Identification of Social Interaction Effects in Financial Data

Abstract

In this paper, we present a stochastic agent-based 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, simulation-based 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 %).

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:

$$\begin{aligned} \ln p_{t+\varDelta t} - \ln p_{t} = \beta [N_{F} T_{F} ( \ln p_{F} - \ln p_{t} ) + N_{C}T_{C}x_{t}] \varDelta t, \end{aligned}$$

(19)

where $\varDelta t$ denotes the step size for a short discrete-time interval.

To approximate the above equation over a small time step, we take one step back in time:

$$\begin{aligned} \ln p_{t} - \ln p_{t-\varDelta t} = \beta [N_{F} T_{F} ( \ln p_{F-\varDelta t} - \ln p_{t-\varDelta t} ) + N_{C}T_{C}x_{t-\varDelta t}] \varDelta t. \end{aligned}$$

(20)

By subtracting the above from Eq. (19), we arrive at:

$$\begin{aligned} \ln p_{t+\varDelta t} - \ln p_{t}&= (1- \beta N_{F}T_{F} \varDelta t) (\ln p_{t} - \ln p_{t - \varDelta t})\nonumber \\&+ \beta N_{F}T_{F} \varDelta t(\ln p_{F,t} - \ln p_{F,t-\varDelta t}) + \ \beta N_{C}T_{C}\varDelta t(x_{t} - x_{t-\varDelta t}),\nonumber \\ \end{aligned}$$

(21)

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 discrete-time Langevin equation:

$$\begin{aligned} x_{t+\varDelta t} - x_{t} = \sqrt{\frac{\nu \varDelta t (2 - \nu \varDelta t)}{N}[1-x^{2}_{t}] } \ \eta _{t+\varDelta t}, \end{aligned}$$

(22)

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:

$$\begin{aligned}&\!\!\!\!P(\ln p_{t+\varDelta t} | \ln p_{t}, \ln p_{t-\varDelta t}) {\sim } N \Big ( (2{-}\beta N_{F}T_{F})\varDelta t \ln p_{t} {\,-\,} (1 {\,-\,} \beta N_{F}T_{F} \varDelta t) \ln p_{t-\varDelta t},\nonumber \\&\quad (\beta N_{C}T_{C})^{2} \frac{\nu \varDelta t (2 - \nu \varDelta t)}{N} [1 - x_{t}^{2}] + (\beta N_{F}T_{F} \varDelta t)^{2} \delta _{0}^{2} \Big ), \end{aligned}$$

(23)

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:

$$\begin{aligned} \omega _{x} (x_{t+\varDelta t}|x_{t}; \theta ) = N \frac{1- x^{2}}{4} \nu \varDelta t. \end{aligned}$$

(24)

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., $x=\frac{N-1}{N}$):

  $$\begin{aligned} w_{x}(x_{t+\varDelta t}|x_{t}; \theta )&= \nu \varDelta t \Big [ 1 - \Big ( \frac{N-1}{N} \Big )^{2} \Big ]\frac{N}{4}\\&= \nu \varDelta t \Big ( \frac{2N-1}{4N} \Big ); \end{aligned}$$

• (ii) the pessimistic sentiment of $x=-1$ is equivalent to the case of $N_{P}=N$ & $N_{O}=1$ (i.e., $x=\frac{1-N}{N}$):

  $$\begin{aligned} w_{x}(x_{t+\varDelta t}|x_{t}; \theta )&= \nu \varDelta t \Big [ 1 - \Big ( \frac{1 - N}{N} \Big )^{2} \Big ]\frac{N}{4}\\&= \nu \varDelta t \Big ( \frac{2N-1}{4N} \Big ). \end{aligned}$$

3: Choice of Moments

In our study, FX data are treated as second-order 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 ($m_{1},\cdots ,m_{7}$, $m_{ID,1}, m_{ID,2}, m_{ID,3}$):

$$ \begin{aligned} m_{1}&= E[r_{t}] - \mu _r\\ m_{2}&= E\left[ r_{t}^{2}\right] - \sigma ^{2}\\ m_{3}&= E\left[ \frac{ (r_{t}-\mu _r) ( r_{t-1}-\mu _r )}{\sigma ^{2}_r} \right] - \rho _{1, r}\\ m_{3+i}&= E\left[ \frac{ ( |r_{t}| - \mu _{|r|}) (|r_{t-\tau }| - \mu _{|r|})}{\sigma ^{2}_{|r|}} \right] - \rho _{\tau , |r|} , \ i {=} 1, 2, 3, 4 \quad \& \quad \tau {=} 1, 5, 10, 20. \end{aligned}$$

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) :

  $$\begin{aligned} r_{t}&= \gamma _{0} + \epsilon _{i}, \quad \epsilon _{t} \sim N(0, h_{t})\\ h_{t}&= \alpha _{0} + \alpha _{1} \epsilon _{t-1}^{2} + \beta _{1} h_{t-1}. \end{aligned}$$

• Tail index; see Hill (1975):

  $$\begin{aligned} \xi = \frac{1}{k-1} \overset{k-1}{\underset{i=1}{\sum }} \ \text{ ln } \ r_{i,n} - \text{ ln } \ r_{k,n},\quad \quad \quad \text{ for } \quad k \ge 2. \end{aligned}$$

From this, we obtain the additional moment conditions for SMM:

$$\begin{aligned} m_{ID,1}&= \alpha _{1}\\ m_{ID,2}&= \beta _{1}\\ m_{ID,3}&= \alpha \left( :=\frac{1}{\xi }\right) . \end{aligned}$$

4: Summary Statistics of FX Data

See Figs. 9, 10, 11 and 12; Tables 12, 13, 14 and 15.

Fig. 9

Prices (upper) and returns (lower) of CHF/US data

Fig. 10

Prices (upper) and returns (lower) of FRF/US data

Fig. 11

Prices (upper) and returns (lower) of GBP/US data

Fig. 12

Prices (upper) and returns (lower) of JPY/US data

Table 12 Empirical moments and block bootstrap samples: CHF/US

Table 13 Empirical moments and block bootstrap samples: FRF/US

Table 14 Empirical moments and block bootstrap samples: GBP/US

Table 15 Empirical moments and block bootstrap samples: JPY/US