ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA

Transcription

1 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA DANIEL PREVE AND YIU-KUEN TSE AUGUST 26, 2 Abstract. Recently Duarte and Young (29) extended the probability of informed trading (PIN) proposed by Easley et al. (22) and decomposed it into two components: the adjusted PIN (APIN) as a measure of asymmetric information and the probability of symmetric order-flow shock (PSOS) as a measure of illiquidity. They provided some cross-section estimates of these measures using daily data over annual periods and argued that the APIN is not priced. In this paper we propose a method to estimate daily APIN and PSOS as an extension of Tay et al. (29) using high-frequency transaction data. Our empirical results indicate that daily APIN is much more stable than daily PIN. In contrast to PIN, daily APIN is not positively correlated with daily variance, while daily PSOS is. Moreover, in comparison with the daily APIN, the daily PSOS exhibits clustering and sporadic bursts over time. Key words and phrases. autoregressive conditional duration, market microstructure, probability of informed trading, probability of symmetric order-flow shock, transaction data. Uppsala University. Singapore Management University. Address correspondence to Yiu-Kuen Tse, School of Economics, Singapore Management University, 9 Stamford Road, Singapore 7893, Singapore; yktse@smu.edu.sg. We are grateful to conference participants at the 4th International conference on Computational and Financial Econometrics (2, London), SMU-ESSEC Symposium on Empirical Finance & Financial Econometrics (2, Singapore) and seminar participants at Uppsala University and Helsinki Center of Economic Research for their comments and suggestions. All remaining errors are our own. The authors gratefully acknowledge research support from the Singapore MOE AcRF Tier 2 fund, research grant T26B43-RS. The first author is thankful to the Sim Kee Boon Institute for Financial Economics, and the Institute s Centre for Financial Econometrics, at Singapore Management University for partial research support. Tao Yang provided excellent research assistance.

2 2. Introduction Since the seminal work of Easley et al. (996), Easley et al. (997) and Easley et al. (22, EHO), many studies in the finance literature have used the probability of informed trading (PIN) to analyze the effects of asymmetric information on asset pricing and volatility. Easley and O Hara (24) argued that the effect of asymmetric information is undiversifiable and is thus priced. Hence, as a proxy for information asymmetry, PIN is expected to be significantly positively correlated with average stock returns. Recently Duarte and Young (29, DY) extended the EHO framework to analyze PIN as a measure of asymmetric information. Apart from relaxing the assumption that the arrival rate of informed sellers is the same as the arrival rate of informed buyers, as was imposed by EHO, they introduce what they call a symmetric order-flow shock to the model. They argue that traders may disagree on the interpretation of a public news event, which may cause both buy- and sell-orders to arrive at higher rates. As a result, DY propose a modification of PIN to measure the probability of informed trading, called the adjusted PIN (APIN). More importantly, they introduce a measure called the probability of symmetric order-flow shock (PSOS), which is the unconditional probability that a given trade comes from a shock to both buyand sell-order flows. The authors show that high PSOS firms are usually firms with low trading volumes on most days, but who experience large increases in both buy- and sell-orders on days with public news. To this extent, they argue that PSOS is effectively a proxy for illiquidity, which is supported by their empirical finding that high PSOS firms tend to have high? measures. Furthermore, they find that APIN is not priced, while PSOS is priced. The empirical results of DY are based on the analysis of daily stock data over annual subperiods, for which the parameters of their APIN model are assumed to be constant within each year. Indeed, the empirical literature on PIN typically assumes constant probabilities of news and buy-sell intensity parameters over the sample data. The commonly adopted methodology is to estimate PIN using daily aggregates of buy- and sell-orders, which are assumed to be statistically independent. In addition to the assumption of constant probabilities of no news, good news and bad news, trade volume is not taken into account. These limitations have been criticized recently as possible causes for the anomalous behavior of PIN in some studies (see, e.g., Aktas et al., 27). To overcome these difficulties Tay et al. (29, TTTW) consider the estimation of PIN using transaction data. Their model allows the probabilities of the state of news to vary daily, and they incorporate the use of covariates such as volume and duration of trade for the determination of PIN. In this paper we consider the estimation of APIN and PSOS using high-frequency transaction data by extending the methodology of TTTW. Following TTTW, we model transaction duration using the asymmetric autoregressive conditional duration (AACD) model proposed by Bauwens and Giot (23).

3 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 3 We allow the expected duration of buy- and sell-orders to be dependent on covariates such as lagged duration, lagged conditional expected duration, lagged trade direction (buy- or sell-order) and lagged trade volume. Also, we incorporate into our model time varying probabilities of no news, good news, bad news and common shocks, as featured in the DY model. The model parameters are estimated using the maximum likelihood estimation (MLE) method, from which we obtain daily estimates of APIN and PSOS. The results provide an enhanced methodology to study the effects of asymmetric information and illiquidity on asset pricing. Note that the DY methodology produces only cross-sectional analysis of the relation of APIN and PSOS with size, spread and other illiquidity measures of a sample of stocks. In contrast, our daily APIN and PSOS estimates can be used to trace the time-varying relation between asymmetric information, illiquidity and the price dynamics of each stock. The remainder of the paper is organized as follows. In Section 2 we briefly review the PIN model of EHO and the APIN model of DY. In Section 3 we review the PIN-AACD model of TTTW and outline our extension, the APIN-AACD model. In doing so, we also briefly review the AACD model of Bauwens and Giot (23). Section 4 reports the results of an empirical study. Section 5 concludes. 2. The PIN and APIN Models In this section we briefly summarize the PIN model of EHO and its extension the APIN model of DY. A more extensive review can be found in the Appendix. 2.. The PIN Model. Let B d and S d denote the aggregate number of buy- and sell-orders on day d, respectively. In the PIN model, B d and S d are assumed to be independent Poisson random variables, with different intensities for days with bad news (B), good news (G) and no news (N). Let θ E denote the probability of news being released on day d and let θ B denote the probability of bad news, conditional on the release of news. Thus, the daily state probabilities are π B = θ E θ B, π G = θ E ( θ B ) and π N = θ E, for a day with bad news, good news and no news, respectively. For a day with no news, the means of B d and S d are λ and λ, respectively. For a day with bad news the sell intensity increases by a constant δ, while the buy intensity remains the same as for a day with no news. Similarly, for a day with good news the buy intensity increases by δ, while the sell intensity stays the same as for a no-news day. EHO compute the PIN as the relative intensity of informed trades to the intensity of all trades, so that where P = λ + λ and P 2 = θ E δ. PIN = P 2 P + P 2, (2.) 2.2. The APIN Model. DY extended the PIN model of EHO by allowing for the arrival rate of informed sellers to be different from the arrival rate of informed buyers and, more importantly, by allowing both

4 4 DANIEL PREVE AND YIU-KUEN TSE buy- and sell-order flows to increase on certain days even when there is no news. In the APIN model B d and S d have different intensities for days with bad news and a common shock (CB), good news and a common shock (CG) and no news and a common shock (CN). Let θ C denote the daily probability of a common shock. In the event of a common shock, the buy intensity increases by and the sell intensity by. DY compute the APIN as and introduce PSOS as APIN = PSOS = P 2 P + P 2 + P 3, (2.2) P 3 P + P 2 + P 3, (2.3) where P = λ +λ, P 2 = θ E [( θ B )δ +θ B δ ] and P 3 = θ C ( + ). Note that APIN in Equation (2.2) reduces to the original PIN measure in Equation (2.) whenever δ = δ and θ C =, as expected. 3. The PIN-AACD and APIN-AACD Models In this section we review the PIN-AACD model of TTTW and outline our extension, the APIN-AACD model, analogous to the extension by DY of the PIN model of EHO. In doing so, we first review the AACD model of Bauwens and Giot (23). 3.. The AACD Model of Trade Direction. TTTW model trade direction (buy- and sell-initiated order) and duration between trades (waiting time) jointly using an AACD model, and compute PIN from this model. We denote x i as the (diurnally adjusted) waiting time between trade i at time t i and trade i at time t i so that x i = t i t i. In addition, we denote y i as the trade direction of the ith trade, which takes on values and representing a sell- and buy-order, respectively. Φ i denotes the information upon the (i )th trade, which may include trade direction y i, transaction volume v i, waiting time x i, as well as their lagged values. Conditional on Φ i, TTTW assume that both potential trade directions (buy or sell) at time t i follow latent point processes. More specifically, given Φ i, {B i (s i ), s i } and {S i (s i ), s i } are latent Poisson processes, representing buy- and sell-orders, with common start time t i, i.e. s i = s i (t) = t t i, and intensities λ i and λ,i. The observed trade direction at time t i is the outcome of the competition between the two Poisson processes to be the first arrival. Conditional on Φ i, let d ji be the latent waiting time (duration) until the first occurrence for trade direction j and suppose that d i is independent of d,i. Let x i = min {d i, d,i } and y i = j, where j = if d,i = x i and j = if d i = x i. Under the Poisson process assumption d ji is exponentially distributed with mean ψ ji = /λ ji given Φ i. It can be shown that x i is conditionally exponential with DY also consider models for which the probability of common shock varies with the state of news or no news. However, they argue empirically that the restriction of imposing invariance is innocuous. In this paper we adopt this restriction.

5 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 5 pdf f(x i Φ i ) = (λ i + λ,i )e (λi+λ,i)xi, x i, and y i is conditionally two-point distributed with probability function Pr(y i = j Φ i ) = λ ji λ i + λ,i, j =,. Moreover, it can be shown that x i and y i are independent conditional on Φ i. Hence, the conditional joint distribution of duration x i and direction y i is given by f(x i, y i Φ i ) = λ ji e (λi+λ,i)xi (3.) ( ) ( {yi =} = ψ i where { } is the indicator function. ψ,i ) {yi = } [ ( exp + ) ] x i, ψ i ψ,i 3.2. The PIN-AACD Model. Like the PIN model of EHO, the PIN-AACD model of TTTW has states corresponding to no news, good news and bad news. However, unlike the EHO approach, TTTW allow for interactions between consecutive buy- and sell-orders, and account for the duration between trades and the volume of the trade. Due to its specification, the PIN-AACD model allows for the PIN to be computed for a specific day as well as over intraday intervals using high-frequency transaction data. In the PIN-AACD model, the conditional expected duration ψji s of ds ji for s S = {G, B, N} is based on ψ N ji (the conditional expected duration on a no-news day), where ψn ji is assumed to follow the extended logarithmic ACD(,) model ln ψ N ji = ν j {yi =} + ν j, {yi = } + β j ln ψ N j,i + α j ln x i + ς j y i ln v i, (3.2) for j =,, where v i is the volume of the trade at time t i. 2 Thus, the base equation ψ N ji depends on whether the previous transaction is a buy- or sell-initiated order, y i, the lagged conditional expected duration, ψj,i N, the previous duration, x i, and the lagged signed logarithmic volume, y i ln v i. Hence, in contrast to the PIN model of EHO, the PIN-AACD model allows volume to impact trade intensity. Analogous to the PIN model of EHO, on a bad-news day ln ψ,i B = ln ψn,i µ B and on a good-news day ln ψi G = ln ψn i µ G. 3 The equations for the implied conditional Poisson intensities, λ s ji, are λ G i = λn i (eµ G ) and λ B,i = λn,i (eµ B ), where λ N ji = /ψn ji with ψn ji as in Equation (3.2). 2 Tay et al. (29) use the Log-ACD(, ) model of Bauwens and Giot (2) as a basis for (3.2), rather than the standard ACD(,) model of Engle and Russell (998), as it is flexible for including additional explanatory variables in the autoregressive equation. 3 In fact, Tay et al. (29) assume that µb = µ G. Here we allow µ B to be different from µ G, as might be justified due to short-selling restrictions (see Diamond and Verrecchia, 99).

6 6 DANIEL PREVE AND YIU-KUEN TSE We expect the parameters µ B and µ G to be positive so that on a bad-news (good-news) day, ψ,i B (ψg i ) decreases and λ B,i (λg i ) increases. In their general specification, TTTW model θ E and θ B rather than assuming them to be constant, thus, allowing the probabilities of news to vary over time. More specifically, they assume logistic models in which the arrival of bad news, good news and no news on day d depend on the aggregate volume of buy and sell orders. This is motivated by recent empirical work reporting positive correlation between (public) information and trading volume. TTTW denote the average number of lots traded per day initiated by buy orders by V B. Similarly, they denote the average number of lots traded per day initiated by sell orders by V S. The numbers of lots traded on day d initiated by buy and sell orders are denoted by V B d and V d S, respectively. The probability of news on day d is assumed to be θ Ed = [ ( + exp {γ + γ 2 ln Vd B + V S d ) ln (V B + V S)]}, where γ 2 is expected to be strictly positive. Given news on day d, the probability of bad news is assumed to be θ Bd = ( + exp [γ 3 ln Vd S ln V S) ( γ 4 ln Vd B B)], ln V where γ 3 and γ 4 are expected to be strictly positive. The arrival of bad news, good news and no news on day d are given by π Bd = θ Ed θ Bd, π Gd = θ Ed ( θ Bd ) and π Nd = θ Ed, respectively. The PIN-AACD model is estimated using the ML method. With N d = B d + S d orders on day d, its likelihood function is given by [ ( D Nd )] π sd f s (x i, y i Φ i ), for S = {B, G, N}, (3.3) d= s S i= where ( ) ( {yi =} f s (x i, y i Φ i ) = ψ s i ψ s,i ) {yi = } exp [ ( ψ s i ) ] + ψ,i s x i, whenever x i, y i =, and zero otherwise (cf. Equation 3.). Because of the Poisson process assumption, it can be shown that conditional on Φ i the expected number of trades due to all traders in the fixed interval (t i, t i ] on day d is E[B i (x i ) + S i (x i ) Φ i, x i ] = π s E[B i (x i ) + S i (x i ) Φ i, x i, s] = ( λ N i + λ N G,i + π Gdλi + π Bd λ B ),i xi, }{{}}{{} s S P i P 2i

7 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 7 where P i and P 2i are due to uninformed and informed trades, respectively. 4 Similar to the original PIN measure of EHO, TTTW compute the daily AACD PIN as PIN d = Nd i= P 2ix i Nd i= (P i + P 2i )x i, (3.4) emphasizing that Equation (3.4) can be modified to compute AACD PIN measures over intraday intervals, in which case the summations are over trades in specific intraday intervals The APIN-AACD Model. Like the APIN model of DY, our proposed APIN-AACD model, outlined in Figure, has six different daily states, allowing for symmetric order-flow shock trades. Similar to the PIN-AACD model, and in contrast to the APIN model, the APIN-AACD model allows for the APIN, and the PSOS, to be computed daily as well as over intraday intervals. In contrast to the PIN-AACD model, the APIN-AACD model has three additional states representing trading days in which the conditional intensities of both B i and S i increase due to common shocks. These days occur with probabilities θ Cd. Analogous to TTTW, we assume a logistic model for the daily probability of a common shock such that θ Cd = ( exp [γ 5 ln Vd B ln V B) ( ln Vd S ln V S) ], + + where (u) + equals u if u > and zero otherwise. Note that, for θ Cd to lie between an, we must have γ 5. Furthermore, θ Cd = unless Vd B > V B d and Vd S > V S d. Thus, there is no symmetric order-flow shock on day d unless both the buy- and sell-orders on that day are larger than their corresponding sample average. This assumption appears to be reasonable given that a symmetric order-flow shock induces both buy and sell orders. In practice θ Cd is frequently zero. For example, for the IBM data in our empirical study 55 out of 754 probabilities (i.e., θ Cd ) are zero. Like the PIN-AACD model, the conditional expected duration ψ s ji for each state s S is based on ψn ji in (3.2). Equations for the remaining ψji s are given in Table. Analogous to the APIN model of DY, on a bad-news day with a common shock ln ψ CB i on a good-news day with a common shock ln ψ CG i Finally, on a no-news day with a common shock ln ψ CN ji = ln ψ N ji µ CN. 5 = ln ψ N i µ CB and ln ψ CB,i = ln ψn,i µ B µ CB. Similarly, = ln ψ N i µ G µ CG and ln ψ CG,i = ln ψn,i µ CG. 4 By definition, λ G i = /ψ G i λn i and λb,i = /ψb,i λn,i. 5 We also experimented with a more parsimonious model specification (µb = µ G and µ CB = µ CG ) which yielded empirical results similar to those reported in Section 4.

8 8 DANIEL PREVE AND YIU-KUEN TSE It can be shown that conditional on Φ i the expected number of trades due to all traders in the fixed interval (t i, t i ] on day d is E[B i (x i ) + S i (x i ) Φ i, x i ] = [ λ N i + λ N,i }{{} P i G + π Gdλ i + π Bd λ B,i }{{} P 2i ( + π CBd λ CB i + λ CB ) (,i + πcgd λ CG i + λ CG ) (,i + πcnd λ CN i + λ CN ) ],i xi, }{{} P 3i where P i, P 2i and P 3i are the expected numbers of trades due to uninformed, informed and symmetric order-flow shock trades, respectively. 6 The daily AACD APIN and PSOS are given by and APIN d = PSOS d = Nd i= P 2ix i Nd i= (P, (3.5) i + P 2i + P 3i )x i Nd i= P 3ix i Nd i= (P. (3.6) i + P 2i + P 3i )x i Note that the APIN-AACD measure in Equation (3.5) reduces to the PIN-AACD measure in Equation (3.4) whenever θ Cd =, as expected. 4. Empirical Results 4.. Data. The intraday data used in this section was extracted and compiled from the New York Stock Exchange (NYSE) Trade and Quote (TAQ) Database provided through the Wharton Research Data Services. We retrieved data from the Consolidated Trade (CT) file as well as the Consolidated Quote (CQ) file. From the CT file we downloaded the data for the date, trading time, price and number of shares traded for each stock in our study. From the CQ file we downloaded the data for the offer and bid prices, as well as the time of the quote revisions. The data sets used consist of high-frequency transaction data for the IBM, GE (General Electric), PG (Procter and Gamble) and WMT (Walmart) stocks over the period Jan, 25 through Dec 3, 27, covering 754 trading days. Due to opening effects, the first 2 minutes (9:3 am to 9:5 am) of each trading day were removed. All transactions after 4: pm were also deleted. Days where the opening transaction occurred after the first 2 minutes of the trading day or where there were insufficient (less than ) transactions between 9:5 am and : am to obtain meaningful initial values for the ML estimation were also removed. The frequency of zero trade durations (simultaneous transactions) in the data sets is high. For example, about 35% of the observations for the IBM data are of zero durations. We deal with the zero durations 6 Analogous to TTTW, we define λ G i = /ψi G λn i and λb,i = /ψb,i λn,i. In addition, we define λcb i = /ψ CB λ CB,i = /ψcb,i λn,i λb,i, λcg i λ CN,i = /ψcn,i λn,i. = /ψ CG i λ N i λg i, λcg,i = /ψcg,i λn,i, λcn i i λ N i, = /ψi CN λ N i and

9 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 9 in the following way. For transactions with the same time stamp we aggregate the transaction volumes and compute an average price weighted by volume, as described in Pacurar (28). We compute the diurnal factors, which are linked to the trading habits and intraday seasonality, by applying a smoothing cubic spline to the average raw duration at each time point with available data. We use the Matlab function spap2 to estimate the spline by least-squares. The cubic spline is made up of 6 polynomial pieces, with knots set on each hour (: am to 4: pm). Following Engle and Russell (998), we set the mean of the computed diurnal factors equal to the sample mean of the raw durations. Note that, in practice, this implies that the sample mean of our diurnally adjusted durations is approximately. Like DY, we classify trade direction according to the Lee and Ready (99) algorithm. Trades for which the algorithm does not apply were further classified as buyer- or seller-initiated based on a tick test. Some summary statistics of the resulting data sets are given in Table 2. The average number of trades per day ranges from 4, (PG) to 5,49.49 (GE). More than 5% of the trades for all stocks are sell orders Maximum Likelihood Estimation of the Models. ML estimation of the PIN- and APIN- AACD models with time varying probabilities was performed using the Matlab function fmincon with the interior-point algorithm and numerical derivatives. The values ψ N j used to initialize each day were computed as follows. Let n d denote the number of transactions between 9:5 am and : am on trading day d. As initial values for day d we use ψ N j = nd i= {y i=j}x i nd i=, for j =,. {y i=j} To search for a global optimum, we use a random starting point for the numerical method and run the likelihood optimization times for each data set. We then select the maximum of these optimizations. The estimation procedure converges for all data sets. The ML estimation results for the PIN- and APIN-AACD models with time varying probabilities are presented in Table 3. It can be seen that the parameter estimates exhibit a remarkable resemblance across the four stocks. We note that ˆγ is negative for all stocks, implying that the estimated probability of news ˆθ Ed is less than.5 on an average day (when the buy and sell volumes are equal to the sample average). As expected, estimates of γ 2 through γ 5 are all positive. Both ˆβ + ˆα and ˆβ + ˆα are less than. The persistence of the latent processes, however, appears to be quite high. Similar to TTTW, we observe that ˆς > and ˆς < for both models and all stocks (although these estimates are not statistically significant for WMT in both models), implying that large buy orders induce shorter conditional expected durations for subsequent buy orders but longer conditional expected durations for sell orders. The opposite goes

10 DANIEL PREVE AND YIU-KUEN TSE for large sell orders. Thus, the results suggest that volume plays an explicit part in predicting trade direction. Note that the standard regularity conditions that ensure that the limiting null distribution of the likelihood ratio test statistic is chi-square are not satisfied when testing the restricted (γ 5 = ) PIN-AACD model against the unrestricted APIN-AACD model. This is because the parameter lies on the boundary of the parameter space under the null hypothesis. Consequently we do not report any likelihood ratio or Wald test results. Finally we note that all estimates are statistically significant at the 5 percent level, except for ˆγ 2 of PG for the APIN-AACD model as well as ˆς and ˆς for WMT for both models. The statistical insignificance of ˆγ 2 implies that the probability of news does not vary over different days. However, as ˆγ 3 and ˆγ 4 are statistically nonzero, the probabilities of good news and bad news are still time varying Estimates of Daily PIN, APIN and PSOS. Figures 2 and 3 present the plots of the estimated daily probabilities of good news, no news and bad news for the PIN-AACD model applied to the IBM and GE data sets, respectively. 7 For both stocks, the model implied probabilities of bad news seem to be quite stable throughout the sample period and are less than.2 more than half of the days. In contrast, the estimated probabilities of good news for these stocks are more volatile, with values exceeding.8 for a few days. Figures 4 and 5 show the plots of the estimated daily state probabilities for the APIN-AACD model applied to the IBM and GE data sets. For this model, the estimated probabilities of good and bad news without common shocks (ˆπ G and ˆπ B ) appear to be more stable over time compared to the estimates of the probabilities of both good and bad news in the PIN-AACD model. In particular, for both stocks ˆπ G is less than.5 for all days. In contrast, estimates of the probabilities of events with common shock (π CG, π CN and π CB ) are irregular and sporadic. The estimated probabilities are zero for most days, but may be quite large (exceeding.5) on some days. This result shows that the volatile pattern of ˆπ G in the PIN-AACD model may be due to common-shock trading. Daily AACD PIN, APIN and PSOS estimates were computed using equations (3.4), (3.5) and (3.6), respectively. Figures 6 and 7 present the plots of the estimated PIN, APIN and PSOS for the IBM and GE data sets. For both stocks, APIN appears to be much more stable than PIN. While APIN is less than.5 for both stocks on almost all days, PIN fluctuates a lot with quite a few days exceeding.. PSOS behaves quite differently from PIN and APIN. In particular, while PSOS is zero for many days, it also fluctuates to above.2 for quite a few days for both stocks. We also note that PSOS may remain zero for an extended period, in which common-shock traders are absent from the market. Overall, our results are consistent with DY s comment that high PSOS firms tend to be firms with low volume on most days 7 The plots for the remaining two stocks of the PIN-AACD and APIN-AACD models are visually similar and are not presented in this paper.

11 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA but who experience large increase in both buy and sell order flow on days associated with public news events. DY reported the correlations between PIN, APIN and PSOS computed over a period of time with some variables of interest such as spread and firm size across a cross section of NYSE stocks. Our empirical results, however, show that the correlations between daily estimates of PIN, APIN and PSOS versus Kyle s lambda and opening spread are mixed and mostly insignificant. In Table 4 we present the correlations of the daily estimates of PIN, APIN and PSOS with the daily variance and return for the four stocks in our study. 8 Our correlation coefficients computed over time series of daily data provide some complementary results to the cross section analysis of DY. The following can be observed from the table. First, there seems to be no regular pattern of contemporaneous correlation between PIN/APIN/PSOS and return. This may be due to the noisy nature of the daily measures of return. Second, PIN and PSOS are positively correlated with variance, while APIN is negatively correlated with variance. This suggests that on a day characterized by high APIN, the market may receive strong information signal which reduces the intraday volatility. On the other hand, high PSOS characterizes disagreement in the interpretation of information and may induce higher volatility. Third, APIN is negatively contemporaneously correlated with PIN and PSOS, while PIN and PSOS are positively contemporaneously correlated. Although DY reported positive pairwise correlations between PIN, APIN and PSOS in the cross-section context, our result shows different contemporaneous correlation patterns. On a daily basis, strong information signal increases APIN and reduces common shocks caused by disagreement in information interpretation, hence causing negative contemporaneous correlation between APIN and PSOS. On the other hand, PSOS is a component of the PIN measure, as argued by DY, hence inducing negative contemporaneous correlation between APIN and PIN. 5. Conclusions In this paper we propose a method to estimate time varying APIN and PSOS suggested by DY as measures of asymmetric information and illiquidity, respectively. Our method is an extension of TTTW using high-frequency transaction data, which is based on an AACD model of expected durations of buyand sell-orders. We allow the expected duration of buy and sell orders to be dependent on covariates such as lagged duration, lagged conditional expected duration, lagged trade direction and lagged trade volume. Also, we incorporate into our model time varying probabilities of no news, good news, bad news and symmetric order-flow shocks. The model parameters are estimated using MLE, from which we obtain daily estimates of APIN and PSOS. The results provide an enhanced methodology to study the effects of asymmetric information and illiquidity on asset pricing. Our empirical results indicate that daily APIN 8 The daily variance is estimated using the ACD-ICV method proposed by Tse and Yang (2).

12 2 DANIEL PREVE AND YIU-KUEN TSE is much more stable than daily PIN and that, in contrast to PIN, APIN is negatively correlated with variance over time. Furthermore, PSOS is the component that is positively correlated with daily variance. We observe the interesting result that the daily PSOS series exhibit a sporadic pattern of extended periods of no common shocks intermingled with clustered periods of active common-shock trading. References Aktas, N., de Bodt, E., Declerck, F. and H.V. Oppens, 27, The PIN anomaly around M&A announcements. Journal of Financial Markets : Amihud, A., 22, Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5: Bauwens, L. and P. Giot, 2, The logarithmic ACD model: an application to the bid-ask quote process of three NYSE stocks. Annales d Économie et de Statistique 6: Bauwens, L. and P. Giot, 23, Asymmetric ACD models: introducing price information in ACD models. Empirical Economics 28: Diamond, D.W. and R. Verrecchia, 99, Disclosure, liquidity, and the cost of capital. Journal of Finance 66: Duarte, J. and L. Young, 29, Why is PIN priced? Journal of Financial Economics 9: Easley, D., Hvidkjaer, S. and M. O Hara, 22, Is information risk a determinant of asset returns? Journal of Finance 57: Easley, D., Kiefer, N.M. and M. O Hara, 996, Cream-skimming or profit-sharing? The curious role of purchased order flow. Journal of Finance 5: Easley, D., Kiefer, N.M. and M. O Hara, 997, The information content of the trading process. Journal of Empirical Finance 4: Easley, D. and M. O Hara, 24, Information and the cost of capital. Journal of Finance 59: Engle, R.F. and J.R. Russell, 998, Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66: Lee, C.M.C. and M.J. Ready, 99, Inferring trade direction from intraday data. Journal of Finance 46: Pacurar, M., 28, Autoregressive conditional duration models in finance: a survey of the theoretical and empirical literature. Journal of Economic Surveys 22: Tay, A., Ting, C., Tse, Y.K. and M. Warachka, 29, Using high-frequency transaction data to estimate the probability of informed trading. Journal of Financial Econometrics 7: Tse, Y.K. and T. Yang, 2, Estimation of high-frequency volatility: an autoregressive conditional duration approach. Working paper. Singapore Management University.

13 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 3 Table. Conditional expected durations for the APIN-AACD model with ψji N as in Equation (3.2). The positive constants µ CB, µ B, µ CG, µ G and µ CN are unknown parameters. Buy-initiated trade Sell-initiated trade ψi CB = ψi Ne µ CB ψ,i CB,i e (µ B+µ CB ) ψi B = ψn i ψ,i B = ψn,i e µ B ψi CG = ψi N e (µ G+µ CG ) ψ,i CG,i e µ CG ψi G = ψn i e µ G ψ,i G = ψn,i ψi CN = ψi Ne µ CN ψ,i CN,i e µ CN Table 2. Summary statistics: Serial correlation of trade direction is the sample autocorrelation at lag and Runs test of trade direction is the p-value of the Wald-Wolfowitz test for randomness, for trade direction. IBM GE PG WMT Frequency of buy-orders (%) Frequency of sell-orders (%) Serial correlation of trade direction Runs test of trade direction.... Average trade volume ,49.2,38.7, Average logarithmic trade volume Average daily number of trades 4, , , ,82.37 Average daily number of buy-orders 2, , , ,33.77 Average daily number of sell-orders 2,58.6 2, ,33.9 2,85.6 Number of observations 3,623,936 4,86,293 3,369,4 3,97,59

14 4 DANIEL PREVE AND YIU-KUEN TSE Table 3. ML estimation results for the PIN- and APIN-AACD models: The data sets used for parameter estimation consist of high-frequency transaction data for the IBM, GE, PG and WMT stocks over the period Jan, 25-Dec 3, 27, covering 754 trading days. Numbers within parentheses are standard errors. IBM GE PIN-AACD APIN-AACD PIN-AACD APIN-AACD γ (.85) (.532) (.95) -.95 (.729) γ (.396).5243 (.257) 2.92 (.353).2 (.296) γ (.46).889 (.2) (.365) (.249) γ (.726).7954 (.9274) (.727).367 (.274) γ (.437) (.6659) ν,.25 (.22).222 (.9).263 (.24).267 (.23) ν,.989 (.22).93 (.2).22 (.26).228 (.25) β.8989 (.3).929 (.).7773 (.5).7738 (.4) α.73 (.7).72 (.6).222 (.6).22 (.6) ς -.52 (.3) -.47 (.3) -.36 (.3) -.35 (.3) ν,.46 (.). (.2).925 (.4).267 (.25) ν,.554 (.).73 (.8).2 (.4).88 (.2) β.9565 (.5).8954 (.3).936 (.8).7929 (.6) α.423 (.4).665 (.6).843 (.6).35 (.6) ς.2 (.2).6 (.3).63 (.2).48 (.3) µ B.842 (.32).799 (.2).232 (.35).2358 (.29) µ G.26 (.2).945 (.24).39 (.8).732 (.22) µ CB.2334 (.34).362 (.24) µ CG.2793 (.3).27 (.39) µ CN.987 (.23).2768 (.6) PG WMT PIN-AACD APIN-AACD PIN-AACD APIN-AACD γ (.748) (.734) (.884) (.295) γ (.4).65 (.836).4795 (.297).5497 (.5) γ (.337).792 (2.78) 2.85 (5.424) (.7743) γ (.367) (2.67) 3.57 (8.4268) (.6473) γ (.3776) (.4223) ν,.9 (.25).833 (.27).277 (.98).2233 (.37) ν,.458 (.27).37 (.39).823 (.2).947 (.32) β.8429 (.6).855 (.26).8679 (.58).8559 (.39) α.95 (.7).92 (.).93 (.7).944 (.38) ς -.75 (.3) -.7 (.4) -. (.76) -.4 (.24) ν,.429 (.).2 (.4).583 (.93).75 (.24) ν,.82 (.).44 (.92).858 (.23).6 (.27) β.9449 (.6).88 (.68).9283 (.2).97 (.2) α.5 (.5).743 (.24).625 (.2).689 (.8) ς.5 (.2).52 (.3).23 (.574).3 (.9) µ B.888 (.28).993 (.2).877 (.39).943 (.29) µ G.252 (.22).6 (.3).2225 (.37).82 (.2) µ CB.322 (.63).577 (.43) µ CG.346 (.4).843 (.22) µ CN.239 (.42).8 (.3)

15 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 5 Table 4. Empirical correlations for the PIN-AACD and APIN-AACD models: Sample correlations between the daily variance/return and PIN/APIN/PSOS for the IBM, GE, PG and WMT stocks, covering 754 trading days. Numbers within parentheses are p- values. IBM GE PIN APIN PSOS PIN APIN PSOS Variance.3786 (.) (.4).3822 (.).4928 (.) (.).4897 (.) Return (.57) (.39) (.4) (.288) (.) (.7543) PSOS.8842 (.) (.).8777 (.) (.) APIN (.) (.) PG WMT PIN APIN PSOS PIN APIN PSOS Variance.46 (.) -.26 (.5).239 (.).233 (.).296 (.465).348 (.2) Return (.6) (.6) (.5665) (.3) (.782) (.2744) PSOS.7395 (.) (.).7994 (.) (.) APIN (.) (.)

16 6 DANIEL PREVE AND YIU-KUEN TSE stock no news no common shock {Bi(si), si }, λi = λ N i {Si(si), si }, λ,i = λ N,i θcd θed θcd news good news no common shock {Bi(si), si }, λi = λ N i + λg i {Si(si), si }, λ,i = λ N,i common shock {Bi(si), si }, λi = λ N i + λcn i {Si(si), si }, λ,i = λ N,i + λcn,i θcd θ Ed θbd common shock {Bi(si), si }, λi = λ N i + λg i + λcg i {Si(si), si }, λ,i = λ N,i + λcg,i θcd bad news no common shock {Bi(si), si }, λi = λ N i {Si(si), si }, λ,i = λ N,i + λb,i θcd θ Bd θ Cd common shock {Bi(si), si }, λi = λ N i + λcb i {Si(si), si }, λ,i = λ N,i + λb,i + λcb,i Figure. Trading tree for the APIN-AACD model: {Bi(si), si } and {Si(si), si } are the latent Poisson processes of buy and sell orders initiated at time ti on trading day d, respectively, given the information Φi. In contrast to the PIN-AACD model, the APIN-AACD model has three additional states representing trading days in which both the conditional intensity of buy and sell orders increase due to common shocks. These days occur with probabilities θcd, irrespective of no news, good news or bad news.

17 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 7 Estimated Daily State Probabilities for the PIN AACD Model and IBM πg πn πb Figure 2. Model implied probabilities for the PIN-AACD model of IBM. Estimated Daily State Probabilities for the PIN AACD Model and GE πg πn πb Figure 3. Model implied probabilities for the PIN-AACD model of GE.

18 8 DANIEL PREVE AND YIU-KUEN TSE Estimated Daily State Probabilities for the APIN AACD Model and IBM πg πn πb.5 πcg πcn πcb Figure 4. Model implied probabilities for the APIN-AACD model of IBM. Estimated Daily State Probabilities for the APIN AACD Model and GE πg πn πb.5 πcg πcn πcb Figure 5. Model implied probabilities for the APIN-AACD model of GE.

19 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 9 Estimated Daily PIN, Adjusted PIN and PSOS for IBM. PIN APIN PSOS Figure 6. Estimated Daily PIN, APIN and PSOS for IBM: The top panel displays the PIN measure of the PIN-AACD model. The bottom panels display the APIN and PSOS measures of the APIN-AACD model, respectively. Estimated Daily PIN, Adjusted PIN and PSOS for GE. PIN APIN PSOS Figure 7. Estimated Daily PIN, APIN and PSOS for GE: The top panel displays the PIN measure of the PIN-AACD model. The bottom panels display the APIN and PSOS measures of the APIN-AACD model, respectively.

20 2 DANIEL PREVE AND YIU-KUEN TSE APPENDIX. PIN and APIN Models EHO propose a market microstructure model to derive a measure of asymmetric information reflecting the relative intensity of informed versus uninformed (liquidity) trades, called the probability of informed trading, PIN. As described in Figure 8, the PIN model assumes that each trading day may be classified as one with news or no news. Furthermore, a day with news can be one with good news or bad news. The daily aggregate number of buyer- and seller-initiated trades (buy and sell orders) are assumed to follow independent Poisson distributions with intensities dependent on whether the trading day is one with good news, bad news or no news. In the model there are two types of traders, informed traders who trade based on relevant news or information, and uninformed traders who trade for reasons not accounted for by relevant information, such as portfolio rebalancing and liquidity needs. Let B d and S d denote the aggregate number of buy and sell orders on day d, respectively. In the PIN model, B d and S d are assumed to be independent Poisson random variables, with different intensities for days with bad news (B), good news (G) and no news (N). Let θ E denote the probability of news being released on day d and let θ B denote the probability of bad news, conditional on the release of news. Thus, the daily state probabilities are π B = θ E θ B, π G = θ E ( θ B ) and π N = θ E, for a day with bad news, good news and no news, respectively. The means of B d and S d (the intensity parameters) vary according to whether the trading day is one with good news, bad news or no news. In particular, for a day with no news, the means of B d and S d are λ and λ, respectively. For a day with bad news the sell intensity increases by a constant δ, while the buy intensity remains the same as for a day with no news. Similarly, for a day with good news the buy intensity increases by δ, while the sell intensity stays the same as for a no-news day. The PIN model assumes that orders due to informed traders and uninformed traders are independent. For each trading day d, the joint distribution of B d and S d is given by f(b d, S d ) = s S f(b d, S d, s) = s S π s f(b d, S d s), for S = {B, G, N}, implying that the daily expected total number of trades is E(B d + S d ) = s S π s E(B d + S d s) = θ E θ B (λ + λ + δ) + θ E ( θ B )(λ + λ + δ) + ( θ E )(λ + λ ) = λ + λ + θ E δ, }{{}}{{} P P 2

21 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 2 where P and P 2 are the expected numbers of trades due to uninformed and informed traders, respectively. EHO compute the PIN as the relative intensity of informed trades to the intensity of all trades, so that PIN = P 2 P + P 2. The parameters in the PIN model can be estimated using the MLE. With D days of data, the likelihood function is [ D d= λ B d π B B d! e λ as given by EHO. (λ + δ)sd S d! e (λ +δ) + π G (λ + δ) B d B d! e (λ +δ) λs d ] λ Bd S d! e λ + π N B d! e λ λsd S d! e λ, DY extended the PIN model of EHO by allowing for the arrival rate of informed sellers to be different from the arrival rate of informed buyers and, more importantly, by allowing both buy- and sell-order flows to increase on certain days even when there is no news. Their APIN model, outlined in Figure 9, has three additional states representing days in which both the numbers of buys and sells increase due to symmetric order-flow shocks, or common shocks for short. The motivation for the first extension is to improve the ability of the PIN model to account for the fact that buy-order flow has a larger variance than sell-order flow, for almost all firms, in their empirical study. The second, more important, extension allows for increased buy and sell variations, and a positive correlation between buys and sells, as each day a common shock may occur that causes both buy- and sell-order flows to increase. In the APIN model B d and S d have different intensities for days with bad news and a common shock (CB), good news and a common shock (CG) and no news and a common shock (CN). The occurrence of a common shock is assumed to be independent of the arrival of news (good, bad or no news). Let θ C denote the daily probability of a common shock. The state space S then represents cases of no common shocks, and the extended state space is S = {CB, CG, CN, B, G, N}. In the event of a common shock, the buy intensity increases by and the sell intensity by. Possible causes for common shocks include the arrival of public news the implications of which traders disagree, and coordinated trading on certain days in order to reduce trading costs (Duarte and Young, 29). The APIN model also allows the arrival rate of informed sellers to be different from the arrival rate of informed buyers. On a day with bad news the sell intensity increases by δ, while the buy intensity remains the same as for a day with no news. On a day with good news the buy intensity increases by δ, while the sell intensity stays the same as for a no-news day.

22 22 DANIEL PREVE AND YIU-KUEN TSE The likelihood function for the APIN model is given by [ D d= π CB (λ + ) B d B d! λ B d + π (λ + δ ) Sd B B d! e λ S d! + π CG (λ + δ + ) B d B d! + π CN (λ + ) B d B d! e (λ+ ) (λ + δ + ) S d e (λ +δ + ) S d! e (λ +δ ) e (λ+δ+ ) (λ + ) S d S d! e (λ+ ) (λ + ) S d S d! e (λ + ) (λ + δ ) B d + π G B d! ] e (λ + ) + π λ B d N B d! λsd e λ S d! e λ e (λ+δ) λs d. S d! e λ For this model it is straightforward to show that the expected value of all trades for day d can be decomposed into three parts E(B d + S d ) = λ + λ + θ E [( θ B )δ + θ B δ ] + θ C ( + ) }{{}}{{}}{{} P P 2 P 3. TTTW estimated the PIN model assuming the latent trade directions follow the AACD model. The conditional intensities of buy- and sell-orders under different news environments are illustrated in Figure. bad news B d P(λ ) θ B S d P(λ + δ) news θ E θ B good news stock B d P(λ + δ) θ E no news B d P(λ ) S d P(λ ) S d P(λ ) Figure 8. Trading tree for the PIN model: B d and S d are the total number of buy and sell orders on trading day d, respectively. We write B d P(λ ) to indicate that B d is Poisson distributed with intensity parameter (mean and variance) λ. On each trading day news arrive with probability θ E. On a no-news day, B d is Poisson distributed with intensity λ and S d is Poisson distributed with intensity λ. Bad news causes an increase in the intensity of S d, consequently S d is Poisson distributed with intensity λ + δ on a bad-news day. Similarly, B d is Poisson distributed with intensity λ +δ on a good-news day.

23 ESTIMATION OF TIME VARYING APIN AND PSOS USING HIGH-FREQUENCY TRANSACTION DATA 23 common shock bad news Bd P(λ + ) Sd P(λ + δ + ) no common shock θc θb Bd P(λ) θ C news Sd P(λ + δ ) common shock good news Bd P(λ + δ + ) Sd P(λ + ) no common shock θc θe θ B stock common shock Bd P(λ + δ) θ C Bd P(λ + ) Sd P(λ ) Sd P(λ + ) θc θ E no news no common shock θ C Bd P(λ) Sd P(λ ) Figure 9. Trading tree for the APIN model: Bd and Sd are the total number of buy and sell orders on trading day d, respectively. We write Bd P(λ) to indicate that Bd is Poisson distributed with intensity parameter (mean and variance) λ. In contrast to the PIN model, the APIN model has three additional states representing trading days in which both the intensity of Bd and Sd increase due to common shocks. These days occur with probability θc, irrespective of no news, good news or bad news.

24 24 DANIEL PREVE AND YIU-KUEN TSE bad news {B i (s i ), s i }, λ i = λ N i θ Bd {S i (s i ), s i }, λ,i = λ N,i + λb,i news θ Ed θ Bd good news stock {B i (s i ), s i }, λ i = λ N i + λg i θ Ed no news {B i (s i ), s i }, λ i = λ N i {S i (s i ), s i }, λ,i = λ N,i {S i (s i ), s i }, λ,i = λ N,i Figure. Trading tree for the PIN-AACD model: {B i (s i ), s i } and {S i (s i ), s i } are the latent Poisson processes of buy and sell orders initiated at time t i on trading day d, respectively, given the information Φ i. On each trading day news arrive with probability θ Ed. On a no-news day the conditional intensity of the buy orders is λ N i and the conditional intensity of the sell orders is λ N,i. On a bad-news day the conditional intensity of sell orders increase by λ B,i, while that of buy orders remains the same as on a no-news day. Similarly, on a good-news day the conditional intensity of buy orders increase by λ G i, while that of sell orders remains the same as on a no-news day.