Financial information and diverging beliefs

Abstract

Standard Bayesians’ beliefs converge when they receive the same piece of new information. However, when agents initially disagree and have uncertainty about the precision of a signal, their disagreement might instead increase, despite receiving the same information. We demonstrate that this divergence of beliefs leads to a unimodal effect of the absolute surprise in the signal on trading volume. We show that this prediction is consistent with the empirical evidence using trading volume around earnings announcements of U.S. firms. We find evidence of elevated volume following moderate surprises and depressed volume following more extreme surprises, a pattern that is more pronounced when investors hold more distant prior beliefs and are more uncertain about earnings’ precision. The evidence is consistent with the model where investors disagree about stocks’ expected returns and do not know the precision of earnings as a signal about the firm’s value.

1 Introduction

Information plays a key role in financial markets. Conventional wisdom in the finance and accounting literature suggests that, when investors receive a common signal, their beliefs converge. For example, for a signal in between two investors’ priors, an optimistic investor revises her belief downward while a pessimistic investor updates upward. Despite this perception, there exists ample evidence (e.g., Gentzkow and Shapiro 2006; Filipowicz et al. 2018; Fryer et al. 2019; Kartik et al. 2021) that individuals often reject information that does not conform to their priors, believing that the news is not informative. In this case, new information can actually cause their beliefs to diverge more. We theoretically and empirically investigate the effect of information on belief divergence. First, we develop a theoretical model of investors in a firm and show that, in a perfectly rational setting, when investors disagree about the firm’s value and are uncertain about the precision of a signal about that value, investors’ beliefs can diverge more after they observe the signal. Because beliefs cannot be observed in the data directly, we derive predictions about an observable variable – trading volume. We choose it as our research setting because trading volume is "a product of the extent to which investors hold diverse opinions ... and the extent to which these opinions change on average" at the time when new information arrives (Verrecchia 2001), in contrast to the price change, which is a function only of the average evolution of investors’ beliefs. Finally, we empirically test the predictions from our model using trading volume around quarterly earnings disclosures of publicly traded U.S. firms.

In our model, investors have different expectations about the firm’s cash flow, are uncertain about the precision of a signal about the firm, and trade both before and after the signal is disclosed. We first demonstrate why difference-of-beliefs and signal-precision uncertainty are the two key features that explain belief divergence after investors receive the same signal. When investors only disagree about the value of the firm but know the precision of the signal about the firm, their beliefs strictly converge following the disclosure. When investors initially agree but do not know the signal’s precision, no signal can create a belief divergence that did not exist before the signal was realized. However, with signal-precision uncertainty and difference-of-beliefs together, investors’ beliefs can be driven further away by certain realizations of the disclosed signal.

Investors’ beliefs can diverge after investors receive the same signal for the following reason. An investor who is uncertain about the signal’s precision rationally assigns a higher precision to signals that are closer to her prior, or she believes that information that is consistent with her view of the world is more credible and thus updates her belief more strongly based on that information. Because different investors have different priors, they assign different levels of precision to different signals. As a result, investors’ (posterior) beliefs diverge more after some realizations of the signal. This feature makes the model realistic: in everyday life, we often observe how people, even if they are professionals, may disagree more after they receive the same piece of information.¹

Because investors’ beliefs in financial markets are unobservable, to test our model, we derive predictions about an observable statistic – trading volume. We numerically demonstrate that the divergence of beliefs after a signal is released results in trading volume that is an M-shaped function of the signal itself and unimodal in the absolute value of its information content (or "surprise"). In other words, trading volume, at first, increases with a signal that produces a moderate surprise but then decreases as the signal generates a more extreme surprise. This M-shaped pattern is more pronounced when investors have greater prior disagreement and are more uncertain about the signal’s precision.

We next assess the validity of our model by developing several empirical tests of its implications based on firms’ quarterly earnings disclosures and systematic patterns in the trading volume surrounding the disclosures. We formulate and test three predictions: (1) trading volume increases for intermediate levels of earnings surprise and dampens for extreme levels, (2) the M-shape of trading volume is more pronounced under higher prior investor disagreement, and (3) the M-shape of trading volume is more pronounced under high signal-precision uncertainty. To test our first prediction, we analyze nonparametric and parametric shapes of the trading volume. We show that trading volume around earnings announcements indeed weakens for extreme levels of earnings surprises. If we impose a polynomial shape on trading volume, we find that trading volume as a function of the earnings surprise is most closely fitted by fourth-order polynomial regression. The estimated parameters exhibit a clear M-shape. We also estimate quantile regressions and show that trading volume increases for moderate absolute surprise levels but does not increase for the upper 5% of surprises.

To test our second prediction, we use a measure of investor disagreement based on the StockTwits platform developed by Cookson and Niessner (2020) and Cookson and Niessner (2023). We find support for our theory: the M-shape of trading volume is indeed more pronounced if investors disagree more about a firm’s stock prior to the firm’s earnings announcement.

To test our final prediction, we develop a new measure of earnings-announcement-precision uncertainty that allows us to separate precision uncertainty from the overall market uncertainty. Namely, we measure earnings-announcement precision uncertainty as residuals from the regression of analyst forecast spread on the Chicago Board Options Exchange (CBOE) Volatility Index (VIX). Conditional on the VIX picking up fundamental uncertainty, the residual component of analyst forecast spread is an estimate of the market’s uncertainty about the precision of the reported earnings number. We verify our measure by showing that an S-shaped earnings response coefficient (ERC)² is more pronounced for observations with high earnings-precision uncertainty. As predicted by our model, trading volume gradually becomes more M-shaped when earnings-precision uncertainty increases.

We rely on and aim to extend multiple streams of literature. First, we contribute to the broad literature in economics and psychology that demonstrates the evidence and studies the implications of confirmation bias (Gentzkow and Shapiro 2006; Filipowicz et al. 2018; Fryer et al. 2019; Kartik et al. 2021; Martel and Schneemeier 2021). We demonstrate that this behavior is present in financial markets and leads to unusual patterns in trading volume, which are more pronounced as individuals become more uncertain about the precision of the information they receive. We also propose a way to measure the degree of this signal-precision uncertainty.

Researchers have proposed several explanations for why investors’ views diverge after they receive the same information (e.g., Kondor 2012; Atmaz and Basak 2018; Banerjee and Kremer 2010; Bordalo et al. 2021). Atmaz and Basak 2018 provide a dynamic model with a continuum of investors that have different beliefs. As in our paper, the authors obtain the M-shaped trading volume as a function of information surprise when there are two types of investors. In their paper, the M-shape arises as a result of assumptions about the distribution of types and evolution of news in the stock market. In our paper, investors’ different perceived precision arises endogenously because investors perceive signals that are closer to their priors as more precise. Our second set of tests provides evidence that precision uncertainty plays a role in the M-shaped trading volume. Banerjee and Kremer (2010) develop a difference-of-beliefs model, where investors exogenously disagree about their interpretations of a public signal. This gives rise to "belief-convergence" trading, when investors’ beliefs converge after a prior disagreement, and "idiosyncratic" trading, when investors disagree on the interpretation of new information. Banerjee et al. (2024) demonstrate how empirically descriptive deviations from the rational expectations framework arise endogenously when investors use wishful thinking – choose subjective beliefs to make themselves happier about the future. According to Bordalo et al. (2021), investors overreact to positive news if they have observed stock growth in the past. Our paper differs from Bordalo et al. (2021) and Banerjee and Kremer (2010) in that, in our model, the differential reaction to signals arises endogenously, as opposed to being exogenously imposed. In our model, investors only underreact (namely when they observe very surprising signals) and never overreact. The greatest divergence of beliefs occurs when one type of investor underreacts and the other type reacts as if there were no signal-precision uncertainty. Another study that resembles ours is the work of Jia et al. (2017), who provide evidence that market segmentation may increase after an analyst recommendation because of the social connection between analysts and local investors: local investors react more strongly to local analysts’ recommendations than foreign investors. If local investors’ and local analysts’ priors are closer or if local investors believe that local analysts are more precise than foreign analysts, the mechanism that we propose would also explain their evidence of increased market segmentation after analyst recommendations. Finally, Kondor (2012) theoretically shows and Gallo (2017) empirically confirms that investors’ beliefs about firm fundamentals may converge but their beliefs about the firm’s price may diverge at the same time. The mechanism is higher-order beliefs: short-horizon investors on the market combine their private information with an informative public announcement and generate even greater disagreement about the short-term stock price. Our main contribution to this literature is to show, in a perfectly rational framework, how precision uncertainty interacts with the divergence of beliefs and to provide empirical evidence that US investors are uncertain about the precision of disclosed earnings.

Our paper extends the theoretical literature on the trading volume effects of public signals (e.g., Karpoff 1986; Kandel and Pearson 1995; Kim and Verrecchia 1991; Kondor 2012; Banerjee 2011). Early work in this area suggests that trading volume monotonically increases with a signal’s surprise, which is attributable to either differential information or information precision among traders before the signal’s release (Kim and Verrecchia 1991). Subsequent studies introduced differences-in-beliefs models, and they typically provide inferences that generally resemble those of their predecessors (see Bamber et al. 2011 for a review). Kandel and Pearson (1995) introduce investors’ disagreement about the function used to update beliefs about the value of a stock and show how this framework explains empirically documented increased trading volume around quarterly earnings announcements. To our knowledge, our study is the first to incorporate uncertainty about public signals’ precision and show that it can result in a nonmonotonic pattern between the magnitude of a signal’s surprise and subsequent trading volume.

Finally, our study contributes to the parallel literature that studies trading volume, its determinants, and its functional form. Bamber (1987) finds that trading volume increases with the magnitude of an earnings surprise, and Choi (2019) shows that this relationship is amplified when markets are more volatile. We build on the latter, which studies uncertainty about the value of the signal (i.e., the first moment) by instead focusing on uncertainty about its precision (i.e., the second moment). Our paper also logically extends work by Bamber et al. (1997), Irvine and Giannini (2012), and Booker et al. (2023), who show that abnormal trading volume exhibits a positive relationship with changes in beliefs. Another related study by Giannini et al. (2019) suggests that both convergence and divergence of beliefs lead to increased trading volume around earnings announcements. Our paper differs from these by showing – both theoretically and empirically – a previously unknown effect of precision uncertainty: namely that trading volume exhibits different behavior across environments with low and high signal-precision uncertainty. On the functional form of trading volume, several studies document a U-shaped pattern of trading volume throughout the day (e.g., Jain and Joh 1988; Foster and Viswanathan 1993). Atiase and Bamber 1994 empirically confirm the prediction of Kim and Verrecchia 1991 that trading volume around earnings announcements increases with price reaction and the level of pre-announcement information asymmetry. We contribute by documenting a robust M-shape of trading volume around earnings announcements and providing cross-sectional evidence on its distinctiveness.

The rest of our paper is as follows. Section 2 develops a new model that incorporates both difference-of-beliefs and signal-precision-uncertainty and shows that the introduction of the latter can cause investors’ beliefs to become more divergent after they receive the same signal. We then characterize the resulting equilibrium and derive empirical predictions for trading volume patterns. Section 3 describes our research design and develops multiple empirical tests of the main predictions implied by our model. Section 4 concludes.

2 The model

We first introduce the assumptions and timeline of the model that features investors with different beliefs and uncertainty about the precision of the signal that the investors receive. Next we present two benchmark models: with only difference-of-beliefs and with only signal-precision uncertainty. These benchmarks help us understand how the two main forces of the model work and why both of them are needed to explain the divergence of investors’ beliefs after the investors receive the signal. Finally, we discuss the full model and its empirical implications.

2.1 Model setup

A continuum of investors competes for shares in two assets. The first asset is risk free, has infinite supply, and a rate of return that is normalized to zero without loss of generality. The second asset — the shares of a firm — is risky and yields a random return of $\tilde{x}$. The entire firm is sold to new investors in the first trading period; supply of the risky asset at date $t=1$ is 1. We assume that there are two types of investors, each type aware of the other type, and denote the type of investor by $i \in \{1,2\}$. A fraction $\lambda _1$ of investors are of type 1 and a fraction $\lambda _2=1-\lambda _1$ are of type 2. We assume that investors are risk-averse and have mean-variance utility over their terminal wealth:

$$\begin{aligned} U_i=E[W_i]-\frac{1}{2}r_iVar[W_i], \end{aligned}$$

(2.1)

where $W_i$ and $r_i$ are investor i’s terminal wealth and coefficient of risk aversion, respectively.³ Investors are initially endowed with wealth $W_{i0}=0$.

There are three periods:

$t=1$.:

Pre-announcement Period. Investors trade in anticipation of the disclosure at $t=2$.

$t=2$.:

Post-announcement Period. The signal, $\tilde{y}=\tilde{x}+\tilde{u}$, is disclosed and investors trade for the second time.

$t=3$.:

Realization Period. The risky asset’s return, x, is realized and investors consume their terminal wealth, which is given by:

$$\begin{aligned} W_{i3}=d_{i2}x+q_{i2}, \end{aligned}$$

(2.2)

where $d_{i2}$ and $q_{i2}$ are the amounts of risky and riskless assets held in $t=2$, respectively.

We assume that investors have heterogeneous prior beliefs about the risky asset’s expected return:

$$\tilde{x}\sim N\left( m_i,\frac{1}{\nu }\right) ,$$

where $m_{i}$ is investor i’s expected return of the risky asset and $\frac{1}{\nu }$ is the variance of the return. Furthermore, we assume that $\tilde{u}$, the noise term of the signal $\tilde{y}=\tilde{x}+\tilde{u}$, is independent of $\tilde{x}$ and normally distributed with zero mean and unknown precision. This implies that investors have heterogeneous prior beliefs about the signal:

$$\tilde{y}\sim N\left( m_i,\frac{1}{\tilde{w}}\right) , $$

where $\tilde{w}$ is a random variable. Following (Subramanyam 1996), we assume that the true signal-precision, w, follows a truncated gamma distribution with support $[0, \nu ]$ and parameters $\alpha$ and $\beta$.⁴ Investors are uncertain about both the mean of the return, $\tilde{x}$, and the precision of the signal about the return, $\tilde{w}$, and they use the only piece of information – the realization of the signal, y, – to update their beliefs about both $\tilde{x}$ and $\tilde{w}$.

2.2 Benchmark cases: difference-of-beliefs or signal-precision uncertainty

In this section, we first present the case where investors only disagree about the risky asset’s return, $\tilde{x}$, but know the precision of the signal, w, and agree on what this precision is. We will see that, in this setup, after signal y is revealed, all investors update their beliefs about the risky asset’s return and trade. However, different investors’ beliefs always converge following the disclosure of the signal.

Proposition 1

When investors have different prior beliefs about the risky asset’s return and know and agree upon the precision of the signal about the asset’s return, the investors’ beliefs will always weakly converge after the disclosure of the signal: $|E_1[x]-E_2[x]|\ge |E_1[x|y]-E_2[x|y]|$.

The proposition above suggests that simple disagreement about first moments is not enough to explain the real-world pattern of beliefs divergence even when economic agents receive the same piece of information. When agents disagree about the expected asset return and agree on the precision of the signal about the assets that they receive, the agents’ beliefs always converge following the release of the signal.

Next we analyze the case where investors agree on the first moment of the distribution of the asset’s return but are uncertain about the precision of the signal about the asset’s return that they receive. In this benchmark, all investors will always hold the same beliefs, implying no disagreement and no trading volume either before or after the disclosure of the signal y, consistent with the traditional framework with homogeneously informed investors (e.g., Kim and Verrecchia 1991).

Proposition 2

When investors agree about the risky asset’s return before the disclosure of the signal, are uncertain about the precision of the signal about the asset’s return, and agree upon the mean of the precision of the signal, they will agree about the asset’s return after the disclosure of the signal.

The two benchmarks considered in this section imply that neither different beliefs nor uncertainty about signal-precision alone can generate the phenomenon that we seek to describe: divergence of economic agents’ beliefs after they receive the same signal. When the agents disagree about the mean of the return but receive the signal with known precision, they move their beliefs closer to each other. When the agents do not disagree about the mean of the return, uncertainty about the second moment of the signal cannot make the agents disagree about the first moment. In the next section, we analyze a model where difference-of-beliefs is combined with signal-precision uncertainty.

2.3 Full model: convergence and divergence of beliefs

This section describes how investors’ beliefs evolve in the model where the investors (1) disagree about the expected return of the risky asset, x, and (2) are uncertain how precise is the signal about this asset’s return.

When an investor i is uncertain about both the asset’s return, $\tilde{x}$, and the precision of the signal about the return, $\tilde{w}$, she uses the realization of the signal y to update her beliefs in the following way. First, she uses y to re-evaluate the precision of this signal y – $\hat{w}_i$. Next she uses the updated precision $\hat{w}_i$ as the precision of the signal y when updating her beliefs about the asset’s return $\tilde{x}$. More precisely, the joint distribution of the signal $\tilde{y}$ and its precision $\tilde{w}$ for an investor i is given by

$$\begin{aligned} f_i\left( w, y\right) =\sqrt{\frac{w}{2\pi }}exp\left[ -\frac{w\left( y-m_i\right) ^2}{2}\right] \frac{\beta ^\alpha w^{\alpha -1}exp\left[ -\beta w\right] }{\Gamma (\alpha )}. \end{aligned}$$

(2.3)

After receiving the signal realization y, the investor i forms the conditional distribution of the precision $\tilde{w}_i$ given the signal y, $f_i\left( w|y\right)$ and uses the mean of this distribution $\hat{w}_i=E_i\left[ \tilde{w}|y\right]$ to update her beliefs about the asset’s return $\tilde{x}$ as in standard Bayesian updating for normal distributions.

Proposition 3

At $t=2$, after the investors observe the realization of the signal y they update their beliefs about the precision of the signal y, w⁵:

$$\begin{aligned} \hat{w_i}=E_i[\tilde{w}|y]=\frac{\Gamma (\alpha +1.5,[\frac{(y-m_i)^2}{2}+\beta ]\nu )[\frac{(y-m_i)^2}{2}+\beta ]^{-1}}{\Gamma (\alpha +0.5,[\frac{(y-m_i)^2}{2}+\beta ]\nu )}, \end{aligned}$$

(2.4)

and use these updated beliefs about precision to update their beliefs about the mean and the variance of the firm’s return x:

$$\begin{aligned} E_i[\tilde{x}|y]=m_i+\hat{w_i}(y-m_i)\nu ^{-1},\end{aligned}$$

(2.5)

$$\begin{aligned} Var_i[\tilde{x}|y]=\nu ^{-1}\left( 1-\hat{w_i}\nu ^{-1}\right) . \end{aligned}$$

(2.6)

Investors’ beliefs can diverge after they receive the signal y, or $|m_1-m_2|$ can be less than $|m_1+\hat{w_1}(y-m_1)\nu ^{-1}-\left( m_2+\hat{w_2}(y-m_2)\nu ^{-1}\right) |$.

Proposition 3 shows that the combination of different prior beliefs and precision uncertainty is sufficient to yield cases when investors disagree about the risky asset’s return more after they receive the same piece of new information. When investors observe signal realization y, first, they update their respective beliefs about the precision of the observed signal, w. Because the normal distribution of the signal y is bell-shaped, a rational investor would more likely attribute to noise a signal more distant from the mean of this distribution. Therefore signal realizations y that are further from a given investor’s prior $m_i$ are perceived by this investor as less precise – given a lower precision $\hat{w}_i$. Next investors use their believed precisions $\hat{w}_i$ to update their beliefs about the firm’s return. Since investors have different prior means $m_i$, each investor will infer different precision of the signal $\hat{w}_i$ and assign a different weight to the signal realization y in her updating about the firm’s mean return. Such differential updating can sometimes result in increased disagreement. For example, if a signal realization y is higher than both investors’ priors $m_1$ and $m_2$, $m_1>m_2$, both investors will update their beliefs upward. However, investor 1 will assign a higher precision $\hat{w}_1$ to the signal because it is closer to her prior, and thus will update her belief upward more strongly than investor 2, increasing the disagreement.

Fig. 1

$\lambda _H=\lambda _L=0.5$, $r_H=r_L=4$, $\alpha =10$, $m_H=6$, $m_L=0$, $\nu =1$, $n=1$

Figure 1a plots investors’ ex ante beliefs about the mean and variance of the firm’s cash flows, $E_i[\tilde{x}]$ and $Var_i[\tilde{x}]$, Figure 1b plots the ex-post counterparts of these two moments, conditional on the realization of the signal y, $E_i[\tilde{x}|y]$ and $Var_i[\tilde{x}|y]$, and Fig. 1c plots the difference between investors’ ex-ante and ex-post beliefs to illustrate the magnitude and direction in which the signal alters their beliefs. Before the signal is realized (Fig. 1a), investors disagree: the first type of investor (call her investor H) expects firm cash flow to be 6; the second type of investor (call her investor L) expects it to be 0. After observing the same piece of information — or public signal — investors’ beliefs can become more divergent. The reason is that investors infer that the signal has a low precision when the signal does not correspond to their priors. Consequently, an optimistic investor (type H) will exhibit a stronger response to an optimistic signal than will a pessimistic investor (type L), who will exhibit a more muted response. Figure 1b demonstrates this phenomenon. In this case, for $y=3$, investors agree more about the firm’s expected cash flows after observing the signal. The solid line in Fig. 1c is below the dashed line. Here both investors come to the same estimate of the signal precision and their beliefs converge toward 3. However, for $y=10$, the signal increases investors’ disagreement: H now expects the firm’s cash flows to be around 9, whereas L expects cash flows of roughly 2. The solid line in Fig. 1c is above the dashed line. In other words, while both investors update their beliefs toward the realized signal realization, the optimistic investor does so more strongly than does the pessimistic investor, which results in greater divergence.

Investors’ beliefs about the variances also diverge for some values of y. Expected variances are equal before the signal (Fig. 1a), but, after the signal, they may differ (Fig. 1b). After the signal is realized, investors revise their expectations in the same direction. However, an investor whose ex-ante expectation of cash flow is closer to the signal moves her beliefs more than another investor. In our example, $y=9$ is closer to investor H’s prior expectation. As a result, the ex-post expectations are driven further away.

The model describes common real-world scenarios. When researchers get a result of a drug test, they may further debate the drug’s effectiveness. Specifically, a researcher who does not expect that a drug should work and is uncertain about the conditions of the experiment may dismiss the results. Similarly, political opinions in the news are often dismissed by people with opposing views. Finally, even more generic news in the media can easily be dismissed by appealing to the credibility of the source. In all these cases, people disagree even more because they (1) disagree in the first place and (2) are not sure how precise the signal is (Jaynes 2003). The same logic holds for investors: they have different prior beliefs and do not know exactly how precise the signal by a firm is.

2.4 Full model: equilibrium prices and trading volume

Because market participants’ beliefs are not directly observable to researchers, to test the validity of our theory, we need to develop a number of predictions about observed variables. In this section, we solve for equilibrium prices and trading volume in the model.

We solve the model by backward induction. First, we derive investors’ demands and the market-clearing price at $t=2$. Next, anticipating their choices in $t=2$, we solve for investors’ demands and the price at $t=1$. We measure trading volume as the absolute difference between demands at $t=1$ and $t=2$.

Let $P_t$ denote the price of the risky asset at time t.

Proposition 4

At $t=2$, after the signal y is disclosed, the equilibrium price is set at

$$\begin{aligned} P^*_2=\left[ \psi _1(\hat{w_1})+\psi _2(\hat{w_2})\right] ^{-1}\left[ E_1[\tilde{x}|y]\psi _1(\hat{w_1})+E_2[\tilde{x}|y]\psi _2(\hat{w_2})-1\right] , \end{aligned}$$

(2.7)

where $\psi _i(\hat{w_i})=\frac{\lambda _i\nu }{r_i(1-\frac{\hat{w_i}}{\nu })}$, $E_i[\tilde{x}|y]=m_i+\hat{w_i}(y-m_i)\nu ^{-1}.$

Subscript i denotes investor i’s expectation or variance, respectively. The function $\psi _i(\hat{w_i})$ represents an investor i’s confidence in the quality of the signal. It increases as an investor perceives the signal as more precise. Because investors are risk averse, a higher value of $\psi _i(\hat{w_i})$ increases investor i’s demand for the risky asset.

Proposition 5

The equilibrium price at time $t=1$ is:

$$\begin{aligned} P^*_1=(\phi _1+\phi _2)^{-1}\left( E_1[P_2^*]\phi _1+E_2[P_2^*]\phi _2-1\right) , \end{aligned}$$

(2.8)

where $\phi _i=\frac{\lambda _i}{r_iVar_i[P^*_2]}$

$\phi _i=\frac{\lambda _i}{r_iVar_i[P^*_2]}$ resembles $\psi _i(\hat{w_i})$. Because the signal is not yet realized, investors cannot estimate its precision. Instead they rely on the expected next-period price variance, $Var_i[P_2^*]$.

Following (Subramanyam 1996), we plot the return, $\frac{P_2^*-P_1^*}{P_1^*}$, as a function of the signal realization. Our model predicts an S-shaped form of returns. Figure 2 shows the return, defined as the difference between prices at times 2 and 1 scaled by price at time 1, as a function of the firm’s signal, y, predicted by the model. The figure also shows that the S-shape of the return is less pronounced for higher values of the parameter $\beta$. That is, when the precision uncertainty decreases, the S shape in returns becomes less pronounced.

Fig. 2

Return, $\frac{P^*_2-P^*_1}{P^*_1}$ as a function of y, for different levels of $\beta$. $\lambda _1=\lambda _2=0.5$, $r_1=r_2=4$, $\alpha =10$, $m_1=6$, $m_2=0$, $\nu =1$, $n=1$

Fig. 3

Trading volume and investors’ disagreement for different realizations of the signal

Fig. 4

A type-1 investor’s trading volume as a function of the public signal, y, for different levels of investors’ disagreement. $\lambda _1=\lambda _2=0.5$, $r_1=r_2=4$, $m_2=0$, $\nu =1$, $n=1$, $\alpha =10$, $\beta =1$

Our model contrasts with the traditional framework with homogeneously informed investors (e.g., Kim and Verrecchia 1991), where there is no trading volume following the disclosure of a signal unless investors disagree about the signal precision in the first place. In our case, while investors initially have homogeneous beliefs about the signal precision, they have heterogeneous beliefs following the disclosure. This divergence of beliefs, in turn, generates trading following the disclosure.

Investor i’s trading volume is defined as the absolute difference between demands in pre- and post-announcement periods:

$$\begin{aligned} V_i=|d^*_{i2}-d^*_{i1}| \end{aligned}$$

(2.9)

In our model with two investor types, $V_i=-V_j$ such that total trading volume is given by $2V_i$. Because of the updating over the uncertain precision, it is difficult to analyze $V_i$ analytically. To develop some intuition, Figure 3a displays a numerically computed graph of trading volume, $V_i$, as a function of the signal realization y. As the figure shows, trading volume is an M-shaped function of the signal.⁶ Specifically, trading volume first increases as the signal moves away from zero, but, as the signal gets extreme, trading volume starts to decrease.⁷

To get intuition for the shape of the trading volume, we plot the difference in investors’ beliefs before and after the signal realization in Fig. 3b on the same scale. Post-announcement trading volume is the greatest when the divergence in investors’ post-announcement beliefs increases. In contrast, trading volume is smallest when investors’ beliefs converge after the disclosure. The forces that drive the M-shape in trading volume are the heterogeneous priors in combination with uncertainty about the signal-precision. As a result, even when investors receive the same public signal, disagreement about the firm’s future cash flow may increase, which leads to more trading.

2.5 How does trading volume vary with disagreement and precision uncertainty?

Next we analyze how the degree of disagreement between investors and signal-precision uncertainty affect trading volume.

First, we vary the extent of investors’ disagreement. Figure 4 shows numerically computed trading volume for different distances between investors’ priors. We keep the prior of the pessimistic investor constant and increase the prior of the optimistic investor. As investors’ initial beliefs get more distant, the overall level of trading volume increases, and the humps center around a new average belief point. The M-shape of trading volume also becomes more pronounced.

Fig. 5

A type-1 investor’s trading volume as a function of the public signal, y.$\lambda _1=\lambda _2=0.5$, $r_1=r_2=4$, $m_1=6$, $m_2=0$, $\nu =1$, $n=1$

Second, we examine what effect the parameters of the distribution of signal-precision, $\alpha$ and $\beta$, have on the nonlinearity of trading volume (see Fig. 5). As $\alpha$ increases ($\beta$ decreases), the nonlinearity of trading volume is more pronounced, and for sufficiently low levels of $\alpha$ (high levels of $\beta$), the humps disappear. To understand this finding, note that the expectation and variance of signal precision are proportional to $\alpha$ and inversely proportional to $\beta$⁸. On the one hand, as $\alpha$ increases ($\beta$ decreases), the expected precision increases. If the signal is perceived as more precise on average, the market reacts to the signal more strongly: the increase in trading volume is more pronounced. On the other hand, as $\alpha$ increases ($\beta$ decreases), the market is also more uncertain about the precision. As in the work of Subramanyam (1996), a higher signal-precision uncertainty causes investors to use the signal more to update their beliefs about the signal precision. This implies that, for example, a positive signal is more easily dismissed by a pessimistic investor, and therefore the humps in trading volume are more pronounced.

3 Empirical testing

3.1 Model predictions

In this section, we empirically test the results of the model. While we cannot observe investors’ beliefs, we can observe an outcome: trading volume. We choose earnings announcements as the signal that investors receive because, first, earnings are one of the most important statistics used in inferring the value of a firm, and, second, investors are likely uncertain how good of a signal of a firm’s value earnings are (due to uncertain persistence of earnings, unknown components of economic performance, or reporting noise). We test three model predictions. The first is about the general functional form of trading volume as a function of an earnings surprise:

Prediction 1. Trading volume is an M-shaped function of the earnings surprise.

The second prediction describes how trading volume changes when investors hold more distant initial beliefs. We have demonstrated that greater disagreement implies a more pronounced M-shape of trading volume:

Prediction 2. The M-shape in trading volume is more pronounced when investors have greater prior disagreement about a firm.

The third prediction is about how trading volume’s form changes with the parameters of the signal-precision uncertainty. As discussed above, when the variance of the signal precision increases, the M-shaped pattern of trading volume becomes more pronounced. Interpreting a high variance of the signal precision as a high uncertainty about this precision, we formulate the third empirical prediction as follows:

Prediction 3. The M-shape in trading volume is more pronounced when investors are more uncertain about the precision of an earnings announcement.

3.2 Data and measurement

To test our theoretical predictions, we gather data on quarterly earnings announcements of US firms from the first quarter of 1990 to the fourth quarter of 2019. We obtain data on released earnings, analyst EPS forecasts, and analyst price targets from IBES; prices and trading volume from CRSP; and firm characteristics from Compustat. Because research has shown that negative earnings have negligible information content (Hayn 1995; Lipe et al. 1998), we delete observations with negative actual EPS.⁹ We keep only firms with prices at the end of a previous fiscal quarter greater than $5 to minimize the effect of market frictions (Ball et al. 1995).

Following (Landsman and Maydew 2002; Truong 2012; Choi 2019), we measure excessive trading volume in [0,1] days event window around the earnings announcement as:

$$\begin{aligned} AVOL_{i,q}=\sum _{t=0}^{t=1}\frac{VOL_{i,q,t}-mVOL_{i,q}}{\sigma (VOL)_{i,q}}, \end{aligned}$$

(3.1)

where $VOL_{i,q,t}$ is the trading volume, $mVOL_{i,q}$ and $\sigma (VOL)_{i,q}$ are the mean and the standard deviation of the daily trading volume in $[-240, -5]$ days before the earnings announcement (Truong 2012; Choi 2019); i, q, and t denote the firm, the quarter, and the day after the earnings announcement, respectively.¹⁰

We measure the earnings surprise following (Conrad et al. 2002) and (Choi 2019):

$$\begin{aligned} Surp_{i,q}=\frac{\text {Actual EPS}_{i,q}-\text {Med. forecast}_{i,q}}{PRC_{i,q-1}}, \end{aligned}$$

(3.2)

where $\text {Actual EPS}_{i,q}$ is the actual value announced by the firm, $\text {Med. forecast}_{i,q}$ is the median analyst forecast of firm’s EPS, $PRC_{i,q-1}$ is the firm’s price at the end of a previous fiscal quarter. We use only the most recent forecasts by each analyst to calculate the median.

Table 1 Sample selection procedure

Table 2 Summary statistics

We include firm size and market-to-book ratio to control for differences in risk that are not already captured by the excess return (Fama and French 1992, 1993). We measure firm size as the natural logarithm of the market value of equity and the market-to-book ratio as the market value of equity divided by the book value of equity. Because our model predicts the M-shaped trading volume for a given level of ex ante disagreement, we also include analyst forecast dispersion before the announcement as a control variable. We control for market-wide liquidity levels by including (Pástor and Stambaugh 2003) liquidity factor. We include an indicator for a firm having a Big Four auditor to control for the mean signal-precision uncertainty: we assume that firms with a Big Four auditor firm are perceived by investors as having more precise earnings numbers. Finally, we include year fixed effects to control for potential correlation of earnings surprises with business cycles. To minimize the effect of outliers, we truncate the earnings surprise variable at the 5% level and all the other variables at the 1% level. As a result, we have a dataset of 87,944 firm-years from the first quarter of 1990 to the fourth quarter of 2019. The total number of firms in the sample is 4739. Table 1 describes how the sample size changed at each stage.

We present summary statistics in Table 2. Mean excessive trading volume equals 0.75 with the standard deviation 1.15. Earnings surprises are 0.001 on average and vary from -0.01 to 0.01.

3.3 Functional form of trading volume

The model predicts that trading volume is an M-shaped function of the earnings surprise. Trading volume increases for medium levels of surprises and decreases for extreme levels. We test this result in three ways. First, we look at scatterplots of trading volume as a function of earnings surprise with fitted nonparametric curves. Second, we estimate polynomial regressions of excessive trading volume on earnings surprise and use an analysis-of-variance statistical test to choose the model that fits the data in the best possible way. Third, we estimate quantile regressions for different quantiles of an absolute earnings surprise.

Fig. 6

Scatterplot of residuals of excessive trading volume with LOESS smoother

3.3.1 Nonparametric analysis

We begin our analysis of the functional form of trading volume by looking at scatterplots of trading volume as a function of earnings surprise with locally estimated scatterplot smoothing (LOESS) curves. LOESS is a local regression method that combines elements of simple linear least squares regression with elements of nonlinear regression. The method builds simple models for localized subsets of the data and then combines them into a function describing full data support. The advantage of this approach is that it does not require a researcher to pre-specify any functional form.

To control for other important factors that might affect trading volume, we first run a regression of excessive trading volume on the set of control variables:

$$\begin{aligned} AVOL_{i,q}=a_0+a_1\times Size_{i,q-1}+a_2\times Market/Book_{i,q-1}+a_3\times Dispersion_{i,m-1}\nonumber \\ +a_4\times PSLiquidity_{i,q-1}+a_5\times Big4+a_6\times Year\text { }FE. \end{aligned}$$

(3.3)

Next we analyze scatterplots of the residuals of the regression above, $(AVOL_{i,q}-\hat{AVOL}_{i,q}).$ This procedure allows us to concentrate on the part of trading volume that is orthogonal to other factors.

Table 3 Polynomial regressions of excessive trading volume on the earnings surprise

Figure 6 demonstrates the scatterplot of the residuals of excessive trading volume as a function of an earnings surprise. While the classic V-shape is pronounced for intermediate levels of earnings surprises, trading volume does not increase but rather stays flat for more extreme surprises. The nonparametric analysis provides preliminary evidence that trading volume reactions modestly increase with earnings surprises.

3.3.2 Parametric analysis: polynomial regression

For our second test, we estimate the following regression:

$$\begin{aligned} ln(AVOL_{i,q})=a_0+B'\times poly(Surp_{i,q})+C'Controls, \end{aligned}$$

(3.4)

where $poly(Surp_{i,q})$ denotes the polynomials of $Surp_{i,q}$ from first to fifth order,¹¹$B'$ is a vector of the polynomial’s coefficients, $C'$ is a vector of coefficients in front of control variables.

Table 4 Analysis of variance: comparison of polynomial models

The results of the estimation of the polynomials up to fifth order are shown in Table 3. To choose the model that most closely describes the relation between trading volume and the earnings surprise, we conduct an analysis-of-variance test. The statistics are shown in Table 4. Adding the third and the fifth order summands to the polynomial does not improve the predictive power, whereas the second and the fourth order summands improve the fit of the model to the data significantly. We conclude that the best model is the one that includes first-, second-, and fourth-order summands.

Fig. 7

Functional form of the equation from the regression of residual excessive trading volume on the earnings surprise

Table 5 Quantile regressions of excessive trading volume for different levels of earnings surprise

While the fourth-order polynomial suggests a nonlinear effect of the earnings surprise on trading volume, the M-shape is not obvious. To illustrate the shape of the polynomial, we run a regression of excessive trading volume on all the control variables, take residuals from this regression, and regress the residuals on the earnings surprise. We plot the equation from the regression of the trading volume’s residuals after controlling for various factors on the earnings surprise in Fig. 7. The plotted curve has two humps and a pronounced M-shape, similar to the model plots in Fig. 3a.

Fig. 8

Residuals of excessive trading volume as a function of earnings surprise for zero prior investor disagreement

Fig. 9

Residuals of excessive trading volume as a function of earnings surprise for prior investor disagreement above zero

3.3.3 Parametric analysis: quantile regression

As an alternative way to show the nonmonotonicity in trading volume, we estimate quantile regressions. Specifically, we estimate the following regression:

$$\begin{aligned} ln(AVOL_{i,q})=a_0+b_1\times |Surp_{i,q}|+C'Controls, \end{aligned}$$

(3.5)

separately for the upper $5\%$ of Surp variable and the rest of the sample. Table 5 shows the results. Earnings surprise is positively associated with the trading volume for the lower 95% of the absolute earnings surprise. For the upper 5%, that is, for the 2.5% of the lowest and 2.5% of the highest non-absolute earnings surprise, the association disappears. Extreme levels of earnings surprise do not invoke high reactions of trading volume, as predicted by the model (Fig. 8).

Our evidence from the three empirical tests speak in favor of the first model’s prediction. When we do not impose any restrictions on the functional form, excessive trading volume increases for small earnings surprises but less so for larger surprises. Imposing a particular functional form further corroborates this finding: trading volume as a function of the earnings surprise is most closely approximated by the fourth-order polynomial, which is M-shaped. The estimated polynomial pattern resembles its theoretical counterpart: trading volume is around zero for no earnings surprise, increases for modest surprises, and then decreases for extreme surprises (Fig. 9).

3.4 Cross-sectional analysis

Next we test cross-sectional predictions about the M-shape of trading volume: we expect the M-shape to be more pronounced when investors have greater prior disagreement about a firm and when investors are more uncertain about the precision of earnings.

3.4.1 Disagreement and the M-shape of trading volume: measure of disagreement

To measure the level of stock market investors’ disagreement, we use the measure developed by Cookson and Niessner (2020) and Cookson and Niessner (2023). The authors develop a company-level daily measure of investors’ disagreement using the StockTwits platform for 2010-2021. The papers analyze the language of investors’ posts and couple it with investors’ characteristics to measure disagreement among all investors, within each investor group based on investment approach (fundamental, technical, value, momentum, and growth) and across investor groups. Since we investigate overall trading volume, we take the average daily disagreement among all investors one month before an earnings announcement as a proxy for investors’ prior disagreement about a company.

Our second theoretical prediction is that the M-shape of trading volume is more pronounced when investors disagree more before the signal is disclosed. We test whether the M-shape is more pronounced for greater investor disagreement (Cookson and Niessner 2020, 2023) in nonparametric and parametric ways.

3.4.2 Disagreement and the M-shape of trading volume: nonparametric analysis

For the nonparametric analysis, we partition our sample into groups based on the levels of the disagreement measure and scatter plot the residuals of excessive trading volume as a function of earnings surprise for different groups. Since disagreement is above zero only for the 62$^{\text {nd}}$ quantile in our sample, we consider two groups: in the first group, investor disagreement is zero, and, in the second, investor disagreement is positive.

Figures and show scatter plots for subsamples with zero and nonzero investor disagreement, respectively. For the subsample with disagreement, the M-shape appears more pronounced than for the subsample without disagreement.

We corroborate our observational evidence with statistical tests in the next section.

3.4.3 Disagreement and the M-shape of trading volume: parametric analysis

In this section, we parametrically test whether the M-shape of trading volume is more pronounced for greater investor disagreement before earnings announcements. Since measuring the concavity of a fourth-order polynomial is nontrivial, we use absolute values of earnings surprises and run the following regression:

$$\begin{aligned} ln(AVOL_{i,q})=a_0+b_1\times |Surp_{i,q}|+b_2\times Surp_{i,q}^2+b_3\times |Surp_{i,q}|^2\times Disag_{m-1}\nonumber \\+b_4\times Disag_{m-1}+B'Controls. \end{aligned}$$

(3.6)

We interpret the coefficient in front of the interaction of the disagreement measure, $Disag_{m-1}$, and the quadratic term, $b_3$, as the change in the concavity of absolute trading volume with a one unit increase in prior investor disagreement. We present the estimates in Tables 6 and 7.

Table 6 Validation of earnings-precision uncertainty measure: S-shaped ERC interacted with the earnings-precision uncertainty measure

Table 7 Regression of excessive trading volume on the earnings surprise interacted with the disagreement measure

The coefficient in front of the interaction term, $|Surp_{i,q}|^2\times Disag_{m-1}$ is negative and statistically significant, confirming our prediction that the M-shape of trading volume is more pronounced when investors have greater disagreement prior to an earnings release.

3.4.4 Earnings-precision uncertainty and the M-shape of trading volume: measure of earnings-precision uncertainty

Our third prediction is the M-shaped pattern becomes more pronounced when uncertainty about the earnings precision is higher. To test this implication, we need to measure earnings-announcement-precision uncertainty in the sample. It is difficult to separate the earnings-announcement-precision uncertainty from the overall market uncertainty. We cannot use standard market uncertainty measures, such as VIX, analyst forecast volatility, or market returns volatility, because they may include both fundamental uncertainty and uncertainty about the earnings-announcement-precision.

The measure of earnings-precision uncertainty needs to capture investors’ uncertainty about the amount of noise in the earnings number. The investors’ perceived total variance of an upcoming earnings number can be thought of as a sum of the variance of the actual earnings (fundamental variance) and the variance of the noise term in the earnings report (precision uncertainty). We therefore need to, first, choose a proxy for investors’ total variance of upcoming earnings and, second, find a way to measure fundamental variance and remove this portion of variance from the total investors’ perceived variance.

We choose to proxy investors’ perceived variance of upcoming earnings with analyst forecast spread. When analysts are not certain about a firm’s earnings, the range for each analyst’s forecasts widens, and different analysts’ forecasts are more distant from each other. To measure the fundamental volatility component, we use the VIX. The earnings-precision uncertainty is measured as residuals from regressing analyst forecast spreads on the VIX. We calculate the measure in the following steps. First, we run the regression:

$$\begin{aligned} \hat{Analyst \quad forecast \quad spread}_{i,q}=\gamma _0+\gamma _1\times VIX_{m-1}, \end{aligned}$$

(3.7)

where $Analyst \quad forecast \quad spread_{i,q}$ is the difference between the highest and the lowest analyst forecasts of the EPS of the firm i for the quarter q. $VIX_{m-1}$ is the average monthly VIX from the daily data from the Chicago Board of Exchange website. The index is taken in the month before the disclosure month (Choi 2019). Next we take the residuals of this regression as the measure of earnings-announcement-precision uncertainty:

$$\begin{aligned} PrecUnc_{i,q}\!=\!|Analyst \quad forecast \quad spread_{i,q}\!-\!\hat{Analyst \quad forecast \quad spread}_{i,q}|. \end{aligned}$$

(3.8)

To validate our measure, we ask ourselves whether the common S-shaped ERC (e.g., Freeman and Tse 1992; Das and Lev 1994) is more pronounced for high levels of our measure, which would be consistent with the theory by Subramanyam (1996) (see Figure 2(b)). We calculate cumulative abnormal returns in the [0, 1] event window around earnings announcements in the following way:

$$\begin{aligned} CAR_{i,q}=\sum _{t=0}^{1}AR_{i,q,t}, \end{aligned}$$

(3.9)

where $AR_{i,q,t}$ is the abnormal return calculated from the Fama-French three-factor model:

$$\begin{aligned} \hat{R}_{i,t}=\gamma _0+R_{f,t}+\gamma _1\times (R_{m,t}-R_{f,t})+\gamma _2\times SMB_t+\gamma _3\times HML_t,\end{aligned}$$

(3.10)

$$\begin{aligned} AR_{i,q,t}=R_{i,q,t}-\hat{R}_{i,t}, \end{aligned}$$

(3.11)

where $R_{i,q,t}$ is daily return on a stock, $R_{f,t}$ is a risk-free interest rate, $R_{m,t}$ is a market rate of return, $SMB_t$ is excess returns of small capitalization firms over large capitalization firms, and $HML_t$ is excess returns of value stocks over growth stocks on day t. To estimate normal levels of returns, we use stock returns in the $\left[ -150, -50\right]$ day window before the announcement date. We require at least 70 days of returns to be available for the stock to remain in the sample.

Fig. 10

Residuals of excessive trading volume as a function of earnings surprise, first quartile of earnings-precision uncertainty

First, we confirm (untabulated) that the S-shaped functional form of the earnings response coefficient holds for our sample of quarterly earnings announcements. If our measure is indeed capturing the underlying theoretical construct – earnings-precision uncertainty – we expect the S-shape to be more concave for higher levels of our measure. For a quadratic function, the level of concavity can be measured by looking at the coefficient in front of a quadratic term. We run the following regression:

$$\begin{aligned} CAR_{i,q}=a_0+b_1\times Surp_{i,q}+b_2\times Surp_{i,q}^2+c_1\times PrecUnc_{i,q}+c_2\times PrecUnc_{i,q}\times Surp_{i,q}^2, \end{aligned}$$

(3.12)

separately for positive and negative values of earnings surprises, $Surp_{i,q}$. For positive (negative) levels of earnings surprise, we expect $c_2<0$ ($c_2>0$). Table 6 presents the results. For positive (negative) levels of earnings surprise, the coefficient in front of the quadratic term ($Surp^2$) is more negative (positive) for higher levels of earnings precision uncertainty, suggesting that our signal-precision uncertainty measure is capturing the theoretical construct to a certain extent.

Having validated the measure, we proceed to estimate the functional form of trading volume for different levels of the earnings-announcement-precision uncertainty.

Fig. 11

Residuals of excessive trading volume as a function of earnings surprise, second quartile of earnings-precision uncertainty

Fig. 12

Residuals of excessive trading volume as a function of earnings surprise, third quartile of earnings-precision uncertainty

3.4.5 Earnings-precision uncertainty and the M-shape of trading volume: nonparametric analysis

If the third model prediction holds and the measure of the earnings-announcement-precision uncertainty that we develop accurately captures the underlying theoretical construct, then the trading volume’s M-shape should be more pronounced for observations with high PrecUnc. We conduct both nonparametric and parametric tests of this prediction.

Fig. 13

Residuals of excessive trading volume as a function of earnings surprise, fourth quartile of earnings-precision uncertainty

For our nonparametric analysis, we partition our sample into four groups based on the quartiles of our earnings-announcement-precision uncertainty measure. The scatterplots of the residuals of excessive trading volume as a function of the earnings surprise for different quartiles of the earnings-announcement-precision uncertainty are presented in Figs. 10, 11, 12, and 13. The pictures demonstrate a transition from the typical V-shape to the M-shape, as we move from the first to the third quartile of the earnings-announcement-precision uncertainty. However, in the fourth quartile, the M-shape is not pronounced. The evidence largely supports our theoretical prediction: the depressed trading volume at the extremes is more pronounced when there is greater market uncertainty about the signal precision.

Table 8 Regression of excessive trading volume on the earnings surprise interacted with the earnings-precision uncertainty measure

3.4.6 Earnings-precision uncertainty and the M-shape of trading volume: parametric analysis

Finally, we parametrically test whether the more pronounced M-shape of trading volume is associated with higher signal-precision uncertainty in the financial market. We run the regression similar to 3.6:

$$\begin{aligned} ln(AVOL_{i,q})=a_0+b_1\times |Surp_{i,q}|+b_2\times Surp_{i,q}^2+b_3\times |Surp_{i,q}|^2\times PrecUnc_{i,q}\nonumber \\+b_4\times PrecUnc_{i,q}+B'Controls. \end{aligned}$$

(3.13)

We interpret the coefficient in front of the interaction of the signal-precision uncertainty measure and the quadratic term, $b_3$, as the change in the concavity of absolute trading volume with a 1 unit increase of signal-precision uncertainty. We present the estimates in Table 8.

The coefficients in front of the quadratic term, $Surp_{i,q}^2$, and the interaction term, $Surp_{i,q}^2\times PrecUnc_{i,q}$, are negative and significant. This result implies that as signal-precision uncertainty increases, the U-shape of trading volume becomes more concave, suggesting that the M-shape is more pronounced when investors are more uncertain about how precise a firm’s signal is.

3.4.7 Interaction between disagreement and earnings-precision uncertainty

In the final section of the paper, we study how trading volume behavior changes with the interaction of the two proxies for the two theoretical constructs – investor initial disagreement and earnings-precision uncertainty. The purpose of this analysis is twofold. First, it allows us to jointly validate our theory and how well our empirical proxies capture the theoretical constructs. As we discussed in Section 2.3, to obtain beliefs divergence after investors receive the same information and thus the M-shape of trading volume, both disagreement and signal-precision uncertainty have to be present. Therefore, in our empirical tests, we should find that the trading volume’s functional form is only affected by the double-interaction term between our two proxies and not by coefficients in front of each individual interaction term.

Second, the double-interaction analysis helps address potential concerns that our two empirical measures proxy for the same theoretical construct. In particular, one might worry that our earnings-precision uncertainty measure is instead capturing investor disagreement because it is based on analyst forecast spread. If in the regression that includes both disagreement and earnings-precision uncertainty measures each individual metric is significant, we would conclude that the two proxies capture at least some different forces that affect the shape of trading volume.

We run the following regression:

$$\begin{aligned} ln(AVOL_{i,q})=a_0+a_1\times |Surp_{i,q}|+a_2\times Surp_{i,q}^2\nonumber \\+b_1\times |Surp_{i,q}|^2\times Disag_{m-1}+c_1\times |Surp_{i,q}|^2\times PrecUnc_{i,q}\nonumber \\+d_1\times |Surp_{i,q}|^2\times Disag_{m-1}\times PrecUnc_{i,q}\nonumber \\ +b_2\times Disag_{m-1}+c_2\times PrecUnc_{i,q}+A'Controls. \end{aligned}$$

(3.14)

The estimated coefficients are presented in Table 9. The coefficients in front of the squared earnings surprise, $Surp_{i,q}^2$, and in front of the double-interaction term, $Surp_{i,q}^2\times Disag_{m-1}\times PrecUnc_{i,q}$, are negative but not significant. The coefficients in front of individual interaction terms, $Surp_{i,q}^2\times Disag_{m-1}$ and $Surp_{i,q}^2\times PrecUnc_{i,q}$, are negative and statistically significant.

Table 9 Regression of excessive trading volume on the earnings surprise interacted with the disagreement measure, earnings-precision uncertainty measure, and the interaction between the two measures

We interpret the findings as follows. First, setting both investors’ disagreement, Disag, and earnings-precision uncertainty, PrecUnc, to zero, implies that trading volume is not M-shaped ($a_2$ is indistinguishable from zero). This result confirms our theory that disagreement and precision uncertainty have to be present to obtain the M-shaped trading volume. Second, setting either investors’ disagreement, Disag, or earnings-precision uncertainty, PrecUnc, to zero does not remove the M-shape of trading volume ($a_2+b_1$ and $a_2+c_1$ are negative and statistically significant), suggesting that each individual measure (Disag and PrecUnc) may be capturing both theoretical constructs to some extent. Finally, because coefficients in front of individual interaction terms ($b_1$ and $c_1$) are statistically significant, we argue that our measures must also be capturing some different economic forces. Overall we acknowledge that our measures of disagreement and earnings-precision uncertainty may not be capturing respective theoretical constructs in isolation. However, they do capture different constructs, each of which affects the functional form of trading volume.

4 Conclusion

We provide initial evidence that investors’ beliefs can further diverge, even when they receive the same public signal. The source of this phenomenon lies in uncertainty about the precision of the financial information that investors receive, coupled with their differential beliefs. We develop a model where investors with different beliefs about a firm’s future cash flow trade in the firm’s shares before and after the realization of a public signal. The novelty in the model is that investors are uncertain about the precision of this signal. Because of this uncertainty, their beliefs further diverge for some signal realizations. As a result of investors’ posterior beliefs, trading volume is increasing for intermediate levels of signal surprise but is damped for extreme levels. The M-shape of trading volume is more pronounced when the uncertainty about the signal precision rises.

We test the predictions of our model using trading volume around quarterly earnings announcements of public U.S. firms. As a starting point in our empirical tests, we nonparametrically and parametrically show that total trading volume is described by a function that increases for the intermediate levels of the absolute earnings surprise and decreases (or stays flat) for the extreme levels. We further corroborate our theory by testing cross-sectional predictions of our model and confirm that the M-shape of trading volume is more pronounced when investors hold more different prior beliefs and are more uncertain about the precision of earnings as a signal about the firm’s value.

Overall we believe our paper strongly suggests that investors in capital markets are uncertain about the accounting quality underlying firms’ earnings reports. Our paper provides a small yet significant piece of evidence that investors’ beliefs diverge due to this uncertainty about the quality of financial information.

5 Appendix

5.1 A.1 Proof of Propositions 1 and 2

When investors initially disagree about the asset’s return and know the precision of the signal about the asset’s return, the difference in investors’ beliefs before the disclosure of the signal y is $|m_1-m_2|$, and after the signal is released, the difference in investors’ beliefs is

$$\begin{aligned} |E_1[x|y]-E_2[x|y]|=|m_1-m_2|\frac{w-\nu }{w}\le |m_1-m_2|. \end{aligned}$$

(A1)

When investors are uncertain about the precision of the signal but agree on the mean of the asset return (i.e., $m_1=m_2\equiv m$), the disagreement before the signal realization is $|m_1-m_2|=0$. After the signal realization, the investors perceived signal-precision is

$$\begin{aligned} \hat{w}_1= & {} \frac{\Gamma (\alpha +1.5,[\frac{(y-m_1)^2}{2}+\beta ]\nu )[\frac{(y-m_1)^2}{2}+\beta ]^{-1}}{\Gamma (\alpha +0.5,[\frac{(y-m_1)^2}{2}+\beta ]\nu )} \nonumber \\= & {} \frac{\Gamma (\alpha +1.5,[\frac{(y-m_2)^2}{2}+\beta ]\nu )[\frac{(y-m_2)^2}{2}+\beta ]^{-1}}{\Gamma (\alpha +0.5,[\frac{(y-m_2)^2}{2}+\beta ]\nu )}=\hat{w}_2\equiv \hat{w}, \end{aligned}$$

(A2)

and the disagreement is

$$\begin{aligned} |E_1\left[ \tilde{x}|y\right] -E_2\left[ \tilde{x}|y\right] |=|m+\hat{w}(y-m)\nu ^{-1}-m-\hat{w}(y-m)\nu ^{-1})|=0 \end{aligned}$$

(A3)

5.2 A.2 Proof of Proposition 3

This section is based on Subramanyam (1996). Conditional on precision w, $\tilde{y}$ is normally distributed, $\tilde{y} \sim N(m_i, w^{-1}).$ Then

$$\begin{aligned} h(y|w)=\sqrt{\frac{w}{2\pi }}exp\left[ -\frac{w(y-m_i)^2}{2}\right] . \end{aligned}$$

(A4)

Compute the conditional expectation of $\tilde{w}$:

$$\begin{aligned} E[\tilde{w}|y]=\int wh(w|y)dw, \end{aligned}$$

(A5)

$$\begin{aligned} =\int \frac{1}{f(y)}wh_1(y|w)f(w)dw, \end{aligned}$$

(A6)

$$\begin{aligned} = \frac{\int wh_1(y|w)f(w)dw}{\int h_1(y|w)f(w)dw}, \end{aligned}$$

(A7)

$$\begin{aligned} = \frac{\int w^{1.5}(2\pi )^{-0.5}exp[-\frac{w(y-m_i)^2}{2}]f(w)dw}{\int w^{0.5}(2\pi )^{-0.5}exp[-\frac{w(y-m_i)^2}{2}]f(w)dw}. \end{aligned}$$

(A8)

Recall that $\tilde{w}\in [0, \nu ]$ and substitute for $f(\tilde{w})$ to get:

$$\begin{aligned} E[\tilde{w}|y]=\frac{\Gamma (\alpha +1.5,[\frac{(y-m_i)^2}{2}+\beta ]\nu )[\frac{(y-m_i)^2}{2}+\beta ]^{-1}}{\Gamma (\alpha +0.5,[\frac{(y-m_i)^2}{2}+\beta ]\nu )}. \end{aligned}$$

(A9)

5.3 A.3 Proof of Proposition 4

Investor i’s budget constraint at time $t=2$ is:

$$\begin{aligned} P_2d_{i2} +q_{i2} =q_{i1}^* +P_2d_{i1}^*, \end{aligned}$$

(A10)

where $q_{i1}^*$ and $d_{i1}^*$ are the amounts of riskless and risky assets held in equilibrium in $t=1$, respectively. $q_{i2}$ and $d_{i2}$ are the amounts of riskless and risky assets held in $t=2$. Investor i solves:

$$\begin{aligned} max_{d_{i2},q_{i2}} \quad E_i[\tilde{x}d_{i2}+q_{i2}|y]-\frac{1}{2}r_iVar_i[\tilde{x}d_{i2}+q_{i2}|y], \end{aligned}$$

(A11)

subject to (A9). The only random variable in the investor’s utility is the return of the risky asset, $\tilde{x}$. Therefore one can write $E_i[\tilde{x}d_{i2}+q_{i2}|y]=E_i[\tilde{x}|y]d_{i2}+q_{i2}$, $Var_i[\tilde{x}d_{i2}+q_{i2}|y]=Var_i[\tilde{x}|y]d_{i2}^2$. Rewrite the problem:

$$\begin{aligned} max_{d_{i2},q_{i2}}E_i[\tilde{x}|y]d_{i2}+q_{i2}-\frac{1}{2}r_iVar_i[\tilde{x}|y]d_{i2}^2 \end{aligned}$$

(A12)

$$\begin{aligned} \text {s.t. }P_2d_{i2} +q_{i2} =q_{i1}^* +P_2d_{i1}^* . \end{aligned}$$

(A13)

Using the budget constraint (A13), express $q_{i2}$:

$$\begin{aligned} q_{i2}=q^*_{i1}+P_2d^*_{i1}-P_2d_{i2}. \end{aligned}$$

(A14)

Plug this expression into (A12):

$$\begin{aligned} max_{d_{i2}}E_i[\tilde{x}|y]d_{i2}+q^*_{i1}+P_2d^*_{i1}-P_2d_{i2}-\frac{1}{2}r_iVar_i[\tilde{x}]|yd_{i2}^2. \end{aligned}$$

(A15)

$q^*_{i1}$ and $d^*_{i1}$ are chosen in $t=1$ and are constant in our problem. Take the derivative of the (A15) with respect to $d_{i2}$ and set it equal to zero:

$$\begin{aligned} E_i[\tilde{x}|y]-P_2-r_iVar_i[\tilde{x}|y]d_{i2}=0. \end{aligned}$$

(A16)

Express $d_{i2}$:

$$\begin{aligned} d_{i2}=\frac{E_i[\tilde{x}|y]-P_2}{r_iVar_i[\tilde{x}|y]}, \end{aligned}$$

(A17)

or

$$\begin{aligned} d_{i2} =\dfrac{m_i+\hat{w_i}(y-m_i)\nu ^{-1} -P_2}{r_i\frac{1}{\nu }(1-\frac{\hat{w_i}}{\nu })}. \end{aligned}$$

(A18)

Use the market clearing condition to find an equilibrium price:

$$\begin{aligned} \lambda _1d_{12}+\lambda _2d_{22}=1, \end{aligned}$$

(A19)

$$\begin{aligned} \lambda _1\dfrac{m_1+\hat{w_1}(y-m_1)\nu ^{-1} -P_2}{r_1\frac{1}{\nu }(1-\frac{\hat{w_1}}{\nu })}+\lambda _2\dfrac{m_2+\hat{w_2}(y-m_2)\nu ^{-1} -P_2}{r_2\frac{1}{\nu }(1-\frac{\hat{w_2}}{\nu })}=1. \end{aligned}$$

(A20)

Solve for the price:

$$\begin{aligned} P^*_2=\left[ \frac{\lambda _1\nu }{r_1(1-\frac{\hat{w_1}}{\nu })}+\frac{\lambda _2\nu }{r_2(1-\frac{\hat{w_2}}{\nu })}\right] ^{-1} \nonumber \\ \times \left[ (m_1+\hat{w_1}(y-m_1)\nu ^{-1})\frac{\lambda _1\nu }{r_1(1-\frac{\hat{w_1}}{\nu })}+(m_2+\hat{w_2}(y-m_2)\nu ^{-1})\frac{\lambda _2\nu }{r_2(1-\frac{\hat{w_2}}{\nu })}-1\right] . \end{aligned}$$

(A21)

The investor i’s demand in equilibrium:

$$\begin{aligned} d^*_{i2} =\frac{\psi _i(\hat{w_i})}{\lambda _i}\left[ E_i[\tilde{x}|y]-c\left( E_i[\tilde{x}|y]\psi _i(\hat{w_i})+E_j[\tilde{x}|y]\psi _j(\hat{w_j})-(\psi _i(\hat{w_i}))^{-1} \right) \right] , \end{aligned}$$

(A22)

where $c=\frac{\psi _i(\hat{w_i})}{\psi _i(\hat{w_i})+\psi _j(\hat{w_j})}$.

5.4 A.4 Proof of Proposition 5

Investor i’s problem:

$$\begin{aligned} maxE_i[W_{i3}]-\frac{1}{2}r_iVar_i[W_{i3}] \end{aligned}$$

(A23)

$$\begin{aligned} \text {s.t. }W_{i3}=xd_{i2}^*+q_{i2}^* . \end{aligned}$$

(A24)

From the budget constraint of the announcement period problem:

$$\begin{aligned} q_{i2}^*=q_{i1}^*+P_2^*d_{i1}^*-P_2^*d_{i2}^*. \end{aligned}$$

(A25)

The budget constraint in $t=1$ is

$$\begin{aligned} P_1d_{i1}+q_{i1}=W_{i0}=0, \end{aligned}$$

(A26)

where $q_{i1}$ and $d_{i1}$ are the amounts of riskless and risky assets hold in $t=1$. Plug (A25) and (A26) into (A24), express the terminal wealth:

$$\begin{aligned} W_{i3}=(P_2^*-P_1)d_{i1}+(x-P_2^*)d_{i2}^*. \end{aligned}$$

(A27)

Rewrite the problem of investor i:

$$\begin{aligned} max_{d_{i1}}E_i[W_{i3}]-\frac{1}{2}r_iVar_i[W_{i3}] \end{aligned}$$

(A28)

$$\begin{aligned} \text {s.t. }W_{i3}=(P_2^*-P_1)d_{i1}+(x-P_2^*)d_{i2}^*. \end{aligned}$$

(A29)

Plug (A29) into (A28):

$$\begin{aligned} max_{d_{i1}}E_i[(P_2^*-P_1)d_{i1}+(x-P_2^*)d_{i2}^*]-\frac{1}{2}r_iVar_i[(P_2^*-P_1)d_{i1}+(x-P_2^*)d_{i2}^*]. \end{aligned}$$

(A30)

Take the derivative with respect to $d_{i1}$ and set it equal to zero:

$$\begin{aligned} E_i[P^*_2]-P_1-r_iVar_i[P^*_2]d_{i1}=0. \end{aligned}$$

(A31)

Express $d_{i1}$:

$$\begin{aligned} d_{i1} =\dfrac{E_i[P_2^*] -P_1}{r_iVar_i[P_2^*]}. \end{aligned}$$

(A32)

Use the market clearing condition to find an equilibrium price:

$$\begin{aligned} \lambda _1d_{11}+\lambda _2d_{21}=1, \end{aligned}$$

(A33)

$$\begin{aligned} \lambda _1\dfrac{E_1[P_2^*] -P_1}{r_1Var_1[P_2^*]}+\lambda _2\dfrac{E_2[P_2^*] -P_1}{r_2Var_2[P_2^*]}= 1. \end{aligned}$$

(A34)

The equilibrium price is:

$$\begin{aligned} P^*_1=\frac{E_1[P_2^*]\frac{\lambda _1}{r_1Var_1[P^*_2]}}{\frac{\lambda _1}{r_1Var_1[P^*_2]}+\frac{\lambda _2}{r_2Var_2[P^*_2]}}+\frac{E_2[P_2^*]\frac{\lambda _2}{r_2Var_2[P^*_2]}}{\frac{\lambda _1}{r_1Var_1[P^*_2]}+\frac{\lambda _2}{r_2Var_2[P^*_2]}}-\frac{1}{\frac{\lambda _1}{r_1Var_1[P_2^*]}+\frac{\lambda _2}{r_2Var_2[P_2^*]}}, \end{aligned}$$

(A35)

where investor i’s expectation and variance of $P^*_2$, $E_i[P_2^*]$ and $Var_i[P_2^*]$, are of the following form:

$$\begin{aligned} E_i[P_2^*]=\int ^{\infty }_{-\infty } \frac{1}{\sqrt{2\pi \frac{\nu +n}{\nu n}}}exp(-\frac{1}{2\frac{\nu +n}{\nu n}}(y-m_i)^2)P_2^*dy, \end{aligned}$$

(A36)

$$\begin{aligned} Var_i[P_2^*]=\int ^{\infty }_{-\infty } \frac{1}{\sqrt{2\pi \frac{\nu +n}{\nu n}}}exp(-\frac{1}{2\frac{\nu +n}{\nu n}}(y-m_i)^2)[P_2^*]^2dy-[E_i[P_2^*]]^2. \end{aligned}$$

(A37)

The investor i’s demand in equilibrium:

$$\begin{aligned} d^*_{i1}=\frac{E_i[P^*_2]-P^*_1}{r_iVar_i[P^*_2]}. \end{aligned}$$

(A38)

5.5 A.5 Alternative specifications

In this section, we examine the robustness of our findings to various specification choices. Specifically, we run our tests for larger trading windows, normalize earnings surprise by the median analyst forecast rather than by the firm’s price, and consider a different data truncation.

Table 10 Polynomial regressions of excessive trading volume on the earnings surprise for different data specifications

Table 10 presents estimation results for the main specification (regression (1)) and alternative specifications that use different trading windows ([0,3] and [0,5] days) or normalize earnings surprise by median analyst forecast instead of the firm’s price. The polynomial that includes linear, quadratic, and fourth-order terms of earnings surprise generally is consistent across all specifications, and the coefficients in front of earnings surprise have consistent signs and relative magnitudes.

Next we examine the behavior of trading volume when including more extreme values of earnings surprise in the sample. In the main part of the paper, we truncate all the variables at the 1% level and earnings surprise at the 5% level. We choose to truncate earnings surprises to a greater extent because our model predicts an M-shape of trading volume (which is straightforward to test) for intermediate levels of earnings surprises and then trading volume goes to zero for more extreme surprises. This particular functional form is not obvious to test, as potential terms in the trading volume as a function of earnings surprise might include not only higher-order polynomials but also the inverse of earnings surprise or other nonlinear terms. To provide some evidence of the predicted shape of trading volume for the extended sample of earnings surprises, we plot a scatterplot of trading volume with a LOESS curve (Fig. 14) for the sample where we truncate all the variables, including earnings surprises, at the 1% level. The general pattern remains the same: trading volume increases for medium levels of earnings surprises and dampens for greater surprise levels.

Fig. 14

Scatterplot of residuals of excessive trading volume with LOESS smoother for a larger sample of earnings surprises