDelta t \frac{N}{4}\). The opposite case shows that the boundary points (\(N_{O}=0\) or \(N_{P}=0\)) provide very small conditional probability for the opinion index: \(\nu \varDelta t \frac{2N  1}{4N}\), instead of no movement.
The intermediate steps can be shown below. First, we rewrite the transition probabilities of Eq. (2) in terms of the opinion index \(x_{t}\). After some algebra, we then arrive at:
In the case of \(x=0\), the above equation suggests that \(w_{x} = \frac{N}{4} \nu \varDelta t\). In the cases involving extreme opinions, the transition probability becomes zero (\(w_{x} = 0\)), but the model avoids this likelihood by using the reflecting conditions in Eq. (5). From this, the probability for \(x=+1\) and \(x=1\) can be expressed as follows:

(i)
the optimistic sentiment of \(x=+1\) is equivalent to the case of \(N_{O}=N\) & \(N_{P}=1\) (i.e., \(x=\frac{N1}{N}\)):
$$\begin{aligned} w_{x}(x_{t+\varDelta t}x_{t}; \theta )&= \nu \varDelta t \Big [ 1  \Big ( \frac{N1}{N} \Big )^{2} \Big ]\frac{N}{4}\\&= \nu \varDelta t \Big ( \frac{2N1}{4N} \Big ); \end{aligned}$$ 
(ii)
the pessimistic sentiment of \(x=1\) is equivalent to the case of \(N_{P}=N\) & \(N_{O}=1\) (i.e., \(x=\frac{1N}{N}\)):
$$\begin{aligned} w_{x}(x_{t+\varDelta t}x_{t}; \theta )&= \nu \varDelta t \Big [ 1  \Big ( \frac{1  N}{N} \Big )^{2} \Big ]\frac{N}{4}\\&= \nu \varDelta t \Big ( \frac{2N1}{4N} \Big ). \end{aligned}$$
3: Choice of Moments
In our study, FX data are treated as secondorder stationary. \(\mu _r\), \(\mu _{r}\), and \(\mu _{r^2}\) denote the first moment for the raw (\(r\)), absolute (\(r\)), and square (\(r^2\)) returns, respectively. Their second moments are \(\sigma _r\), \(\sigma _{r}\), and \(\sigma _{r^2}\). Autocorrelation coefficient \(\rho _{\tau }\) measures the correlation of the returns with its own past values after time lag \(\tau \). The chosen moment conditions are as follows (\(m_{1},\cdots ,m_{7}\), \(m_{ID,1}, m_{ID,2}, m_{ID,3}\)):
In SMM, we include the parameters of auxiliary models such as univariate GARCH (1,1) and the Hill estimator; see also the indirect estimation method.

GARCH (1,1) :
$$\begin{aligned} r_{t}&= \gamma _{0} + \epsilon _{i}, \quad \epsilon _{t} \sim N(0, h_{t})\\ h_{t}&= \alpha _{0} + \alpha _{1} \epsilon _{t1}^{2} + \beta _{1} h_{t1}. \end{aligned}$$ 
Tail index; see Hill (1975):
$$\begin{aligned} \xi = \frac{1}{k1} \overset{k1}{\underset{i=1}{\sum }} \ \text{ ln } \ r_{i,n}  \text{ ln } \ r_{k,n},\quad \quad \quad \text{ for } \quad k \ge 2. \end{aligned}$$
From this, we obtain the additional moment conditions for SMM:
4: Summary Statistics of FX Data
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Jang, TS. Identification of Social Interaction Effects in Financial Data. Comput Econ 45, 207–238 (2015). https://doi.org/10.1007/s1061401394156
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DOI: https://doi.org/10.1007/s1061401394156