Estimating heterogeneous agents behavior in a two-market financial system

Abstract

In this paper, we propose a two-market empirical model with heterogeneous agents based on Chiarella et al. (J Econ Behav Organ 83(3):446–460, 2012). Using monthly data of French and US stock markets, the regression shows that individual markets have features of a two-regime switching process. By including inter-market traders whose trading decision is based on fundamental value of foreign market, the two-market model has a better capability in explaining both markets with domestic fundamental traders turning to be significant. The existence of inter-market traders implies that the two markets impact each other through their fundamentals and hence share some common set of factors, which provides foundation of market interactions, such as market co-movement.

Appendix

In the main test, the transition probabilities are assumed to be constant. In this section, we evaluate the setting of time varying transition probabilities as a robustness check. Following the method of Perez-Quiros and Timmermann (2000), we assume that the state transition probabilities between periods $t-1$ and t, $P_{lk,t}$ for l, $k=1,2$ are functions of $y_{t-1}$ which is a vector of state variables:

$$\begin{aligned} P_{lk,t}=\varPhi \left( y_{t-1}\right) , \end{aligned}$$

(12)

where $\varPhi$ is the cumulative density function of a standard normal variable. The use of $\varPhi$ is to make sure $0\le P_{lk,t}\le 1$. In our model, the state variable is the relative fundamental price deviation with respect to price for market j, $x_{t-1}^{j}$:

$$\begin{aligned} x_{t-1}^{j}=\frac{\left| p_{t-1}^{j}-u_{t}^{j}\right| }{p_{t-1}^{j}}. \end{aligned}$$

Table 4 Estimation result under two-market framework with setting of time varying transition probabilities, sample period from January 2000 to April 2013

Depending on the market, the state variable $y_{t-1}$ can be $x_{t-1}^{ Fr }$, $x_{t-1}^{ US }$ or a combination of $x_{t-1}^{ Fr }$ and $x_{t-1}^{ US }$ as shown in Table 4 which reports the estimation results using the setting of time varying transition probabilities. The regressions are run using the method of Ding (2012). The regression models are the same as columns 4 and 5 of Table 3. For the French market, the estimation results are qualitatively the same as the ones with the setting of constant transition probabilities, regardless of the choices of state variable $y_{t-1}$. When $x_{t-1}^{ Fr }$ acts as the state variable, coefficient of $x_{t-1}^{ Fr }$ is significant in $P_{11,t}$. The transition probability of remaining in the low volatility regime for French market increases with the magnitude of the fundamental price deviation of the market itself. The coefficient of $x_{t-1}^{ Fr }$ is insignificant with a negative value in $P_{12,t}$, indicating that the transition probability of switching from high to low volatility regimes decreases with $x_{t-1}^{ Fr }$. The interesting point is that when we utilize $x_{t-1}^{ US }$ as the state variable for French market, its coefficients have the same sign and significance as $x_{t-1}^{ Fr }$ in both $P_{11,t}$ and $P_{12,t}$. This means the US market has influence on the regime-switching of French market. However, when we include both $x_{t-1}^{ Fr }$ and $x_{t-1}^{ US }$ together as components of state variables, their coefficients become highly insignificant in both $P_{11,t}$ and $P_{12,t}$. This might be due to the collinearity relationship between $x_{t-1}^{ Fr }$ and $x_{t-1}^{ US }$ as the cross-correlation between them is as high as 0.785. For the US market, with $x_{t-1}^{ US }$ as the state variable, the estimation results are still comparable to those with the setting of constant transition probabilities. However, only a few coefficients are significant. Even for $P_{11,t}$ and $P_{12,t}$, the coefficients of $x_{t-1}^{ US }$ are also insignificant with a positive value in $P_{11,t}$ and a negative one in $P_{12,t}$. $x_{t-1}^{ US }$ plays a role in US similar to $x_{t-1}^{ Fr }$ in the transition probabilities of French in the way that the transition probability of remaining in the low volatility regime increases with $x_{t-1}^{ US }$ while the transition probability of switching from high to low volatility regimes decreases with $x_{t-1}^{ US }$. We further evaluate other choices of state variable by using $x_{t-1}^{ Fr }$ as well as the joint inclusion of $x_{t-1}^{ Fr }$ and $x_{t-1}^{ US }$, the estimations encounter a singular problem with huge error terms and therefore we don’t report these results.

Overall, the estimation results based on the setting of time varying transition probabilities are quite close to the ones based on the constant transition probabilities. The result that $x_{t-1}^{ US }$ influence the transition probabilities of both US and France indicates that the switching effect might spread from US to France. As $x_{t-1}^{ Fr }$ is not a state variable to US, we cannot conclude that there is a similar switching effect spreading from France to US.