Skip to main content

Business cycle and herding behavior in stock returns: theory and evidence

Abstract

This study explains the role of economic uncertainty as a bridge between business cycles and investors’ herding behavior. Starting with a conventional stochastic differential equation representing the evolution of stock returns, we provide a simple theoretical model and empirically demonstrate it. Specifically, the growth rate of gross domestic product and the power law exponent are used as proxies for business cycles and herding behavior, respectively. We find stronger herding behavior during recessions than during booms. We attribute this to economic uncertainty, which leads to strong behavioral bias in the stock market. These findings are consistent with the predictions of the quantum model.

Introduction

In the twenty-first century, stock trading has predominantly been conducted through electronic platforms rather than on the floor. By the end of 2014, approximately 15% of the total trading on the New York Stock Exchange was conducted on the floor, with the rest handled electronically (Hiltzik 2014). The dominance of anonymous electronic trading implies that more trades are independent of others. However, it has been widely documented that participants in financial markets mimic other traders’ actions, termed “herding” in literature. Herding in the stock market often leads to higher volatility, implying greater fluctuations in stock returns (Cont and Bouchaud 2000; Orĺean 1995; Banerjee 1993; Topol 1991). Thus, understanding the origins of herding behavior in financial markets is critically important for regulators and practitioners.

Our definition of “herding” differs slightly from the conventional one because we focus on the tail of the distribution with a scaling exponent. In other words, although investors are likely to herd within the center to follow the larger group of investors, “local” herding among investors who are at the extremes of the distribution is also possible with a certain regularity. Previously, most literature has explained “herding” as the general tendency of market participants to cluster around the center. For example, some studies have focused on the co-movement of stock returns using dynamic correlations and have defined herding as a high correlation among investors across different markets (Chiang et al. 2007; Boyer et al. 2006). Other studies use financial assets following extreme market returns to capture herding behavior in financial markets (Sibande et al. 2021; Kumar et al. 2021; Bouri et al. 2021; Bouri et al. 2019; Demirer et al. 2019; Balcilar et al. 2017; Galariotis et al. 2015; Chiang and Zheng 2010; Chang et al. 2000; Christie and Huang 1995). Moreover, prior studies on herding in financial markets have mostly focused on statistical tests of the relationship between herding behavior and business cycles but have failed to explain business cycles as the origin of herding behavior in the marketplace.

This study investigates herding behavior in stock returns based on concepts pioneered by the physics community. Stock markets exhibit universal characteristics similar to physical systems with considerable interacting units, for which several microscopic models have been developed (Shalizi 2001; Lux and Marchesi 1999). For example, the return distribution presents pronounced tails that are thicker than those of the Gaussian distribution (Shalizi 2001; Lux 1996; Mantegna and Stanley 1995). Several models have been proposed that phenomenologically show fat-tail distributions induced by investors’ herding behavior (Banerjee 1993; Topol 1991). Furthermore, Cont and Bouchaud (2000), Orĺean (1995), Banerjee (1993), and Topol (1991) showed that market participants’ interactions through imitation can lead to large fluctuations in aggregate demand and heavy tails in the distribution of returns. This approach had been formalized as a power law exponent at the tail of the distribution with a smaller magnitude associated with stronger herding behavior in stock returns (Nirei et al. 2020; Gabaix et al. 2005; Plerou et al. 1999; Gopikrishnan et al. 1999), trading volumes (Gabaix et al. 2006; Gopikrishnan et al. 2000), and commodity returns (Joo et al. 2020), which have been empirically investigated. Another stream of literature theoretically explains the power law in firm size distribution (Ji et al. 2020; Luttmer 2007) and trading volume (Nirei et al. 2020). However, these studies are limited to providing a connection between the power law exponent and other external factors, such as the business cycles and economic uncertainty.

We contribute to literature by explaining the role of economic uncertainty as a bridge between business cycles and investors’ herding behavior. Specifically, we propose a parsimonious model that employs quantum mechanics as an intermediate step to obtain the final solution and justify the power law distribution in stock returns. We start with the Fokker–Planck (FP) equation to model the dynamics of stock return distribution and derive the Schrödinger equation for a particular external potential (Ahn et al. 2017). The form of the potential is postulated based on empirical evidence of the evolution of stock returns in the marketplace. The solution suggests the existence of a power law for the tail distribution of stock returns. This also predicts a positive association between business cycles and the power law exponent. Our model provides new insights into existing research that models stock prices using random walks (Bartiromo 2004; Ma et al. 2004), quantum oscillators (Ahn et al. 2017; Ye and Huang 2008), quantum wells (Pedram 2012; Zhang and Huang 2010), and quantum Brownian motions (Meng et al. 2016).

We provide further empirical evidence on whether herding behavior in stock returns is negatively associated with business cycles. Furthermore, business cycles, which are often used as proxies for economic growth, are closely related to economic uncertainty, whereby it is believed that recessions are accompanied by higher economic uncertainty (Bloom 2014). Moreover, greater economic uncertainty leads to higher levels of uncertainty in the stock market. With greater uncertainty in the stock market, investors are more likely to mimic others because increased information asymmetry leads to fewer investors having confidence in their valuations (Alhaj-Yaseen and Yau 2018; Park and Sabourian 2011; Devenow and Welch 1996), amplifying investors’ herding behavior in the tail. As hypothesized, we find that herding behavior is stronger during recessions than booms and that economic uncertainty causes significant herding behavior.

Data and methodology

Data description

Our sample includes 137 US firms that were continuously included in the Standard & Poor’s 500 (S&P 500) index from January 1992 to December 2021.Footnote 1 We exclude firms that either entered or exited the index during our sample period to avoid the influence of abnormal trading around entry or exit events (Chen et al. 2004; Lynch and Mendenhall 1997; Beneish and Whaley 1996; Harris and Gurel 1986; Shleifer 1986). We obtain the daily stock return data from the Center for Research in Security Prices with 1,031,914 firm-day observations for our sample firms. We normalize the return of each firm by subtracting its mean and dividing it by its standard deviation over the entire sample period to remove heterogeneity in stock return volatility among different stocks (Feng et al. 2012; Gabaix et al. 2003). Furthermore, we obtain yearly recession indicators from the National Bureau of Economic Research (NBER) and seasonally adjusted US real Gross Domestic Product (GDP) growth rates from the Federal Reserve Economic Data. As a proxy for economic uncertainty, we adopt Bloom’s (2014) definition of forecaster uncertainty as the median of forecasters’ subjective variances. It measures the annual average uncertainty of each forecaster. Data for the forecaster probability distribution of the GDP growth rate were obtained from the Survey of Professional Forecasters at the Federal Reserve Bank of Philadelphia.

Table 1 Descriptive statistics

Table 1 presents the descriptive statistics of the main variables of this study. The mean daily stock return was 0.060%. During our sample period (January 1992 to December 2021), four years were recessionary periods, including the Dot-com Crash, Global Financial Crisis, and Coronavirus disease (COVID-19) pandemic. The average annual US real GDP growth rate is 2.518%. The forecast uncertainty ranged between 0.290 and 0.538 with a sample mean of 0.418 and a standard deviation of 0.064, indicating a symmetric distribution around the mean.

This table summarizes the descriptive statistics of our sample data. The sample period is from January 1992 to December 2021. We use daily returns of S&P 500 constituent stocks. The annual GDP growth rate is the percentage change in US real GDP from the preceding year. Forecaster uncertainty is defined as the median of forecasters’ subjective variances according to Bloom (2009).

Power law exponent

The universal nature of the power law of returns is widely recognized in financial markets (Gabaix 2009). An implication of the presence of a power law in economics is the increased occurrence of extreme events compared to what would be expected in a Gaussian distribution. In other words, according to Gabaix et al. (2005), stock market crashes are not outliers of a power law; therefore, analyzing tail distributions can provide valuable insights into the regular behavior of the market within the tails and the occurrence of extreme events such as herding behavior (Gabaix 2016, 2009). The key to comprehending the stock market as a whole can be unlocked by striving to understand the power law phenomenon.

Typically, a power law distribution is defined by its counter cumulative density function, known as the survival function, which is characterized by the scaling exponent \({\upzeta }\). Our analysis of herding in stock returns \(\left( x \right)\) is based on the literature on power law distributions, expressed as

$${\text{P}}\left( {X \ge x} \right) = 1 - F\left( x \right) = kx^{ - \zeta } ,$$
(1)

where \({\text{P}}\left( {X \ge x} \right)\) is the probability that a random variable \(X\) is greater than \(x\), \(F\left( x \right)\) is the cumulative distribution function, \(k\) is a constant, and \({\upzeta }\) is the power law exponent. By taking the logarithms of both sides of Eq. (1), the following linear regression model is obtained:

$$\log {\text{P}}\left( {X > x} \right) = c - \zeta \cdot {\text{log}}x + \varepsilon ,$$
(2)

where \(c\) is a constant, and \(\varepsilon\) is the error term following the independent and identically distributed normal distribution. The power law exponent is normally obtained as the slope \(\zeta\) of the linear function. Due to the autocorrelation of residuals \(\varepsilon\), \(\zeta\) has an asymptotic standard error of \(\widehat{\zeta }{\left(n/2\right)}^{-1/2}\), for which \(n\) is the number of observations.

We fit the stock return data in our sample to a power law distribution for each year in the sample period. Specifically, for the power law exponents, we take the absolute value of the normalized daily stock return in each year to analyze extreme negative and positive returns together (Gabaix et al. 2003). We definethe tail of the distribution as a region with more than two standard deviations from the mean of the distribution (Gabaix et al. 2006; Plerou et al. 1999). We then estimated the power law exponent \(\zeta\) using Eq. (2).

Theory development

Quantum model

This subsection derives the power law distribution of stock returns from the Schrödinger equation, which originates from the FP equation. The model also predicts the relationship between the power law exponent and business cycles. Stock return is defined as

$$x = \ln p_{t} - \ln p_{t - \Delta t} ,$$

where \(p\) and \(x\) are the stock price and its log return, respectively. The dynamics of stock returns are then modeled using the following stochastic differential equation:

$$dx = v\left( {x,t} \right)dt + \sigma \left( {x,t} \right)dW_{t} ,$$

where \(v\left( {x,t} \right)\) denotes drift, \(\sigma \left( {x,t} \right)\) represents volatility, and \(W_{t}\) is the standard Wiener process.

We assume that the drift of stock returns arises from an external potential \(V\left( {x,t} \right)\) and define

$$v\left( {x,t} \right) = - \frac{{\partial V\left( {x,t} \right)}}{\partial x} = - V_{x} ,$$

which is analogous to classical kinetics. We further define the diffusion coefficient \(D\left( {x,t} \right)\) as

$$D\left( {x,t} \right) = \frac{1}{2}\sigma^{2} \left( {x.t} \right).$$

The probability density function of \(x\) is denoted as \(\rho \left( {x, t} \right)\). According to the FP equation, we have

$$\frac{\partial }{\partial t}\rho \left( {x,t} \right) = \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {D\left( {x,t} \right)\rho \left( {x,t} \right)} \right) + \frac{\partial }{\partial x}\left( {V_{x} \rho \left( {x,t} \right)} \right).$$
(3)

For simplicity, we assume that the diffusion coefficient is constant, that is, \(D\left( {x,t} \right) = D\).

Furthermore, \(\Psi \left( {x,t} \right)\) and a Hermitian operator \(\hat{H}\) are introduced as follows:

$$\begin{aligned} \Psi \left( {x,t} \right) & = \frac{{\rho \left( {x,t} \right)}}{{\sqrt {\rho_{s} \left( x \right)} }}, \\ \hat{L}\rho \left( {x,t} \right) & = - \sqrt {\rho_{s} \left( x \right)} \hat{H}\Psi \left( {x,t} \right), \\ \rho_{s} \left( x \right) & = \frac{1}{C}\exp \left( { - \frac{V\left( x \right)}{D}} \right), \\ \end{aligned}$$

where \(C\) is the normalization constant and \(\hat{H}\) is the Hermitian operator,

$$\begin{aligned} C & = \mathop \smallint \limits_{ - \infty }^{ + \infty } \exp \left( { - \frac{V\left( x \right)}{D}} \right)dx, \\ \hat{H} & = \frac{1}{2}V_{xx} + \frac{1}{4D}V_{x}^{2} - D\frac{{\partial^{2} }}{{\partial x^{2} }}. \\ \end{aligned}$$

Then, we define imaginary time \(\tau = - i\hslash t\) and a mass \(m = \frac{{\hslash ^{2} }}{2D}\) and Eq. (3) can be rearranged into the well-known Schrödinger equation,

$$i\hslash \frac{\partial }{\partial \tau }\Psi \left( {x, \tau } \right) = \hat{H}\Psi \left( {x, \tau } \right) \equiv \left( { - \frac{{\hslash ^{2} }}{2m}\frac{{\partial^{2} }}{{\partial x^{2} }} + U\left( x \right)} \right)\Psi \left( {x,\tau } \right),$$
(4)

where \(U\left( x \right)\) is the effective potential:

$$U\left( x \right) = - \frac{{V_{xx} }}{2} + \frac{{V_{x}^{2} }}{4D}.$$

We chose the functional form of the external potential \(V\left( x \right)\) based on empirical evidence. Some studies show a contrarian effect on stock markets worldwide (Shi and Zhou 2017; Clare et al. 2014; De Bondt and Thaler 1985). Relatively high or low stock returns revert, indicating a market force that always draws short-run fluctuations back to the long-run equilibrium. Thus, we define the potential as \(V\left( x \right) = \alpha \left| {x - a} \right|\). If stock returns deviate from the equilibrium return \(a\), the market force from the potential will draw back stock returns at the speed of \(\alpha\). Given \(V\left( x \right) = \alpha \left| {x - a} \right|\), we have

$$U\left( x \right) = - \alpha \delta \left( {x - a} \right) + \frac{{\alpha^{2} }}{4D},$$

where the extra drift \(\frac{{\alpha^{2} }}{4D}\) does not affect the wave function (Bracewell 2000). Following this, we can solve the time-independent Schrödinger equation which is given by:

$$E\psi \left( x \right) = \hat{H}\psi \left( x \right) = - \frac{{\hslash ^{2} }}{2m}\frac{{\partial^{2} }}{{\partial x^{2} }}\psi \left( x \right) - \alpha \delta \left( {x - a} \right)\psi \left( x \right).$$

The solution is well known with energy \(E = - \frac{{m\alpha^{2} }}{{2\hslash ^{2} }}\) (Griffiths 2005):

$$\psi \left( x \right) = \frac{{\sqrt {m\alpha } }}{\hslash }\exp \left( { - \frac{m\alpha }{{\hslash ^{2} }}} \right)\left| {x - a} \right|.$$

Hence, the general solution of Eq. (4) is

$$\Psi \left( {x, \tau } \right) = A\psi \left( x \right)e^{{ - \frac{iE\tau }{\hslash }}} = A\exp \left( { - \frac{m\alpha }{{\hslash ^{2} }}\left| {x - a} \right| + \frac{{m\alpha^{2} \tau }}{{2\hslash ^{3} }}i} \right).$$

Using \(\Psi \left( {x, \tau } \right)\), \(\rho_{s} \left( x \right)\), \(\tau = - i\hslash t\), and \(m = \frac{{\hslash ^{2} }}{2D}\), we obtain

$$\rho \left( {x, \tau } \right) = \sqrt {\rho_{s} \left( x \right)} {\text{A}}\psi \left( x \right)e^{ - Et} = A\sqrt {\frac{\alpha }{2D}} {\text{exp}}\left( { - \frac{\alpha }{D}\left| {x - \alpha } \right| + \frac{{\alpha^{2} t}}{4D}} \right),$$

where \(A\) is the normalization multiplier. After normalization, the final form of the solution is a Laplace distribution:

$$\rho \left( x \right) = \frac{\alpha }{2D}e^{{ - \frac{\alpha }{D}\left| {x - \alpha } \right|}} .$$
(5)

From Eq. (5), we obtain the tail distribution of log returns. We define the gross return as

$$Y = \frac{{p_{t} }}{{p_{t - \Delta t} }} = e^{x} .$$

In the right tail satisfying \(y > e^{a}\), we have

$$P\left( {Y \ge y} \right) = P\left( {x \ge {\text{ln}}y} \right) = \mathop \smallint \limits_{{{\text{ln}}y}}^{ + \infty } \frac{\alpha }{2D}e^{{ - \frac{\alpha }{D}\left| {x - \alpha } \right|}} dx \propto y^{{ - \frac{\alpha }{D}}} .$$

Hence, our result follows power law distribution in the tail, and the power law exponent is \(\frac{\alpha }{D}\).

The formula for the power law exponent could be useful for connecting herding behavior in stock returns to business cycles. Recessions are accompanied by economic uncertainty (Bloom 2014). Therefore, the market return moves slowly toward equilibrium (a smaller \(\alpha\)) and becomes more volatile (a larger \(D\)). Hence, recession leads to strong herding behavior, resulting in a smaller \(\alpha /D\). On the contrary, during booms, the market return reverts quickly toward equilibrium with a larger \(\alpha\) with less volatility, implying a smaller \(D\). Thus, a boom leads to a larger power law exponent and, thus, weak herding behavior. Therefore, our model predicts a positive association between business cycles and power law exponent.

Hypothese

Most studies document asymmetric market movements with respect to economic cycles. The literature on return volatility documents substantial volatility clusters during economic downturns (Choudhry et al. 2016; Corradi et al. 2013). Additionally, it is widely accepted that analysts’ forecasts are more dispersed during economic troughs than during peaks (Amiram et al. 2018; Hope and Kang 2005). Thus, our first testable hypothesis is as follows:

Hypothesis 1

The herding behavior in stock returns is stronger during recessions than in booms.

To test this hypothesis, we first calculated the power law exponents during booms and recessions and compared their magnitudes. To examine the relationship between business cycles and herding behavior further, we ran the following regression model:

$$\zeta_{t} = \alpha + \beta g_{t} + \varepsilon_{t} ,$$
(6)

where \({\zeta }_{t}\) and \({g}_{t}\) are the power law exponent and GDP growth rate, respectively, in year \(t\). A significantly positive \(\beta\) indicates that GDP growth rate has explanatory power for herding behavior, as hypothesized.

Herding assumes a certain degree of coordination between groups of agents. This coordination may arise in different ways, either because agents share the same information or follow the same rumor (Cont and Bouchaud 2000). Herding may be stronger when financial markets experience extreme uncertainty (Bouri et al. 2019). When information asymmetry is minimal, market participants do not necessarily need to observe or follow other participants’ transactions. However, with severe information asymmetry, traders are more inclined to imitate other participants to compensate for missing information through the behavior of their counterparts (Alhaj-Yaseen and Yau 2018; Park and Sabourian 2011; Devenow and Welch 1996). Moreover, extant literature has documented a significant relationship between economic growth and economic uncertainty. Specifically, a low economic growth rate is associated with high economic uncertainty (Bloom 2014). Therefore, we conjecture that economic uncertainty (higher volatility) is an intermediary linking the business cycle (lower GDP growth rate) and herding behavior (smaller power law exponent). Our second hypothesis is as follows:

Hypothesis 2

Economic uncertainty is the origin of counter-cyclical herding behavior in stock returns.

To examine whether economic uncertainty is the intermediary, we test the following models:

$$u_{t} = \alpha + \beta g_{t} + \varepsilon_{t} ,$$
(7)
$$\zeta_{t} = \alpha + \beta u_{t} + \varepsilon_{t} ,$$
(8)

As a robustness test, we employ the following models:

$$\zeta_{t} = \alpha + \beta D1_{t} + \gamma D2_{t} + \varepsilon_{t} ,$$
(9)
$$\zeta_{t} = \alpha + \beta g_{t} + \gamma D1_{t} + \delta D2_{t} + \varepsilon_{t} ,$$
(10)
$$\zeta_{t} = \alpha + \beta u_{t} + \gamma D1_{t} + \delta D2_{t} + \varepsilon_{t} ,$$
(11)

where \(u_{t}\) stands for economic uncertainty in time \(t\). In the first model, we run a regression of economic uncertainty on the GDP growth rate. In the second model, we use economic uncertainty as an explanatory variable for the power law exponent, that is, as a proxy for herding behavior. For the remaining models, we test whether business cycles and herding behavior are connected through economic uncertainty by examining factor loading on the dummy variable: (i) \(D1_{t} = 1\) when \(g_{t} > \overline{g}\) and \(u_{t} < \overline{u}\), and otherwise equals zero; and (ii) \(D2_{t} = 1\) when \(g_{t} < \overline{g}\) and \(u_{t} > \overline{u}\), and otherwise equals zero.

Empirical results

The annual power law exponent is shown in Fig. 1. The Gray-shaded areas indicate years of economic recessions based on the NBER recession indicator. As evident by the figure, the power law exponents are generally smaller in recession years than in non-recession years.

Fig. 1
figure 1

The power law exponent, economic uncertainty, and business cycle. The solid line is the annual power law exponent calculated by aggregating daily normalized S&P 500 stock returns, and the dashed line is the annual forecaster uncertainty according to Bloom (2009). The shaded areas indicate recession periods identified by the NBER recession indicator

We begin our analysis by confirming prior findings in literature. Overall, most studies on the power law distribution of stock returns, such as those of Feng et al. (2012) and Gopikrishnan et al. (1999), report a range of power law exponents between two and four. Our estimated power law exponent was approximately 3.138 with an \(R^{2}\) of 97.75% for the entire sample period, which is consistent with the findings of prior studies. It has been suggested that the degree of herding in the tail is stronger when the power law exponent is smaller in magnitude (Feng et al. 2012; Cont and Bouchaud 2000; Eguiluz and Zimmermann 2000).

Further we examined the link between herding behavior and business cycles by comparing power law exponents during recessions with those in booms. For this purpose, power law exponents are divided into two groups, booms and recessions, according to the NBER recession indicator. As is evident in Table 2, the power law exponents are significantly larger during booms than during recessions. The difference in power law exponents during booms and recessions is significant at the 1% level for the mean based on the t-test and the median based on the Wilcoxon rank-sum test. As a smaller power law exponent indicates stronger herding, we can firmly conclude that there is stronger herding in stock returns during recessions than during booms.

Table 2 Power law exponents in booms and recessions

We then estimated the regression model from Eq. (6) using power law exponents and GDP growth rates. The results are shown in Model (1) of Table 3, where we use the heteroskedasticity and autocorrelation consistent estimator for the standard error (Newey and West 1987). The GDP growth rate is significantly and positively associated with the power law exponent at the 1% significance level. If the GDP growth rate decreases by one percentage point, the corresponding power law exponent drops by 0.159, intensifying herding behavior in stock returns.

Table 3 GDP growth rate, economic uncertainty, and the power law exponent

Additionally, we tested whether economic uncertainty links business cycles and herding behavior. The results are summarized in Table 3. Models (2) and (3) present the estimation results of Eqs. (7) and (8), respectively. In Model (2), the GDP growth rate is significant and negatively associated with forecaster uncertainty, which is our proxy for economic uncertainty. The adjusted \({R}^{2}\) is approximately 30%, indicating that GDP growth rate explains a significant portion of economic uncertainty. In Model (3), forecaster uncertainty is significant and negatively associated with the power law exponent. Combining the results from Models (2) and (3), economic uncertainty appears to be the link between the GDP growth rate and herding in stock returns.

In Models (4)–(6), we present the results from the estimations of Eqs. (9)–(11). In particular, the factor loading on the dummy variable indicates the importance of economic uncertainty on top of business cycles in explaining herding behavior in stock returns. In Models (5) and (6), Dummy2 (\({D2}_{t}=1\) when \({g}_{t}<\overline{g }\) and \({u}_{t}>\overline{u }\), and otherwise equals zero) is highly significant. Accordingly, we conclude that GDP growth rate explains herding behavior in stock returns through economic uncertainty. In other words, rising uncertainty accompanied by low economic growth significantly accelerates herding behavior in stock returns.Footnote 2

Conclusion

This study examined the relationship between business cycles and herding behavior in the US stock market. The recession indicator and GDP growth rate are used as proxies for business cycles, whereas herding behavior is represented by the power law exponent in stock returns. First, we propose a theoretical model of stock returns employing quantum mechanics. Our model predicts a positive association between business cycles and the power law exponent and economic uncertainty links business cycles and herding behavior. We then tested these predictions using empirical data. We find evidence of stronger herding during recessions than booms. Specifically, the GDP growth rate can significantly explain the herding behavior in stock returns. Using forecaster uncertainty as a proxy for economic uncertainty, we confirm that economic uncertainty links business cycles with herding behavior in stock returns. Greater economic uncertainty is accompanied by a recession, which leads to increased information asymmetry, for example, higher dispersion in analysts' forecasts. Accordingly, investors are more likely to mimic others because of lower confidence in their valuations. Finally, information asymmetry leads to greater volatility in firms’ activities and drives more extreme stock returns, resulting in smaller power law exponents and implying stronger herding behavior in stock returns.

The findings of this study provide a clear link between herding behavior and business cycles. The results underscore the importance of monitoring herding activities during periods of increased uncertainty accompanied by low economic growth. At the macro level, an increase in policy uncertainty can lead to a decline in economic growth (Baker et al. 2016). Therefore, policymakers should consider how policy uncertainty influences investors’ decision-making (Ahn et al. 2021) in the financial market. For individual investors, our results provide a way to formulate hedging strategies to mitigate downside risk in their investment portfolios during a recession. Our empirical setting is designed to confirm the results of the theoretical (toy) model. Future studies can extend our empirical setting to show (i) the robustness of our main results by adding various control variables and (ii) the robustness of the transmission channel.

Availability of data and materials

We use data from three sources: Center for Research in Stock Prices (CRSP), National Bureau of Economic Research (NBER), Federal Reserve Economic Data (FRED), and Federal Reserve Bank of Philadelphia. Restrictions apply to the availability of data from CRSP, which were used under license for this study. Data are available at https://www.crsp.org/ with license. Data from NBER are available to public at https://www.nber.org/research/business-cycle-dating. Data from FRED are available at https://fred.stlouisfed.org/. Data on forecaster uncertainty from Federal Reserve Bank of Philadelphia is available at https://www.philadelphiafed.org/surveys-and-data/real-time-data-research/survey-ofprofessionalforecasters#:~:text=The%20Survey%20of%20Professional%20Forecasters,over%20the%20survey%20in%201990.

Notes

  1. To check the representativeness of 137 US firms, we calculate the correlation coefficient between the S&P 500 index and our sample index. The composite sample index is derived from the average value of all ticker price data. As a result, the correlation coefficient is close to 1 (0.947 at the 1% significant level); this result provides supporting evidence for the similarity between the S&P 500 and our index. We subsequently investigate the market cap of our sample companies to that of the S&P 500. Throughout the test period, our sample’s aggregate market capitalization surpasses half of the total market capitalization of the benchmark on average (see Table A1 in the Appendix).

  2. We consider an inflation rate and federal funds rate as a control variable and still obtain results consistent with Table 3 (see Table 5 in the Appendix). We also deal with media coverage as a control variable using the “Economic Policy Uncertainty Index,” which uses news coverage about policy-related economic uncertainty (Baker et al. 2016). We find that it has a strong correlation coefficient with our uncertainty index, the forecaster uncertainty index; therefore, we do not include this index as a control variable (see Table 6 in the Appendix).

Abbreviations

FP:

Fokker–Planck

GDP:

Gross domestic product

NBER:

National Bureau of Economic Research

S&P 500:

Standard and Poor’s 500

VIF:

Variance inflation factors

References

  • Ahn K, Choi MY, Dai B, Sohn S, Yang B (2017) Modeling stock return distributions with a quantum harmonic oscillator. Europhys Lett 120:38003

    Article  Google Scholar 

  • Ahn K, Chu Z, Lee D (2021) Effects of renewable energy use in the energy mix on social welfare. Energy Econ 96:105174

    Article  Google Scholar 

  • Alhaj-Yaseen YS, Yau SK (2018) Herding tendency among investors with heterogeneous information: evidence from China’s equity markets. J Multinatl Financ Manag 47–48:60–75

    Article  Google Scholar 

  • Amiram D, Landsman WR, Owens EL, Stubben SR (2018) How are analysts’ forecasts affected by high uncertainty? J Bus Financ Acc 45:295–318

    Article  Google Scholar 

  • Baker SR, Bloom N, Davis SJ (2016) Measuring economic policy uncertainty. Quart J Econ 131:1593–1636

    Article  Google Scholar 

  • Balcılar M, Demirer R, Ulussever T (2017) Does speculation in the oil market drive investor herding in emerging stock markets? Energy Econ 65:50–63

    Article  Google Scholar 

  • Banerjee AV (1993) The economics of rumours. Rev Econ Stud 60:309–327

    Article  Google Scholar 

  • Bartiromo R (2004) Dynamics of stock prices. Phys Rev E 69:067108

    Article  Google Scholar 

  • Beneish MD, Whaley RE (1996) An anatomy of the “S&P Game”: the effects of changing the rules. J Finance 51:1909–1930

    Google Scholar 

  • Bloom N (2009) The impact of uncertainty shocks. Econometrica 77:623–685

    Article  Google Scholar 

  • Bloom N (2014) Fluctuations in uncertainty. J Econ Perspect 28:153–176

    Article  Google Scholar 

  • Bouri E, Gupta R, Roubaud D (2019) Herding behaviour in cryptocurrencies. Financ Res Lett 29:216–221

    Article  Google Scholar 

  • Bouri E, Demirer R, Gupta R, Nel J (2021) COVID-19 pandemic and investor herding in international stock markets. Risks 9:168

    Article  Google Scholar 

  • Boyer BH, Kumagai T, Yuan K (2006) How do crises spread? Evidence from accessible and inaccessible stock indices. J Finance 61:957–1003

    Article  Google Scholar 

  • Bracewell R (2000) The Fourier transform and its applications. McGraw-Hill, New York

    Google Scholar 

  • Chang EC, Cheng JW, Khorana A (2000) An examination of herd behavior in equity markets: an international perspective. J Bank Finance 24:1651–1679

    Article  Google Scholar 

  • Chen H, Noronha G, Singal V (2004) The price response to S&P 500 index additions and deletions: Evidence of asymmetry and a new explanation. J Finance 59:1901–1930

    Article  Google Scholar 

  • Chiang TC, Jeon BN, Li H (2007) Dynamic correlation analysis of financial contagion: Evidence from Asian markets. J Int Money Finance 26:1206–1228

    Article  Google Scholar 

  • Chiang TC, Zheng D (2010) An empirical analysis of herd behavior in global stock markets. J Bank Finance 34:1911–1921

    Article  Google Scholar 

  • Choudhry T, Papadimitriou FI, Shabi S (2016) Stock market volatility and business cycle: Evidence from linear and nonlinear causality tests. J Bank Finance 66:89–101

    Article  Google Scholar 

  • Christie WG, Huang RD (1995) Following the pied piper: Do individual returns herd around the market? Financ Anal J 51:31–37

    Article  Google Scholar 

  • Clare A, Seaton J, Smith PN, Thomas S (2014) Trend following, risk parity and momentum in commodity futures. Int Rev Financ Anal 31:1–12

    Article  Google Scholar 

  • Cont R, Bouchaud JP (2000) Herd behavior and aggregate fluctuations in financial markets. Macroecon Dyn 4:170–196

    Article  Google Scholar 

  • Corradi V, Distaso W, Mele A (2013) Macroeconomic determinants of stock volatility and volatility premiums. J Monet Econ 60:203–220

    Article  Google Scholar 

  • De Bondt WFM, Thaler R (1985) Does the stock market overreact? J Finance 40:793–805

    Article  Google Scholar 

  • Demirer R, Leggio KB, Lien D (2019) Herding and flash events: evidence from the 2010 Flash Crash. Financ Res Lett 31:476–479

    Article  Google Scholar 

  • Devenow A, Welch I (1996) Rational herding in financial economics. Eur Econ Rev 40:603–615

    Article  Google Scholar 

  • Eguıluz VM, Zimmermann MG (2000) Transmission of information and herd behavior: an application to financial markets. Phys Rev Lett 85:5659–5662

    Article  Google Scholar 

  • Feng L, Li B, Podobnik B, Preis T, Stanley HE (2012) Linking agent-based models and stochastic models of financial markets. Proc Natl Acad Sci USA 109:8388–8393

    Article  Google Scholar 

  • Gabaix X, Gopikrishnan P, Plerou V, Stanley HE (2005) Are stock market crashes outliers? Working Paper, Massachusetts Institute of Technology

  • Gabaix X (2009) Power laws in economics and finance. Annu Rev Econ 1:255–294

    Article  Google Scholar 

  • Gabaix X, Gopikrishnan P, Plerou V, Stanley HE (2003) A theory of power-law distributions in financial market fluctuations. Nature 423:267–270

    Article  Google Scholar 

  • Gabaix X, Gopikrishnan P, Plerou V, Stanley HE (2006) Institutional investors and stock market volatility. Quart J Econ 121:461–504

    Article  Google Scholar 

  • Gabaix X (2016) Power laws in economics: an introduction. J Econ Perspect 30:185–206

    Article  Google Scholar 

  • Galariotis EC, Rong W, Spyrou SI (2015) Herding on fundamental information: a comparative study. J Bank Finance 50:589–598

    Article  Google Scholar 

  • Gopikrishnan P, Plerou V, Nunes Amaral LAN, Meyer M, Stanley HE (1999) Scaling of the distribution of fluctuations of financial market indices. Phys Rev E 60:5305–5316

    Article  Google Scholar 

  • Gopikrishnan P, Plerou V, Gabaix X, Stanley HE (2000) Statistical properties of share volume traded in financial markets. Phys Rev E 62:R4493

    Article  Google Scholar 

  • Griffiths DJ (2005) Introduction to quantum mechanics. Pearson Education International

  • Harris L, Gurel E (1986) Price and volume effects associated with changes in the S&P 500 list: new evidence for the existence of price pressures. J Finance 41:815–829

    Article  Google Scholar 

  • Hiltzik M (2014) End of an era: the NYSE floor isn’t even good for PR photos anymore. Los Angeles Times

  • Hope OK, Kang T (2005) The association between macroeconomic uncertainty and analysts’ forecast accuracy. J Int Account Res 4:23–38

    Article  Google Scholar 

  • Ji G, Dai B, Park SP, Ahn K (2020) The origin of collective phenomena in firm sizes. Chaos Soliton Fract 136:109818

    Article  Google Scholar 

  • Joo K, Suh JH, Lee D, Ahn K (2020) Impact of the global financial crisis on the crude oil market. Energ Strat Rev 30:100516

    Article  Google Scholar 

  • Kumar A, Badhani KN, Bouri E, Saeed T (2021) Herding behavior in the commodity markets of the Asia-Pacific region. Financ Res Lett 41:101813

    Article  Google Scholar 

  • Luttmer EG (2007) Selection, growth, and the size distribution of firms. Quart J Econ 122:1103–1144

    Article  Google Scholar 

  • Lux T (1996) The stable Paretian hypothesis and the frequency of large returns: An examination of major German stocks. Appl Financ Econ 6:463–475

    Article  Google Scholar 

  • Lux T, Marchesi M (1999) Scaling and criticality in a stochastic multi-agent model of a financial market. Nature 397:498–500

    Article  Google Scholar 

  • Lynch AW, Mendenhall RR (1997) New evidence on stock price effects associated with changes in the S&P 500 index. J Bus 70:351–383

    Article  Google Scholar 

  • Ma WJ, Hu CK, Amritkar RE (2004) Stochastic dynamical model for stock-stock correlations. Phys Rev E 70:026101

    Article  Google Scholar 

  • Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376:46–49

    Article  Google Scholar 

  • Meng X, Zhang JW, Guo H (2016) Quantum Brownian motion model for the stock market. Physica A 452:281–288

    Article  Google Scholar 

  • Newey WK, West KD (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708

    Article  Google Scholar 

  • Nirei M, Stachurski J, Watanabe T (2020) Trade clustering and power laws in financial markets. Theor Econ 15:1365–1398

    Article  Google Scholar 

  • Orĺean A (1995) Bayesian interactions and collective dynamics of opinion: herd behavior and mimetic contagion. J Econ Behav Organ 28:257–274

    Article  Google Scholar 

  • Park A, Sabourian H (2011) Herding and contrarian behavior in financial markets. Econometrica 79:973–1026

    Article  Google Scholar 

  • Pedram P (2012) The minimal length uncertainty and the quantum model for the stock market. Physica A 391:2100–2105

    Article  Google Scholar 

  • Plerou V, Gopikrishnan P, Nunes Amaral LAN, Meyer M, Stanley HE (1999) Scaling of the distribution of price fluctuations of individual companies. Phys Rev E 60:6519–6529

    Article  Google Scholar 

  • Shalizi C (2001) An introduction to econophysics: correlations and complexity in finance. Quant Finance 1:391–392

    Article  Google Scholar 

  • Shleifer A (1986) Do demand curves for stocks slope down? J Finance 41:579–590

    Article  Google Scholar 

  • Shi HL, Zhou WX (2017) Time series momentum and contrarian effects in the Chinese stock market. Physica A 483:309–318

    Article  Google Scholar 

  • Sibande X, Gupta R, Demirer R, Bouri E (2021) Investor sentiment and (anti) herding in the currency market: evidence from Twitter feed data. J Behav Financ 24:56–72

    Article  Google Scholar 

  • Topol R (1991) Bubbles and volatility of stock prices: effect of mimetic contagion. Econ J 101:786–800

    Article  Google Scholar 

  • Ye C, Huang JP (2008) Non-classical oscillator model for persistent fluctuations in stock markets. Physica A 387:1255–1263

    Article  Google Scholar 

  • Zhang C, Huang L (2010) A quantum model for the stock market. Physica A 389:5769–5775

    Article  Google Scholar 

Download references

Acknowledgments

Not applicable.

Funding

This work was supported by the National Research Foundation of Korea grant funded by the Korean government (No. 2022R1A2C100425811, Kwangwon Ahn).

Author information

Authors and Affiliations

Authors

Contributions

KA: Supervision, Writing the manuscript, and Reviewing. LC: Data collection, Writing the manuscript, and Data analysis. HJ: Data collection, Writing the manuscript, and Data analysis. DSK: Supervision, Writing the manuscript, and Reviewing.

Corresponding author

Correspondence to Daniel Sungyeon Kim.

Ethics declarations

Competing interests

The authors declare no competing financial interests nor competing non-financial interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See the Tables 4, 5, 6 and 7.

Table 4 Market capitalization
Table 5 GDP growth rate, economic uncertainty, and power law exponent with control variables (inflation and federal funds rates)
Table 6 Correlation between news-based policy uncertainty index and forecaster uncertainty
Table 7 VIF tests

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahn, K., Cong, L., Jang, H. et al. Business cycle and herding behavior in stock returns: theory and evidence. Financ Innov 10, 6 (2024). https://doi.org/10.1186/s40854-023-00540-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s40854-023-00540-z

Keywords