Government intervention model based on behavioral heterogeneity for China’s stock market

Abstract

Active government intervention is a striking characteristic of the Chinese stock market. This study develops a behavioral heterogeneous agent model (HAM) comprising fundamentalists, chartists, and stabilizers to investigate investors’ dynamic switching mechanisms under government intervention. The model introduces a new player, the stabilizer, into the HAM as a proxy for the government. We use the model to examine government programs during the 2015 China stock market crash and find that it can replicate the dynamics of investor sentiment and asset prices. In addition, our analysis of two simulations, specifically the data-generating processes and shock response analysis, further corroborates the key conclusion that our intervention model not only maintains market stability but also promotes the return of risk asset prices to their fundamental values. The study concludes that government interventions guided by the new HAM can alleviate the dilemma between reducing price volatility and improving price efficiency in future intervention programs.

Introduction

Among the major stock markets in the world, the Chinese stock market has the highest turnover rate and relatively high price volatility (Hu et al. 2018). Government intervention in maintaining market stability is a striking feature of the stock market. The Chinese government implements this through administrative bans (e.g., bans on short selling), monetary policies (e.g., changes in interest rates and deposit reserves), and direct trading in the asset market (e.g., purchases of underlying stocks). For instance, during the 2015 stock market crash, the Chinese government organized a “National Team” fund to buy stocks directly to backstop the stock market meltdown (Huang et al. 2019). Since the 2008 Global Financial Crisis, governments in the Organisation for Economic Co-operation and Development countries have intervened in the market after episodes of large fluctuations or severe market dysfunction. Government intervention in the stock market is becoming increasingly common.

Although government intervention can reduce excess price volatility in financial markets, such interventions should be organized and implemented to achieve the objective of maintaining market stability without distorting financial markets. Answering this question requires intensive research on investors behavioral patterns under government intervention and the development of corresponding theoretical models to guide the intervention. Although several empirical studies have investigated the effects of government intervention,¹ to the best of our knowledge, research on a theoretical model to characterize direct government intervention is scarce. The goal of our study is to address this gap in the extant literature.

Government intervention faces a dilemma between reducing price volatility and improving price efficiency, because investors may ignore economic fundamentals and instead follow the trading behaviors of government intervention (Brunnermeier et al. 2021). This study developed a theoretical model to address this dilemma. This model focuses on government intervention through direct trading against heterogeneous investors in the stock market. We built on the heterogeneous agent asset pricing framework proposed by Brock and Hommes (1997, 1998).

Brock and Hommes (1997, 1998) assumed that conventional agents adapt their beliefs over time by choosing from a finite set of expectations of future prices, and revise their beliefs in each period with bounded rationality according to past realized profits. The introduction of the heterogeneous agent asset pricing framework has contributed to expanding the literature on HAMs.² For example, Boswijk et al. (2007) estimated a behavioral HAM for the stock market using annual US stock market data from 1871 to 2003. de Jong et al. (2009) estimated a dynamic HAM with fundamentalists, chartists, and internationalists for the Asian equity market during the Asian crisis. Chiarella et al. (2012) proposed a simple Markov-switching HAM with fundamentalists, chartists, and noise traders and demonstrate that the estimation results match the excess volatility of the S &P 500 index fairly well. HAMs display strong explanatory power for bubbles, crashes, and excess volatility in the stock market (see He and Li (2012), Chiarella et al. (2014), Lof (2015), He and Li (2015a), Hommes and in ’t Veld (2017), and Zhang et al. (2019)).³ The above-mentioned empirical literature employs two types of traditional heterogeneous agents in HAMs: fundamentalists and chartists.⁴ Fundamentalists believe that asset prices will mean-revert to the fundamental value determined by economic fundamentals such as supply and demand and money supply, and hence, trade based on a mean-reversion strategy. In contrast, chartists believe that asset prices are not determined by fundamentals alone and that the trend of prices will continue in the next period according to the simple predictions of technical trading rules, and therefore, trade based on a trend-following strategy. Chen et al. (2012) indicated that these two types of HAMs have an approximate explanatory power for the stylized facts compared with multi-type HAMs. In our model, fundamentalists and chartists illustrate the basic behavior of investors in the stock market.

Fundamentalists provide negative feedback to asset prices, whereas chartists offer positive feedback. Negative feedback causes asset prices to converge to their fundamental values, whereas positive feedback extrapolates prices and causes persistent price deviations from fundamental values. Akerlof and Shiller (2009) considered positive feedback as one of the primary factors for market fluctuations. De Grauwe (2012) defined an animal spirits index measured by the fraction of optimistic extrapolators in an economic activity, and argued that Keynesian animal spirits provide an endogenous explanation of business cycle movements (booms and busts in economic activity).⁵ Brock and Hommes (1998) believed that the interaction between fundamentalists (stabilizing force) and chartists (destabilizing force) causes asset price fluctuations. When the destabilizing force is far stronger than the stabilizing force, the market booms or busts. Admittedly, these large fluctuations are detrimental to the fundamentals of an economy. An intuitive way to prevent excess volatility is to introduce an external force, such as the government, that can balance the destabilizing force.⁶ Therefore, we introduce a government proxy into the traditional HAM and use the new HAM to examine investors behavioral characteristics under government intervention.

The new HAM comprises two conventional heterogeneous agents (fundamentalists and chartists) and a government agent (stabilizer). The behavioral pattern of conventional heterogeneous agents is in line with the framework proposed by Brock and Hommes (1997, 1998) which introduced a simple present discounted value asset-pricing model with heterogeneous beliefs. The stabilizer, as a proxy for the government, trades based on the belief that market participants should be guided by the government during periods of booms and busts. This perspective is motivated by Keynes (1936), who argued that most booms and busts of economic activities are induced by participants’ animal spirits and insisted that the government should moderately intervene in these activities. The novel player, the stabilizer, together with fundamentalists and chartists, constitutes the new HAM. We estimated the HAM using China Securities Index (CSI) 300 daily data via nonlinear least squares and compared the HAM to a benchmark model without a stabilizer through data-generating processes and shock response analysis.⁷ The estimation results indicate that the main parameters of our HAM are statistically and economically significant under multiple intervention scenarios; that is, the HAM can appropriately characterize the main features of government intervention. Furthermore, the results of the data-generating processes and shock response analysis illustrate that the stabilizer can significantly reduce excess volatility and promote the return of asset prices to their fundamental values.

The remainder of this study is organized as follows. In "Behavioral HAM" section builds a behavioral HAM using a government agent. In "Model estimation" section estimates the HAM using CSI 300 daily data. In "Simulations" section compares the HAM with a fundamentalist–chartist benchmark model through data-generating processes and shock response analysis. Finally, "Conclusions" section concludes the paper.

Behavioral HAM

In this section, we built an asset pricing model with heterogeneous agents, following the frameworks of Brock and Hommes (1998), Boswijk et al. (2007), and Hommes and in ’t Veld (2017). The model considers a market with three types of heterogeneous agents: fundamentalists, chartists, and stabilizers, and assumes that all agents have perfect knowledge of the fundamental value but disagree on how the next period’s price will deviate from the fundamental price. Fundamentalists believe that the trend of deviation from the fundamental value is not persistent and the deviation will shrink, whereas chartists are generally positive feedback traders who believe that the trend of deviation from the fundamental value is persistent and the deviation will increase. Prior studies on heterogeneous agents have frequently introduced fundamentalists and chartists; however, we are the first to introduce a novel type of agent, the stabilizer. To simplify the model and focus on dynamic switching among these three types of agents, we ignored the cost of government intervention.⁸

Model

The asset pricing model comprises three types of investors, all of whom have perfect knowledge of the fundamental value of the risk asset. The pricing equation is specified in a standard manner:⁹

$$\begin{aligned} x_{t}&=\frac{1}{R^{*}}\left( n_{f,t}E_{f,t}[x_{t+1}]+n_{c,t}E_{c,t}[x_{t+1}]+n_{s,t}E_{s,t}[x_{t+1}]\right) \nonumber \\&=\frac{1}{R^{*}}\sum _i{n_{i,t}E_{i,t}[x_{t+1}]}, \quad i=f,c,s, \end{aligned}$$

(1)

where $x_{t}$ is the deviation of the price-dividend (PD) ratio from its fundamental value, $1/R^*$ is the expected discount factor that we assume is common to all agents in the market and determined by economic fundamentals, $n_{i,t}$ is the fraction of agents having belief $E_{i,t}$, and E is the mathematical expectations operator. $E_{f,t}$, $E_{c,t}$, and $E_{s,t}$ denote the beliefs of fundamentalists, chartists, and stabilizers, respectively. Subsequently, we specify the variables.

Fundamentalists

Fundamentalists are assumed to have full knowledge of their fundamental value, which is determined by their real economic activity. They trade on the expectation that the trend of deviation from the fundamental value is not persistent and that stock prices will converge to their fundamental values. Consistent with Hommes and in ’t Veld (2017) and Zhang et al. (2019), we used the simplest rule to forecast future deviation, which is linear in the last observation:

$$\begin{aligned} E_{f,t}[x_{t+1}]=\phi _{f}x_{t-1}, \quad 0<\phi _{f}<1, \end{aligned}$$

(2)

where $\phi _{f}$ is the belief parameter of the fundamentalists and measures the speed of the mean reversion of the deviation x from the fundamental value. However, this forecasting rule has a timing problem. Investors do not know the present value $x_{t}$ in period t, which is the period in which they predict period $t+1$. In fact, agents cannot use $x_{t}$ to forecast the next period value $x_{t+1}$ in period t. This rule and its explanation are also applicable to chartists and stabilizers.

Chartists

Chartists believe that prices will continue into the next period. Let $\phi _{c}$ be the belief parameter of chartists that measures the strength of the trend. We can write the belief of chartists as

$$\begin{aligned} E_{c,t}[x_{t+1}]=\phi _{c}x_{t-1}, \quad \phi _{c}>1, \end{aligned}$$

(3)

where $\phi _{c}$ denotes the belief parameter of chartists and implies a positive feedback dynamic process. In other words, chartists play a destabilizing role in the heterogeneous asset pricing model. De Grauwe (2012) considers the fraction of chartists in the market as the indicator variable of animal spirits and finds that the variations in animal spirits correspond fairly well with the booms and busts in economic activity. Motivated by De Grauwe (2012), we developed a new type of agent that can induce investors’ animal spirits in the financial market, which we call a stabilizer.

Stabilizers

Chartists play a destabilizing role in the market, whereas stabilizers act as a steadying force. Stabilizers can induce animal spirits when chartists are strong enough to contribute to a sharp rise or fall in the market. Thus, stabilizers enter the market and attempt to maintain market stability through negative feedback on price deviations. We obtain their expectations as follows:

$$\begin{aligned} E_{s,t}[x_{t+1}]=\phi _{s}x_{t-1}, \quad 0<\phi _{s}<1, \end{aligned}$$

(4)

where $\phi _{s}$ is the belief parameter of the stabilizers and takes a value between 0 and 1 like $\phi _{f}$. However, while stabilizers are akin to fundamentalists in their expectations for $x_{t+1}$, their motivations for trading decisions differ. The stabilizer, as a proxy for the government, is an external factor in the financial market. In contrast, the fundamentalist, as the basic investor in the financial market, is an internal factor. These two types of agents have different effects on price dynamics because of their different trading motivations and strategies. In "Trading strategies and switching mechanism" section, we present the trading strategies of all agents.

Trading strategies and switching mechanism

Brock and Hommes (1998) argue that switching between the stabilizing and destabilizing forces causes asset price fluctuations. In our model, the fundamentalists and stabilizers are stabilizing forces, whereas the chartist is the destabilizing force. Both fundamentalists and stabilizers work to reduce the price deviation from (or drive the price close to) its fundamental value, whereas chartists serve to increase the price deviation (or drive the price away) from its fundamental value. Corresponding to their expectations, the fundamentalist and stabilizer apply a “centripetal force” of “near-fundamental” to asset pricing, whereas the chartist applies the “centrifugal force” of “far-from-fundamental”. In other words, fundamentalists and stabilizers present two types of negative feedback, whereas chartists provide positive feedback in asset pricing. Brock and Hommes (1997) considered negative and positive feedback as an intuitive behavioral interpretation of the agents’ interaction mechanism.

Figure 1 presents the switching among fundamentalists, chartists, and stabilizers. The most basic switching in stock markets is the interaction between fundamentalists and chartists, which several studies, such as Boswijk et al. (2007), Franke and Westerhoff (2012), Chiarella et al. (2014), He and Li (2015b), and Hommes and in ’t Veld (2017) have investigated. We illustrate this switching using the heterogeneous strategy-switching framework of Brock and Hommes (1997, 1998). The key idea of the Brock–Hommes model is that agents learn through reinforcement learning or evolutionary selection, and tend to switch to strategies that perform better based on the realized profits of their forecasting rules.¹⁰ Unlike the two basic agents, stabilizers, which aim to maintain market stability, enter the market when it fluctuates highly (big $|x_t|$). The larger the price deviation, the larger the percentage of stabilizers in the market. Stabilizers do not interact separately with fundamentalists or chartists. The change in the overall percentage of basic investors is related to the percentage of stabilizers.

Fig. 1

Switching among fundamentalists, chartists, and stabilizers. Solid arrows indicate switching and dashed arrows indicate entering and leaving the market. As per the discrete choice theory, the most basic switching in the stock market is the interaction between fundamentalists and chartists

According to the switching mechanism discussed above, we can formulate the agents’ switching strategies as

$$\begin{aligned} n_{f,t}&=\frac{\exp (\beta U_{f,t-1})}{\exp (\beta U_{f,t-1})+\exp (\beta U_{c,t-1})}(1-n_{s,t}), \end{aligned}$$

(5)

$$\begin{aligned} n_{c,t}&=\frac{\exp (\beta U_{c,t-1})}{\exp (\beta U_{f,t-1})+\exp (\beta U_{c,t-1})}(1-n_{s,t}), \end{aligned}$$

(6)

$$\begin{aligned} n_{s,t}&=n_{s,max}\left( 1-\exp \left( -\ln {2}\left( \frac{x_{t-1}}{x_{s}}\right) ^2\right) \right) . \end{aligned}$$

(7)

The first multiplier in Eqs. (5)–(6) is the same as Boswijk et al. (2007) and Hommes and in ’t Veld (2017), which is obtained according to discrete choice theory (Brock and Hommes 1997).¹¹ This means that the fraction of fundamentalists or chartists is updated using a multinomial logit model with the intensity of choice $\beta$ and the performance (utility) of beliefs $U_{f,t}$ and $U_{c,t}$.¹² According to Hommes and in ’t Veld (2017), fundamentalists and chartists compute the performance of beliefs as follows:

$$\begin{aligned} U_{f,t}&=\left( E_{f,t-1}[x_{t}]-R^{*}x_{t-1}\right) \left( x_{t}-R^{*}x_{t-1}\right) , \end{aligned}$$

(8)

$$\begin{aligned} U_{c,t}&=\left( E_{c,t-1}[x_{t}]-R^{*}x_{t-1}\right) \left( x_{t}-R^{*}x_{t-1}\right) . \end{aligned}$$

(9)

Equation (8) shows that the realized profit is proportional to the fundamentalists’ demand, which depends on the expectation $(E_{f,t-1}[x_{t}]-R^{*}x_{t-1})$ times the realized excess return, which, in turn, depends on the realization $(x_{t}-R^{*}x_{t-1})$. Equation (9) has a similar interpretation.

Examining Eqs. (5)–(6), the second multiplier $(1-n_{s,t})$ introduces the influence of stabilizers on the fraction of fundamentalists and chartists, which reflects the switching between stabilizers, fundamentalists, and chartists. Here, the formula $n_{f,t}+n_{c,t}+n_{s,t}=1$ is satisfied.

As a key contribution of this study, Eq. (7) reveals the nature of the stabilizers, which differs markedly from standard agents’ beliefs. Intuitively, a financial market should not be dominated by stabilizers, implying that a fraction of stabilizers should have a limited maximum value of $n_{s,max}$ and satisfy $n_{s,max}<1$. In "Fraction of stabilizers" section presents more details of Eq. (7).

Fraction of stabilizers

To obtain the formula for the fraction of stabilizers, we need to understand their motivations. As agents of the government, stabilizers aim to maintain market stability. Based on the above analysis of trading strategies and the switching mechanism, we can intuitively depict the fraction of stabilizers as a Gaussian curve (see Fig. 2).¹³

Figure 2 illustrates that more stabilizers participate in the financial market to guide investors when the deviation from fundamental value goes beyond the limit of $x_{s}$. The larger the absolute value of x, the greater the number of stabilizers in the market. In contrast, stabilizers leave the market gradually when the deviation is smaller than the threshold value $x_{s}$; that is, the smaller the absolute value of x, the fewer the number of stabilizers in the market. Given that stabilizers behave symmetrically, we considered identical amplitudes for their entry and exit behaviors.

Fig. 2

Fraction of stabilizers $n_{s,t}$ varies with the deviation x. $n_{s,max}$ is the limited maximum fraction of stabilizers in the market and $x_s$ is the threshold for stabilizers. When |x| is greater than $x_s$, stabilizers enter the market; when |x| is smaller than $x_s$, stabilizers exit the market

Because $n_{s,t}$ satisfies the Gaussian distribution, it follows a general pattern:

$$\begin{aligned} n_{s,t}(x)=\alpha +\eta \exp \left( -\frac{(x-\mu )^2}{2\sigma ^2}\right) . \end{aligned}$$

(10)

In line with Fig. 2, we have:

$$\begin{aligned} n_{s,t}(-x)&=n_{s,t}(x), \end{aligned}$$

(11)

$$\begin{aligned} n_{s,t}(0)&=0, \end{aligned}$$

(12)

$$\begin{aligned} n_{s,t}(x_{s})&=\frac{1}{2}n_{s,max}, \end{aligned}$$

(13)

$$\begin{aligned} n_{s,t}(\infty )&=n_{s,max}. \end{aligned}$$

(14)

After solving the above equations, we obtained $\alpha =n_{s,max}$, $\eta =-n_{s,max}$, $\mu =0$, and $\sigma ^2=\frac{x_{s}^2}{2\ln 2}$. We can then rewrite Eq. (10) as

$$\begin{aligned} n_{s,t}&=n_{s,max}-n_{s,max}\exp \left( -\frac{x_{t-1}^2}{x_{s}^2/\ln 2}\right) \nonumber \\&=n_{s,max}\left( 1-\exp \left( -\ln 2\left( \frac{x_{t-1}}{x_{s}}\right) ^2\right) \right) . \end{aligned}$$

(15)

Clearly, we have $n_{s,t}\in [0,n_{s,max}]$.

Fundamental value and deviations

The models used to calculate the fundamental value include the Gordon model based on dividends and the Campbell–Cochrane model based on consumption habits. The former is commonly employed to examine the fundamental value of stocks.¹⁴ Following the approach in Fama and French (2002), let $P_{t}^*$ be the fundamental value, $D_{t}$ the dividend flow, $g_{t}$ the growth rate of dividends, and $r_{t}$ the expected return.¹⁵ According to the Gordon model, we have:

$$\begin{aligned} P_{t}^*&=E_{t}\left[ \sum _{j=1}^{\infty }\left( \prod _{k=1}^{j}\frac{1+g_{t+k}}{1+r_{t+k}}\right) D_{t}\right] \nonumber \\&\approx \left( \frac{1+g}{r-g} -\frac{c_{r}(1+g)(r_{t}-r)}{(r-g)(1+r-c_{r}(1+g))} +\frac{c_{g}(1+r)(g_{t}-g)}{(r-g)(1+r-c_{g}(1+g))}\right) D_{t}, \end{aligned}$$

(16)

where $c_{g}$ and $c_{r}$ are the first-order autocorrelation coefficients of the time series $g_{t}$ and $r_{t}$, respectively, and g and r are their average values.¹⁶ Therefore, we can obtain the fundamental PD ratios as

$$\begin{aligned} \delta _{t}^*&=\frac{P_{t}^*}{D_{t}} =\left( \frac{1+g}{r-g} -\frac{c_{r}(1+g)(r_{t}-r)}{(r-g)(1+r-c_{r}(1+g))} +\frac{c_{g}(1+r)(g_{t}-g)}{(r-g)(1+r-c_{g}(1+g))}\right) . \end{aligned}$$

(17)

Based on the fundamental PD ratios $\delta _{t}^*$, we calculated the deviation $x_{t}$ in Eq. (1) as follows:

$$\begin{aligned} x_{t}=\delta _{t}-\delta _{t}^*. \end{aligned}$$

(18)

Here, $\delta _{t}$ denotes the realized PD ratio at time t, obtained by dividing the realized price $P_{t}$ by dividend $D_{t}$.

Thus far, we have defined all of the main parameters in our heterogeneous beliefs model. We can rewrite the HAM in Eq. (1) as an econometric model, after adding the error term:

$$\begin{aligned} x_{t}&=\frac{1}{R^{*}}\left( n_{f,t}\phi _{f}+n_{c,t}\phi _{c}+n_{s,t}\phi _{s}\right) x_{t-1}+\epsilon _{t}\nonumber \\&\equiv \Phi _{t}x_{t-1}+\epsilon _{t}, \quad \epsilon _{t}\sim N(0,\sigma ^{2})\quad IID. \end{aligned}$$

(19)

Here, we defined the time-varying coefficient $\Phi _{t}$ as the average market sentiment calculated as $\Phi _{t}=\frac{1}{R^{*}}(n_{f,t}\phi _{f}+n_{c,t}\phi _{c}+n_{s,t}\phi _{s})$.¹⁷ The error term $\epsilon _{t}$ captures the unobserved exogenous fundamental shocks. We calculated the expected discount rate $R^*$ as $\frac{1+r}{1+g}$ following Hommes and in ’t Veld (2017).

The key parameters in the HAM include three belief parameters: $\phi _{f}$, $\phi _{c}$, and $\phi _{s}$, the intensity of choice $\beta$, the threshold value $x_{s}$, and the maximum fraction $n_{s,max}$ for stabilizers. In "Model estimation" section, we estimated HAM using daily CSI 300 data from the Chinese stock market.

Model estimation

To estimate our model, we used daily CSI 300 price and dividend data from July 1, 2015, to December 31, 2017, which we collected from the China Stock Market and Accounting Research database. Our sample included 613 daily observations.¹⁸ This database (or others) does not provide daily but annual dividend data for CSI 300. Considering that the increase in dividends is a smooth process, we obtained the daily dividends through linear interpolation.¹⁹ The fundamental PD ratio $\delta _{t}^*$ was calculated using Eq. (17) remains close to the static value of $\delta _{t}^*=39.1$, which can be explained by the almost constant economic fundamentals during our sample period. Thus, we used the static value of $\delta _{t}^*=39.1$ to calculate $x_{t}$. Figure 3 depicts the CSI 300 with its fundamental values, corrected for inflation (top panel), and the deviations from its fundamental values (bottom panel). Table 1 presents the descriptive statistics.

To obtain the key parameters, we used nonlinear least squares to estimate the HAM using time series $x_{t}$ of price deviations from the Gordon benchmark. Here, we focused on the belief coefficients of fundamentalists, chartists, and stabilizers, that is, $\phi _{f}$, $\phi _{c}$, and $\phi _{s}$, respectively. We also needed to investigate other parameters, such as the intensity of choice $\beta$, threshold value $x_{s}$, and maximum fraction $n_{s,max}$ of stabilizers.

Table 1 Descriptive statistics

Fig. 3

CSI 300 with its fundamental value, corrected for inflation (top panel), and the deviation from the fundamental value (bottom panel)

Admittedly, we do not know the exact values of $x_{s}$ and $n_{s,max}$ in the sample period. According to the definition of the threshold value $x_{s}$, this is an indicator of large deviation from the fundamental value. Obviously, the period in which the deviations $x_{t}$ exceed the threshold value $x_{s}$ should be short, and we consider the maximum $x_{s}$ to be no more than q(0.3), that is, the period with large deviations accounts for no more than 30% of the sample period. Intuitively, the maximum fraction $n_{s,max}$ of stabilizers cannot exceed 0.5, because the stabilizer cannot be a major agent in the market. To investigate the effect of $x_{s}$ and $n_{s,max}$ on the estimation results, we estimated 15 specifications: three values of $x_{s}$ (q(0.1), q(0.2), and q(0.3)) and five values of $n_{s,max}$ (0.1, 0.2, 0.3, 0.4, and 0.5).

Table 2 HAM estimation results on CSI 300 for the July 1, 2015 to December 31, 2017 period

Table 2 presents the estimation results for different model specifications. In all estimations, the belief coefficients $\phi _{f}$, $\phi _{c}$, and $\phi _{s}$ and the intensity of choice $\beta$ are highly significant. This means that our HAM is robust to parameters $x_{s}$ and $n_{s,max}$. Panels A, B, and C illustrate that the estimation results vary slightly according to the value of $x_{s}$, whereas they vary significantly according to the value of $n_{s,max}$. For example, given a certain value of $x_{s}$, $\phi _{f}$ decreases ($\phi _{c}$ increases) as $n_{s,max}$ increases. In other words, an increase in the fraction of stabilizers causes the fraction of fundamentalists and chartists to decrease; hence, the strengths of the beliefs of these two agents measured through the distance from 1 ($|\phi _{f}-1|$ and $|\phi _{c}-1|$) increase to satisfy the unchanged sample in the different model specifications. In contrast, the strength of the beliefs of stabilizers ($|\phi _{s}-1|$) decreases as $n_{s,max}$ increases for the same sample in different model specifications.

Fig. 4

Realized price (top panel) and the corresponding average market sentiment (bottom panel) for the model specification $x_{s}=q(0.2)$, $n_{s,max}=0.3$. The average market sentiment is given by $\Phi _{t}=\frac{1}{R^{*}}(n_{f,t}\phi _{f}+n_{c,t}\phi _{c}+n_{s,t}\phi _{s})$. The shaded regions in the bottom panel that captures five high fluctuations over the sample period duly match the shadow regions in the top panel

To investigate price dynamics, we calculated the average market sentiment according to Eq. (19). The time-varying market sentiment combines the fractions and belief coefficients of all heterogeneous agents. Figure 4 illustrates the realized price and the corresponding market sentiment for the model specification $x_{s}=q(0.2)$, $n_{s,max}=0.3$.²⁰ We shaded the five fluctuations in the top panel and found the corresponding changes in sentiment in the bottom panel. The time-varying market sentiment based on our model duly matches the realized price. Our model can accurately reproduce the price dynamics during the sample period.

Fig. 5

Realized deviation (top panel) and the corresponding market fraction of stabilizers (second panel), fundamentalists (third panel), and chartists (bottom panel), respectively, for the model specification $x_{s}=q(0.2)$, $n_{s,max}=0.3$. The shadow regions in the first two panels capture characterized periods that indicate that the fraction of stabilizers duly matches with the deviation

Figure 5 demonstrates the realized deviation and the corresponding market fraction of traders. We focused on the fraction $n_{s,t}$ of stabilizers (second panel). When the deviation $x_{t}$ (top panel) is close to the fundamental value, $n_{s,t}$ is close to 0 ( shaded area). $n_{s,t}$ increases as $|x_{t}|$ increases and vice versa. This is consistent with the trading motivation of stabilizers. In Fig. 5, the fraction $n_{f,t}$ of fundamentalists (third panel) and the fraction $n_{c,t}$ of chartists (bottom panel) present completely opposite changes, which is in line with Eqs. (5)–(6).

Simulations

This section attempts to determine whether the government can reduce excess volatility and maintain market stability. In "Comparing the HAM to a fundamentalist–chartist benchmark model" section, we compared the simulation results based on behavioral HAMs with and without stabilizers. Second, in "Response to a fundamental shock" section, we investigated the responses to positive and negative shocks corresponding to overvalued and undervalued assets compared to fundamentals. To focus more on the impact of the government on the market, we discussed only one version of the model ($x_{s}=q(0.2)$, $n_{s,max}=0.3$).

Comparing the HAM to a fundamentalist–chartist benchmark model

We compared the simulation results for HAM2 and HAM3. Here, HAM2 includes two types of agents: fundamentalists and chartists, whereas HAM3 includes fundamentalists, chartists, and stabilizers. For these two HAMs, we generated innovations using the estimated residual $s^2$ in "Model estimation" section. The data-generating equation is as follows:

$$\begin{aligned} \mathrm{HAM3}: x_{t}=\frac{1}{R^{*}}\left( 0.908n_{f,t}+1.113n_{c,t}+0.906n_{s,t}\right) x_{t-1}+\epsilon _{t}, \quad \epsilon _{t}\sim N(0,0.048), \end{aligned}$$

(20)

where $n_{f,t}$, $n_{c,t}$, and $n_{s,t}$ are updated according to Eqs. (5–7), with $\beta =7.552$. We obtained the HAM2 data-generating equation by setting $n_{s,t}=0$ in the HAM3 equation.

$$\begin{aligned} \mathrm{HAM2}: x_{t}=\frac{1}{R^{*}}\left( 0.908n_{f,t}+1.113n_{c,t}\right) x_{t-1}+\epsilon _{t}, \quad \epsilon _{t}\sim N(0,0.048). \end{aligned}$$

(21)

We ran these two data-generating equations for a certain number of periods T and compared the outcomes of these processes. Table 3 illustrates the average variances used to measure the excess volatility of the stock market for over 10,000 simulations. Evidently, the average variances of HAM3 in all period specifications are significantly smaller than the corresponding variances of HAM2. In other words, government intervention can significantly reduce excess stock market volatility. Figure 6 provides a visual explanation.

Our simulations exhibit an interesting phenomenon: when the number of periods exceeds 80 in HAM2, the generated data display extreme instability with increasing probability. Extremely unstable simulations produced invalid average variances. In Table 3, we denoted the number of invalid simulations in the parentheses. HAM3 does not even exhibit a single extremely unstable simulation for all period specifications.

Table 3 Simulation results

Fig. 6

Average variances of generated data based on HAM2 and HAM3 for a certain number of periods T from 20 to 100. Every period specification is run for 10,000 simulations. HAM2: $x_{t}=\frac{1}{R^{*}}(0.908n_{f,t}+1.113n_{c,t})x_{t-1}+\epsilon _{t}$, $\epsilon _{t}\sim N(0,0.048)$; HAM3: $x_{t}=\frac{1}{R^{*}}(0.908n_{f,t}+1.113n_{c,t}+0.906n_{s,t})x_{t-1}+\epsilon _{t}$, $\epsilon _{t}\sim N(0,0.048)$

Response to a fundamental shock

In this section, we investigated the market response to good news with HAM2 and HAM3.²¹ Assume that at $T=1$, the deviation value is $x=0$ and good news at $T=3$ drives x to 5. The threshold value is $x_{s}=3$ and the maximum fraction is $n_{s,max}=0.3$ for HAM3. Figure 7 depicts the deviation value dynamics in response to the positive shock for HAM2 and HAM3 and indicates a clear difference between the two. For HAM3, the positive shock exceeds the threshold value, and more stabilizers enter the market, which reduces the shock below the threshold value and maintains stability. In contrast, in the HAM2 case, the positive shock drives the deviation from fundamentals to increasing values and contributes to instability.

Fig. 7

Responses to a positive shock, corresponding to overvalued assets compared to fundamentals. We assumed that at $T=1$, the deviation value is $x=0$ and good news at $T=3$ drives x to 5, and the threshold value is $x_{s}=3$

Fig. 8

Dynamic response to a positive shock based on HAM2: $x_{t}=\frac{1}{R^{*}}(0.908n_{f,t}+1.113n_{c,t})x_{t-1}+\epsilon _{t}$, $\epsilon _{t}\sim N(0,0.048)$

Fig. 9

Dynamic response to a positive shock based on HAM3: $x_{t}=\frac{1}{R^{*}}(0.908n_{f,t}+1.113n_{c,t}+0.906n_{s,t})x_{t-1}+\epsilon _{t}$, $\epsilon _{t}\sim N(0,0.048)$

To determine the dynamic responses to the exogenous positive shock based on HAM2 and HAM3, we investigated the variability in the fractions of heterogeneous agents and the average market sentiment corresponding to the change in deviation $x_{t}$. Figure 8 illustrates that the fraction of chartists increases quickly from the initial value of 0.5 to near 1, and the market is dominated by chartists as the deviation rises suddenly from 0 to 5 in the HAM2 case. The deviation thereafter increases exponentially because extrapolation belief dominates the market, and the exponential growth rate is equal to $\phi _{c}$ ($0\times \phi _{f}+1\times \phi _{c}=\phi _{c}>1$). Thus, without an agent guiding or balancing the extrapolation belief, a large deviation contributes to high excess volatility.

In contrast, for HAM3, Fig. 9 indicates that the stabilizers can bring the deviation back to a relatively stable value below the threshold value of 3 by guiding the extrapolation belief. When the deviation from fundamentals exceeds the threshold $x_{s}$, as soon as the stabilizers enter the market, according to Eq. (7), chartists switch to fundamentalists, and the deviation is pushed back toward fundamentals. The stabilizers then exit the market as the deviation decreases and remain in a relatively steady proportion in the market.

The discussion above demonstrates that stabilizers can reduce excess volatility in the stock market by guiding heterogeneous beliefs. On the one hand, the average variance of HAM3 is significantly smaller than the corresponding variance of HAM2 based on the data-generating processes. We also noted an increasing probability of extreme deviations as the data-generating period increased. In contrast, HAM3 can promote the return of asset prices to their fundamental values.

Conclusions

In this study, we developed a behavioral HAM to investigate the behavioral characteristics of investors under government intervention in the Chinese stock market. Our study contributes to the literature on government intervention and behavioral asset pricing in three ways.

First, this study can be viewed as the first attempt to introduce a government agent into behavioral HAMs. We developed a HAM with three types of agents by defining a government agent called a stabilizer, which plays a stabilizing role in financial markets. Second, we formulated the behavioral pattern of government intervention and the interaction of stabilizers with investors, which allows us to analyze the trading behaviors of investors and stock price fluctuations under government intervention. Finally, our model provides important insight by describing how the government can effectively intervene in a dysfunctional market, which has practical implications not only for government intervention in China but also for intervention programs in other countries.

However, this study is deficient in its design of investors’ belief parameters. Government intervention in the stock market inevitably affects investor expectations. Does a change in investors’ expectations affect their belief parameters? If so, how should this be considered in modeling? This is not discussed in this study, and to simplify the model, all investors’ belief parameters are set as constants.

Finally, our results raise an interesting question. In the process of direct intervention in the stock market, the government needs to trade the underlying assets with investors. What is the minimum amount of capital the government needs to invest in a given intervention program to achieve the goal of maintaining market stability? Future research should be able to confirm this.