Abstract We propose and estimate a model of endogenous informed trading that is a hybrid of the PIN and Kyle models. When an informed trader trades optimally, both returns and order flows are needed to identify information asymmetry parameters. Empirical relationships between parameter estimates and price impacts and between parameter estimates and stochastic volatility are consistent with theory. We illustrate how the estimates can be used to detect information events in the time series and to characterize the information content of prices in the cross-section. We also compare the estimates to those from other models on various criteria. Received April 5, 2017; editorial decision September 21, 2017 by Editor Itay Goldstein. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. Information asymmetry is a fundamental concept in economics, but its estimation is challenging because private information is generally unobservable. Many proxies for information asymmetry exist including bid/ask spreads, price impacts, and estimates from structural models. In this paper, we study the identification of information asymmetry parameters in structural models. Structural modeling allows the econometrician to capture parameters related to the underlying economic mechanisms such as the probability and magnitude of private information events or the intensity of liquidity trading. Demand for plausible measures of information asymmetry is high because private information plays a key role in so many economic settings. Evidence of this demand is the large literature in finance and accounting that utilizes the probability of informed trade (PIN) measure of Easley et al. (1996) to proxy for information asymmetry.1 Our first contribution is to propose and solve a model of informed trading in securities markets that shares many features of the PIN model of Easley et al. (1996) but in which informed trading is endogenous like in Kyle (1985). We call this a hybrid PIN-Kyle model. In the paper, we study a binary signal following Easley et al. (1996), but the model can accommodate more general signal distributions. An important implication of the model is that order flows alone cannot identify information asymmetry. The intuition is quite simple. Consider, for example, a stock for which there is a large amount of private information and another for which there is only a small amount of private information. If it is anticipated that private information is more of a concern for the first stock than for the second, then the first stock will be less liquid, other things being equal. The lower liquidity will reduce the amount of informed trading, possibly offsetting the increase in informed trading due to greater private information. In equilibrium, the amount of informed trading may be the same in both stocks, despite the difference in information asymmetry. In general, the distribution of order flows need not reflect the degree of information asymmetry when liquidity providers react to information asymmetry and informed traders react to liquidity. Thus, we provide the first theoretical explanation of why methodologies that use order flows alone to estimate information asymmetry parameters, like PIN and Adjusted PIN (Duarte and Young, 2009), may not identify private information.2 Our second contribution is to develop novel estimates characterizing the information environment in financial markets. We structurally estimate our theoretical model for a panel of stocks and provide several validation checks that the estimated parameters are plausibly related to information asymmetry. First, reduced-form estimates of price impact are increasing in our structural estimates of the probability and magnitude of information events, as implied by theory. Second, the model implies that the magnitude of price changes is proportional to Kyle’s lambda, which depends on order flows and parameters of the model. Empirically, volatility over the latter part of a trading day is increasing in the conditional model-implied lambda, where the conditioning is based on cumulative order flows over the first part of the day and our estimated parameters. This phenomenon of stochastic volatility occurs in both the model and the data.3 To demonstrate potential applications of the estimates, we revisit two settings in which PIN estimates have been employed. One application of PIN has been to attempt to capture time-series variation in information asymmetry.4 We show that conditional probabilities of information events calculated using order flows and our parameter estimates rise on average around earnings announcements and are higher both pre- and post-announcement for announcements with larger absolute earnings surprises. Private information is more likely to be present around such announcements. Conditional probabilities are also elevated during block accumulations by Schedule 13D filers, which existing information asymmetry measures fail to detect (Collin-Dufresne and Fos, 2015). These results indicate that the model does capture time-series variation in information asymmetry. The second application illustrates how estimates of the information asymmetry parameters from our model can be used to augment studies concerned with cross-sectional differences in the information content of prices. To do so, we consider the hypothesis of Chen, Goldstein, and Jiang (2007) that corporate investment is more sensitive to market prices when there is more private information in prices. Our model allows us to measure the amount of private information alternatively by the frequency of private information events, by the magnitude of private information, and by the fraction of total price movement that is due to private information. We show that corporate investment is more sensitive to prices when any of these measures is higher. These measures of private information should prove useful in other settings in which researchers are interested in capturing distinct facets of the information environment (e.g., the amount of liquidity trading or the magnitude of private information). Related structural models of informed trading include the Adjusted PIN (APIN) model of Duarte and Young (2009), the Volume-Synchronized PIN (VPIN) model of Easley, López de Prado, and O’Hara (2012), and the modified Kyle model of Odders-White and Ready (2008). The APIN model allows for time variation in liquidity trading (with positively correlated buy and sell intensities), which provides a better fit to the empirical distribution of buys and sells. The VPIN model estimates buys and sells within a given time interval by assigning a fraction of total volume to buys and the remaining fraction to sells based on standardized price changes during the time interval.5Odders-White and Ready (2008; OWR) analyze a Kyle model in which the probability of an information event is less than 1, as it is in our model. However, they analyze a single-period model, whereas we study a dynamic model. Unlike our dynamic model in which prices equal conditional expectations, market makers in their model only match unconditional means of prices to unconditional means of asset values.6 Our estimate of the probability of an information event is not positively correlated in the cross-section with estimates from the other models. The divergence between the estimates is not surprising, because the models have different assumptions/implications regarding what data is required to identify the probability of an information event.7 We also calculate a composite measure of information asymmetry in our model: the expected average lambda. This measure incorporates both the probability and the magnitude of information events, as well as the amount of liquidity trading. Unlike the probability of an information event, the expected average lambda from our model is positively correlated with similar measures from other models (PIN, APIN, VPIN, and the OWR lambda). Each of these measures should be increasing in the probability of an information event, so it is surprising that they are all positively correlated, given the lack of correlation of the ‘probability of an information event’ estimates. However, the measures are also decreasing in the amount of liquidity trading, and we present evidence in Section 4 that the measurement of liquidity trading is quite positively correlated across models, resulting in the positive correlation of the composite measures. Of course, applications of the measures generally assume that they are correlated with private information, not just inversely correlated with liquidity trading. Theory predicts that orders have larger price impacts and quoted spreads when information asymmetry is more severe.8 This is true in both the Kyle (1985) model, on which the hybrid and OWR models are based, and the Glosten and Milgrom (1985) model, on which PIN models are based. To test this implication of theory, we examine reduced-form price impacts for our sample as well as quoted spreads. Empirically, expected average lambda from the hybrid model is positively correlated with price impacts and quoted spreads both in the time series and cross-sectionally. While the same is also true for PIN, APIN, VPIN, and the OWR lambda, expected average lambda has a higher correlation with price impacts and spreads in the time series than do the other composite measures. Expected average lambda also adds explanatory power relative to the other measures in cross-sectional regressions of price impacts or quoted spreads on the composite measures. Other related theoretical work includes Rossi and Tinn (2010), Foster and Viswanathan (1995), Chakraborty and Yilmaz (2004), Goldstein and Guembel (2008), Banerjee and Breon-Drish (2017), and Wang and Yang (2017). Rossi and Tinn solve a two-period Kyle model in which there are two large traders, one of whom is certainly informed and one of whom may or may not be informed. In their model, unlike ours, there are always information events. Foster and Viswanathan (1995) consider a series of single-period Kyle models in which traders choose in each period whether to pay a fee to become informed. There may be periods in which there are no informed traders. However, in their model, it is always common knowledge how many traders choose to become informed, so, in contrast to our model, there is no learning from orders about whether informed traders are present. Chakraborty and Yilmaz (2004) and Goldstein and Guembel (2008) study discrete-time Kyle models in which there may or may not be an information event. The main result in Chakraborty and Yilmaz (2004) is that the informed trader will manipulate (sometimes buying when she has bad information and/or selling when she has good information) if the horizon is sufficiently long. The primary difference between their model and ours is that they assume that the liquidity trade distribution has finite support, so market makers may incorrectly rule out a type of trader if the horizon is sufficiently long. In contrast, market makers in our model can never rule out any type of the informed trader until the end of the model, so it does not strictly pay for a low type to pretend to be a high type or vice versa. The primary focus of Goldstein and Guembel (2008) concerns the incentives for an uninformed strategic trader to manipulate if information in financial markets feeds back into managers’ investment decisions. In their benchmark equilibrium with no feedback, the uninformed speculator behaves as a contrarian but does not manipulate, which is the case in our equilibrium. Banerjee and Breon-Drish (2017) and Wang and Yang (2017) study continuous-time Kyle models (specifically, the model of Back and Baruch (2004) in which there is a random announcement date) in which an informed trader may not be present. Banerjee and Breon-Drish study the information acquisition decision, treating it as a real option. In one version of their model, the timing of information acquisition is publicly observed. In that version, the market is infinitely deep before information is acquired, and the model is essentially the same as in Back and Baruch after information is acquired. In a second version of their model, the timing of information acquisition is not publicly observed, and the market tries to learn from orders whether information has been acquired. For that version, they establish a nonexistence result: In the class of pricing rules they consider, there is no equilibrium. Wang and Yang also study the Back-Baruch version of the Kyle model. In their model, nature chooses at date 0 whether there is an information event (and all information events are “good news” events). Unlike in our model or the model of Banerjee and Breon-Drish, the strategic trader is not present in their model when there is no information event.9 They also show the nonexistence of equilibria (though they have an existence result for a second version of their model in which the market maker is a monopolist). 1. The Hybrid Model The hybrid model includes two important features of PIN models—a probability less than 1 of an information event and a binary asset value conditional on an information event—and it also includes an optimizing (possibly) informed trader, like in the Kyle (1985) model. Denote the time horizon for trading by $$[0,1]$$. Assume there is a single risk-neutral strategic trader. Assume this trader receives a signal $$S \in \{L,H\}$$ at time 0 with probability $$\alpha$$, where $$L<0<H$$.10 Let $$p_L$$ and $$p_H=1-p_L$$ denote the probabilities of low and high signals, respectively, conditional on an information event. With probability $$1-\alpha$$, there is no information event, and the trader also knows when this happens. Let $$\xi$$ denote an indicator for whether an information event has occurred ($$\xi=1$$ if yes and $$\xi=0$$ if no). In addition to the private information, public information can also arrive during the course of trading, represented by a martingale $$V$$. The possible private information—whether there was an information event and, if so, whether the signal was low or high—becomes public information after the close of trading at date 1, producing an asset value of $$V_1 + \xi S$$. Without loss of generality, we take the signal $$S$$ to have a zero mean. We can always do this by taking the signal mean to be part of the public information $$V_0$$. In addition to the strategic trades, there are liquidity trades represented by a Brownian motion $$Z$$ with zero drift and instantaneous standard deviation $$\sigma$$. Let $$X_t$$ denote the number of shares held by the strategic trader at date $$t$$ (taking $$X_0=0$$ without loss of generality), and set $$Y_t=X_t+Z_t$$. The processes $$Y$$ and $$V$$ are observed by market makers. Denote the information of market makers at date $$t$$ by $$\mathcal{F}^{V,Y}_t$$. One requirement for equilibrium in this model is that the price equal the expected value of the asset conditional on the market makers’ information and given the trading strategy of the strategic trader: \begin{equation} \label{eq1} P_t = {\mathsf{E}} \left[V_{1} + \xi S \mid \mathcal{F}_t^{V,Y}\right] = V_t + {\mathsf{E}} \left[\xi S \mid \mathcal{F}_t^{V,Y}\right]\,. \end{equation} (1) We will show that there is an equilibrium in which $$P_t = V_t + p(t,Y_t)$$ for a function $$p$$. This means that the expected value of $$\xi S$$ conditional on market makers’ information depends only on cumulative orders $$Y_t$$ and not on the entire history of orders. The other requirement for equilibrium is that the strategic trades are optimal. Let $$\theta_t$$ denote the trading rate of the strategic trader (i.e., $$\mathrm{d} X_t = \theta_t\,\mathrm{d} t$$). The process $$\theta$$ has to be adapted to the information possessed by the strategic trader, which is $$V$$, $$\xi S$$, and the history of $$Z$$ (in equilibrium, the price reveals $$Z$$ to the informed trader). The strategic trader chooses the rate to maximize \begin{equation}\label{expectedprofit} {\mathsf{E}} \int_0^1 \left[V_{1} + \xi S - P_t\right]\theta_t\,\mathrm{d} t = {\mathsf{E}} \int_0^1 \left[\xi S - p(t,Y_t)\right]\theta_t\,\mathrm{d} t\,, \end{equation} (2) with the function $$p$$ being regarded by the informed trader as exogenous. In the optimization, we assume that the strategic trader is constrained to satisfy the “no doubling strategies” condition introduced in Back (1992), meaning that the strategy must be such that $${\mathsf{E}} \int_0^1 p(t,Y_t)^2 \,\mathrm{d} t < \infty$$ with probability 1. Let $${\rm{N}}$$ denote the standard normal distribution function, and let $${\rm{n}}$$ denote the standard normal density function. Set $$y_L = \sigma{\rm{N}}^{-1}(\alpha p_L)$$ and $$y_H = \sigma{\rm{N}}^{-1}(1-\alpha p_H)$$. This means that the probability mass in the lower tail $$(-\infty,y_L)$$ of the distribution of cumulative liquidity trades $$Z_1$$ equals $$\alpha p_L$$, which is the unconditional probability of bad news. Likewise, the probability mass in the upper tail $$(y_H,\infty)$$ of the distribution of $$Z_1$$ equals $$\alpha p_H$$, which is the unconditional probability of good news. Set \begin{equation} q(t,y,s) = \begin{cases} {\mathsf{E}}[Z_1 -Z_t \mid Z_t = y, Z_1 < y_L] & \text{if $s=L$}\,,\\ {\mathsf{E}}[Z_1 -Z_t \mid Z_t = y, y_L \leq Z_1 \leq y_H] & \text{if $s=0$}\,,\\ {\mathsf{E}}[Z_1 -Z_t \mid Z_t = y, Z_1 > y_H] & \text{if $s=H$}\,. \end{cases} \end{equation} (3) From the standard formula for the mean of a truncated normal, we obtain the following more explicit formula for $$q$$: \begin{equation} \label{thetaformula} \hspace{-0cm}\frac{q(t,y,s)}{\sigma\sqrt{1-t}} = \begin{cases} -{\rm{n}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right)/{\rm{N}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right) & \hspace{-1.2cm}\text{if $s=L$}\,,\\ \left.\left[{\rm{n}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right) - {\rm{n}}\left(\frac{y_H-y}{\sigma\sqrt{1-t}}\right)\right]\right/\left[{\rm{N}}\left(\frac{y_H-y}{\sigma\sqrt{1-t}}\right) - {\rm{N}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right)\right] & \\ & \hspace{-1.2cm} \text{if $s=0$}\,,\\ {\rm{n}}\left(\frac{y-y_H}{\sigma\sqrt{1-t}}\right)/{\rm{N}}\left(\frac{y-y_H}{\sigma\sqrt{1-t}}\right) & \hspace{-1.2cm} \text{if $s=H$}\,. \end{cases} \end{equation} (4) The equilibrium described in Theorem 1 below can be shown to be the unique equilibrium in a certain broad class, following Back (1992). The proof of Theorem 1 is given in Appendix A.11 Theorem 1. There is an equilibrium in which the trading rate of the strategic trader is \begin{equation}\label{thm_trade} \theta_t = \frac{q(t,Y_t,\xi S)}{1-t} \,. \end{equation} (5) Given market makers’ information at any date $$t$$, the conditional probability of an information event with a low signal is $${\rm{N}}\left(\frac{y_L-Y_t}{\sigma\sqrt{1-t}}\right)$$ and the conditional probability of an information event with a high signal is $${\rm{N}}\left(\frac{Y_t-y_H}{\sigma\sqrt{1-t}}\right)$$. The equilibrium asset price is $$P_t = V_t + p(t,Y_t)$$, where the pricing function $$p$$ is given by \begin{equation} \label{thm_price} p(t,y) = L\cdot {\rm{N}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right) + H \cdot {\rm{N}}\left(\frac{y-y_H}{\sigma\sqrt{1-t}}\right)\,. \end{equation} (6) In this equilibrium, the process $$Y$$ is a martingale given market makers’ information and has the same unconditional distribution as does the liquidity trade process $$Z$$; that is, it is a Brownian motion with zero drift and standard deviation $$\sigma$$. The last statement of the theorem implies that the distribution of order flows in the model does not depend on the information asymmetry parameters $$\alpha$$, $$H$$, and $$L$$. Thus, if the model is correct, it is impossible to estimate those parameters using order flows alone. In general, the theorem suggests that it may be difficult to identify information asymmetry parameters using order flows alone, as discussed in the Introduction and Section 1.1. When we estimate the hybrid model, we use both order flows and returns, in contrast to related models that only use order flows. Empirically, we test the relationship between $$\alpha$$ and price impacts of trades. Figure 1 plots the equilibrium price as a function of $$Y_t$$ for two different values of $$\alpha$$. It shows that the price is more sensitive to orders when $$\alpha$$ is larger. To investigate further how the sensitivity of prices to orders depends on $$\alpha$$ in the hybrid model, we calculate the price sensitivity—that is, we calculate Kyle’s lambda. Figure 1 View largeDownload slide The equilibrium price $$V_t + p(t,Y_t)$$ as a function of the order imbalance $$Y_t$$ The parameter values are $$t=0.5$$, $$V_t=50$$, $$H=10$$, $$L=-10$$, $$\sigma=1$$, and $$p_H=p_L=1/2$$. Figure 1 View largeDownload slide The equilibrium price $$V_t + p(t,Y_t)$$ as a function of the order imbalance $$Y_t$$ The parameter values are $$t=0.5$$, $$V_t=50$$, $$H=10$$, $$L=-10$$, $$\sigma=1$$, and $$p_H=p_L=1/2$$. Theorem 2. In the equilibrium of Theorem 1, the asset price evolves as $$\mathrm{d} P_t = \mathrm{d} V_t + \lambda (t,Y_t) \,\mathrm{d} Y_t$$, where Kyle’s lambda is \begin{equation} \label{thm_lambda} \lambda(t,y) = -\frac{L}{\sigma\sqrt{1-t}}\cdot {\rm{n}}\left(\frac{y_L-y}{\sigma\sqrt{1-t}}\right) + \frac{H}{\sigma\sqrt{1-t}}\cdot {\rm{n}}\left(\frac{y_H-y}{\sigma\sqrt{1-t}}\right)\,. \end{equation} (7) Furthermore, Kyle’s lambda $$\lambda(t,Y_t)$$ is a martingale with respect to market makers’ information on the time interval $$[0,1)$$. Kyle’s lambda is a stochastic process in our model, but we can easily relate the expected average lambda to $$\alpha$$. Because lambda is a martingale, the expected average lambda is $$\lambda(0,0)$$. Substitute the definitions of $$y_L$$ and $$y_H$$ in (7) to compute12 \begin{equation} \label{exp_avg_lambda} \lambda(0,0) = -\frac{L}{\sigma}{\rm{n}}\left({\rm{N}}^{-1}(\alpha p_L)\right) + \frac{H}{\sigma}{\rm{n}}\left({\rm{N}}^{-1}(1-\alpha p_H)\right)\,. \end{equation} (8) Figure 2 plots the expected average lambda as a function of $$\alpha$$ for two values of $$H$$, taking $$L=-H$$. Doubling the signal magnitudes doubles lambda. Furthermore, the expected average lambda is increasing in $$\alpha$$. Figure 2 View largeDownload slide Expected average lambda (8) as a function of $$\alpha$$ The parameter values are $$\sigma = 1$$, $$p_L=p_H=1/2$$, and $$L=-H$$. Figure 2 View largeDownload slide Expected average lambda (8) as a function of $$\alpha$$ The parameter values are $$\sigma = 1$$, $$p_L=p_H=1/2$$, and $$L=-H$$. 1.1 Nonidentifiability using order flows alone A key result of Theorem 1 is that the aggregate order imbalance $$Y_1$$ has the same distribution as the liquidity trades $$Z_1$$ and is invariant with respect to the information asymmetry parameters.13 Further insight into this identification issue can be gained by noting that the unconditional distribution of the order imbalance in our model is a mixture of three conditional distributions. With probability $$\alpha p_L$$, $$Y_1$$ is drawn from the distribution conditional on a low signal; with probability $$\alpha p_H$$, $$Y_1$$ is drawn from the distribution conditional on a high signal; and with probability $$1-\alpha$$, $$Y_1$$ is drawn from the distribution conditional on no information event. The first two distributions have nonzero means—there is an excess of sells over buys in the first and an excess of buys over sells in the second. One might conjecture that changing $$\alpha$$—thereby changing the likelihood of drawing from the first two distributions—will alter the unconditional distribution of $$Y_1$$. If so, then one could perhaps identify $$\alpha$$ from the distribution of $$Y_1$$. In other models with a potential information event, it is indeed true that changing $$\alpha$$, holding other parameters constant, alters the unconditional distribution of the order imbalance. However, it is not true in our model, because the distribution of informed trades in our model endogenously depends on $$\alpha$$ due to liquidity depending on $$\alpha$$. With a larger alpha, the market is less liquid (see the comparative statics in Figure 2) and the informed trader trades less aggressively. Furthermore, with endogenous informed orders, the arrival rate of informed orders depends on prior price changes as shown in Figure 3, which is not the case in other models with a potential information event. In particular, when prices have moved in the direction of the news, informed orders slow down, and, when prices have moved in the opposite direction, informed orders speed up. Figure 3 shows that these changes in intensity depend on the ex ante probability $$\alpha$$ of an information event. Thus, the distributions over which we are mixing change when the mixture probabilities change, leaving the unconditional distribution of $$Y_1$$ invariant with respect to $$\alpha$$. Figure 3 View largeDownload slide The equilibrium informed trading rate $$\theta_t$$ as a function of the price $$V_t + p(t,Y_t)$$ The parameter values are $$t=0.5$$, $$\xi S = H$$, $$V_t=50$$, $$H=10$$, $$L=-10$$, $$\sigma=1$$, and $$p_H=p_L=1/2$$. Figure 3 View largeDownload slide The equilibrium informed trading rate $$\theta_t$$ as a function of the price $$V_t + p(t,Y_t)$$ The parameter values are $$t=0.5$$, $$\xi S = H$$, $$V_t=50$$, $$H=10$$, $$L=-10$$, $$\sigma=1$$, and $$p_H=p_L=1/2$$. The change in the conditional distributions is illustrated in Figure 4. The top and bottom panels of Figure 4 show that the strategic trader trades more aggressively when an information event occurs if an information event is less likely ($$\alpha=0.1$$ versus $$\alpha=0.5$$). The unconditional distribution of $$Y_1$$ is standard normal for both $$\alpha=0.1$$ and $$\alpha=0.5$$ in Figure 4, so we cannot hope to use the unconditional distribution to recover $$\alpha$$. Figure 4 View largeDownload slide The conditional density function of the net order flow $$Y_1$$ The density is conditional on a low signal, no information event, or a high signal. The parameter values are $$\sigma=1$$ and $$p_L=p_H=1/2$$. Figure 4 View largeDownload slide The conditional density function of the net order flow $$Y_1$$ The density is conditional on a low signal, no information event, or a high signal. The parameter values are $$\sigma=1$$ and $$p_L=p_H=1/2$$. Of course, identifying the information asymmetry parameters from the distribution of order imbalances is a very different issue from using order imbalances to update the probability of an information event in a particular instance of the model. Conditional on knowledge of the parameters, the order imbalance does help in estimating whether an information event occurred in a particular instance of the model; in fact, the market makers in the model update their beliefs regarding the occurrence of an information event based on the order imbalance. So, we can compute $$\text{prob} (\text{info event} \mid Y_t, \text{parameters})\,,$$ and this probability does depend on the information asymmetry parameters. We could use this to identify the information asymmetry parameters if we had data on order imbalances and data on whether information events occurred. Of course, we generally do not have data of the latter type. Theorem 1 shows that the likelihood function of the information asymmetry parameters given only data on order imbalances is a constant function of those parameters; hence, the order imbalances alone cannot identify them. In our empirical work, we estimate the model parameters using prices and order flows. Armed with these parameter estimates and order flow observations, we can compute conditional probabilities of an information event. We examine their time-series properties around earnings announcements and around Schedule 13D filer trades in Section 3.1. 1.2 The contrarian trader assumption One way in which our model departs from related models like the PIN model is that the strategic trader is present in our model even when there is no information event. When there is no information event, this trader behaves as a contrarian, selling on price increases and buying on price declines.14 The existence of such a contrarian trader seems likely if there are always some traders who are best informed—corporate managers, for example. This would be the case if information were truly idiosyncratic to the firm. If, on the other hand, there is an industry or other aggregate components to the information, then it is possible that no one knows when no one else has information. In that case, the contrarian trader that we posit would not exist. In Internet Appendix B, we solve a variant of the PIN model in which contrarian traders arrive at the market when there is no information event. The contrarian traders condition their trading direction on the prevailing bid and ask quotes and the intrinsic value of the asset. The distribution of order imbalances in that model is shown in Figure 5 for three different values of $$\alpha$$ (the probability of an information event). The figure shows that the distribution depends on $$\alpha$$; thus, order imbalances can be used to identify information asymmetry in the PIN model even when a contrarian trader is present. Thus, the contrarian trader assumption is not the main driving force behind our nonidentifiability result. Instead, the result depends on market makers reacting to information asymmetry and on strategic traders reacting both to liquidity and to price changes. That is, order flows depend on market liquidity, which depends on information asymmetry. This creates an indirect dependence of order flows on information asymmetry that is countervailing to the direct relation. Figure 5 View largeDownload slide The simulated distribution of order imbalances for a variant of the Easley et al. (1996) model in which contrarian traders arrive in the event of no information The model is described in Internet Appendix B. Order imbalance is the number of buys minus number of sells. The histograms plot 50,000 instances of the model. The parameter values are $$\alpha \in \{0.25,0.5,0.75\}$$, $$p_L=0.5$$, $$\varepsilon=10$$, $$\mu=10$$, $$L = -1$$, $$H = 1$$, and $$V^* = 0$$. Figure 5 View largeDownload slide The simulated distribution of order imbalances for a variant of the Easley et al. (1996) model in which contrarian traders arrive in the event of no information The model is described in Internet Appendix B. Order imbalance is the number of buys minus number of sells. The histograms plot 50,000 instances of the model. The parameter values are $$\alpha \in \{0.25,0.5,0.75\}$$, $$p_L=0.5$$, $$\varepsilon=10$$, $$\mu=10$$, $$L = -1$$, $$H = 1$$, and $$V^* = 0$$. 2. Estimation of the Model We estimate the hybrid model using trade and quote data from TAQ for NYSE firms from 1993 through 2012.15 We sign trades as buys and sells using the Lee and Ready (1991) algorithm: trades above (below) the prevailing quote midpoint are considered buys (sells). If a trade occurs at the midpoint, then the trade is classified as a buy (sell) if the trade price is greater (less) than the previous differing transaction price.16 We sample prices and order imbalances hourly and at the close and define order imbalances as shares bought less shares sold (denoted in thousands of shares). We estimate the model by maximum likelihood, maintaining the standard assumptions in the literature that each day is a separate realization of the model and that parameters are constant within each year for each stock. We assume that the dispersion of the possible signals on each day $$i$$ is proportional to the observed opening price on day $$i$$, $$P_{i0}$$. Specifically, we assume that, for each firm-year, there is a parameter $$\kappa$$ such that the low signal value each day is $$L=-2p_H\kappa P_{i0}$$ and the high signal value is $$H=2p_L\kappa P_{i0}$$. This construction ensures that the signal has a zero mean and $$(H-L)/P_{i0} = 2\kappa$$. Thus, $$\kappa$$ measures the signal magnitude. We also assume that the public information process $$V$$ is a geometric Brownian motion on each day with a constant volatility $$\Delta$$. The likelihood function for the hybrid model depends on the signal magnitude $$\kappa$$, the probability $$\alpha$$ of information events, the probability $$p_L$$ of a negative signal conditional on an information event, the standard deviation $$\sigma$$ of liquidity trading, and the volatility $$\Delta$$ of public information. We derive the likelihood function for the model in Appendix B. Dropping constants, the log-likelihood function $$\mathcal{L}$$ for an observation period of $$n$$ days satisfies \begin{align}\label{-L} - \mathcal{L} &= n(k+1)\log \sigma + \frac{1}{2\sigma^2\Delta} \sum_{i=1}^n Y_i'\Sigma^{-1}Y_i + n(k+1) \log \Delta \notag\\ &\quad+ \frac{1}{2\Delta^2\Delta } \sum_{i=1}^n U_i'\Sigma^{-1}U_i + \frac{n\Delta^2}{8}+ \sum_{i=1}^n \left(\sum_{j=1}^k U_{ij} + \frac{3}{2}U_{i,k+1}\right)\,, \end{align} (9) where $$k$$ is the number of intraday observations sampled at regular intervals of length $$\Delta$$. We sample every hour and at the close, so $$k=6$$ and $$\Delta = 1/6.5$$. $$Y_i$$ is the vector of cumulative order flows for day $$i$$. $$U_i$$ is the vector $$(U_{i1},\ldots, U_{i,k+1})'$$ of log pricing differences \begin{equation} \label{Uij} U_{ij} = \log\left(\frac{P_{ij}}{P_{i0}} - p(t_j,Y_{ij})\right) \end{equation} (10) between the observed return and the model’s pricing function. $$\Sigma$$ is a $$(k+1)\times (k+1)$$ matrix that depends on $$\Delta$$ as described in Appendix B. We minimize (9) in $$\alpha$$, $$\kappa$$, $$p_L$$, $$\sigma$$, and $$\Delta$$. The private information parameters $$\alpha$$, $$\kappa$$, and $$p_L$$ enter the likelihood function via the log pricing errors $$U_i$$, because the parameters affect the pricing function $$p(t,Y_t)$$. As can be seen from (9), $$\alpha$$, $$\kappa$$, and $$p_L$$ are estimated by minimizing a quadratic function of the log pricing errors. In the model, the pricing errors are due to public information. In minimizing the quadratic function, the estimation procedure tries to maximize the fit of the model prices $$p(t_j,Y_{ij})$$ to the observed returns and thereby to minimize how much we have to rely on public information to explain the returns. Figure 6 illustrates how the pricing errors depend on the private information parameters. For simplicity, Figure 6 treats the case $$k=0$$; that is, it only uses daily order imbalances and returns. The pricing error each day is the difference between the daily return $$P_1/P_0$$ and the model price $$p(1,Y_1)$$. The price function $$p(1,\cdot)$$ is a step function,17 with steps at $$y_L$$ and $$y_H$$ defined in Section 1 as $$y_L = \sigma{\rm{N}}^{-1}(\alpha p_L)$$ and $$y_H = \sigma{\rm{N}}^{-1}(1-\alpha p_H)$$. Thus, $$\alpha$$ and $$p_L$$ affect the step locations. If $$\alpha$$ is larger, the step locations are closer together. If $$p_L$$ is increased, both step locations shift to the right. The parameter $$\kappa$$ determines the height of the steps. Notice that $$\sigma$$ and $$\alpha$$ play similar roles in determining the step locations; either increasing $$\sigma$$ or decreasing $$\alpha$$ will spread out the steps. However, maximizing the likelihood function also involves fitting the order imbalances to a Brownian motion with standard deviation $$\sigma$$. Table 2 (see Section 2.1) shows that our empirical estimates of $$\sigma$$ are almost entirely determined by the standard deviations of order imbalances—likewise, the estimates of $$\Delta$$ (the standard deviation of the public information process) are almost entirely determined by the standard deviations of returns. Figure 6 View largeDownload slide Returns, order flows, and log pricing differences for various parameters Simulations of 1,000 instances of the hybrid model. The data-generating parameters are $$\alpha=0.5$$, $$\kappa=0.015$$, $$p_L=0.5$$, $$\sigma=0.1$$, $$\Delta=0.01$$. Standardized order flows are on the horizontal axis. The left column plots end-of-day net returns, $$P_1/P_0 - 1$$, and the pricing function, $$p(1,Y_1)$$. The right column plots log pricing differences, $$U_1=\ln(P_1/P_0 - p(1,Y_1))$$. The pricing function $$p(1,Y_1)$$ depends on the indicated hatted parameters in each panel caption. Each row plots the pricing function and log pricing differences for different parameter estimates (hatted values). The vertical lines indicate the thresholds $$y_L/\sigma$$ and $$y_H/\sigma$$ for the true parameters. The first row uses parameter estimates in which $$\alpha$$ and $$\kappa$$ are too low relative to the true parameters. These generate log pricing differences that are positively correlated with order flows. The second row uses the data-generating parameters. The log pricing differences are uncorrelated with order flows. The third row uses parameter estimates in which $$\alpha$$ and $$\kappa$$ are too high relative to the true parameters. These generate log pricing differences that are negatively correlated with order flows. Figure 6 View largeDownload slide Returns, order flows, and log pricing differences for various parameters Simulations of 1,000 instances of the hybrid model. The data-generating parameters are $$\alpha=0.5$$, $$\kappa=0.015$$, $$p_L=0.5$$, $$\sigma=0.1$$, $$\Delta=0.01$$. Standardized order flows are on the horizontal axis. The left column plots end-of-day net returns, $$P_1/P_0 - 1$$, and the pricing function, $$p(1,Y_1)$$. The right column plots log pricing differences, $$U_1=\ln(P_1/P_0 - p(1,Y_1))$$. The pricing function $$p(1,Y_1)$$ depends on the indicated hatted parameters in each panel caption. Each row plots the pricing function and log pricing differences for different parameter estimates (hatted values). The vertical lines indicate the thresholds $$y_L/\sigma$$ and $$y_H/\sigma$$ for the true parameters. The first row uses parameter estimates in which $$\alpha$$ and $$\kappa$$ are too low relative to the true parameters. These generate log pricing differences that are positively correlated with order flows. The second row uses the data-generating parameters. The log pricing differences are uncorrelated with order flows. The third row uses parameter estimates in which $$\alpha$$ and $$\kappa$$ are too high relative to the true parameters. These generate log pricing differences that are negatively correlated with order flows. Table 2 Hybrid model parameter estima tes and moments of order flow and returns A. Standardized Regression $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) –0.129*** 0.007 –0.089*** 0.986*** –0.000 (–5.57) (0.38) (–6.17) (135.67) (–0.02) sd($$R$$) 0.155*** 0.460*** 0.016 –0.007 0.963*** (5.15) (7.89) (1.39) (–1.46) (138.47) skew(OIB) 0.007 0.003 –0.058*** 0.003 0.006* (1.02) (0.39) (–6.11) (0.79) (1.69) skew($$R$$) –0.008 0.009 0.047*** –0.001 0.005* (–1.05) (1.51) (4.33) (–0.41) (1.95) $$\text{corr}(R_1,\text{OIB}_1)$$ 0.258*** 0.484*** –0.018 0.009 0.039*** (5.40) (17.25) (–0.80) (1.26) (2.96) $$\text{corr}(R_1,\text{OIB}^2_1)$$ –0.039*** –0.018 0.185*** –0.003 –0.008* (–3.16) (–1.29) (5.73) (–1.12) (–1.92) $$\text{corr}(R_2,\text{OIB}_2)$$ 0.218*** 0.314*** –0.034 –0.012** –0.022** (6.10) (14.92) (–1.26) (–2.14) (–1.97) $$\text{corr}(R_2,\text{OIB}^2_2)$$ –0.049*** –0.028** 0.099*** –0.001 –0.009** (–5.79) (–2.04) (4.19) (–0.41) (–2.52) # right tail OIB & $$R$$ –0.122*** –0.103*** –0.128*** 0.011* –0.074*** (–4.17) (–5.59) (–3.86) (1.76) (–5.95) # left tail OIB & $$R$$ –0.163*** –0.063*** 0.029 0.005 0.012* (–7.39) (–6.66) (1.38) (0.65) (1.67) Constant 2.159*** –0.482*** 3.439*** 0.068*** 0.118*** (17.04) (–4.53) (60.66) (3.56) (5.39) Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^{2}$$ 0.152 0.680 0.040 0.978 0.938 B. Variance Decomposition $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) 0.125 0.000 0.127 1.000 0.000 sd($$R$$) 0.237 0.636 0.005 0.000 0.997 skew(OIB) 0.000 0.000 0.075 0.000 0.000 skew($$R$$) 0.001 0.000 0.047 0.000 0.000 $$\text{corr}(R_1,\text{OIB}_1)$$ 0.221 0.240 0.002 0.000 0.001 $$\text{corr}(R_1,\text{OIB}^2_1)$$ 0.009 0.001 0.458 0.000 0.000 $$\text{corr}(R_2,\text{OIB}_2)$$ 0.159 0.101 0.008 0.000 0.000 $$\text{corr}(R_2,\text{OIB}^2_2)$$ 0.016 0.002 0.137 0.000 0.000 # right tail OIB & $$R$$ 0.055 0.012 0.128 0.000 0.002 # left tail OIB & $$R$$ 0.176 0.008 0.012 0.000 0.000 Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^2$$ 0.152 0.680 0.040 0.978 0.938 A. Standardized Regression $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) –0.129*** 0.007 –0.089*** 0.986*** –0.000 (–5.57) (0.38) (–6.17) (135.67) (–0.02) sd($$R$$) 0.155*** 0.460*** 0.016 –0.007 0.963*** (5.15) (7.89) (1.39) (–1.46) (138.47) skew(OIB) 0.007 0.003 –0.058*** 0.003 0.006* (1.02) (0.39) (–6.11) (0.79) (1.69) skew($$R$$) –0.008 0.009 0.047*** –0.001 0.005* (–1.05) (1.51) (4.33) (–0.41) (1.95) $$\text{corr}(R_1,\text{OIB}_1)$$ 0.258*** 0.484*** –0.018 0.009 0.039*** (5.40) (17.25) (–0.80) (1.26) (2.96) $$\text{corr}(R_1,\text{OIB}^2_1)$$ –0.039*** –0.018 0.185*** –0.003 –0.008* (–3.16) (–1.29) (5.73) (–1.12) (–1.92) $$\text{corr}(R_2,\text{OIB}_2)$$ 0.218*** 0.314*** –0.034 –0.012** –0.022** (6.10) (14.92) (–1.26) (–2.14) (–1.97) $$\text{corr}(R_2,\text{OIB}^2_2)$$ –0.049*** –0.028** 0.099*** –0.001 –0.009** (–5.79) (–2.04) (4.19) (–0.41) (–2.52) # right tail OIB & $$R$$ –0.122*** –0.103*** –0.128*** 0.011* –0.074*** (–4.17) (–5.59) (–3.86) (1.76) (–5.95) # left tail OIB & $$R$$ –0.163*** –0.063*** 0.029 0.005 0.012* (–7.39) (–6.66) (1.38) (0.65) (1.67) Constant 2.159*** –0.482*** 3.439*** 0.068*** 0.118*** (17.04) (–4.53) (60.66) (3.56) (5.39) Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^{2}$$ 0.152 0.680 0.040 0.978 0.938 B. Variance Decomposition $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) 0.125 0.000 0.127 1.000 0.000 sd($$R$$) 0.237 0.636 0.005 0.000 0.997 skew(OIB) 0.000 0.000 0.075 0.000 0.000 skew($$R$$) 0.001 0.000 0.047 0.000 0.000 $$\text{corr}(R_1,\text{OIB}_1)$$ 0.221 0.240 0.002 0.000 0.001 $$\text{corr}(R_1,\text{OIB}^2_1)$$ 0.009 0.001 0.458 0.000 0.000 $$\text{corr}(R_2,\text{OIB}_2)$$ 0.159 0.101 0.008 0.000 0.000 $$\text{corr}(R_2,\text{OIB}^2_2)$$ 0.016 0.002 0.137 0.000 0.000 # right tail OIB & $$R$$ 0.055 0.012 0.128 0.000 0.002 # left tail OIB & $$R$$ 0.176 0.008 0.012 0.000 0.000 Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^2$$ 0.152 0.680 0.040 0.978 0.938 The dependent variables are the estimated parameters from the hybrid model. The explanatory variables are various moments of order flows and returns. The unit of observation is a firm-year. OIB denotes the cumulative order flow over the full day. OIB$$_1$$ and OIB$$_2$$ are the order flows over the first 3 and last 3.5 hours of the trading day. Similarly, $$R$$ is the return over the full day, and $$R_1$$ and $$R_2$$ are returns over the first 3 and last 3.5 hours of the trading day. The indicated moments of these variables are calculated across days for each firm-year. # Right Tail OIB &$$R$$ is the fraction of days where both OIB $$> \text{sd}(\text{OIB})$$ and $$R - 1 > \text{sd}(R)$$. # Left Tail OIB &$$R$$ is the fraction of days where both OIB $$< - \text{sd}(\text{OIB})$$ and $$R - 1 < - \text{sd}(R)$$. Panel A reports estimates where all variables are standardized to have a unit standard deviation. Standard errors are clustered by firm and year. $$t$$-statistics are in parentheses, and statistical significance is represented by * $$p<0.10$$, ** $$p<0.05$$, and *** $$p<0.01$$. Panel B reports variance decompositions. Each number in panel B represents the fraction of the model’s total partial sum of squares corresponding to the moment in the row. The sum of each column is thus one. Table 2 Hybrid model parameter estima tes and moments of order flow and returns A. Standardized Regression $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) –0.129*** 0.007 –0.089*** 0.986*** –0.000 (–5.57) (0.38) (–6.17) (135.67) (–0.02) sd($$R$$) 0.155*** 0.460*** 0.016 –0.007 0.963*** (5.15) (7.89) (1.39) (–1.46) (138.47) skew(OIB) 0.007 0.003 –0.058*** 0.003 0.006* (1.02) (0.39) (–6.11) (0.79) (1.69) skew($$R$$) –0.008 0.009 0.047*** –0.001 0.005* (–1.05) (1.51) (4.33) (–0.41) (1.95) $$\text{corr}(R_1,\text{OIB}_1)$$ 0.258*** 0.484*** –0.018 0.009 0.039*** (5.40) (17.25) (–0.80) (1.26) (2.96) $$\text{corr}(R_1,\text{OIB}^2_1)$$ –0.039*** –0.018 0.185*** –0.003 –0.008* (–3.16) (–1.29) (5.73) (–1.12) (–1.92) $$\text{corr}(R_2,\text{OIB}_2)$$ 0.218*** 0.314*** –0.034 –0.012** –0.022** (6.10) (14.92) (–1.26) (–2.14) (–1.97) $$\text{corr}(R_2,\text{OIB}^2_2)$$ –0.049*** –0.028** 0.099*** –0.001 –0.009** (–5.79) (–2.04) (4.19) (–0.41) (–2.52) # right tail OIB & $$R$$ –0.122*** –0.103*** –0.128*** 0.011* –0.074*** (–4.17) (–5.59) (–3.86) (1.76) (–5.95) # left tail OIB & $$R$$ –0.163*** –0.063*** 0.029 0.005 0.012* (–7.39) (–6.66) (1.38) (0.65) (1.67) Constant 2.159*** –0.482*** 3.439*** 0.068*** 0.118*** (17.04) (–4.53) (60.66) (3.56) (5.39) Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^{2}$$ 0.152 0.680 0.040 0.978 0.938 B. Variance Decomposition $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) 0.125 0.000 0.127 1.000 0.000 sd($$R$$) 0.237 0.636 0.005 0.000 0.997 skew(OIB) 0.000 0.000 0.075 0.000 0.000 skew($$R$$) 0.001 0.000 0.047 0.000 0.000 $$\text{corr}(R_1,\text{OIB}_1)$$ 0.221 0.240 0.002 0.000 0.001 $$\text{corr}(R_1,\text{OIB}^2_1)$$ 0.009 0.001 0.458 0.000 0.000 $$\text{corr}(R_2,\text{OIB}_2)$$ 0.159 0.101 0.008 0.000 0.000 $$\text{corr}(R_2,\text{OIB}^2_2)$$ 0.016 0.002 0.137 0.000 0.000 # right tail OIB & $$R$$ 0.055 0.012 0.128 0.000 0.002 # left tail OIB & $$R$$ 0.176 0.008 0.012 0.000 0.000 Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^2$$ 0.152 0.680 0.040 0.978 0.938 A. Standardized Regression $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) –0.129*** 0.007 –0.089*** 0.986*** –0.000 (–5.57) (0.38) (–6.17) (135.67) (–0.02) sd($$R$$) 0.155*** 0.460*** 0.016 –0.007 0.963*** (5.15) (7.89) (1.39) (–1.46) (138.47) skew(OIB) 0.007 0.003 –0.058*** 0.003 0.006* (1.02) (0.39) (–6.11) (0.79) (1.69) skew($$R$$) –0.008 0.009 0.047*** –0.001 0.005* (–1.05) (1.51) (4.33) (–0.41) (1.95) $$\text{corr}(R_1,\text{OIB}_1)$$ 0.258*** 0.484*** –0.018 0.009 0.039*** (5.40) (17.25) (–0.80) (1.26) (2.96) $$\text{corr}(R_1,\text{OIB}^2_1)$$ –0.039*** –0.018 0.185*** –0.003 –0.008* (–3.16) (–1.29) (5.73) (–1.12) (–1.92) $$\text{corr}(R_2,\text{OIB}_2)$$ 0.218*** 0.314*** –0.034 –0.012** –0.022** (6.10) (14.92) (–1.26) (–2.14) (–1.97) $$\text{corr}(R_2,\text{OIB}^2_2)$$ –0.049*** –0.028** 0.099*** –0.001 –0.009** (–5.79) (–2.04) (4.19) (–0.41) (–2.52) # right tail OIB & $$R$$ –0.122*** –0.103*** –0.128*** 0.011* –0.074*** (–4.17) (–5.59) (–3.86) (1.76) (–5.95) # left tail OIB & $$R$$ –0.163*** –0.063*** 0.029 0.005 0.012* (–7.39) (–6.66) (1.38) (0.65) (1.67) Constant 2.159*** –0.482*** 3.439*** 0.068*** 0.118*** (17.04) (–4.53) (60.66) (3.56) (5.39) Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^{2}$$ 0.152 0.680 0.040 0.978 0.938 B. Variance Decomposition $$\alpha$$ $$\kappa$$ $$p_L$$ $$\sigma $$ $$\Delta$$ sd(OIB) 0.125 0.000 0.127 1.000 0.000 sd($$R$$) 0.237 0.636 0.005 0.000 0.997 skew(OIB) 0.000 0.000 0.075 0.000 0.000 skew($$R$$) 0.001 0.000 0.047 0.000 0.000 $$\text{corr}(R_1,\text{OIB}_1)$$ 0.221 0.240 0.002 0.000 0.001 $$\text{corr}(R_1,\text{OIB}^2_1)$$ 0.009 0.001 0.458 0.000 0.000 $$\text{corr}(R_2,\text{OIB}_2)$$ 0.159 0.101 0.008 0.000 0.000 $$\text{corr}(R_2,\text{OIB}^2_2)$$ 0.016 0.002 0.137 0.000 0.000 # right tail OIB & $$R$$ 0.055 0.012 0.128 0.000 0.002 # left tail OIB & $$R$$ 0.176 0.008 0.012 0.000 0.000 Observations 19,965 19,965 19,965 19,965 19,965 Adjusted $$R^2$$ 0.152 0.680 0.040 0.978 0.938 The dependent variables are the estimated parameters from the hybrid model. The explanatory variables are various moments of order flows and returns. The unit of observation is a firm-year. OIB denotes the cumulative order flow over the full day. OIB$$_1$$ and OIB$$_2$$ are the order flows over the first 3 and last 3.5 hours of the trading day. Similarly, $$R$$ is the return over the full day, and $$R_1$$ and $$R_2$$ are returns over the first 3 and last 3.5 hours of the trading day. The indicated moments of these variables are calculated across days for each firm-year. # Right Tail OIB &$$R$$ is the fraction of days where both OIB $$> \text{sd}(\text{OIB})$$ and $$R - 1 > \text{sd}(R)$$. # Left Tail OIB &$$R$$ is the fraction of days where both OIB $$< - \text{sd}(\text{OIB})$$ and $$R - 1 < - \text{sd}(R)$$. Panel A reports estimates where all variables are standardized to have a unit standard deviation. Standard errors are clustered by firm and year. $$t$$-statistics are in parentheses, and statistical significance is represented by * $$p<0.10$$, ** $$p<0.05$$, and *** $$p<0.01$$. Panel B reports variance decompositions. Each number in panel B represents the fraction of the model’s total partial sum of squares corresponding to the moment in the row. The sum of each column is thus one. Figure 6 depicts simulated data and three different sets of possible estimates for the parameters $$\alpha$$ and $$\kappa$$. The fit of the price function $$p(1,Y_1)$$ to the daily returns is shown in the left column. The log pricing errors in all three cases are shown in the right column. The parameters that were used in the simulation are shown in the middle row. Of the three sets of parameters shown in the figure, the parameters in the middle row give the largest value for the likelihood function. The parameters in the top row produce steps that are too far apart and too small, generating a price function that is too flat compared to the data. Consequently, the log pricing errors shown in the top row of the right column are positively correlated with order imbalances. The parameters in the bottom row produce steps that are too close together and too large, generating a price function that is too steep compared to the data. Consequently, the log pricing errors in the bottom row are negatively correlated with order imbalances. 2.1 Estimates of the hybrid model Table 1 reports summary statistics of the parameter estimates for the panel of firm-years (summary statistics by year are plotted in Figure 7 in Section 2.5). To see which aspects of the data determine the parameter estimates, Table 2 reports regressions of the parameter estimates on various moments of order flows and returns. The table also reports variance decompositions. The moments include correlations of order flows and returns split into two subperiods of the day: the first 3 hours and the last 3.5 hours. The price function in the model is nonlinear, so we also include nonlinear measures of the comovement of returns and order imbalances. Specifically, we include correlations of returns with squared order imbalances for the two subperiods. We also include the fraction of the days on which returns and order imbalances are both in the right tails of their distributions and the fraction in which they are both in their left tails, defining a tail as a standard deviation away from zero (a zero order imbalance or a zero rate of return). Figure 7 View largeDownload slide The annual cross-sectional mean and 25th and 75th percentiles of parameter estimates for the hybrid model The model is estimated on a stock-year basis for NYSE stocks from 1993 to 2012 using prices and order imbalances in six hourly intraday bins and at the close. The mean and the 25th and 75th percentiles are shown. The model parameters are $$\alpha = $$ probability of an information event, $$\kappa = $$ signal scale parameter, $$\sigma = $$ standard deviation of liquidity trading, $$\Delta = $$ volatility of public information, and $$p_L = $$ probability of a negative event. $$\lambda_{\text{hybrid}}$$ is the expected average lambda $$\lambda(0,0)$$ based on Equation (8). Figure 7 View largeDownload slide The annual cross-sectional mean and 25th and 75th percentiles of parameter estimates for the hybrid model The model is estimated on a stock-year basis for NYSE stocks from 1993 to 2012 using prices and order imbalances in six hourly intraday bins and at the close. The mean and the 25th and 75th percentiles are shown. The model parameters are $$\alpha = $$ probability of an information event, $$\kappa = $$ signal scale parameter, $$\sigma = $$ standard deviation of liquidity trading, $$\Delta = $$ volatility of public information, and $$p_L = $$ probability of a negative event. $$\lambda_{\text{hybrid}}$$ is the expected average lambda $$\lambda(0,0)$$ based on Equation (8). Table 1 Hybrid model parameter estimate summary statistics $$\alpha$$ $$\kappa$$ $$ p_L $$ $$\sigma$$ $$\Delta$$ Mean 0.64 0.0068 0.51 0.12 0.0213 SD 0.25 0.0050 0.15 0.11 0.0087 First quartile 0.54 0.0032 0.46 0.05 0.0149 Median 0.68 0.0058 0.50 0.08 0.0197 Third quartile 0.81 0.0095 0.56 0.16 0.0258 N 19,965 19,965 19,965 19,965 19,965 $$\alpha$$ $$\kappa$$ $$ p_L $$ $$\sigma$$ $$\Delta$$ Mean 0.64 0.0068 0.51 0.12 0.0213 SD 0.25 0.0050 0.15 0.11 0.0087 First quartile 0.54 0.0032 0.46 0.05 0.0149 Median 0.68 0.0058 0.50 0.08 0.0197 Third quartile 0.81 0.0095 0.56 0.16 0.0258 N 19,965 19,965 19,965 19,965 19,965 The model is estimated on a stock-year basis for NYSE stocks from 1993 to 2012 using prices and order imbalances in 6 hourly intraday bins and at the close. The model parameters are $$\alpha =$$ probability of an information event, $$\kappa = $$signal scale parameter, $$\sigma = $$standard deviation of liquidity trading, $$\Delta =$$ volatility of public information, and $$p_L =$$ probability of a negative event. Table 1 Hybrid model parameter estimate summary statistics $$\alpha$$ $$\kappa$$ $$ p_L $$ $$\sigma$$ $$\Delta$$ Mean 0.64 0.0068 0.51 0.12 0.0213 SD 0.25 0.0050 0.15 0.11 0.0087 First quartile 0.54 0.0032 0.46 0.05 0.0149 Median 0.68 0.0058 0.50 0.08 0.0197 Third quartile 0.81 0.0095 0.56 0.16 0.0258 N 19,965 19,965 19,965 19,965 19,965 $$\alpha$$ $$\kappa$$ $$ p_L $$ $$\sigma$$ $$\Delta$$ Mean 0.64 0.0068 0.51 0.12 0.0213 SD 0.25 0.0050 0.15 0.11 0.0087 First quartile 0.54 0.0032 0.46 0.05 0.0149 Median 0.68 0.0058 0.50 0.08 0.0197 Third quartile 0.81 0.0095 0.56 0.16 0.0258 N 19,965 19,965 19,965 19,965 19,965 The model is estimated on a stock-year basis for NYSE stocks from 1993 to 2012 using prices and order imbalances in 6 hourly intraday bins and at the close. The model parameters are $$\alpha =$$ probability of an information event, $$\kappa = $$signal scale parameter, $$\sigma = $$standard deviation of liquidity trading, $$\Delta =$$ volatility of public information, and $$p_L =$$ probability of a negative event. The R-squareds and the variance decomposition show that the estimates of the standard deviation $$\sigma$$ of order imbalances from the model are almost entirely determined by the empirical standard deviations of order imbalances. Likewise, the estimates of the volatility $$\Delta$$ of the public news process are almost entirely determined by the standard deviations of returns. The private information parameters $$\kappa$$, $$\alpha$$, and $$p_L$$ are naturally more complex. The moments have little explanatory power for the $$p_L$$ estimates. As shown in Table 1, the distribution of the $$p_L$$ estimates is fairly tight around 50%, so there is not too much variation to explain. The $$\kappa$$ and $$\alpha$$ estimates are the most interesting. The magnitude $$\kappa$$ of private information is fairly well explained by the moments, with the most important moments being the standard deviation of returns and the correlations between order imbalances and returns. The variance decomposition shows that all of the moments except skewness affect the estimated probability $$\alpha$$ of information events. The nonlinear specification is important for $$\alpha$$. More than 20% of the R-squared comes from the tail variables. 2.2 Testing whether an information event is always present in the hybrid model Our hybrid model relaxes the assumption in Kyle (1985) that an information event occurs in each instance of the model (in each day in our implementation). A natural question is whether this relaxation is supported in the data. The Kyle framework is nested in our model by the restriction that $$\alpha=1$$. Accordingly, we estimate the model with this restriction. The standard likelihood ratio test of the null that $$\alpha=1$$ against the alternative that $$\alpha \in [0,1]$$ is rejected for 73% of the firm-years (with a test size of 10%). However, the usual regularity conditions for the likelihood ratio test require that the restriction not be at the boundary of the parameter space. To address this issue, we bootstrap the distribution of the likelihood ratio statistic for a random sample of 100 firm-years like in Duarte and Young (2009). Specifically, for a given firm-year, we estimate the restricted model ($$\alpha=1$$) and then simulate 500 firm-years under the null using the estimated (restricted) parameters. We then estimate the restricted and unrestricted models for each simulated firm-year to obtain the distribution of the likelihood ratio under the null. The 90th percentile of this distribution is the critical value to evaluate the empirical likelihood ratio. These bootstrapped likelihood ratio tests reject the restricted Kyle model in favor of the hybrid model for 62 of the 100 randomly selected firm-years. The data thus supports the conclusion that the probability of an information event is less than 1. 2.3 Estimated parameters and reduced-form price impacts The model places structure on the price and order flow data, allowing the econometrician to identify components of Kyle’s lambda. Of course, one can estimate a reduced-form price impact as well. As an initial test of whether our estimates relate to price impact as implied by theory, we test the comparative statics from Figure 2 that price impacts are increasing in both the probability and magnitude of information events. We employ three estimates of the price impact of orders. The first is the 5-minute percent price impact of a given trade $$k$$ as \begin{equation} \label{eq_priceimpact} \textit{5-minute price impact}_k = \frac{2D_k(M_{k+5} - M_k)}{M_k}, \end{equation} (11) where $$M_k$$ is the prevailing quote midpoint for trade $$k$$, $$M_{k+5}$$ is the quote midpoint five minutes after trade $$k$$, and $$D_k$$ equals 1 if trade $$k$$ is a buy and $$-1$$ if trade $$k$$ is a sell. Goyenko, Holden, and Trzcinka (2009) use this measure as one of their high-frequency liquidity benchmarks in a study assessing the quality of various liquidity measures based on daily data.18 For a given stock-day, the estimate of the percent price impact is the equal-weighted average price impact over all trades on that day. We average these daily price impact estimates for each stock-year. We also estimate the cumulative impulse response function (Hasbrouck, 1991), which captures the permanent price impact of an order. The cumulative impulse response is calculated from a vector autoregression of log price changes and signed trades. Finally, we estimate another price impact measure (denoted $$\widehat{\lambda}_{\text{intraday}}$$) using a regression of 5-minute returns on the square root of signed volume following Hasbrouck (2009) and Goyenko, Holden, and Trzcinka (2009). We estimate these for each stock day, taking the median estimate across days as the stock-year estimate. The first panel of Table 3 reports panel regressions of the three price impact measures on the hybrid model parameters that measure private information (the probability $$\alpha$$ of an information event
