We thank participants at the 2026 North American Summer Meeting of the Econometric Society at Emory University and at the 20th CIREQ-CIRANO Conference in Montréal for helpful comments. Hounyo gratefully acknowledges financial support from the 2025–2026 Faculty Research Award Program, which funded the purchase of the Context Analytics sentiment data from Dewey Data. We thank the Massry School of Business for access to CRSP and Compustat data through Wharton Research Data Services (WRDS). The views expressed herein are those of the authors.
© 2026 by Piyush Akimitsu and Ulrich Hounyo. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.
Introduction
Data vendors now sell firm-level social media sentiment to investors. A large branch of literature finds that text-based sentiment predicts returns [8 , 71 , 17 , 12 ] . The reliability of these signals depends on who writes the posts. A sentiment index is derived from a feed to which people with information about the firm contribute, along with fads and automated content that carry no information about the firm. The index itself does not reveal the share of each contributor type, the information content of the feed, or the precision of the signal generated by such a feed. We study how much the composition of the contributor pool affects the reliability of sentiment as a signal about prices. For this, we need contributor pools that differ in a meaningful way. The literature on home bias shows that geographical proximity gives investors real informational advantages, so the investors who know a firm best are concentrated in its home market. When a firm is headquartered in a foreign country but listed in the U.S., the home-market investors who know it best are underrepresented among those who post about it on English-language Twitter.1
Using a noisy rational expectations framework, this paper models the composition of the contributor pool and the size of the noise the pool produces, and estimates both using the generalized method of moments (GMM). Specifically, for the composition of the contributor pool, we estimate the share of repeat authors (the accounts which post about the same stock more than once on a day). Additionally, for the noise, we estimate the scale of the fad component (the transient component of the Twitter feed which carries no information about the firm). Our hypothesis is that the foreign feed is less informative than the domestic feed. The results show that both the share of the repeat authors and the scale of the fad component are larger for foreign stocks than for domestic stocks. The headline result is that the reliability of information in the social media sentiment for foreign stocks differs systematically from that of the domestic stocks. An economically coherent explanation for this difference comes from analyzing the contributor composition, which differs systematically between feeds for foreign and domestic stocks. Our structural model disciplines this interpretation. A firm’s geographical location, whether domestic or foreign, is observable in the data. Thus, sorting on such geographical location lets us compare the contributor pools that differ in composition and lets us measure the difference in reliability between the two groups.
Our hypothesis about the mechanism behind this difference is the mismatch between contributors who write the tweet and people who hold information about the firm (the producer-knowledge mismatch hypothesis). Under this hypothesis, if a firm has headquarters in a foreign country (foreign firm), then most of the contributors who post about such a firm on English-language Twitter are likely not the ones who know the firm best. Even though Twitter has been a global social media platform, under our hypothesis English-language discussion of the U.S.-listed stocks is dominated disproportionately by U.S.-based users.2 [64 , 63 , 69 ] . The home-market investors of a firm with headquarters in a foreign country tend to possess an informational advantage about such a firm [31 , 9 ] . The advantage is visible in prices, since home markets lead price discovery for cross-listed stocks on average [33 ] .3 [68 ] .
This matters for three main reasons. First, investors who trade using Twitter sentiment get a signal whose reliability varies with firm geography. The variation in reliability is concealed if the sentiment signals from the feed of foreign and domestic stocks are weighted equally. Second, the conventional sentiment indices are aggregates of sentiment scores across tweets for a firm’s stock. This aggregate doesn’t reveal either the composition of the pool by contributor type (specialists, attention-driven contributors posting information about the firm along with the day’s fads, and automated accounts whose feed carries no information about the firm) or how much noise that composition adds. Third, the question bears on a large class of listings. Foreign firms list in the U.S. to reach U.S. investors. The recognition they gain lowers their cost of raising capital [61 , 53 ] . A noisier feed could move the prices of the foreign listings for reasons unrelated to the true firm value.
We formalize this idea in a noisy rational expectations model in which the Twitter feed acts as a public signal [47 , 1 , 35 ] . We add three features to the standard setup. The first is the repeat-author share. For a given posting rate, the vendor’s dispersion measure (distinct accounts per tweet) gives the share of repeat authors. The second one is the short-sale constraint [62 ] which allows part of the overpricing caused by optimistic noise to carry over into the next quarter. The third one is a daily dealer market [55 ] in which quoted spreads rise with informed trading. Since informed trading is not observed, we use the spread to rank stocks by it and compare domestic and foreign stocks within terciles of that rank. The model gives closed-form expressions for the moments of sentiment, returns, gains and losses, liquidity and price informativeness.
We estimate the model using GMM on the 2018 to 2024 primary sample, drawn from a panel of 471,880 stock-quarters of U.S.-listed firms covering 2010 to 2024, with firm-level Twitter sentiment from Context Analytics.4
The paper has three main results. The first one is obtained from reduced-form quarterly regressions. The temporal stability across quarters of the predictive relation between measured sentiment noise and next-quarter losses, the signal’s reliability, differs systematically between domestic and foreign stocks. In these regressions of next-quarter losses on sentiment noise,5 t = − 2.99 t = -2.99 t = − 2.99 p p p 6 − 2.00 -2.00 − 2.00 t = − 3.72 t = -3.72 t = − 3.72 − 0.74 -0.74 − 0.74 t = − 1.29 t = -1.29 t = − 1.29
The second result comes mainly from the structural estimation. The foreign feed’s contributor pool has a higher share of repeat authors than that of the domestic feed, and the foreign feed also carries a larger fad component than the domestic feed. The estimated share of repeat authors is higher for foreign stocks than for domestic stocks in all terciles of informed trading intensity. Thus, the domestic-foreign gap in repeat-author share doesn’t merely reflect the domestic-foreign gap in informed trading intensity. The domestic-foreign gap in repeat-author share is the widest among stocks with the most informed trading intensity. In the top tercile, the repeat-author share for foreign and domestic are 0.170 and 0.145, respectively (Table I). The fad scale, which is a standard deviation in the vendor’s per-tweet sentiment units, is 0.0293 for foreign stocks against 0.0273 for domestic stocks, so the variance of that component, which is noise by construction, is about 15 percent larger for foreign stocks. A joint Wald test rejects that the four parameters, the repeat-author share in each of the three informed-trading-intensity terciles and the fad scale, are equal for domestic and foreign stocks.7 Table XXI ). Thus, the foreign feed carries a less precise signal than the domestic feed at every level of informed trading intensity, which shows that the precision gap is not a byproduct of the domestic-foreign gap in informed trading. The gap narrows where informed trading is heaviest.
Table XXI. Implied price-informativeness data term by group and horizon against the median signal-quality index. Notes: This table is descriptive and has no test statistics. In each cell we divide the signed price-informativeness estimate at horizon s s s ( T q / V A Q S V f q 2 ) (T_q/\mathrm{VAQSV}^{2}_{fq}) ( T q / VAQSV f q 2 )
Our third result pertains to the mechanism through which the feed’s noise goes into next-quarter losses. In the model, the noise goes into the losses in two ways. The first way is through volatility. A feed with higher noise makes a day’s price more volatile, which produces larger daily losses and daily gains equally. This raises losses without changing the balance between losses and gains. The second way is through overpricing. When the feed’s sentiment is optimistic, the price rises above the firm’s value. The investors who don’t hold any stock can lower the prices back only by selling short. When short selling is closed to them, a part of the overpricing survives at the end of the quarter. In the next earnings announcement, the true value of the firm is revealed, which corrects the overpricing. This manifests as losses in the next quarter without an equivalent amount of gain to match it. This second way can make the losses exceed the gains. We ask which of the two the data support, and we use how much the price leans on the Twitter feed, which depends on feed precision. For each stock-quarter, we have a model-motivated signal-noise index that indicates how much overpricing the feed’s noise would create once the price leans on the Twitter feed. This index is the product of the weight the price puts on the feed (which rises with the feed’s precision) and the size of the noise the feed carries (which falls with the feed’s precision). So, the index is highest at an intermediate precision level where the feed is precise enough for the market to give it real weight but still imprecise enough that it carries sizeable noise. When the price leans more on the feed than on outside knowledge (what investors know about the firm from other sources), the noise from the feed goes into the price. In the model, the more overpricing the feed’s noise creates in a quarter, the larger the excess of losses over gains in the next quarter. As predicted by the overpricing correction channel, the excess of next-quarter losses over gains increases with this index (t = 9.63 t = 9.63 t = 9.63
The model gives some additional predictions which are tested using a battery of reduced-form checks.8 − 0.317 -0.317 − 0.317 − 0.612 -0.612 − 0.612 t = − 6.02 t = -6.02 t = − 6.02 − 2.11 -2.11 − 2.11 t = − 40.91 t = -40.91 t = − 40.91 t = 3.15 t = 3.15 t = 3.15
The interpretation of the structural results depends on two features. First, each stock’s signal precision is recovered as its observable signal-quality index divided by a conversion constant common to all stocks. This constant absorbs the scale of the vendor’s score, the serial dependence across days and the dilution of the signal by automated content, which the data do not separate (§ 5 ). The estimate of this conversion constant is about 400, with a standard error of 34 (Table I). In the domestic-foreign comparisons of signal precision, the constant cancels and the comparisons depend on the observable index alone. Second, about 85 percent of the weighted distance between model predicted and data computed moments comes from the 24 moments of the four liquidity measures, namely the quoted spread, the high-low spread, the Amihud ratio and turnover. None of the main results depend on the levels which the model recovers from those objects. The paper uses these objects mainly to rank the stocks by informed-trading-intensity terciles within which the domestic-foreign comparisons are made.
The paper makes three contributions. First, we show that the reliability of a social media sentiment signal, the stability across quarters of its link to next-quarter losses, depends on firm geography. The studies that link text-based sentiment to returns leave out of their scope who produces the sentiment and for which stocks it is reliable (§ 2 ). The domestic-foreign gap in reliability holds among the stocks with the least informed trading, which shows that such a gap is not an artifact of the domestic-foreign difference in informed trading. The home-bias literature documents a home-market informational advantage in holdings and returns. We interpret this domestic-foreign gap in reliability as that advantage showing up in the feed for domestic stocks, whose home-market investors dominate English-language Twitter, and not in the feed for foreign stocks, whose home-market investors are underrepresented there. For an investor who trades on Twitter sentiment, the sentiment index is a less reliable signal for a foreign stock than for a domestic stock. Second, the paper gives the composition of the Twitter feed contributor pool and the precision of the sentiment signal from the vendor’s aggregate metrics alone, without account-level data. Specifically, we get the share of repeat authors from contributor dispersion at a given posting rate (Proposition 1), the fad from the part of sentiment volatility that does not fall as tweet volume rises, and the precision of the signal from that volatility as a whole (Proposition 2). Prior work backs that precision out of price informativeness, where it is a residual that cannot be traced to a source. In this paper, precision is the observable signal-quality index which is computed from the feed’s daily readings, divided by one constant common to all stocks. So, the domestic-foreign comparison of precision comes from the feed alone, can be traced to the composition of the pool that produces it and can be tested against how the feed moves prices. Our hypothesis of a mismatch between who posts about a firm and who knows it is bolstered by the structural estimates which show a higher share of repeat authors and a lower signal-quality index for foreign stocks in each informed-trading-intensity tercile, and a larger fad component for the foreign feed. A part of the literature links sentiment with the feed contributors and their information. For instance, Cookson et al. (2024) relate differences in sentiment across platforms to differences in their users, while Huang et al. [50 ] find that local posters on a Chinese message board are better informed than non-local ones. This paper, instead, measures contributor composition within a single platform, Twitter, and relates it to firm geography. The paper provides closed-form moment conditions which other studies and future research pertaining to other social media platforms can use to estimate the same parameters. Third, we document a regularity in next-quarter returns whose sign matches the correction of overpricing left by short-sale constraints (Miller [62 ] ), and we find no support for that channel. Within each of the groups, domestic and foreign, and informed-trading-intensity tercile, the excess of next-quarter losses over gains rises with the model-motivated signal-noise index, which indicates how much overpricing the feed’s noise would create. The estimated short-sale friction lies at its zero lower bound, and the loss response to noise is no stronger after quarters with more positive-sentiment days than after other quarters (row R8, Table IV). Since return volatility raises gains and losses alike (Proposition 5), neither channel of the fitted model produces the rise, and we report it as a reduced-form pattern. Thus, a loss-gain asymmetry with the sign that a mispricing channel predicts is not by itself evidence of that channel.
The rest of the paper proceeds as follows. § 2 reviews the literature. § 3 presents the model. Section 4 describes the data. § 5 reports the structural estimation. § 6 reports the reduced-form evidence. § 7 concludes.
Related Literature
Prior work in this area shows that investors overweigh domestic and nearby securities [38 ] , while investment funds hold more domestic than foreign stocks [56 ] . This geographical proximity is more than just familiarity. It also gives real informational advantages to the investors [52 , 2 , 23 ] (Coval and Moskowitz 2001). Some other related work shows that domestic investors outperform foreign investors in the same securities [31 ] , and that local analysts forecast more accurately in less transparent environments [9 ] . Additionally, the investors rationally deepen initial local advantages and Van Nieuwerburgh and Veldkamp [64 ] provide a theoretical basis for that. In addition to geography, language shapes the attention which foreign firms receive. Boulland et al. [14 ] show that investor attention rose when continental European firms began disseminating company news through an English-language electronic wire service. We study this information geography pertaining in the context of Twitter and use the domestic-foreign comparison to identify the information content of the sentiment of its feed.
Our producer-knowledge mismatch hypothesis is based on the premise that the coverage of foreign stocks on the English-language feed by the specialists is lesser than it is for the domestic stocks. Assets which get lower recognition from the investors trade below their full-information value [61 ] . Investors with finite capacity direct their attention to the signals with the highest informational return [69 , 66 ] . This leads to endogenous home bias [63 , 64 ] . Retail investors are net buyers of stocks that get more attention [11 ] . At the market level, daily aggregate retail attention predicts lower returns over the following week, while institutional attention ahead of major announcements predicts higher returns (Da et al., 2025). Evidence from surveys and experiments shows that households and finance professionals read good news about a firm as a sign of higher returns and often neglect the price adjustment that news has already caused [7 ] . English-language Twitter aggregates these attention decisions. Thus, the crowd-sourced signal should be more precise for domestic than for foreign stocks.
The literature establishes the connection between text-based sentiment and returns. For instance, internet message boards help predict volatility [8 ] . Pessimistic coverage in a Wall Street Journal column predicts downward price pressure which reverses later [71 ] , while the predictability from newspaper content is concentrated in recessions [40 ] . This holds for social media as well. Seeking Alpha articles and their reader commentaries predict returns and earnings surprises [17 ] . The aggregate opinion in tweets posted just before an earnings announcement predicts the firm’s earnings and its announcement return [12 ] . Similarly, a comparison between global and local news sentiment shows that the former moves returns more, and more persistently, than the latter [37 ] . Allen et al. [4 ] show that the discussion of the meme stocks of January 2021 on Reddit, Twitter and StockTwits strengthened the relation between retail trading and subsequent returns, and that the short squeezes which followed impaired market quality for those stocks and for their competitors. While these studies show that text-based sentiment carries price-relevant information, it is out of their scope to reflect on who produces this sentiment and for which stocks it is reliable.
Gómez-Cram et al. [44 ] reflect on this question for the prediction markets where prices are probabilities and contracts pay off on the outcome of the event. In their Polymarket data, about 3% of accounts produce most of the price discovery, while the rest supply the volume of trade but little information. For social media, Gil-Bazo and Imbet [41 ] show one producer whose posts move money without carrying information. The posts of U.S. mutual fund families on Twitter raise the flows into their funds, and the authors reject lower search costs, higher attention and information about fund performance as the channels. We ask a similar question for social media, whether the predictive content of the feed comes from the crowd or the smaller subset of informed contributors and whether, for foreign stocks, the contributor pool tilts toward the types that carry no new information about the firm.
A recent strand of literature links this sentiment to the contributor pool’s composition.
Cookson et al. (2024) trace the differences in the sentiment signal of various platforms to their user population composition and platform design. Huang et al. [50 ] show the informational advantage of local posters over non-local posters on a Chinese message board. Our paper focuses on English-language Twitter, links the heterogeneity in sentiment to firm geography and then estimates the contributor composition and nonfundamental component of the Twitter feed.
Our paper also relates to the literature on price informativeness. We use a noisy rational expectations model and adopt the Farboodi et al. [35 ] decomposition of the Bai et al. [10 ] measure of price informativeness. In our model the precision of the sentiment signal enters that decomposition (§ 3.3 ). Dugast and Foucault [30 ] show that cheaper low-precision signals can reduce price informativeness by lowering the demand for more precise signals. The noise we study originates in the public signal and reaches prices through the weight that investors put on the feed. Its source is the part of the feed’s day-to-day variation that carries no information about the firm, and that part is informative about the size of the fad and the tweet shares of the contributor types (§ 3.2 ). (The related work on heterogeneous beliefs and on cross-listing is discussed in E.)
A Model of Sentiment, Information, and Liquidity
Three types of contributors produce a stock’s Twitter feed. (i) Specialists who post about a firm repeatedly. (ii) Attention-driven contributors who engage with any one firm occasionally. Their posts have the firm-specific signal and a transient nonfundamental component that the specialists filter out. (iii) The automated accounts which generate content mechanically. The composition of the feed based on these types determines how informative the aggregated feed is. That informativeness is given by a precision parameter in the model. A more precise public signal implies less uncertainty about firm value for the investors. Lower uncertainty implies a higher information component of price informativeness and lower return volatility. The signal noise enters prices in proportion to the weight the market rationally gives the signal. With short-sale constraints, the noise that overstates value leads to overpricing. This overpricing is corrected over the next quarter, resulting in losses. The model shows this effect is largest at an intermediate level of signal precision. A more precise signal narrows the informational advantage of informed investors over dealers, which compresses bid-ask spreads and market-impact costs. Geography, foreign and domestic, enters the model through the shares of types and the fad and bot variances. Holding the other group parameters fixed, the domestic-foreign differences in volatility, losses, informativeness, and liquidity can all be traced to differences in that signal precision.
Our framework combines three standard ingredients and a new layer: 1. A noisy rational expectations equilibrium in the tradition of Hellwig [47 ] , Admati [1 ] , and Farboodi et al. [35 ] : In such an equilibrium, prices aggregate dispersed information.9 [62 ] : The public signal about a firm’s value has noise which makes the signal either overstate or understate that value. An understatement doesn’t persist since it is removed within the quarter by ordinary buying which no investor is barred from. An overstatement raises the price and causes overpricing. Correcting it requires selling. An investor who holds no shares can sell only through a short sale. When short selling is closed to some of those investors, part of the overstatement remains in the price as overpricing. The surviving overpricing is corrected at the next earnings announcement, when the firm’s realized value becomes public and the price falls to that value. Thus, overpricing today predicts low returns over the following quarter. We refer to the correction of the surviving overpricing as the drift channel (Proposition 6 ). The second channel through which the signal’s noise reaches gains and losses is the volatility of returns, which increases the magnitudes of gains and losses by the same amount such that the difference between those magnitudes remains unchanged (Proposition 5). In the model, only the correction of surviving overpricing can make the magnitude of losses exceed the magnitude of gains. 3. A daily dealer market in the spirit of Kyle [55 ] and Glosten and Milgrom [43 ] : Trading costs are the dealers’ compensation for adverse selection (trading against better-informed investors). As the informed trading increases, the observed spreads and market-impact costs increase. Thus, given the feed’s precision, a stock’s quoted spread reveals how much informed trading it attracts. We sort stocks, within geography and quarter, into terciles of informed-trading intensity on the lagged quoted spread multiplied by the square root of prior precision plus lagged signal precision (§ 3.5 ). This sort needs no records of which types of investor hold each stock. 4. The contributor model: The model links the vendor-provided dispersion metric to the concentration of repeat authors. It also splits the day-to-day variation in sentiment per tweet into two parts. For each stock and day, the data have a sentiment total which is an exponentially time-weighted sum of the sentiment scores of the tweets about the stock over the day’s 24-hour window (S t S_t S t N t N_t N t S t / N t S_t/N_t S t / N t § 3.2 ). We derive how the metrics in the data link to the precision of a public signal. The nonfundamental variance is recovered from these vendor metrics. The signal precision for a quarter is recovered as the stock’s observable signal-quality index (the number of days in the quarter with at least one tweet about the stock divided by the squared within-quarter volatility of daily sentiment per tweet), divided by a conversion constant that the estimation holds common across all stocks. Thus, the feed for a stock or group of stocks with a higher signal-quality index is more precise than the feed for a stock or group with a lower signal-quality index, irrespective of the value of the constant.10 § 6 , Table I).
The economic environment
3.1. The economic environment: Time is a sequence of quarters. We describe one quarter and suppress the quarter index except where needed. Within the description, periods s = 0 , 1 , 2 , … s = 0, 1, 2, \ldots s = 0 , 1 , 2 , … s = 0 s = 0 s = 0 f f f j j j f f f s s s d f s ∗ d^*_{fs} d f s ∗ 11 d f 0 ∗ d^*_{f0} d f 0 ∗ f f f
d f 1 ∗ = v j d f 0 ∗ + ϵ ˉ f 1 + ϵ f 1 , ϵ f 1 ∼ N ( 0 , Σ j d ) , d f s ∗ = v j s − 1 d f 1 ∗ ( s ≥ 2 ) , (1) d^*_{f1} = v_j d^*_{f0} + \bar{\epsilon}_{f1} + \epsilon_{f1}, \qquad\epsilon_{f1} \sim N(0,\Sigma_{jd}), \qquad d^*_{fs} = v_j^{s-1}d^*_{f1} \quad(s \ge2), \tag*{(1)} d f 1 ∗ = v j d f 0 ∗ + ϵ ˉ f 1 + ϵ f 1 , ϵ f 1 ∼ N ( 0 , Σ j d ) , d f s ∗ = v j s − 1 d f 1 ∗ ( s ≥ 2 ) , (1) where v j v_j v j r r r 0 ≤ v j < r 0 \le v_j < r 0 ≤ v j < r v j v_j v j d f , s + 1 ∗ = v j d f s ∗ d^*_{f,s+1} = v_j d^*_{fs} d f , s + 1 ∗ = v j d f s ∗ s ≥ 1 s \ge1 s ≥ 1 v j v_j v j ϵ ˉ f 1 \bar{\epsilon}_{f1} ϵ ˉ f 1 ϵ f 1 \epsilon_{f1} ϵ f 1 Σ j d \Sigma_{jd} Σ j d [35 ] , all investors observe the common component, so all learning concerns ϵ f 1 \epsilon_{f1} ϵ f 1 P f 1 ∗ P^*_{f1} P f 1 ∗ V f 1 ∗ = ∑ s ≥ 1 d f s ∗ / r s − 1 V^*_{f1} = \sum_{s \ge1} d^*_{fs}/r^{s-1} V f 1 ∗ = ∑ s ≥ 1 d f s ∗ / r s − 1 d f s ∗ = v j s − 1 d f 1 ∗ d^*_{fs} = v_j^{s-1}d^*_{f1} d f s ∗ = v j s − 1 d f 1 ∗ v j / r v_j/r v j / r V f 1 ∗ = γ j d f 1 ∗ V^*_{f1} = \gamma_j d^*_{f1} V f 1 ∗ = γ j d f 1 ∗ γ j ≡ r / ( r − v j ) \gamma_j \equiv r/(r-v_j) γ j ≡ r / ( r − v j )
regressions do not use the precision, since their regressor is the within-quarter volatility of daily sentiment per tweet. Only the level of a group’s precision depends on the constant. None of the findings quote that level. of a growing perpetuity (D.1). The supply of shares is x ˉ j + x ~ f \bar{x}_j+\tilde{x}_f x ˉ j + x ~ f x ˉ j \bar{x}_j x ˉ j x ~ f ∼ N ( 0 , Σ j x ) \tilde{x}_f \sim N(0,\Sigma_{jx}) x ~ f ∼ N ( 0 , Σ j x ) [59 ] . An investor’s private signal is the firm-specific innovation and noise specific to that investor. The public signal is composed of the innovation and the public noise. The private signals factor into the prices through their average. The investor-specific noise averages out across investors. Hence, the average private signal is equal to the innovation. Thus, the price depends on—innovation, public noise and the number of shares on the market (Proposition 3). If an investor could observe the number of shares on the market then the price and the public signal could reveal the average private signal or the innovation. The random component prevents that so that the price still reflects the private signals without revealing their average.
The model’s events are spread over two quarters. Figure I shows the timing of the four events, the feed, the trading, the announcement, and the correction. The feed and the trading fall in quarter q q q q + 1 q+1 q + 1
Figure I. Timing of the mechanism and the empirical design
Notes: A firm announces its earnings for quarter q q q q + 1 q+1 q + 1 q q q
Alt text: There are five boxes—three for quarter q q q q + 1 q+1 q + 1 q q q
A continuum of investors i i i ρ ˉ \bar{\rho} ρ ˉ q f i q_f^i q f i E [ W i ∣ I i ] − ρ ˉ 2 Var [ W i ∣ I i ] E[W^i\mid\mathcal{I}^i]-\frac{\bar{\rho}}{2}\operatorname{Var}[W^i\mid\mathcal{I}^i] E [ W i ∣ I i ] − 2 ρ ˉ Var [ W i ∣ I i ] I i \mathcal{I}^i I i i i i W 0 i W_0^i W 0 i W i = r W 0 i + ∑ f q f i ( V f 1 ∗ − r P f 1 ∗ ) W^i=rW_0^i+\sum_f q_f^i(V_{f1}^*-rP_{f1}^*) W i = r W 0 i + ∑ f q f i ( V f 1 ∗ − r P f 1 ∗ ) i i i f f f ϵ f 1 \epsilon_{f1} ϵ f 1 τ d ≡ Σ j d − 1 \tau_d\equiv\Sigma_{jd}^{-1} τ d ≡ Σ j d − 1 k i k^i k i i i i η f 1 i ≡ ϵ f 1 + e f 1 i \eta_{f1}^i\equiv\epsilon_{f1}+e_{f1}^i η f 1 i ≡ ϵ f 1 + e f 1 i e f 1 i ∼ N ( 0 , 1 / k i ) e_{f1}^i\sim N(0,1/k^i) e f 1 i ∼ N ( 0 , 1/ k i )
A higher k i k^i k i k i k^i k i i i i P f 1 ∗ P_{f1}^* P f 1 ∗ τ P \tau_P τ P § 3.2 constructs this signal from the feed and states the scope of such modeling choice. The investor’s posterior is made up of a combination of these four items. Let k j i k_j^i k j i i i i j j j j j j K j = ∫ 0 1 k j i d i K_j=\int_0^1 k_j^i\,di K j = ∫ 0 1 k j i d i K j K_j K j
these two margins separately to get the equilibrium price and the precisions behind it. K j K_j K j j j j 12 K j K_j K j § 3.5 ).13 τ d \tau_d τ d τ p \tau_p τ p K j K_j K j τ s \tau_s τ s T j ≡ τ d + τ p + K j + τ s T_j \equiv\tau_d+\tau_p+K_j+\tau_s T j ≡ τ d + τ p + K j + τ s Σ ‾ j = 1 / T j \overline{\Sigma}_j=1/T_j Σ j = 1/ T j τ d \tau_d τ d τ p \tau_p τ p K j K_j K j j j j τ s \tau_s τ s § 3.2 ). Where a group value of τ s \tau_s τ s K j K_j K j T j T_j T j Σ ‾ j \overline{\Sigma}_j Σ j τ d \tau_d τ d τ p \tau_p τ p τ s \tau_s τ s § 3.5 ).
Social media as a public signal
The Twitter feed is public, available through the vendor’s aggregate metrics, and carries the public signal in the model. This paper is about the composition of the types of contributors who generate the feed and the sentiment in it. Proposition 2 shows how it is constructed from the quarter’s daily readings. The estimates specifically characterize the Twitter feed rather than public information in general.14
The signal is s f 1 = ϵ f 1 + u f 1 s_{f1}=\epsilon_{f1}+u_{f1} s f 1 = ϵ f 1 + u f 1 u f 1 u_{f1} u f 1 τ s \tau_s τ s 15 K j K_j K j § 3.1 ).16
Three vendor metrics summarize the feed at the firm-quarter level. The volume-adjusted quarterly sentiment volatility V A Q S V f q \mathrm{VAQSV}_{fq} VAQSV f q 17 Q S V f q = S D t ∈ q ( S t ) \mathrm{QSV}_{fq} = \mathrm{SD}_{t \in q}(S_t) QSV f q = SD t ∈ q ( S t ) V A Q S V f q = S D t ∈ q ( S t / N t ) \mathrm{VAQSV}_{fq} = \mathrm{SD}_{t \in q}(S_t/N_t) VAQSV f q = SD t ∈ q ( S t / N t ) 18 S t S_t S t S t / N t S_t/N_t S t / N t S t S_t S t e e e N t N_t N t N t N_t N t (4) absorbs that factor into the constant c w ≥ 1 c_w \ge1 c w ≥ 1 T q T_q T q T d a y s T_{days} T d a y s T d a y s T_{days} T d a y s D t = A t / N t D_t = A_t/N_t D t = A t / N t A t A_t A t D ‾ f q \overline{D}_{fq} D f q D t D_t D t T q T_q T q D t D_t D t q + 1 q + 1 q + 1 q q q
concentration rather than disagreement of opinion. Additionally, we have the average inverse tweet count, ( 1 / N t ) ‾ f q = 1 T q ∑ t ∈ q 1 / N t \overline{(1/N_t)}_{fq} = \frac{1}{T_q}\sum_{t\in q}1/N_t ( 1/ N t ) f q = T q 1 ∑ t ∈ q 1/ N t 19
For each stock, posts come from three contributor types. 1. The specialists (S): Accounts that post about a firm repeatedly and whose posts carry the firm-specific signal. In the model, a specialist’s tweet encompasses the firm-specific innovation and the tweet’s own noise. Two forces motivate the repeated posting by the specialists. Since learning about a firm is costly, an investor who already knows a firm well gains more from learning more about it than learning about another unfamiliar firm (Van Nieuwerburgh and Veldkamp, [64 ] ). An investor who has traded on private information earns profits when the price moves toward the value tied to that private information. Public posting helps such price movement, so such an investor who has private information has an incentive to post repeatedly. These two forces motivate the assumption that specialists post repeatedly. They are not evidence, and the data doesn’t show how often any account or contributor type posts. Moreover, posting does not make a specialist’s private signal public. A specialist’s tweet consists of the firm-specific innovation and the tweet’s own noise. The feed averages such tweets into the public signal, whose precision is τ s \tau_s τ s k i k^i k i [11 ] ). Their posts have the firm-specific signal along with the day’s fad which is a transient nonfundamental component that specialists filter out and these attention-driven contributors do not. 3. Automated accounts (B): Automated accounts, which post machine-generated content, also called bots (an informal label used for brevity). Such accounts are rare among authors but their tweets occur commonly in the feed.20
A t A_t A t t t t j j j π S , j \pi_{\mathrm{S},j} π S , j π B , j \pi_{\mathrm{B},j} π B , j 1 − π S , j − π B , j 1-\pi_{\mathrm{S},j}-\pi_{\mathrm{B},j} 1 − π S , j − π B , j g g g λ g \lambda_g λ g λ S > 1 \lambda_{\mathrm{S}}>1 λ S > 1 λ B > 1 \lambda_{\mathrm{B}}>1 λ B > 1 λ A = 1 \lambda_{\mathrm{A}}=1 λ A = 1 21 ϵ f 1 + e \epsilon_{f1}+e ϵ f 1 + e β t + e \beta_t+e β t + e ϵ f 1 + b t + e \epsilon_{f1}+b_t+e ϵ f 1 + b t + e b t b_t b t t t t β t \beta_t β t t t t e e e ϵ f 1 \epsilon_{f1} ϵ f 1 § 3.1 ). Both a specialist tweet and an attention-driven tweet contain ϵ f 1 \epsilon_{f1} ϵ f 1 b t b_t b t β t \beta_t β t ϵ f 1 \epsilon_{f1} ϵ f 1
Assumption A1 (distribution). The fad b t ∼ N ( 0 , σ b , j 2 ) b_t\sim N(0,\sigma^2_{b,j}) b t ∼ N ( 0 , σ b , j 2 ) β t ∼ N ( 0 , σ β , j 2 ) \beta_t\sim N(0,\sigma^2_{\beta,j}) β t ∼ N ( 0 , σ β , j 2 ) e ∼ N ( 0 , σ e 2 ) e\sim N(0,\sigma^2_e) e ∼ N ( 0 , σ e 2 ) b t b_t b t β t \beta_t β t e e e ϵ f 1 \epsilon_{f1} ϵ f 1
The specialist type contains the informed specialists and institutional trading desks and investor relations accounts, since these post repeatedly with knowledge of the firm. Investor relations accounts post favorably about their own firm. If the degree of that favorable tilt is the same each day, then it increases the sentiment score by the same amount each day. QSV and VAQSV remain unchanged by it. If the degree of that favorable tilt changes from day to day, it adds day-to-day movement unrelated to the firm’s innovation to the sentiment score. The model categorizes it as a nonfundamental component. The bot type contains automated accounts and also news aggregators, which repost public firm news repeatedly and add no new information about the innovation.
Dispersion depends on the shares of specialists and bots. On a day with A t A_t A t ( π S , j + π B , j ) λ I A t (\pi_{\mathrm{S},j}+\pi_{\mathrm{B},j})\lambda_I A_t ( π S , j + π B , j ) λ I A t ( 1 − π S , j − π B , j ) A t (1-\pi_{\mathrm{S},j}-\pi_{\mathrm{B},j})A_t ( 1 − π S , j − π B , j ) A t π j λ I + ( 1 − π j ) \pi_j\lambda_I+(1-\pi_j) π j λ I + ( 1 − π j ) π j ≡ π S , j + π B , j \pi_j\equiv\pi_{\mathrm{S},j}+\pi_{\mathrm{B},j} π j ≡ π S , j + π B , j
Proposition 1 (Dispersion determines the repeat-author share for a given posting intensity. ) Given the feed structure, daily dispersion in group j j j D = [ π j λ I + ( 1 − π j ) ] − 1 D=[\pi_j\lambda_I+(1-\pi_j)]^{-1} D = [ π j λ I + ( 1 − π j ) ] − 1 π j = π S , j + π B , j \pi_j=\pi_{\mathrm{S},j}+\pi_{\mathrm{B},j} π j = π S , j + π B , j π j \pi_j π j
π j = 1 − D D ( λ I − 1 ) (2) \pi_j=\frac{1-D}{D(\lambda_I-1)} \tag*{(2)} π j = D ( λ I − 1 ) 1 − D (2) recovers the total repeat-author share from observed dispersion for a given λ I > 1 \lambda_I>1 λ I > 1 π j = 0 \pi_j=0 π j = 0 D = 1 D=1 D = 1 π j \pi_j π j
Proposition 2 links the within-quarter volatility of the day’s per-tweet sentiment to the noise in the public signal. The day’s per-tweet sentiment s ˉ t \bar{s}_t s ˉ t N t N_t N t N t = A t / D N_t=A_t/D N t = A t / D N t N_t N t ω S , j ≡ π S , j λ I D \omega_{\mathrm{S},j}\equiv\pi_{\mathrm{S},j}\lambda_I D ω S , j ≡ π S , j λ I D ω B , j ≡ π B , j λ I D \omega_{\mathrm{B},j}\equiv\pi_{\mathrm{B},j}\lambda_I D ω B , j ≡ π B , j λ I D ω A , j ≡ ( 1 − π j ) D \omega_{\mathrm{A},j}\equiv(1-\pi_j)D ω A , j ≡ ( 1 − π j ) D D [ π j λ I + ( 1 − π j ) ] D[\pi_j\lambda_I+(1-\pi_j)] D [ π j λ I + ( 1 − π j )] ϵ f 1 + e ˉ t S \epsilon f_1+\bar{e}^{\mathrm{S}}_t ϵ f 1 + e ˉ t S ϵ f 1 + b t + e ˉ t A \epsilon f_1+b_t+\bar{e}^{\mathrm{A}}_t ϵ f 1 + b t + e ˉ t A β t + e ˉ t B \beta_t+\bar{e}^{\mathrm{B}}_t β t + e ˉ t B e ˉ t g \bar{e}^{g}_t e ˉ t g ω g , j N t \omega_{g,j}N_t ω g , j N t g g g σ e 2 / ( ω g , j N t ) \sigma_e^2/(\omega_{g,j}N_t) σ e 2 / ( ω g , j N t )
s ˉ t = ω S , j ( ϵ f 1 + e ˉ t S ) + ω A , j ( ϵ f 1 + b t + e ˉ t A ) + ω B , j ( β t + e ˉ t B ) = ( 1 − ω B , j ) ϵ f 1 + ω A , j b t + ω B , j β t + u ˉ t . (3) \bar{s}_t=\omega_{\mathrm{S},j}\left(\epsilon f_1+\bar{e}^{\mathrm{S}}_t\right)+\omega_{\mathrm{A},j}\left(\epsilon f_1+b_t+\bar{e}^{\mathrm{A}}_t\right)+\omega_{\mathrm{B},j}\left(\beta_t+\bar{e}^{\mathrm{B}}_t\right)=(1-\omega_{\mathrm{B},j})\epsilon f_1+\omega_{\mathrm{A},j}b_t+\omega_{\mathrm{B},j}\beta_t+\bar{u}_t. \tag*{(3)} s ˉ t = ω S , j ( ϵ f 1 + e ˉ t S ) + ω A , j ( ϵ f 1 + b t + e ˉ t A ) + ω B , j ( β t + e ˉ t B ) = ( 1 − ω B , j ) ϵ f 1 + ω A , j b t + ω B , j β t + u ˉ t . (3) The loading on the innovation is ω S , j + ω A , j = 1 − ω B , j \omega_{\mathrm{S},j}+\omega_{\mathrm{A},j}=1-\omega_{\mathrm{B},j} ω S , j + ω A , j = 1 − ω B , j u ˉ t = ω S , j e ˉ t S + ω A , j e ˉ t A + ω B , j e ˉ t B \bar{u}_t=\omega_{\mathrm{S},j}\bar{e}^{\mathrm{S}}_t+\omega_{\mathrm{A},j}\bar{e}^{\mathrm{A}}_t+\omega_{\mathrm{B},j}\bar{e}^{\mathrm{B}}_t u ˉ t = ω S , j e ˉ t S + ω A , j e ˉ t A + ω B , j e ˉ t B σ e 2 / N t \sigma_e^2/N_t σ e 2 / N t S t / N t S_t/N_t S t / N t s ˉ t \bar{s}_t s ˉ t
Proposition 2 (Volume-adjusted quarterly sentiment volatility and the precision of the public signal. ) Under A1, and when the factor c 1 , t c_{1,t} c 1 , t j j j
E [ V A Q S V f q 2 ∣ ϵ f 1 , { N t } t ∈ q ] = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 + c w σ ϵ 2 ( 1 / N t ‾ ) ≡ v ‾ f q , (4) E\left[\mathrm{VAQSV}_{fq}^{2} \mid\epsilon_{f1}, \{N_t\}_{t\in q}\right] = \omega_{A,j}^{2}\sigma_{b,j}^{2} + \omega_{B,j}^{2}\sigma_{\beta,j}^{2} + c_w\sigma_{\epsilon}^{2}\left(\overline{1/N_t}\right) \equiv\overline{v}_{fq}, \tag*{(4)} E [ VAQSV f q 2 ∣ ϵ f 1 , { N t } t ∈ q ] = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 + c w σ ϵ 2 ( 1/ N t ) ≡ v f q , (4) where Φ j \Phi_j Φ j S t / N t S_t/N_t S t / N t c 1 , t σ ϵ 2 / N t c_{1,t}\sigma_{\epsilon}^{2}/N_t c 1 , t σ ϵ 2 / N t c 1 , t = 1 c_{1,t}=1 c 1 , t = 1 c 1 , t > 1 c_{1,t}>1 c 1 , t > 1 c w ≥ 1 c_w\geq1 c w ≥ 1 ( c w σ ϵ 2 ) (c_w\sigma_{\epsilon}^{2}) ( c w σ ϵ 2 ) ( 1 / N t ‾ ) \left(\overline{1/N_t}\right) ( 1/ N t ) (4) . v ‾ f q \overline{v}_{fq} v f q T q T_q T q V A Q S V f q 2 \mathrm{VAQSV}_{fq}^{2} VAQSV f q 2 v ‾ f q \overline{v}_{fq} v f q s f 1 = ϵ f 1 + u f 1 s_{f1}=\epsilon_{f1}+u_{f1} s f 1 = ϵ f 1 + u f 1 s ‾ t \overline{s}_t s t T q T_q T q 1 − ω B , j 1-\omega_{B,j} 1 − ω B , j u f 1 u_{f1} u f 1 ω A , j b t + ω B , j β t + u ‾ t \omega_{A,j}b_t+\omega_{B,j}\beta_t+\overline{u}_t ω A , j b t + ω B , j β t + u t 1 − ω B , j 1-\omega_{B,j} 1 − ω B , j
τ s = T q κ ⋅ V A Q S V f q 2 , κ > 0 , (5) \tau_s=\frac{T_q}{\kappa\cdot\mathrm{VAQSV}_{fq}^{2}}, \qquad\kappa>0, \tag*{(5)} τ s = κ ⋅ VAQSV f q 2 T q , κ > 0 , (5) where κ \kappa κ T q / V A Q S V f q 2 T_q/\mathrm{VAQSV}_{fq}^{2} T q / VAQSV f q 2 τ s \tau_s τ s 1 − ω B , j 1-\omega_{B,j} 1 − ω B , j κ \kappa κ 22
the innovation to 1 − ω B , j 1-\omega_{B,j} 1 − ω B , j (3) normalizes the specialist loading to one. (Proofs of equations (4) and (5) in C.2.)
The group-level precision of the public signal is τ s , j \tau_{s,j} τ s , j j j j § 3.1 . This precision of public signal differs across groups because the tweet shares ( ω g , j ) (\omega_{g,j}) ( ω g , j ) ( σ b , j 2 (\sigma_{b,j}^{2} ( σ b , j 2 σ β , j 2 \sigma_{\beta,j}^{2} σ β , j 2 Φ j \Phi_j Φ j (4) and (5) ).23 ( T q / V A Q S V f q 2 ) (T_q/\mathrm{VAQSV}_{fq}^{2}) ( T q / VAQSV f q 2 ) τ s , j \tau_{s,j} τ s , j (5) ) varies at the stock-quarter level. The estimation aggregates it to the group level (§ 5 ). The linear regression in Equation (4) recovers the nonfundamental variance ( Φ j ) (\Phi_j) ( Φ j ) ( c w σ ε 2 ) (c_w\sigma_{\varepsilon}^{2}) ( c w σ ε 2 ) § 5 ) uses the equations (4) and (5) under two conventions. The first one sets the bot-content variance σ β , j 2 \sigma_{\beta,j}^{2} σ β , j 2 Φ j \Phi_j Φ j κ \kappa κ
Equilibrium prices and informativeness
This subsection deals with how prices respond when the public signal is added to the standard information structure. The answer is a linear equilibrium, derived by the conjecture-and-verify argument of Hellwig [47 ] and Admati [1 ] . Our conjecture is that the price is affine in the innovation, the signal noise and the supply shock. Each investor then forms a posterior by Bayesian updating on the prior, their private signal, the public signal, and the information in the price after netting out the publicly observed signal. Demands follow from mean-variance optimization. Market clearing determines the coefficients.
Proposition 3 (Equilibrium price. ) There is a linear rational expectations equilibrium in which the price P f 1 ∗ P_{f1}^{*} P f 1 ∗
r P f 1 ∗ = A f + B j ϵ f 1 + G j w f 1 + C j x ~ f (6) rP_{f1}^{*}=A_f+B_j\epsilon_{f1}+G_jw_{f1}+C_j\tilde{x}_f \tag*{(6)} r P f 1 ∗ = A f + B j ϵ f 1 + G j w f 1 + C j x ~ f (6) with
B j = γ j ( 1 − Σ ‾ j Σ j d ) , G j = γ j Σ ‾ j τ s , C j = − ρ ‾ γ j 2 Σ ‾ j τ p + K j K j (7) B_j=\gamma_j\left(1-\frac{\overline{\Sigma}_j}{\Sigma_{jd}}\right),\qquad G_j=\gamma_j\overline{\Sigma}_j\tau_s,\qquad C_j=-\overline{\rho}\gamma_j^{2}\overline{\Sigma}_j\frac{\tau_p+K_j}{K_j} \tag*{(7)} B j = γ j ( 1 − Σ j d Σ j ) , G j = γ j Σ j τ s , C j = − ρ γ j 2 Σ j K j τ p + K j (7) and price-signal precision
τ p = K j 2 γ j 2 ρ ˉ 2 ∑ j x (8) \tau_p = \frac{K_j^2}{\gamma_j^2 \bar{\rho}^2 \sum_j x} \tag*{(8)} τ p = γ j 2 ρ ˉ 2 ∑ j x K j 2 (8) which is independent of τ s \tau_s τ s
Risk-averse investors pay the price P f 1 ∗ P^*_{f1} P f 1 ∗ r r r (6) shows its components). A one-unit increase in the innovation ( ϵ f 1 ) (\epsilon_{f1}) ( ϵ f 1 ) ( d f 1 ∗ ) (d^*_{f1}) ( d f 1 ∗ ) V f 1 ∗ = γ j d f 1 ∗ V^*_{f1} = \gamma_j d^*_{f1} V f 1 ∗ = γ j d f 1 ∗ γ j = r / ( r − υ j ) \gamma_j = r/(r-\upsilon_j) γ j = r / ( r − υ j ) V f 1 ∗ V^*_{f1} V f 1 ∗ γ j \gamma_j γ j υ j \upsilon_j υ j 24 R ≡ V f 1 ∗ − r P f 1 ∗ R \equiv V^*_{f1} - rP^*_{f1} R ≡ V f 1 ∗ − r P f 1 ∗ E [ R ] = ρ ˉ γ j 2 Σ ˉ j x ˉ j E[R] = \bar{\rho}\gamma_j^2\bar{\Sigma}_j\bar{x}_j E [ R ] = ρ ˉ γ j 2 Σ ˉ j x ˉ j x ˉ j \bar{x}_j x ˉ j
Var [ R ] = γ j 2 Σ ˉ j 2 [ τ d + τ s + ( τ p + K j ) 2 τ p ] , (9) \operatorname{Var}[R] = \gamma_j^2\bar{\Sigma}_j^2 \left[\tau_d+\tau_s+\frac{(\tau_p+K_j)^2}{\tau_p}\right], \tag*{(9)} Var [ R ] = γ j 2 Σ ˉ j 2 [ τ d + τ s + τ p ( τ p + K j ) 2 ] , (9) falls as the feed grows more precise. A more informative feed makes the price track the firm’s value more closely. Return volatility therefore falls. This is the volatility channel the loss analysis conditions on (§ 3.4 ).
The second question for this subsection is how sentiment affects what the price reveals about the firm’s future earnings. Following Bai et al. [10 ] and Farboodi et al. [35 ] , stock-level informativeness at horizon s s s s s s 25 Proposition 4 (Price informativeness. ) Stock-level price informativeness at horizon s s s
P I N F j , s = Σ ‾ j d S D ( P f 1 ∗ ) ⋅ v j s r − v j ⋅ ( 1 − Σ ‾ j Σ ‾ j d ) , 1 − Σ ‾ j Σ ‾ j d = τ p + K j + τ s τ d + τ p + K j + τ s , (10) \mathrm{PINF}_{j,s} = \frac{\overline{\Sigma}_{jd}}{\mathrm{SD}(P^{*}_{f1})} \cdot\frac{v_j^{s}}{r-v_j} \cdot\left(1-\frac{\overline{\Sigma}_j}{\overline{\Sigma}_{jd}}\right), \qquad1-\frac{\overline{\Sigma}_j}{\overline{\Sigma}_{jd}} = \frac{\tau_p+K_j+\tau_s}{\tau_d+\tau_p+K_j+\tau_s}, \tag*{(10)} PINF j , s = SD ( P f 1 ∗ ) Σ j d ⋅ r − v j v j s ⋅ ( 1 − Σ j d Σ j ) , 1 − Σ j d Σ j = τ d + τ p + K j + τ s τ p + K j + τ s , (10) so the information component is strictly increasing in τ s \tau_s τ s
The decomposition of price informativeness has three factors. The volatility factor ( Σ ‾ j d / S D ( P f 1 ∗ ) ) (\overline{\Sigma}_{jd}/\mathrm{SD}(P^{*}_{f1})) ( Σ j d / SD ( P f 1 ∗ )) ( v j s / ( r − v j ) ) (v_j^{s}/(r-v_j)) ( v j s / ( r − v j )) s s s ( v j s ) (v_j^{s}) ( v j s ) ( r − v j ) (r-v_j) ( r − v j ) 1 − Σ ‾ j / Σ ‾ j d 1-\overline{\Sigma}_j/\overline{\Sigma}_{jd} 1 − Σ j / Σ j d ( τ p + K j + τ s ) / T j (\tau_p+K_j+\tau_s)/T_j ( τ p + K j + τ s ) / T j τ s \tau_s τ s (10) ). It reaches the volatility factor only through the price’s standard deviation. In Farboodi et al. [35 ] , the information factor, whose underlying precision is not observed, is backed out from measured informativeness, growth, and volatility. We do not face that limitation. Equation (5) measures the signal precision τ s \tau_s τ s T q / V A Q S V f q 2 T_q/\mathrm{VAQSV}^{2}_{fq} T q / VAQSV f q 2 κ \kappa κ τ s \tau_s τ s
Short-sale constraints and the asymmetry between gains and losses
The precision of the public signal should matter for losses specifically rather than through the volatility channel alone. The outcome variables are the quarter’s accumulated gains and losses. The outcomes are the sums of the positive and negative parts of the daily excess return r f t r_{ft} r f t G a i n f q = ∑ t ∈ q max ( r f t , 0 ) \mathrm{Gain}_{fq}=\sum_{t\in q}\max(r_{ft},0) Gain f q = ∑ t ∈ q max ( r f t , 0 ) L o s s f q = ∑ t ∈ q min ( r f t , 0 ) \mathrm{Loss}_{fq}=\sum_{t\in q}\min(r_{ft},0) Loss f q = ∑ t ∈ q min ( r f t , 0 )
Assumption A2 (daily accrual). The quarter contains T d a y s T_{days} T d a y s r f t r_{ft} r f t N ( μ d , σ d 2 ) N(\mu_d,\sigma_d^2) N ( μ d , σ d 2 ) T d a y s μ d = E [ R ] + μ d r i f t T_{days}\mu_d = E[R] + \mu^{drift} T d a y s μ d = E [ R ] + μ d r i f t T d a y s σ d 2 = Var [ R ] T_{days}\sigma_d^2 = \operatorname{Var}[R] T d a y s σ d 2 = Var [ R ] μ d r i f t \mu^{drift} μ d r i f t Proposition 6 ), a predictable component of the mean daily excess return that reverses the previous quarter’s overpricing.26
As per the assumption, the quarter’s excess return is the sum of independent days’ excess returns with the same mean and variance. A day’s mean carries two parts, the group-level risk premium and the correction drift that reverses the previous quarter’s overpricing. A day’s variance is the quarter’s return variance divided across the days. Under A2, the expected tails are
E [ Gain ] = T d a y s [ μ d Φ ( μ d / σ d ) + σ d ϕ ( μ d / σ d ) ] , E[\operatorname{Gain}] = T_{days}\left[\mu_d\Phi(\mu_d/\sigma_d) + \sigma_d\phi(\mu_d/\sigma_d)\right], E [ Gain ] = T d a y s [ μ d Φ ( μ d / σ d ) + σ d ϕ ( μ d / σ d ) ] , E [ ∣ Loss ∣ ] = T d a y s [ − μ d Φ ( − μ d / σ d ) + σ d ϕ ( μ d / σ d ) ] , E[|\operatorname{Loss}|] = T_{days}\left[-\mu_d\Phi(-\mu_d/\sigma_d) + \sigma_d\phi(\mu_d/\sigma_d)\right], E [ ∣ Loss ∣ ] = T d a y s [ − μ d Φ ( − μ d / σ d ) + σ d ϕ ( μ d / σ d ) ] , where Φ \Phi Φ ϕ \phi ϕ E [ Gain ] − E [ ∣ Loss ∣ ] = T d a y s μ d E[\operatorname{Gain}] - E[|\operatorname{Loss}|] = T_{days}\mu_d E [ Gain ] − E [ ∣ Loss ∣ ] = T d a y s μ d σ d \sigma_d σ d E [ Gain ] E[\operatorname{Gain}] E [ Gain ] E [ ∣ Loss ∣ ] E[|\operatorname{Loss}|] E [ ∣ Loss ∣ ] μ d \mu_d μ d Loss f q \operatorname{Loss}_{fq} Loss f q T d a y s μ d T_{days}\mu_d T d a y s μ d μ d r i f t \mu^{drift} μ d r i f t Proposition 6 ). The quarterly sum also contains the group-level risk premium (E [ R ] E[R] E [ R ] § 6 ).
Proposition 5 (Volatility channel). At fixed mean daily excess return μ d \mu_d μ d E [ Gain ] E[\operatorname{Gain}] E [ Gain ] E [ ∣ Loss ∣ ] E[|\operatorname{Loss}|] E [ ∣ Loss ∣ ] σ d \sigma_d σ d T d a y s σ d 2 = Var [ R ] T_{days}\sigma_d^2 = \operatorname{Var}[R] T d a y s σ d 2 = Var [ R ] Var [ R ] \operatorname{Var}[R] Var [ R ]
is strictly decreasing in τ s \tau_s τ s (9) , so a noisier public signal raises both E [ G a i n ] E[\mathrm{Gain}] E [ Gain ] E [ ∣ L o s s ∣ ] E[|\mathrm{Loss}|] E [ ∣ Loss ∣ ] Proposition 6 . (Proof in C.6.)
The proposition shows separation of the volatility channel from the mean channel. A noisier feed raises return volatility (equation (9) ) on a day. A day with more return volatility produces bigger swings in gains as well as losses which raises the expected gains and expected losses by the same amount so that the magnitude of the loss-gain gap remains unchanged.
Assumption A3 (short-sale constraints). (i i i u f 1 u_{f1} u f 1 i i ii ii δ ∈ [ 0 , 1 ] \delta\in[0,1] δ ∈ [ 0 , 1 ] u f 1 u_{f1} u f 1
A positive realization of the signal noise u f 1 u_{f1} u f 1 (6) ), an optimistic error. A negative realization holds it below, a pessimistic error. The noise u f 1 u_{f1} u f 1 [62 ] . Selling short requires borrowing the stock. Mandates against short sales prohibit it for some investors (Almazan et al. [5 ] ), while borrowing costs make it expensive for others (D’Avolio [27 ] ). Purchasing is not prohibited for any investor. This corrects the pessimistic error. An optimistic error can be corrected by a sale. The selling can come from pessimists who hold shares and from the arbitrageurs who can borrow and short. Short-selling flow is informed and predicts lower future returns (Boehmer et al. [13 ] ). Thus, the surviving overpricing reflects the borrowing friction instead of lack of information among short sellers. The arbitrageurs have limited capital and limited horizons (Shleifer and Vishny [68 ] ). A pessimist who doesn’t hold any shares and who cannot borrow cannot make any sale. Such a pessimist can abstain from purchasing but that cannot bring the price down. Thus, part of the overpricing stays in the price. The surviving fraction is δ \delta δ 27
On the next earnings date, the firm’s true value becomes public and the surviving overpricing is corrected. No buyer would be willing to pay more than the true value. Thus, the price settles at that true value. This fall in price from overpricing to the true value is a negative expected drift in the next quarter’s returns. The daily excess returns scatter around a mean that carries this drift (A2). A lower mean raises the quarter’s losses relative to its gains.
We now compute the expected size of the overpricing that survives the quarter. The price impact of the signal noise is G j u f 1 G_j u_{f1} G j u f 1 (6) . Under A3, the part that survives is δ G j max ( u f 1 , 0 ) \delta G_j \max(u_{f1},0) δ G j max ( u f 1 , 0 ) δ \delta δ E [ max ( u f 1 , 0 ) ] E[\max(u_{f1},0)] E [ max ( u f 1 , 0 )]
Proposition 6 (Overpricing and its correction. ) Under A3, expected overpricing is
O P j = δ G j E [ max ( u f 1 , 0 ) ] = δ γ j 2 π ⋅ τ s τ d + τ p + K j + τ s , (11) \mathrm{OP}_j = \delta G_j E[\max(u_{f1},0)] = \frac{\delta\gamma_j}{\sqrt{2\pi}} \cdot\frac{\sqrt{\tau_s}}{\tau_d+\tau_p+K_j+\tau_s}, \tag*{(11)} OP j = δ G j E [ max ( u f 1 , 0 )] = 2 π δ γ j ⋅ τ d + τ p + K j + τ s τ s , (11) and under A2 this is corrected as next-quarter drift μ d r i f t = − O P j \mu^{drift}=-\mathrm{OP}_j μ d r i f t = − OP j O P j \mathrm{OP}_j OP j τ s → 0 \tau_s \to0 τ s → 0 τ s → ∞ \tau_s \to\infty τ s → ∞
τ s ∗ = τ 0 ≡ τ d + τ p + K j . (12) \tau_s^* = \tau_0 \equiv\tau_d+\tau_p+K_j. \tag*{(12)} τ s ∗ = τ 0 ≡ τ d + τ p + K j . (12) In measured-noise units w ≡ V A Q S V w \equiv\mathrm{VAQSV} w ≡ VAQSV w ∗ = T q / ( κ τ 0 ) w^*=\sqrt{T_q/(\kappa\tau_0)} w ∗ = T q / ( κ τ 0 ) T q T_q T q κ \kappa κ K j K_j K j
Expected overpricing in (11) is the product of two opposing forces, scaled by the constant δ \delta δ G j = γ j τ s / ( τ 0 + τ s ) G_j=\gamma_j\tau_s/(\tau_0+\tau_s) G j = γ j τ s / ( τ 0 + τ s ) (7) with Σ ‾ j = 1 / T j \overline{\Sigma}_j=1/T_j Σ j = 1/ T j τ s \tau_s τ s E [ max ( u f 1 , 0 ) ] E[\max(u_{f1},0)] E [ max ( u f 1 , 0 )] σ u / 2 π \sigma_u/\sqrt{2\pi} σ u / 2 π σ u \sigma_u σ u u f 1 u_{f1} u f 1 1 / τ s 1/\sqrt{\tau_s} 1/ τ s u f 1 u_{f1} u f 1 τ s \tau_s τ s τ 0 = τ d + τ p + K j \tau_0=\tau_d+\tau_p+K_j τ 0 = τ d + τ p + K j (12) . At that point the feed provides exactly half of the market’s total precision. The total is T j = τ 0 + τ s T_j = \tau_0 + \tau_s T j = τ 0 + τ s τ s \tau_s τ s T j T_j T j τ s = τ 0 \tau_s = \tau_0 τ s = τ 0 τ s / T j = τ 0 / ( 2 τ 0 ) = 1 / 2 \tau_s/T_j = \tau_0/(2\tau_0) = 1/2 τ s / T j = τ 0 / ( 2 τ 0 ) = 1/2 τ s \tau_s τ s τ 0 \tau_0 τ 0 τ s \tau_s τ s τ s \tau_s τ s τ 0 \tau_0 τ 0 τ s \tau_s τ s τ s \tau_s τ s τ 0 \tau_0 τ 0 τ 0 \tau_0 τ 0
Under A3(ii), a fraction δ \delta δ q q q O P j \mathrm{OP}_j OP j q + 1 q + 1 q + 1 q + 1 q + 1 q + 1 q q q q q q q q q q + 1 q + 1 q + 1
The model makes two predictions. The first is about shape. Holding the quarter’s daily volatility fixed, the response of the excess of next-quarter losses over gains, ∣ L o s s f , q + 1 ∣ − G a i n f , q + 1 |\mathrm{Loss}_{f,q+1}| - \mathrm{Gain}_{f,q+1} ∣ Loss f , q + 1 ∣ − Gain f , q + 1 Proposition 6 ). For comparisons at a common T q T_q T q κ \kappa κ
The alternative to the model is a symmetric mechanism, one in which optimistic and pessimistic sentiment errors are equally likely, move the price by the same amount, and are corrected to the same degree (C.8). Such a mechanism reaches losses only through volatility, which grows with the amount of noise and raises expected gains and expected losses by the same amount (Proposition 5). It produces neither an interior peak nor a correction that depends on the sign of the noise. Section 6 tests both predictions against it. (The state dependence of the drift channel is derived in C.11.)
The daily dealer market and the ordinal intensity index
The model shows that how much sentiment noise moves a stock’s price (and hence its losses) depends on how much informed trading the stock has. If there is heavy informed trading, the price moves less. Hence, the noise matters less for the price. In such a case, a domestic-foreign gap in losses could then simply reflect a gap in informed-trading intensity rather than a gap in the precision of the public signal. Thus, a domestic-foreign comparison must hold informed-trading intensity fixed. This is achieved with the terciles.28 29 K ^ \widehat{K} K τ d + τ ^ s \sqrt{\tau_d+\widehat{\tau}_s} τ d + τ s τ d \tau_d τ d τ ^ s \widehat{\tau}_s τ s K ^ \widehat{K} K K ^ \widehat{K} K K ^ \widehat{K} K K ^ \widehat{K} K (43) . The recovered K ^ \widehat{K} K K ^ \widehat{K} K
Geography and the producer-knowledge mismatch hypothesis
Geography (domestic and foreign) enters the model as a grouping of stocks (§ 3.1 ). The hypothesis is about the contributor pool composition behind the public signal, specifically, the shares of the three types of contributors and the variances of the fad and of the bot content (equations (3) and (4) ). Nothing else that differs between the two geographies enters the nonfundamental variance (equation (4) ). A domestic-foreign difference in it therefore reflects a difference in the pool composition. The hypothesis states the direction of that difference.
Hypothesis H1 (producer-knowledge mismatch). (a) On English-language Twit-
ter, foreign stocks have a larger nonfundamental variance (Φ j \Phi_j Φ j κ \kappa κ T q / V A Q S V f q 2 T_q/\mathrm{VAQSV}_{fq}^{2} T q / VAQSV f q 2
The hypothesis compares the informativeness of the feed for foreign and domestic stocks. The nonfundamental variance (Φ j \Phi_j Φ j (4) ). It carries no information about the firm-specific innovation and does not change with the tweet count. When the measurement equation (equation (4) ) is fitted as a linear regression (§ 3.2 ), Φ j \Phi_j Φ j τ s \tau_s τ s (5) ). A larger nonfundamental variance makes the day’s per-tweet sentiment average (S t / N t S_t/N_t S t / N t κ \kappa κ (4) and (5) ). Thus, a larger nonfundamental variance (the object of part (a) of H1) lowers the observable signal-quality index (the object of part (b)). Part (b) of the hypothesis is a claim about the precision of the public signal. It is phrased in terms of the signal-quality index rather than τ s \tau_s τ s τ s \tau_s τ s κ \kappa κ (5) ). The conversion constant κ \kappa κ § 5 ). Hence, the ordering of the precision of the public signal τ s \tau_s τ s κ \kappa κ § 3.2 ).
The economic intuition behind the hypothesis is that investors who understand a foreign firm’s regulatory environment, the firm’s management and product markets are underrepresented among its English-language contributors. Thus, the feed for foreign firms would carry a weaker signal than the feed for domestic firms. Part (a) of H1 relates to the nonfundamental variance as a whole, not split by contributor types. The feed has three contributor types (§ 3.2 ). The empirical question is which of the contributor types produces an excess in the nonfundamental variance for foreign stocks as compared to domestic stocks. The excess in nonfundamental variance for stocks of foreign firms over those of domestic firms can reflect a larger fad variance, a larger bot-content variance, a different split of the day’s tweets across the three types or a combination of these (equation (4) ). Although automated accounts
Data
can be identified in ground-truth samples [42 ] , human fads cannot be. The data vendor’s metrics are aggregated over tweets. Hence, these metrics have no account related identities (§ 3.2 ). Thus, we cannot use account labels to attribute the excess in the nonfundamental variance for foreign stocks to a type of contributor. Such attribution rests on two indirect tests which are shown in the reduced-form battery (§ 6 ). The first one uses a regression of squared measured noise on contributor dispersion (the average inverse tweet count) and the foreign indicator (Row R13a, Table IV). Its reported object is the sign of the dispersion coefficient. The second one uses a regression of the NASDAQ trade count on an indicator for below-median dispersion and on log capitalization, on the NASDAQ subsample (Row R13b, Table IV).30
For part (a) of H1, the evidence is the foreign coefficient in the measurement equation (equation (4) ). We regress squared measured noise on the average inverse tweet count and the foreign indicator, with quarter fixed effects and standard errors double clustered by firm and quarter (§ 3.2 ). The foreign coefficient is the difference in the intercept Φ j \Phi_j Φ j § 5 , Table XIV ). Since a larger nonfundamental variance lowers the observable signal-quality index at a given number of scored days and tweet volume (equation (4) ), evidence for part (a) is also evidence for part (b). Part (b) is therefore not tested separately. We nonetheless report, for each informed-trading-intensity tercile, the median of the signal-quality index (T q / V A Q S V f q 2 T_q/\mathrm{VAQSV}_{fq}^{2} T q / VAQSV f q 2 Table XXI ).31
Table XIV. Table XIV: Size-conditioned measurement contrasts, foreign coefficients. Notes: In Panel A, the contributor dispersion is regressed on the foreign indicator, while in Panel B, the squared measured noise is regressed on the average inverse tweet count and the foreign indicator, both with quarter fixed effects and standard errors clustered by firm and quarter, and with and without controls for size. We construct terciles using size within each quarter on log capitalization. Domestic and foreign stocks share the same bins. Lower dispersion means fewer distinct accounts per tweet. A negative coefficient in Panel A means foreign stocks have lower dispersion than domestic stocks, the concentration contrast. A positive coefficient in Panel B means foreign stocks have more nonfundamental variance than domestic stocks, the fad contrast. The Panel B coefficients are small in level. We scale them and their standard errors by 10 5 10^{5} 1 0 5
Sources and panel construction
The panel is composed of the Compustat quarterly fundamentals, the CRSP daily market data aggregated at the quarterly level, and the daily firm-level Twitter sentiment from Context Analytics (licensed through Dewey Data, Context Analytics, 2022) aggregated at the quarterly level. The sentiment data is matched to the CRSP-Compustat merged database by ticker and quarter. The panel has 471,880 stock-quarters from 2010 to 2024, with the stock identified by the nine-digit CUSIP. When one ticker-quarter is attached to more than one security, we keep the larger-capitalization security, which drops the security with smaller capitalization in each of 738 ticker-quarters. The primary estimation sample starts in the first quarter of 2018 due to a change in the vendor’s sampling methodology in 2017. Earlier quarters are used only in the robustness exercises. At construction, the quarter’s gains and losses, the quarterly excess return, the within-quarter volatility of daily excess returns, and the residual-based gains and losses of the robustness checks are winsorized at the first and ninety-ninth percentiles of the full panel. Sentiment and liquidity measures are not winsorized. A firm-quarter enters the estimation only if it has at least twenty sentiment days, which every quantity built from the feed requires, and at least twenty trading days with a valid return, which every quantity built from returns or liquidity requires. The residual-based gains and losses of the robustness checks require at least twenty days with a residual. The primary structural sample has 98,308 firm-quarters. Group assignment covers 96,229 of them, of which 96,228 enter the estimation across 5,583 firms and 28 quarters. Foreign stocks in our sample are listed on U.S. exchanges, with foreign status classified by Compustat headquarters country. Identified by the CRSP share-level flag under a majority-of-days rule, American Depositary Receipts are 36.5% of foreign observations. Directly listed shares, which include Canadian equities, are 63.5% (Table XIII ). (Calibrated constants and dependence conventions are in A.2.)
Table XIII. Notes: This table shows how the stocks in the baseline estimation sample split across their Compustat headquarters countries (the announcement-aligned R1 screens, Table XII; row R15). The foreign stocks make up 13,422 firm-quarters from 51 countries. Each country’s count is its number of distinct firms. The total row shows the number of distinct firms over all countries.
Structural Estimation
The empirical work has two parts. In the structural part, we impose the model’s equations on the data and estimate the model’s parameters. In the validation part, we estimate specifications that we do not derive from the model. In those specifications the model supplies only a regressor or the predicted sign of a coefficient. This section reports the four structural exercises. § 6 reports the validation exercises. Table II lists all the exercises with their classification.
Estimation design
We estimate the model in one step by the generalized method of moments on ninety-one closed-form moments.32 [45 ] . We pre-specify a diagonal weighting matrix. Each diagonal entry is the inverse of the variance of that moment across 999 firm-block bootstrap replications of the data moments.
Standard errors on the parameters come from the sandwich covariance for a fixed weighting matrix, which combines the Jacobian of the model moments, the weighting matrix, and the covariance of the data moments. For that covariance we use the firm-block bootstrap covariance of the moments, shrunk toward its diagonal by the Ledoit-Wolf rule. We set the posting intensity at 1.5. Hence, it is treated as a known constant in the covariance, so that the standard errors cover the forty-five estimated parameters. The assignment of stocks to informed-trading-intensity terciles is itself computed from the sample (§ 3.5 ), while the sandwich covariance takes it as given. To account for the sampling variation in the tercile assignment, we run a firm-block bootstrap with 399 replications. Each replication draws firms with replacement (keeping each firm’s complete time series) and repeats the tercile assignment, the data moments and the estimation, with the weighting matrix held at its point-estimate value ((A)).
Headline estimates and the joint Wald test
Table I shows the parameter estimates. The GMM objective function (the weighted distance between the data moments and the model moments) has more than one local minimum. The headline estimate is the local minimum at which that distance is the smallest among the minima that satisfy the pre-specified saturation criterion and the acceptance checks ((A.3)). The repeat-author shares are identified from contributor dispersion (Proposition 1 ) and the fad scale from the nonfundamental variance (equation (4) ). Thus, the domestic-foreign contrasts in these estimates rest on two contrasts in the data. Contributor dispersion (the number of distinct accounts per unit of posting volume) is lower for the foreign feed than for the domestic feed. The nonfundamental variance (the part of sentiment variance not coming from firm fundamentals) is larger for the foreign feed. In order to check whether these two contrasts reflect geography and not firm size, we re-estimate both with log capitalization held fixed. Contributor dispersion for foreign stocks remains lower than that for domestic stocks (t = − 4.34 t=-4.34 t = − 4.34 t = 6.12 t=6.12 t = 6.12 Table XIV ). To translate the dispersion into repeat-author shares (Proposition 1 ) requires a value for the posting intensity (the number of tweets per account per day that a repeat author posts). The moments identify the posting intensity weakly, because dispersion identifies the repeat-author shares only jointly with the posting intensity. A higher intensity paired with lower shares produces the same dispersion. When the posting intensity is estimated freely, the estimate is 1,102.98 and every repeat-author share is at its zero lower bound (A.4). The freely estimated shares reached a bound, so we set the posting intensity at 1.5. This value lies inside the range of 1.2 to 1.85 implied by the external posting rates of B. At the set posting intensity, the feed for foreign stocks has the higher repeat-author share in each of the three informed-trading-intensity terciles. The fad scale (estimated for the feed of each geography and held equal across its three terciles, A.1) is higher for the foreign feed too. A joint Wald test rejects the null that the three repeat-author shares and the fad scale are equal across geography. The statistic (four degrees of freedom) is 48.56 with standard errors clustered by firm (p = 7.2 × 10 − 10 p = 7.2 \times10^{-10} p = 7.2 × 1 0 − 10 p = 1.9 × 10 − 14 p = 1.9 \times10^{-14} p = 1.9 × 1 0 − 14 p = 1.9 × 10 − 5 p = 1.9 \times10^{-5} p = 1.9 × 1 0 − 5 33
All four domestic-foreign contrasts in Table II are negative, which affirms that the feed for foreign stocks has the higher repeat-author share in each of the three informed-trading-intensity terciles and the higher fad scale. The higher foreign fad scale and the higher foreign top-tercile share hold in each of the 399 bootstrap replications. These contrasts provide supporting evidence for our producer-knowledge mismatch hypothesis.
The two data contrasts of this subsection, the lower contributor dispersion and the higher nonfundamental variance of the foreign feed, also hold within size terciles (Table XIV , row R16, A.12). The dispersion gap keeps its sign in all three size terciles and is significant in the two smaller ones (t = − 2.49 t = -2.49 t = − 2.49 − 3.54 -3.54 − 3.54 t = 3.12 t = 3.12 t = 3.12
Table V. Table V: Joint Wald test of the equality of the repeat-author shares and the fad scale across geography at eight values of the posting intensity. Notes: This table shows the joint Wald test of Table II at eight values of the posting intensity from 1.2 to 1.85. At each value we re-estimate the other forty-five parameters and compute the four domestic-foreign contrasts (in the three repeat-author shares and the fad scale). We test the null that the four contrasts are zero with standard errors clustered by firm, clustered by quarter and double clustered by firm and quarter, as in Table II. At each value we treat the posting intensity as a known constant in the covariance, as we do at 1.5 (§ 5 ). The second column is the GMM objective at each value. The last column shows that all four contrasts are negative at each value, so that foreign stocks have the higher repeat-author share in each tercile and the higher fad scale. We reject the null at each value under each clustering. The largest p p p 1.0 × 10 − 4 1.0 \times10^{-4} 1.0 × 1 0 − 4
Table I. Headline structural parameter estimates with asymptotic sandwich standard errors and firm-block bootstrap intervals
Notes: 45 parameters are estimated using a one-step GMM with 91 moments while the posting intensity is set at 1.5 (§ 5 ). A diagonal weighting matrix is used where there is an inverse of the bootstrap variance of each moment on the diagonal. Standard errors are computed from the sandwich estimator of the asymptotic covariance matrix of the GMM estimator (§ 5 ). The covariance matrix of the data moments in the sandwich estimator is estimated by a firm-block bootstrap with 999 replications and shrunk toward its diagonal by the Ledoit-Wolf method. The 95% intervals in the last column are the 2.5th and 97.5th percentiles of the estimates over 399 firm-block bootstrap replications. Each replication resamples firms with replacement and re-estimates the parameters on the resampled panel with the same weighting matrix as the point estimates. The surviving overpricing fraction (δ = 1 / ( 1 + e − z ) \delta= 1/(1 + e^{-z}) δ = 1/ ( 1 + e − z ) z z z z z z δ = 0.0003728 \delta= 0.0003728 δ = 0.0003728 z = − 7.89 z = -7.89 z = − 7.89 δ \delta δ δ \delta δ § 5 ).
(a narrower contributor pool), with log capitalization as a control (Row R13b, Table IV). Quarters with a narrower contributor pool have 8,121 more trades (t = 10.5 t = 10.5 t = 10.5 1.6 × 10 − 5 1.6 \times10^{-5} 1.6 × 1 0 − 5 t = 0.12 t = 0.12 t = 0.12 = [ − 2.5 × 10 − 4 , 2.8 × 10 − 4 ] = [-2.5 \times10^{-4}, 2.8 \times10^{-4}] = [ − 2.5 × 1 0 − 4 , 2.8 × 1 0 − 4 ] Proposition 1 ). Hence, we put no number on the automated share. Classifying an account as automated or human from its tweets is correct for about 77 percent of accounts at best (B), so a count of automated accounts in our feeds would mislabel about one account in four.
Table II. Domestic-foreign contrasts in the repeat-author shares and the fad scale, with firm-block bootstrap confidence intervals and the joint Wald test of their equality across geography. Notes: The table shows repeat-author shares in the three terciles (of informed trading intensity) and the fad scale for domestic and foreign stocks, with the posting intensity set at 1.5. The 95% confidence intervals are bootstrap (sampling firms with replacement) percentile intervals from 399 replications. The last column is the share of those replications in which the contrast is negative (implying that foreign stocks have the higher repeat-author share in each of the three terciles and the higher fad scale). The joint Wald test examines the null that all four contrasts are zero (with four degrees of freedom) using standard errors clustered by firm, clustered by quarter and double clustered by firm and quarter ((Cameron et al., 2011)).
The conversion constant
The conversion constant (κ \kappa κ τ s \tau_s τ s κ \kappa κ κ \kappa κ Table VI ). Thus, the fit of the model barely depends on the value of κ \kappa κ κ \kappa κ τ s \tau_s τ s § 3.2 ). The serial dependence can be obtained from the autocorrelation of the daily per-tweet series. The median lag-one autocorrelation is near 0.04 in both geographies with a dependence multiplier near 1.1 ((A.6)). The remainder of κ \kappa κ κ \kappa κ τ s \tau_s τ s κ \kappa κ κ \kappa κ τ s \tau_s τ s κ \kappa κ
Table VI. Table VI: GMM objective function at twenty-five values of the conversion constant
Notes: This table shows the GMM objective function at twenty-five values of the conversion constant (κ \kappa κ log κ \log\kappa log κ κ \kappa κ κ \kappa κ κ \kappa κ
The short-sale friction
In the model, a fraction (δ \delta δ Proposition 6 ). The estimate of δ \delta δ δ \delta δ δ \delta δ δ \delta δ § 5 ), or the reliability gap, which is reduced-form evidence (row R14, Table IV).
Fit, misfit, and the informativeness block
The GMM objective is the sum over the 91 moments (45 free parameters) of the squared distance between each data moment and its model-implied counterpart (the misfit of that moment), each distance weighted by the inverse of that moment’s bootstrap variance (§ 5 ). At the headline estimate (Table I), the GMM objective is equal to 4,338. The model imposes 46 restrictions on the data moments beyond those needed to estimate the parameters (the overidentifying restrictions). The Hansen J J J Table XX is an attempt to locate the misfit by splitting the objective at the headline estimate across the nine blocks of moments and reporting (for each block) its share of the objective and its largest moment-level t t t t t t a s a_s a s (43) ). If we let a s a_s a s a s a_s a s a c s a_{cs} a cs a λ a_\lambda a λ § 3.5 ) which doesn’t contain the scale constants. When the terciles are formed using the high-low spread or using the quoted spread (capped at its adverse-selection bounds), the domestic-foreign difference in the slope of next-quarter losses on measured noise in the lowest intensity tercile (β D 1 − β F 1 \beta_{D1} - \beta_{F1} β D 1 − β F 1 (14) ) keeps its sign. When the terciles are formed on the lagged quoted spread alone, the reliability gap (the domestic-foreign difference in the share of the 27 quarters in which the quarterly slope of next-quarter losses on measured noise is negative) is positive in each of the three spread terciles (A.8).
Table XX. Decomposition of the objective by moment family at the headline estimate. Notes: The Hansen J J J
The misfit of the liquidity block also doesn’t affect the parameters which justify the main results (the repeat-author shares, π j \pi_j π j σ b , j \sigma_{b,j} σ b , j a s a_s a s Table XVIII ). The misfit of the liquidity block also shows up in the regressions of the quoted spread and the Amihud illiquidity ratio on the previous quarter’s measured noise and dispersion (log capitalization and per-day turnover as controls and quarter fixed effects, row R5, Table IV). In the quoted spread regression, the coefficient on lagged measured noise is − 0.034 -0.034 − 0.034 t = − 5.41 t = -5.41 t = − 5.41 − 6.30 -6.30 − 6.30 t = − 3.88 t = -3.88 t = − 3.88 Table XX ) has a misfit of its own. Price informativeness (the covariance of the price with earnings one to four quarters ahead, scaled by the standard deviation of the price, Proposition 4) cannot be negative in the model, since none of the three factors in equation 10 10 10 Table XXI ). The removal of the 12 informativeness moments from the moment system changes the repeat-author shares and the fad scales by less than 10 − 5 10^{-5} 1 0 − 5 Table XVIII ). Thus, neither the misfit of the liquidity block nor the misfit of the informativeness block changes the domestic-foreign contrasts in the repeat-author share and in the fad scale (A.8).
Table XVIII. Pre-specified estimation variants re-estimated from the headline estimate
Table III. Concentration and fad contrasts under liquidity-block robustness
Notes: The first three rows show the domestic-foreign contrasts in the repeat-author share (π \pi π σ b \sigma_b σ b a s a_s a s a c s a_{cs} a cs a λ a_{\lambda} a λ σ x , j \sigma_{x,j} σ x , j κ \kappa κ a s a_s a s 43 43 43 a s a_s a s Table XVIII .
Reduced-Form Evidence and Falsification
The parameter estimates in § 5 are obtained by minimizing the weighted distance between the 91 data moments and their model-implied counterparts. This section is about whether the model’s implications hold in regressions that do not impose such structure. The regressions use quarter fixed effects, standard errors double clustered by firm and quarter, the primary sample from 2018 onward, and the screens of Section 4.34
Y f , q + h = β X f q + Γ ′ Z f q + α q + η f q , (13) Y_{f,q+h} = \beta X_{fq} + \Gamma^\prime Z_{fq} + \alpha_q + \eta_{fq}, \tag*{(13)} Y f , q + h = β X f q + Γ ′ Z f q + α q + η f q , (13) where X f q X_{fq} X f q q q q Y f , q + h Y_{f,q+h} Y f , q + h h h h Z f q Z_{fq} Z f q α q \alpha_q α q h = 1 h = 1 h = 1 h = 0 h = 0 h = 0 Table XVII ).35
Table XVII. Conventions at the seams between data frequencies, with their diagnostics
Notes: Each aggregation seam between daily market data, quarterly fundamentals, and the sentiment feed is a convention we adopt in advance, paired with a diagnostic of its effect. The fiscal and calendar alignment of book assets has zero mismatch by construction because fundamentals enter at the calendar quarter of their statement date. We take the timing convention from the reported announcement date.
The slope of next-quarter total losses on measured noise differs across the six geography-tercile cells (row R1, Table IV). The regressor is interacted with the indicators for geography and for intensity tercile, and each cell has its own intercept,
Y f , q + 1 = ∑ g ∈ { D , F } ∑ k = 1 3 ( β g k X f q + δ g k ) 1 { f ∈ g k } + Γ ′ Z f q + α q + η f q , (14) Y_{f,q+1} = \sum_{g\in\{D,F\}} \sum_{k=1}^{3} (\beta_{gk} X_{fq} + \delta_{gk}) 1\{f \in gk\} + \Gamma^{\prime} Z_{fq} + \alpha_q + \eta_{fq}, \tag*{(14)} Y f , q + 1 = g ∈ { D , F } ∑ k = 1 ∑ 3 ( β g k X f q + δ g k ) 1 { f ∈ g k } + Γ ′ Z f q + α q + η f q , (14) where cell g k gk g k g g g k k k δ g k \delta_{gk} δ g k Z f q Z_{fq} Z f q β g k \beta_{gk} β g k g k gk g k k k k β D k − β F k \beta_{Dk} - \beta_{Fk} β D k − β F k
Reliability differs by geography
For each geography-quarter, we regress next-quarter losses on measured noise across stocks using log capitalization and the FF3 betas as controls. Then, we compare foreign and domestic on two statistics (row R14, Table IV). The first one is the mean of the slope over quarters, which allows us to compare the size of the next-quarter losses-noise relation. The second one is the share of quarters in which the slope is negative, which allows us to compare the stability of the sign of the next-quarter losses-noise relation across quarters. The results show that the mean quarterly slope is more negative for domestic stocks than for foreign stocks by 1.40 (Newey-West t = − 2.99 t = -2.99 t = − 2.99 36 p = 0.016 p = 0.016 p = 0.016 − 1.40 -1.40 − 1.40 − 0.0139 -0.0139 − 0.0139
Table IV. Notes: Each row’s specification is in Table IX. Stars mark the wild-cluster bootstrap p p p p p p p p p p p p
consisting of the lagged quoted spread, the Amihud illiquidity ratio, and turnover to the size and factor-beta controls (− 0.86 -0.86 − 0.86 t = − 2.21 t = -2.21 t = − 2.21 Figure II shows the quarterly slopes and their difference. This instability for the foreign stocks suggests that the next-quarter losses-noise slope for foreign stocks does not have a stable sign from quarter to quarter, implying that the foreign signal becomes diluted.
Figure II. Quarterly reliability of the sentiment-loss relation by geography
Notes: The top panel of the figure shows the quarterly slopes of next-quarter total losses on measured noise, estimated separately by geography with log capitalization and the FF3 betas as controls, over the 27 usable quarters of the primary sample. Each geography-quarter cell requires at least 15 stocks. The bottom panel plots the domestic-foreign difference in these slopes. The dashed line is its mean (− 1.40 -1.40 − 1.40 t = − 2.99 t = -2.99 t = − 2.99
Alt text: The two panels cover the period from 2018 to 2024. The top panel plots the quarterly loss-on-noise slope for domestic stocks (solid line) and for foreign stocks (dashed line). The solid line is negative in more quarters. The bottom panel shows bars for the domestic-foreign difference by quarter. The mean (indicated by the dashed line) is − 1.40 -1.40 − 1.40 t = − 2.99 t = -2.99 t = − 2.99
The reliability gap could reflect a domestic-foreign difference in informed-trading intensity rather than a difference in the feeds (§ 3.5 ). To hold informed-trading intensity fixed, we rerun the quarterly regressions with the same controls within each informed-trading-intensity tercile and compare the share of quarters with a negative slope for foreign and domestic stocks. The domestic-foreign difference in the share of quarters with a negative slope survives in sign in tercile 1 with a difference of 0.296 0.296 0.296 0.852 0.852 0.852 0.556 0.556 0.556 p = 0.004 p = 0.004 p = 0.004 0.148 0.148 0.148 − 0.037 -0.037 − 0.037 0.852 0.852 0.852 0.889 0.889 0.889 0.741 0.741 0.741 0.556 0.556 0.556 0.741 0.741 0.741 0.778 0.778 0.778
We classify the foreign stocks using three categories—by listing type (ADRs and directly listed shares), by the language of the headquarters country (English-speaking and non-English-speaking), and by country (Canada and other foreign). The separation by listing type shows a significant difference between ADRs and directly listed foreign shares. The share of quarters with a negative slope is 0.296 (8 of 27 quarters) for ADRs against 0.593 (16 of 27 quarters) for directly listed foreign shares (a difference of − 0.296 -0.296 − 0.296 p = 0.022 p = 0.022 p = 0.022 37 38
The volatility channel in reduced form
The volatility channel in reduced form: Higher sentiment noise in the present quarter implies larger losses in the next quarter. The slope of next-quarter losses on noise in the headline loss regression is negative in all six geography-tercile cells. However, none of them are individually statistically significant (row R1, Table IV). The slope of next-quarter losses on noise, in a pooled specification with dispersion and the full set of control variables, is negative and significant (row R11, Table IV). After controlling for realized volatility in the regression, the cell slopes move toward zero or turn positive but none of them is statistically distinguishable from zero (row R2, Table IV). Thus, the response of next-quarter losses to noise works through return volatility instead of through a lower mean return. A less precise public signal leads to more fluctuations in price, which show up as larger losses. The coefficients on measured noise and dispersion have opposite signs in the joint regression (row R11, Table IV). Thus, while a signal with higher noise predicts larger next-quarter losses, a wider or more dispersed contributor pool predicts smaller losses, with each effect surviving in the presence of the other. Therefore, the two measures carry different information about the feed. One is about the precision of the feed’s signal and the other is about the span of the feed’s contributors. The paper treats them as distinct channels.
Table XI. Reliability splits, sign stability across quarters. Notes: This table shows the share of quarters in which the slope of next-quarter total losses on measured noise is negative (row R14B; 27 usable quarters). We run this regression for each subset in each quarter, with log capitalization and the FF3 betas as controls. We skip any quarter in which a subset has fewer than 15 firms. We classify a headquarters country as English-speaking when English is its first language ([25 ] ). We classify the stock as an ADR when the CRSP ADR flag holds on a majority of its days. We test each domestic-foreign difference with the R14 circular block bootstrap over quarters (block length four, 999 replications, paired across the two subsets). The shares count only the sign of each slope, not its size. The median number of firms per quarter is 979, 975, and 918 for the domestic terciles, 173, 171, and 153 for the foreign terciles, 231 and 257 for the English and non-English splits, 116 and 371 for Canada and other foreign stocks, and 197 and 306 for ADRs and directly listed shares.
The correction channel in reduced form
6.3. The correction channel in reduced form: In the model, the expected overpricing that survives a quarter is proportional to τ s / ( τ 0 + τ s ) \sqrt{\tau_s}/(\tau_0 + \tau_s) τ s / ( τ 0 + τ s ) (11) ), which varies with the precision of the public signal (equation (12) ). For each stock-quarter we evaluate this factor with τ 0 \tau_0 τ 0 τ d \tau_d τ d τ p \tau_p τ p K j K_j K j τ s \tau_s τ s (5) ). We refer to this as the model-motivated signal-noise index. We then project the excess of next-quarter losses over gains on this index,
∣ L o s s f , q + 1 ∣ − G a i n f , q + 1 = b 3 I n d e x f q + Γ ′ Z f q + α q + γ g + η f q , (15) \lvert\mathrm{Loss}_{f,q+1} \rvert- \mathrm{Gain}_{f,q+1} = b_3 \mathrm{Index}_{fq} + \Gamma' Z_{fq} + \alpha_q + \gamma_g + \eta_{fq}, \tag*{(15)} ∣ Loss f , q + 1 ∣ − Gain f , q + 1 = b 3 Index f q + Γ ′ Z f q + α q + γ g + η f q , (15) where I n d e x f q \mathrm{Index}_{fq} Index f q Z f q Z_{fq} Z f q α q \alpha_q α q γ g \gamma_g γ g t = 9.63 t = 9.63 t = 9.63
Gains and the span of the contributor pool
The results also show that the next-quarter gains rise with the span of the contributor pool as measured by dispersion (the number of distinct posting accounts per tweet), mainly where informed trading intensity is low. The slope of next-quarter gains on dispersion is positive and significant in informed-trading-intensity tercile 1 (row R4, Table IV), while in terciles 2 and 3 it is statistically indistinguishable from zero (t = 0.11 t = 0.11 t = 0.11 0.34 0.34 0.34 (7) ). This direction is seen in the slope of gains on dispersion, which is positive in tercile 1 and indistinguishable from zero in terciles 2 and 3. Holding cash-flow predictability fixed does not remove the pattern, but within tercile 1 it is strongest for the firms with the most predictable cash flows, the opposite of what the model implies (A.12, Table XVI ). A wider contributor pool predicts larger next-quarter gains (row R4, Table IV) and smaller next-quarter losses in the joint regression (row R11, Table IV). Therefore, a wider contributor pool raises the mean of the next-quarter returns. The volatility channel cannot produce such an effect, since it raises expected gains and expected losses by the same amount and leaves the mean return unchanged (Proposition 5). The correction channel cannot produce that pattern either, since it is closed by the zero lower bound on the short-sale friction. We therefore report the relation between the span of the contributor pool and next-quarter gains as a reduced-form regularity which the model doesn’t predict.
Table XVI. Gains on dispersion holding cash-flow predictability fixed (R4B)
The tercile orderings in § 6.4 and in A.7 are subject to the same limitation. The estimated informed-trading intensities (K j K_j K j § 6.1 and § 6.4 and in A.7 agree with the model’s comparative statics only when the other group primitives are held fixed. They do not follow from the estimated values of τ 0 \tau_0 τ 0 K j K_j K j
Conclusion
This paper compares the Twitter feed sentiment of foreign and domestic stocks. Our hypothesis is that the feed for foreign stocks is less informative than the feed for domestic stocks. The main results support the hypothesis. Twitter sentiment provides a more reliable signal for stocks of domestic firms than for stocks of foreign firms. The predictive relation between measured sentiment noise and next-quarter losses is stronger for domestic stocks than for foreign stocks. This relation maintains its sign in four out of five quarters for domestic stocks and about one out of two quarters for foreign stocks. This difference could reflect a gap in informed-trading intensity between domestic and foreign stocks rather than a gap in their feeds, since heavier informed trading makes the price respond less to the feed’s noise. However, the difference survives when domestic and foreign stocks are compared within informed-trading-intensity tercile 1, which has the lowest informed trading.
The composition of contributors provides an explanation for this difference. A sentiment index constructed from the Twitter feed of a U.S.-listed stock consists of two components. The first one consists of the beliefs of investors who hold information about the firm. The second one consists of fads and automated content that carry no information about the firm. The share of each component in the sentiment index depends on the composition of the contributor pool by type. The estimates show that the contributor pool for the feed for foreign stocks has a higher share of repeat authors than that of the feed for domestic stocks, in each of the three informed-trading-intensity terciles. The estimates also show that the foreign feed carries a larger fad component. Thus, the feed for foreign stocks carries a larger nonfundamental component and gives a less precise public signal. Hence, the composition difference between the two feeds explains why the sentiment index is a less reliable signal for foreign stocks than for domestic stocks. Our interpretation of these differences is the producer-knowledge mismatch, which is the mismatch between the contributors who post about the firm and the people who hold information about the firm. For a foreign firm, the investors who understand its regulatory environment, management, and product markets are underrepresented among its English-language contributors. The structural model disciplines this interpretation. We do not observe the geography or the information sets of the individual contributors.
The model also predicts an asymmetry between next-quarter gains and losses. Due to the optimistic error in the public signal, there is overpricing which survives the quarter and manifests as a loss in the next quarter because of the short-sale friction. The results also show that the excess of next-quarter losses over gains rises with the model-motivated signal-noise index within each geography-tercile group. The mechanism of correction of overpricing from the model predicts the direction of this relation. However, as the estimate of the short-sale friction lies at its zero lower bound, the fitted model is not able to produce this rise. Therefore, we report the rise as a reduced-form pattern.
The results from this paper are limited to English-language Twitter and to the stocks listed on the U.S. exchanges. An open question that remains for future research is whether the mismatch between who posts about a firm and who knows about a firm also shows up on the local-language social media platforms of the home markets of the foreign firms. A measurement of the sentiment index on those platforms would allow the testing of a mechanism similar to ours but with the two geographical areas reversed. On a local-language platform in the home market of a foreign firm, investors who know the firm best would be a dominant part of the contributors who post about that firm, while the investors who know a U.S. firm best would be a small part of the contributors who post about a U.S. firm. A mechanism along similar lines to ours would imply that the feed for the firm headquartered in that market is more informative than that for the U.S.-headquartered firm. Such a mechanism would also imply that the relation between measured sentiment noise and next-quarter losses is more stable across quarters for the firm headquartered in that market than for the U.S.-headquartered firm. Our model provides the closed-form moment conditions that such studies on other social media platforms would use to estimate their parameters.
Additional Estimation Detail and Robustness
This section reports the estimation details and the robustness results that the paper refers to. Table I shows each of the model’s objects with its empirical counterpart.
Table I. Model objects and their empirical counterparts
Table 1 (PDF p. 62). Table I, continued
Outcome variables and the common component. In the data, the common component that all investors observe (§ 3.1 ) is measured by the three Fama-French factors (FF3). Excess returns are measured against the riskless rate. The residual from a regression of the stock’s daily excess return on the three factors removes the common component. The quarterly factor realizations are common to all stocks. The quarter fixed effects of the regressions absorb them. The residual-based gains and losses of the robustness checks (Section 4) are the sums over the quarter of the positive and of the negative daily residuals, with each stock’s FF3 betas fixed at their estimates from the end of the previous calendar quarter and with no intercept subtracted. This construction keeps the mean daily excess return, which carries the correction drift that the model predicts (A2). Table VIII reports the drift projection on these residual-based outcomes.
Table VIII. Slope of the loss-gain excess on the signal-noise index under the announcement and the calendar alignment, with and without the group effects
Table II (PDF p. 63). The empirical exercises and their classification
From the measurement propositions to the estimation
A.1. From the measurement propositions to the estimation: We estimate the model’s parameters by the generalized method of moments on ninety-one closed-form moments (§ 5 ). The estimator minimizes a weighted distance between the data moments and their model-implied counterparts, with a pre-specified diagonal weighting matrix. For each geography and informed-trading-intensity tercile, we estimate the informed-trading intensity (K j K_j K j σ x , j \sigma_{x,j} σ x , j υ j \upsilon_j υ j Σ j d \Sigma_{jd} Σ j d π j \pi_j π j μ 0 , j \mu_{0,j} μ 0 , j x ˉ j \bar{x}_j x ˉ j § 3.3 ). For each geography, we estimate the fad scale (σ b , j \sigma_{b,j} σ b , j σ β , j 2 \sigma_{\beta,j}^{2} σ β , j 2 § 3.2 ). For all stocks together, we estimate the conversion constant (κ \kappa κ δ \delta δ s P s_P s P c w σ e 2 c_w\sigma_e^2 c w σ e 2 a s a_s a s a c s a_{cs} a cs a λ a_\lambda a λ λ I \lambda_I λ I § 3.2 ). The gross riskless rate (r r r ρ ˉ \bar{\rho} ρ ˉ [22 ] estimator of the same spread from daily high and low prices), the Amihud ratio (daily absolute return over dollar volume, averaged within the quarter) and turnover (daily share volume over shares outstanding, entered per day as the quarterly sum divided by the number of return days). The moments also include the persistence and the innovation variance of the firm’s cash flows, price informativeness at horizons of one and four quarters (Proposition 4), the mean of dispersion, and the mean of squared measured noise. The last within-group moments are the variance of the quarterly excess return, the means of the quarter’s gains and of its losses, and the slopes of gains and of losses on the lagged signal-quality index. The remaining moment is the slope of squared measured noise on the average inverse tweet count in the pooled sample (A). We conduct two exercises which precede the GMM and supply tercile assignment and starting values for it. The first is the measurement regression, which fits the measurement equation (equation 4) by ordinary least squares across firm-quarters. The second is the microstructure inversion of Proposition 7. Both use estimates of υ j \upsilon_j υ j Σ j d \Sigma_{jd} Σ j d Φ j \Phi_j Φ j c w σ e 2 c_w\sigma_e^2 c w σ e 2 σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j § 3.5 ). We have two equations in two unknowns. We can solve them for exact values of σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j σ I , j \sigma_{I,j} σ I , j (43) ). The quoted spread and the high-low spread both measure that spread. Hence, each is proportional to σ I , j \sigma_{I,j} σ I , j a s a_s a s a c s a_{cs} a cs σ I , j \sigma_{I,j} σ I , j σ I , j \sigma_{I,j} σ I , j σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j (43) hold in the data. We report these over-identifying checks (A.2).
We fit the linear regression of equation (4) across firm-quarters. We read the nonfundamental variance Φ j \Phi_j Φ j c w σ ε 2 c_w\sigma_{\varepsilon}^{2} c w σ ε 2 S t / N t S_t/N_t S t / N t § 3.2 . The conversion constant κ \kappa κ κ \kappa κ
We map Proposition 2 to the data with two conventions (§ 3.2 ). The first is that we set the bot-content variance ( σ β , j 2 ) (\sigma_{\beta,j}^{2}) ( σ β , j 2 ) ( Φ j ) (\Phi_j) ( Φ j ) ( 1 − ω j ) 2 σ b , j 2 (1-\omega_j)^2\sigma_{b,j}^{2} ( 1 − ω j ) 2 σ b , j 2 ω j \omega_j ω j ( π j ) (\pi_j) ( π j ) Proposition 1 ). An automated share that differs between domestic and foreign stocks is the case that would move the domestic-foreign comparison of signal quality (C.3). The second is that we treat the conversion constant κ \kappa κ σ b , j \sigma_{b,j} σ b , j § 5 ). With the fad scale held equal, the differences in the nonfundamental variance across a geography’s three terciles are what identifies the posting intensity (λ I \lambda_I λ I § 5 ).
Conventions that discipline the moments
The gross quarterly riskless rate is 1.0056 (one plus the mean quarterly riskless rate over the primary sample). The risk-aversion coefficient is 0.02 from Farboodi et al. [35 ] . Standard errors on the ordinary least squares estimates are clustered two ways, by firm and by quarter, with the conservative rule that sets degrees of freedom to the smaller number of clusters minus one (Cameron et al. (2011); Petersen [67 ] ). Estimates that depend on generated regressors use a firm-block bootstrap that repeats the generating steps inside each replication.39
Conventions for the liquidity moments. A.1 defines the four liquidity measures. For the moments on the quoted spread and on the high-low spread, we use the firm-quarters whose quarter-end price is at least five dollars. The reason is that the one-cent minimum tick puts a floor under the relative spread of a low-priced stock. The zero-return share, the auxiliary over-identifying check of Proposition 7, is the fraction of the quarter’s trading days with a return of exactly zero, the trading-cost indicator of Lesmond et al. [57 ] . The Amihud ratio in our data has a scale that differs from the values published elsewhere. Therefore, we treat the three scale constants (a s a_s a s a c s a_{cs} a cs a λ a_\lambda a λ (43) does not fix the scale of the noise-trading standard deviations on its own. Multiplying every σ x , j \sigma_{x,j} σ x , j a λ a_\lambda a λ a T a_T a T a T a_T a T
Model dependence of the liquidity identification. C.9 states that the spread does not depend on the amount of noise trading. That property belongs to the Kyle batch market. In a sequential market of the Glosten and Milgrom [43 ] type, the dealer posts quotes for a fixed trade size. There the spread depends on the ratio of informed trades to noise trades as well as on the informed investor’s advantage (σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j
The model moment for the variance of the quarterly excess return is the average over the firm-quarters of the group of the closed-form return variance, with each firm-quarter at its own signal precision (τ s \tau_s τ s (5) . The model moments for the means of the quarter’s gains and losses are the expected gain and the expected loss of § 3.4 , computed once per group at the group averages over those firm-quarters of the expected overpricing and of the return variance. The slopes of gains and of losses on the lagged signal-quality index are estimated with quarter fixed effects, coded to sum to zero, in the data moments and in the model moments alike. The quarter fixed effects remove what moves the index for every stock in a quarter at once. The pooled slope of squared measured noise on the average inverse tweet count is estimated with the same quarter fixed effects and a foreign indicator. Without the quarter fixed effects that pooled slope has the opposite sign in the data. That slope is reported for comparison and is not among the 91 moments. The group-level precision of the public signal is the median of the signal-quality index across the group’s firm-quarters, converted through equation (5) , because the index is heavy tailed. We assign stocks to informed-trading-intensity terciles in two steps. In the first step, we rank stocks within each geography and quarter on the lagged quoted spread alone. In the second, we rank them on the ranking statistic of § 3.5 , the lagged quoted spread multiplied by the square root of the prior precision plus the lagged signal precision. The slopes are estimated with the index lagged one quarter, an approximation to the model’s timing, in which the quarter’s own signal precision moves the quarter’s price (§ 3.3 ). Within a firm, the logarithm of the index has a correlation of 0.32 with its value one quarter earlier in this sample. The group intercepts of the slope regressions absorb the group-level risk premium (μ 0 , j \mu_{0,j} μ 0 , j
The optimizer, the basin table, and the saturation criterion
We start the optimizer from two families of starting values. Table III shows, for each family, the number of starting values, the range of the objective at which they end, how many of them end at a finite objective and whether the lowest point of the family satisfies the saturation criterion. The first family is one starting value built from the two exercises that precede the GMM (A.1), with the conversion constant started at one, together with ten Latin-hypercube draws around it. From these eleven starting values, the optimizer ends at objectives between 76,842.0 and 83,033.8. When we continue the optimization from the lowest of them, the objective falls to 76,752.4. At that point six of the model’s gain and loss moments underflow to exactly zero, so that the objective is flat in the parameters behind the underflow and the optimizer stops there. The saturation criterion excludes any candidate estimate at which one of the model’s gain, loss or variance moments underflows to exactly zero, unless no other candidate ends at a finite objective. Thus, it excludes the point reached from the first family. The second family consists of fourteen starting values set from the data moments at seven values of the conversion constant from 1.05 to 400 and two scales of the informed-trading intensity. From these the optimizer ends at objectives between 4,267.9 and 4,400.2. Continuing the optimization from the lowest of them gives the interior optimum, with an objective of 4,250.3 and no moment at zero. At the interior optimum the posting intensity is free and every repeat-author share is at its zero lower bound (§ 5 ). Hence, six of the twenty acceptance checks are not met there, one for each share. We accept an estimate only if the conversion constant is at least one, the posting intensity exceeds one and, in each of the six groups, the cash-flow persistence lies below the gross riskless rate, the microstructure inversion has a positive denominator and the repeat-author share lies strictly between zero and one. These are the twenty acceptance checks. We set the posting intensity at 1.5 by the rule stated in A.4 and re-estimate the other forty-five parameters starting from the interior optimum. The result is the headline estimate. It satisfies the saturation criterion and the twenty acceptance checks. At the headline estimate the largest component of the gradient with respect to the forty-five free parameters is 0.016. When we restart the optimizer from the headline estimate, the objective changes by less than 10 − 10 10^{-10} 1 0 − 10
The weighting matrix is a further convention. Since the estimate depends on the weighting matrix under misspecification, we fix the weighting in advance rather than choosing it from the fit [45 ] . § 5 states the reason. For the headline estimate we use the diagonal weighting matrix described there. We report the Ledoit-Wolf-shrunk full-matrix weighting as a variant (Table XVIII ).
Table III. Starting values for the GMM estimation and the headline estimate
Notes: This table shows the two families of starting values for the GMM estimation. For each family it shows the number of starting values, the range of the objective at which they end, the number that end at a finite objective, whether the lowest point of the family satisfies the saturation criterion and whether that point is the headline estimate. The indented rows show the point we reach when we continue the optimization from the lowest starting value of each family, which is why their objectives lie below the range of the family. The point reached from the first family has six of the model’s gain and loss moments at exactly zero and its six variance moments above zero but tiny. The saturation criterion excludes it. The point reached from the second family (the interior optimum) satisfies the criterion. Its posting intensity is free and every repeat-author share is at its zero lower bound. Hence, six of the twenty acceptance checks are not met (§ 5 ). In the last row we re-estimate the other forty-five parameters with the posting intensity set at 1.5, starting from the interior optimum (A.4). Its objective is higher than that of the interior optimum, because holding one parameter fixed does not lower the minimum. § 5 states the rule by which we pick the headline estimate among the candidates.
Headline estimates, identification, and the posting-intensity profile
Table I shows the headline estimates and § 5 states the reading. Three diagnostics bear on the identification. The first diagnostic is a comparison of two estimates of the sampling-noise coefficient. The GMM gives 1.64 × 10 − 4 1.64 \times10^{-4} 1.64 × 1 0 − 4 1.95 × 10 − 4 1.95 \times10^{-4} 1.95 × 1 0 − 4 § 5 ). We fixed a rule for weak identification in advance. Whenever a freely estimated repeat-author share reaches a bound, or the Jacobian of the model moments has a near-zero singular value in the direction of the posting intensity, we set the posting intensity at 1.5 and estimate the other parameters at that value. When we estimate the posting intensity freely, the estimate is 1,102.98 and every repeat-author share is below 10 − 4 10^{-4} 1 0 − 4 Proposition 1 , the repeat-author share in a group is a function of the mean of dispersion in that group and the posting intensity. For the share to lie below one, the posting intensity must exceed the inverse of the mean of dispersion. The largest of the six inverses is 1.0974, for foreign stocks in the first tercile. The value 1.5 lies above it. Table V shows the joint Wald test of Table II and the signs of its four domestic-foreign contrasts at each of the eight values of the posting intensity from 1.2 to 1.85.
Table I reports twelve of the forty-five estimated parameters. Table IV reports the remaining thirty-three.
The conversion constant, profile and shape check
A.5. The conversion constant, profile and shape check: Section 5 reports the estimate of the conversion constant (κ \kappa κ κ \kappa κ Table VI shows the profile of the GMM objective function in κ \kappa κ κ \kappa κ κ \kappa κ J J J
Across the 399 firm-block bootstrap replications, the 2.5th and 97.5th percentiles of the estimate of κ \kappa κ
We pre-specified a check of the shape of the loss and variance responses to the signal-quality index. The check is an attempt to bound the conversion constant from the data alone. We look for an interior peak in the excess of losses over gains and for a kink in the response of the return variance (a level of the index around which the slope of the logarithm of the return variance on the logarithm of the index falls from zero toward minus one). Neither is found between the 5th and the 95th percentile of the index in any of the six groups. The slope of losses on the index is negative in the low and middle terciles of the index in each of the six groups. In the top tercile of the index, the slope is smaller in magnitude than in the middle tercile of the index and its sign differs across the six groups. This pattern is consistent with the estimates, in which the surviving fraction of overpricing (δ \delta δ 3.0 × 10 4 3.0 \times10^{4} 3.0 × 1 0 4 κ \kappa κ κ \kappa κ
Table IV (PDF p. 71). Remaining structural parameter estimates
Notes: This table shows the thirty-three parameters estimated jointly with those in Table I, at the headline estimate with the posting intensity set at 1.5. For each of the six geography-tercile groups, it gives the informed-trading intensity (K j K_j K j σ x , j \sigma_{x,j} σ x , j υ j \upsilon_j υ j Σ j d \Sigma_{jd} Σ j d μ 0 , j \mu_{0,j} μ 0 , j a s a_s a s a c s a_{cs} a cs a λ a_\lambda a λ 1.1 × 10 − 7 1.1 \times10^{-7} 1.1 × 1 0 − 7
stocks in the first tercile, the ratio is 0.082. The moments for the means of that group’s gains and losses carry the least information about κ \kappa κ Table VI , in which the objective lies within 1 percent of its minimum from κ \kappa κ Proposition 1 ), which does not depend on κ \kappa κ
One of the three components which is absorbed by the conversion constant (κ \kappa κ S t / N t S_t/N_t S t / N t (5) , § 3.2 and § 5 ). Positive serial dependence raises the dependence multiplier (κ 0 , f q \kappa_{0,fq} κ 0 , f q f q 2 ^{2}_{fq} f q 2 v ˉ f q \bar{v}_{fq} v ˉ f q (23) , C.2). In order to deal with this, we order the days with at least one scored tweet by date and subtract the within-quarter mean (for each firm-quarter), which gives the daily deviations. Then we summarize the dependence across days in these deviations three ways. First, we take the lag-one autocorrelation over the full sequence of these scored days (the first-order piece of the covariance sum in κ 0 , f q \kappa_{0,fq} κ 0 , f q κ 0 , f q \kappa_{0,fq} κ 0 , f q Table VII shows the three statistics for 98,290 firm-quarters.
Table VII. Serial dependence of the daily per-tweet sentiment series by geography
The median lag-one autocorrelation over the days with at least one scored tweet is 0.043 for domestic stocks and 0.040 for foreign stocks (first row, Table VII ). Thus, the serial dependence across days is small. The domestic-foreign gaps are between 0.005 and 0.010 (t t t
Notes: This table shows three statistics of serial dependence in the daily per-tweet sentiment series, for domestic and for foreign stocks. For each firm-quarter, we order the series over the days with at least one scored tweet and subtract the within-quarter mean of the series. The first row is the lag-one autocorrelation over the full sequence of those days. The second row is the lag-one autocorrelation over pairs of calendar-adjacent days (days that are consecutive on the calendar) alone. For the third row, we divide the long-run variance, estimated with the Bartlett kernel at the Newey-West automatic bandwidth ⌊ 4 ( T q / 100 ) 2 / 9 ⌋ \lfloor4(T_q/100)^{2/9}\rfloor ⌊ 4 ( T q /100 ) 2/9 ⌋
foreign gaps of 7 to 17 percent in the median signal-quality index across the informed-trading-intensity terciles (Table XXI ). The sample autocorrelation is biased downward by about 1 / T q 1/T_q 1/ T q T q T_q T q V A Q S V f q 2 \mathrm{VAQSV}^{2}_{fq} VAQSV f q 2 ( T q − κ 0 , f q ) / ( T q − 1 ) (T_q-\kappa_{0,fq})/(T_q-1) ( T q − κ 0 , f q ) / ( T q − 1 ) (23) , which is 0.998 at the median T q T_q T q ( Φ j ) (\Phi_j) ( Φ j ) Table VII ), which is a small part of the estimated conversion constant of 394 (Table I). Hence, with κ \kappa κ § 5 ), the serial dependence overstates the domestic-foreign ratio of τ s \tau_s τ s
The drift projection
A.7. The drift projection: § 6.3 states how we read the projection of the excess of next-quarter losses over gains on the model-motivated signal-noise index (equation (15) ). Table VIII shows the slope b 3 b_3 b 3 A A A § 6.3 ). The gain and loss block holds twelve moments, whose weighted squared misfit makes up 0.4 percent of the objective (Table XX ).
Notes: This table shows the slope b 3 b_3 b 3 (15) (the projection of the excess of next-quarter losses over gains on the model-motivated signal-noise index), with the index evaluated at the headline estimate (§ 6.3 ). We estimate this projection with quarter fixed effects, geography-tercile group effects, and the controls of equation (15) (the factor betas, the lottery control, and log capitalization). The standard errors are clustered by firm and by quarter. The first four rows differ in two choices. We measure the gains and losses either in the quarter of the next earnings announcement (the announcement alignment) or in the next calendar quarter (the calendar alignment). We construct the gains and losses from the daily excess returns or from the residual-based gains and losses of A. The last row is a different composition run where we drop the group effects (81,301 stock-quarters). We construct the gains and losses from the daily excess returns under the announcement alignment, keep the quarter fixed effects but date them at the outcome quarter, and lag the index one quarter more than in the first four rows. The slope becomes negative. We interpret this reversal as an effect of composition (§ 6.3 ). Since this index has no natural scale, we report only the sign of b 3 b_3 b 3 t t t
The model also predicts that the loss-gain excess rises with current-quarter measured noise only up to a certain level of measured noise and falls beyond it (Proposition 6 ). This peak level is lower in groups with more informed trading. To check this, we regress the loss-gain excess on current-quarter measured noise and its square, separately in each informed-trading-intensity tercile. The fitted curve peaks at a lower level of measured noise in the middle tercile than in the low tercile (whether the excess is measured in the next calendar quarter or in the quarter of the next earnings announcement). This is the direction predicted by the model, but the peak levels for the low and middle terciles differ by less than one standard error (Table XIX ). The peak level for the top tercile is too imprecise and is left out of the ordering.
Table XIX. Turning points of the loss-minus-gain response in measured noise units by informed-trading-intensity tercile
Fit and misfit, full diagnostics
The Hansen J J J [46 ] is 5,503 on 46 degrees of freedom at the estimate under the Ledoit-Wolf-shrunk full-matrix weighting (the last column of Table XVIII ), against a 5 percent critical value of 62.8. Thus, the test rejects the null that the 46 overidentifying restrictions hold. § 5 splits the objective across the nine blocks of moments to locate the misfit (Table XX ). The concentration and fad contrasts keep their signs when we exclude the liquidity block and when we let the scale constant a s a_s a s
The liquidity block misfit also appears in the reduced form regressions of the quoted spread and the Amihud illiquidity ratio of quarter q + 1 q + 1 q + 1 q q q − 0.034 -0.034 − 0.034 t = − 5.41 t = -5.41 t = − 5.41 − 6.30 -6.30 − 6.30 t = − 3.88 t = -3.88 t = − 3.88 a s σ I , j a_s\sigma_{I,j} a s σ I , j a λ σ I , j / σ x , j a_\lambda\sigma_{I,j}/\sigma_{x,j} a λ σ I , j / σ x , j − 1.67 × 10 − 5 -1.67 \times10^{-5} − 1.67 × 1 0 − 5 t = − 0.01 t = -0.01 t = − 0.01 − 0.062 -0.062 − 0.062 t = − 1.25 t = -1.25 t = − 1.25
§ 5 states that the liquidity block misfit doesn’t affect the assignment of stocks to informed-trading-intensity terciles. We check this in two ways (rows R10 and R14C). In row R10, we form the terciles using the high-low spread instead of the quoted spread in the ranking statistic, or using the quoted spread capped at each of its four adverse-selection bounds, and re-run the headline regression of row R1 (equation (14) ). The domestic-foreign difference in the tercile 1 slope, β D 1 − β F 1 \beta_{D1}-\beta_{F1} β D 1 − β F 1 + 0.77 +0.77 + 0.77 t = 1.41 t = 1.41 t = 1.41 + 0.74 +0.74 + 0.74 t = 0.91 t = 0.91 t = 0.91 + 0.66 +0.66 + 0.66 + 1.14 +1.14 + 1.14 + 0.037 +0.037 + 0.037 p = 0.901 p = 0.901 p = 0.901 + 0.333 +0.333 + 0.333 p = 0.004 p = 0.004 p = 0.004 + 0.111 +0.111 + 0.111 p = 0.482 p = 0.482 p = 0.482 p p p Table XV ).
Table XV. Table XV: Reliability under a model-free spread sort. Notes: This table shows the share of quarters for which the slope of next-quarter losses on measured noise is negative (row R14C; 27 usable quarters). This regression is run for domestic and the foreign stocks separately, within each tercile of the spread, with log capitalization and the FF3 betas as controls. We construct the terciles using the lagged quoted spread for each geography-quarter which is a sorting not related to the model. We test each domestic-foreign difference with the R14 circular block bootstrap over quarters (block length four, 999 replications, paired across the two subsets).
Price informativeness for foreign stocks as a whole is negative in the primary sample under both price sources (CRSP market value and, as the robustness alternative, Computat market value, Table XVII ). Within the sample of the joint estimation, the estimate at the one-quarter horizon is imprecise for both American Depositary Receipts and directly listed foreign shares. The estimate is more negative at the four-quarter horizon than at the one-quarter horizon. If we take the price one quarter earlier, the four-quarter estimate is more negative for both listing types. Its t t t t t t t = 3.2 t = 3.2 t = 3.2 Table XVIII shows the pre-specified estimation variants. The largest misfit is in the second moments of foreign stocks in the third informed-trading-intensity tercile (the illiquid-foreign group of Table XVIII , whose second moments are the variance of the quarterly excess return and the innovation variance of the firm’s cash flows). That misfit does not pass into the repeat-author shares or the fad scales. We confirm this by dropping those second moments and then the group itself (the two illiquid-foreign columns of Table XVIII ). The repeat-author shares then move by 0.0066 or less and the fad scales by 0.001 or less. The four domestic-foreign contrasts keep their signs.
Table XXI shows the implied price-informativeness data term by group and horizon. We build the term at each horizon as the signed price-informativeness estimate divided by the volatility factor and the growth factor of Proposition 4, both taken from the first-stage estimates that precede the GMM. In the model this term equals the information factor 1 − Σ ˉ j Σ j d 1 - \frac{\bar{\Sigma}_j}{\Sigma_{jd}} 1 − Σ j d Σ ˉ j (10) ). This information factor is at most one. For domestic stocks in the first tercile the term is positive at all four horizons and lies above one, above that model bound. In the domestic second and third terciles it lies between −0.35 and 0.26. For foreign stocks in the first tercile it is positive at two horizons and negative at the other two. In the foreign second and third terciles it is negative at every horizon and most negative at the four-quarter horizon. We interpret the negative sign as evidence that home markets lead price discovery for cross-listed foreign stocks. The U.S. price of such foreign stocks carries less information about their future earnings. If we remove the informativeness block from the moment system, it moves the repeat-author shares and the fad scales by less than 10 − 5 10^{-5} 1 0 − 5 Table XVIII ). Thus, the concentration and fad contrasts do not depend on this block.
Table XXII shows, for each group, the ratio of the variance of the quarterly excess return across the group’s firm-quarters to its value under independent days with the same daily variance (the independence benchmark). If the daily excess returns do not correlate across days, the variance of their quarterly sum equals the sum of the daily variances, even when the daily variance changes from day to day. Thus, day-to-day movement in volatility alone leaves the ratio at one. The ratios instead lie between 0.766 and 0.916, from 8 to 23 percent below one. We check two explanations for the shortfall in the notes to Table XXII . The unwinsorized ratio is lower in each of the six groups. Hence, winsorization asymmetry is not the cause. The measured lag-one autocorrelation lowers the ratio by less than six percentage points. That leaves within-quarter reversal over more than one lag as the remaining explanation. Therefore, we take the data moment for the variance of the quarterly excess return to be that variance itself rather than the independence benchmark.
Table XXII. Bridge ratios between the realized quarterly variance and the independence benchmark, winsorized and unwinsorized
Dependence and the acceptance gate, full detail
Two kinds of dependence enter the inference of § 5 . The first is the dependence within a firm across its quarters, which enters the standard errors and the confidence intervals. The second is the dependence within a quarter across its firms, which enters the joint Wald test. § 5 describes the firm-block bootstrap, whose replications keep each firm’s complete time series and repeat the generating steps. Hence, the dependence across a firm’s quarters and the sampling error of the generated regressors both pass into the confidence intervals. For the dependence across the firms of one quarter, § 5 reports the joint Wald test with two further clusterings of the standard errors, by quarter and two-way by firm and quarter (Table II). We build these tests from three covariances of the data moments. The firm covariance is the covariance of the data moments across the 999 firm-block bootstrap replications of § 5 . For the quarter covariance we instead draw quarters with replacement, again over 999 replications. The two-way covariance (the covariance under both kinds of dependence at once) is the firm covariance plus the quarter covariance minus the covariance across 999 replications that draw firm-quarters with replacement. We shrink each of these three covariances toward its diagonal by the Ledoit-Wolf rule of § 5 before we form the two-way covariance. The measurement regression that precedes the GMM (A.1) has one acceptance gate. We accept the regression only when the sampling-noise coefficient (c w σ e 2 c_w\sigma_e^2 c w σ e 2 t t t 1.95 × 10 − 4 1.95 \times10^{-4} 1.95 × 1 0 − 4 t = 11.7 t = 11.7 t = 11.7 t = 2.2 t = 2.2 t = 2.2
Table IX. Specifications of the reduced-form rows
A.10. Reduced-form battery, specifications and full results: We describe the design of the battery of reduced form regressions in § 6 . For five of the rows (R1, R2, R3FAD, R7 and R8, Table IV), we form the outcome under two alignments. In order to align the outcome with the earnings news, we take it from the quarter of the firm’s next earnings announcement (the announcement alignment, which we report as the headline). To check that this timing does not drive the result, we also take the outcome from the next calendar quarter and repeat these five regressions (the calendar alignment). The outcome quarter is the same under both alignments for 92.4 percent of the firm-quarters that have Notes: This table shows the specification used for each of the rows for the battery of reduced form regressions. Unless stated otherwise, the specification used (equation (13) ) uses quarter fixed effects, double-clustered standard errors (firm and quarter), primary sample (from 2018 onward) and the usual screens (Section 4). In order to test whether sentiment in a quarter predicts an outcome in the next quarter, we take the lag of a regressor by one quarter (true for most of the rows in the table). When we need a within-quarter relation, we don’t take lags (Rows R13a and R13b). Conditioning on the informed-trading-intensity terciles implies interacting the sentiment regressor with the tercile indicators which gives each tercile its own intercept. In some rows, the outcome is taken from the quarter of the firm’s next earnings announcement instead of the next calendar quarter (§ 6 and A.10).
Table IX (PDF p. 80). Specifications of the reduced-form rows (continued)
an announcement-aligned outcome. We have the firm’s next announcement date for 99.0 percent of the primary sample.
Losses and volatility (R1, R2, R3FAD). We give the reading of the loss regression without the volatility control in § 6 (row R1, Table IV). In order to isolate the volatility channel, we add realized volatility as a control (row R2, Table IV). In the base cell (domestic stocks in the lowest tercile), the coefficient on measured noise is not statistically distinguishable from zero whether we take the outcome from the firm’s next earnings announcement quarter or the next calendar quarter (0.211 with t = 1.15 t = 1.15 t = 1.15 t = 1.60 t = 1.60 t = 1.60 − 14.70 -14.70 − 14.70 t = − 40.91 t = -40.91 t = − 40.91 Table XIX ). Before we truncate it at zero, the filtered variance is negative for 8.2 percent of the firm-quarters when we take the outcome from the announcement quarter (7.9 percent from the next calendar quarter).
Gains and dispersion (R4). § 6.4 states the reading.
Volume horse race (R9). We enter the raw quarterly signal and the volume-adjusted signal together in one regression of next-quarter total losses. The coefficient on each is negative and statistically significant (− 0.317 -0.317 − 0.317 t = − 6.02 t = -6.02 t = − 6.02 − 0.612 -0.612 − 0.612 t = − 2.11 t = -2.11 t = − 2.11
Consolidated regression (R11). Table IV shows the headline estimates of the loss regression in row R11. § 6 states the reading. In the gain regression, the coefficient on measured noise is positive and the coefficient on dispersion is negative. By design, we also control for the quoted spread in this specification (Table IX), which is one of the inputs of the ranking statistic by which we sort stocks into informed-trading-intensity terciles (§ 3.5 ).
Similarity-index bridge (R12). To see whether a sort on firm characteristics alone would reproduce our sort into informed-trading-intensity terciles, we compare the two (row R12, Table IV). We rank the stocks by a similarity index built from firm characteristics alone, and again by the informed-precision index behind that sort, which we build from the lagged quoted spread and the lagged signal precision. The rank correlation between the two rankings is negative, at − 0.379 -0.379 − 0.379
Liquidity (R5). § 5 states the reading.
Attribution (R13a, R13b). We check whether automated accounts affect the feeds for domestic and foreign stocks in a different manner (rows R13a and R13b, Table IV and § 5 ) and we find that it does not. However, the result could be because the difference is too small. To get clarity on that, we would need the variance of the automated content, which is not in the data. Instead, we show that the feed for foreign stocks would be the more precise of the two only if it carried 8 percent more weight on the firm’s true signal than the one for domestic stocks (C.3). We also check whether the average dispersion shifts suddenly at any one quarter due to change in characteristics of the social media platform. We treat each quarter as a possible break date, but not the first or last 15 percent of the 28 quarters, since a break that close to the start or end would leave too few quarters on one side to estimate the mean there. The largest Chow statistic is 2.71 (fourth quarter of 2023). The second largest is 2.47 (third quarter of 2023). This is a descriptive scan since we don’t compare the Chow statistics with the critical values. At that break, dispersion decreases more for stocks priced below five dollars or in the smallest size tercile of their quarter than for the others (t = − 8.9 t=-8.9 t = − 8.9 40 (4) ) is the domestic-foreign difference in the nonfundamental variance. It has the same sign and stays significant when we drop the quarters around these events, one at a time and then together (the 2018 removal of fake accounts, the 2022 change of ownership, and the 2023 limits on data access, with t = 6.3 t=6.3 t = 6.3
Reliability (R14). § 6 states the design and the reading. In § 1 and § 6 , we scale the slopes by the standard deviation of measured noise in the estimation sample, 0.0099. Rounded to four decimals, this standard deviation is the same for domestic and for foreign stocks. The sample’s mean quarterly excess return in § 1 is the pooled mean of the quarterly cumulative excess return in the estimation sample, 0.0211.
State dependence and correction after optimism (R7, R8). In row R7 (Table IV), the coefficient on measured noise interacted with demeaned financial uncertainty is − 23.8 -23.8 − 23.8 t = − 1.98 t = -1.98 t = − 1.98 − 11.3 -11.3 − 11.3 t = − 1.63 t = -1.63 t = − 1.63 § 5 states the reading of the correction after optimism (row R8, Table IV). The coefficient on measured noise interacted with the optimism share is 0.214 (t = 0.21 t = 0.21 t = 0.21 t = 0.29 t = 0.29 t = 0.29
Cross-sectional structure on a characteristics-only sort
We wish to know whether our two patterns (gains-dispersion and loss-noise relations) still show up if we sort stocks in different ways. Apart from the sorting of stocks using the informed-trading-intensity (§ 3.5 ), we also sort them using a similarity index built from firm characteristics alone (without the sentiment feed, the quoted spread or the structural model). This similarity index is weighted cosine similarity between a stock’s characteristics and a benchmark stock formed by taking median of all stock features using weights from a random-forest ([48 , 54 ] ). We refer to the stocks closest to the benchmark as typical and those farthest from the benchmark as atypical. The results show that only 0.227 of the stocks are in the same tercile under both types of sorting as opposed to one third which we would get if the two sortings were unrelated. Their rank correlation is − 0.379 -0.379 − 0.379
Table X shows the two patterns by similarity tercile and geography. The gains-dispersion relation is positive for the atypical stocks and near zero for the typical ones. This positive response is marginally significant for domestic stocks and imprecise for foreign stocks. If the similarity sort ran opposite to the intensity sort, the gain-dispersion response would show more in the typical stocks. It shows up in atypical stocks instead. The loss-noise relation doesn’t show any pattern across the similarity terciles. Its sign varies across the cells. The loss-noise relation is weak even when the sorting is by intensity (row R1, Table IV). Such a weak relation would not survive on a sorting such as by similarity which barely overlaps with intensity.
Table X. Cross-sectional structure of the two channels on the characteristics-only similarity sort. Notes: This table shows the two reduced-form channels by similarity tercile and geography, each as the per-cell slope from a foreign-by-tercile interaction with standard errors double clustered by firm and quarter. We form the similarity terciles within geography and quarter. Tercile 1 is the least similar to the benchmark, the atypical end. Tercile 3 is the most similar, the typical end. Panel A is next-quarter total gains on dispersion. Panel B is next-quarter total losses on measured noise. The loss channel has 89,978 firm-quarters in each cell.
Robustness
Reliability splits (R14B). Table XI reports all cells and contrasts. § 6 states the reading.
Mean slopes by listing type and richer controls (R14D). We run two more specifications, with the R14 per-quarter design, screens, and quarters unchanged. In the first, we report the plain difference in mean quarterly slopes for ADRs and for directly listed foreign shares. The difference in the mean quarterly slope between domestic stocks and ADRs is −2.00 (Newey-West t t t t t t Table XI . Most ADRs are non-Canadian. So the split by listing type and the split by country overlap. In the second, we add the lagged quoted spread, the Amihud illiquidity ratio, and turnover to the R14 controls on the full sample. The domestic share of negative-slope quarters stays at 0.778. The foreign share is 0.556. We leave out the firm’s age since its IPO, which we have for only 60.9 percent of the sample, 60.1 percent of the domestic firm-quarters, and 65.7 percent of the foreign firm-quarters. On the subsample where that age exists, with age and the full control set, the difference is −0.11 (t = − 0.25 t = -0.25 t = − 0.25
Headline reliability across sample years (R14E). The headline reliability estimate is the mean across quarters of the domestic-foreign slope of next-quarter total losses on measured noise (−1.40, Newey-West t = − 2.99 t = -2.99 t = − 2.99 t t t t = − 4.26 t = -4.26 t = − 4.26
Additional robustness (R15). We re-estimate row R1, the regression of next-quarter total losses on measured noise (Table IV), on thirty-one alternative samples and specifications (Table XII). The foreign contrast stays negative in all thirty-one re-estimations.
Leave-one-year-out. There is a caveat for the leave-one-year-out re-estimations (Table Table XII. When we drop 2018, the slope for foreign stocks in the first informed-trading-intensity tercile flattens to −0.080, while dropping any other year leaves it between −0.75 and −1.28. So much of that slope in the first tercile comes from the 2018 quarters.
Table XII. Additional robustness of the unconditional loss regression
Size-conditioned measurement contrasts (R 16 R16 R 16 Table XIV shows all the cells. § 5 states the reading.
Reliability under a model-free sort (R 14 C R14C R 14 C We give the design and reading of this exercise in § 5 (row R14C, Table XV ). The lagged quoted spread is only one term of the ranking statistic (§ 3.5 ). It is a noisy proxy for informed-trading intensity. So the gap sits in a different tercile under the two sorts, in spread tercile 2 here and in intensity tercile 1 under the intensity sort. Either way, we draw the same conclusion, that the reliability gap survives the tercile split and does not come from how we build the sort. Table XV shows all the cells and the contrasts.
Gains on dispersion holding cash-flow predictability fixed (R 4 B R4B R 4 B The informed-trading-intensity terciles also differ in the estimated cash-flow innovation variance, which is smallest in tercile 1 and rises across the terciles. So the least informed-traded stocks also have the most predictable cash flows. We ask whether the pattern of row R4 (Table IV), in which dispersion predicts next-quarter gains in tercile 1 only, survives when we hold cash-flow
Table XII. Additional robustness of the unconditional loss regression (continued). Notes: In each row we re-estimate the announcement-aligned specification of row R1 (Table IV), next-quarter total losses on measured noise interacted with the foreign indicator and the informed-trading-intensity tercile (full factorial), with the FF3 betas, MAX, log capitalization, and buzz as controls, quarter fixed effects, and standard errors double clustered by firm and quarter. The foreign contrast is the coefficient on noise interacted with the foreign indicator. The foreign slope, Tercile 1, is the implied foreign loss-noise slope in the lowest informed-trading tercile. The English split classifies headquarters countries by English as a first language following (2003); the ADR split uses the CRSP share-level ADR flag under a majority-of-days rule. Leave-one-country-out covers the seventeen foreign countries with at least 200 estimation-sample observations, with Canada’s exclusion shown in the splits panel; the thirty-four smaller countries, the largest being India with 199 observations, are not dropped individually. The volume rows replace or augment the buzz control with alternatives measured over the outcome window, as the buzz control is (row R15).
predictability fixed. Table XVI shows three results. First, when we add the log innovation variance to the row R4 design as a control, alone or interacted with dispersion, the slope in intensity tercile 1 stays positive and significant, while the slopes in terciles 2 and 3 are indistinguishable from zero. Second, within each cash-flow-predictability tercile, the slope is largest in intensity tercile 1, where it is significant in predictability terciles 1 and 2. Third, within intensity tercile 1 the slope is concentrated in predictability tercile 1, the firms with the most predictable cash flows, and is indistinguishable from zero in predictability tercile 3. The model’s outside-knowledge channel implies a reverse pattern. Firms with more predictable cash flows have a smaller cash-flow innovation variance. Therefore, the investors know more about such a firm before any signal arrives. So the investors give less weight to the feed. Hence, the price responds less to the feed. That implies that the response of next-quarter gains to dispersion should be the weakest among the most predictable firms. Neither
of the model’s return channels (the volatility channel of Proposition 5 and the correction of surviving overpricing of Proposition 6 ) produces that effect.
Notes: Each cell is the slope of next-quarter total gains on dispersion, with the t t t t t t
We repeat the main regressions, each time replacing one variable with an alternative (row R10). We reconstruct the outcomes (losses and gains) from daily returns after removing the three Fama-French factors out of sample. We construct the terciles from the high-low spread instead of the quoted spread. Among the controls used, we have residual idiosyncratic volatility, filtered standard deviation of daily returns instead of realized volatility, sentiment noise from quarterly sentiment volatility (QSV) instead of VAQSV. We also cap the quoted spread at its adverse-selection bounds, standardize the noise within each group, and drop the first quarter of sentiment data. The ranking into terciles is 98.1 percent unchanged under the cap. For all these alternatives, the domestic-foreign difference maintains its sign and the terciles keep their order. The exception is when we use QSV, the slope for foreign stocks becomes positive and the sign of the difference reverses, but is not significant (+ 0.12 +0.12 + 0.12 t = 0.97 t = 0.97 t = 0.97
The tercile pattern of the reliability gap
This subsection gives the full discussion of the tercile pattern summarized in § 6.1 . The difference closes in the tercile where the contributor-pool composition of the two feeds differs most, since the domestic-foreign contrast in the repeat-author share of Table II is widest in tercile 3. The model ties
Notes: We re-estimate each variant from the headline estimate with the posting intensity set at 1.5. The comparison is therefore like with like. The mismatch parameters move by less than 10 − 5 10^{-5} 1 0 − 5 κ \kappa κ Table VI . The shrunk full-matrix weighting is the pre-specified robustness weighting. Under it, the repeat-author shares and the fad scales stay higher for foreign than for domestic.
Notes: We obtain the turning points by fitting, within each tercile, the excess of next-quarter losses over gains on measured noise and its square, with delta-method standard errors. The middle tercile turns left of the low tercile in both cases, which is consistent with the leftward shift the model predicts holding the other group primitives fixed. The high tercile is imprecise. These shape and ordering tests are the drift evidence under the pre-specified design.
Notes: The ratio is the cross-sectional variance of the quarterly accrual sum over the group mean of trading days times the squared daily standard deviation. Values below one mean the independence benchmark overstates the quarterly variance. The unwinsorized ratio is lower in each of the six groups, which rules out winsorization asymmetry as the cause. Measured first-order autocorrelation explains under six percentage points of the shortfall below one. Multi-lag within-quarter reversal is the remaining explanation and ranks correctly with illiquidity.
the size of the loss-on-noise slope to what investors know about a stock from sources other than the feed, the outside knowledge about the stock. The precision of investors’ outside knowledge about the stocks of a group (τ 0 \tau_0 τ 0 (12) ) is the sum of the prior precision (τ d \tau_d τ d τ p \tau_p τ p K j K_j K j G j G_j G j (7) ). The feed’s noise then matters less for the stock’s next-quarter losses (§ 3.5 ). The loss-on-noise slope is closer to zero. The share of quarters with a negative slope is blind to the magnitude of a nonzero slope. A slope that the model places near zero, however, has its sign decided by sampling error in each quarterly cross-section. Its share is then near one half. Thus, the model implies a lower share of quarters with a negative slope for the stocks about which investors have stronger outside knowledge. Which tercile holds those stocks depends on which part of τ 0 \tau_0 τ 0 (12) ). The ranking statistic of § 3.5 forms the terciles by that intensity inside each geography-quarter. Investors’ outside knowledge is then strongest about the stocks in tercile 3. Hence, the share of quarters with a negative slope declines from tercile 1 to tercile 3. However, at the headline estimates, the precision of outside knowledge is almost entirely the prior precision, which is largest for the stocks in tercile 1 and smallest for the stocks in tercile 3 in both geographies. Investors’ outside knowledge is then strongest about the stocks in tercile 1. Hence, the share of quarters with a negative slope rises from tercile 1 to tercile 3. Against the data, the two readings fare as follows. For foreign stocks the share rises at both steps, which agrees with the reading at the estimates at every step and with the intensity-only reading at none. For domestic stocks the share rises from tercile 1 to tercile 2, which agrees with the reading at the estimates, and is lowest in tercile 3, which agrees with the intensity-only reading. Neither reading accounts for the domestic pattern as a whole.
Secondary verification of composition
We calibrate the contributor composition against concentrations reported in independent Twitter samples. The exercise is descriptive, and the results in § 5 and § 6 do not depend on it. The external ratios agree with the ordering we assume in the dispersion identification, with specialists the most concentrated posters and attention-driven posters the least.
Bot prevalence over the sample period. In the platform’s own disclosures, false or spam accounts were below five percent of monetizable daily active users, its reported measure of its active user base. In an independent audit of September 2022, Similarweb found the same figure in its own account-level measurements, and attributed roughly twenty to thirty percent of U.S. content on the platform to bots (similarweb.com/corp/blog/insights/twitter-bot-research , accessed June 2026). These are industry measurements rather than peer-reviewed estimates, and detecting bots is subject to substantial classification error. In Alarfaj et al. [3 ] , the best content-feature classifiers are about seventy-seven percent accurate, and weaker combinations of classifier and features are below fifty percent.
We compare the repeat-author share (Proposition 1 ) with the posting concentration (the ratio of a group’s account share to its tweet share, the population-level analogue of dispersion) reported for the three contributor populations (specialists, attention-driven posters, and bots) in independent data. The ratio is small when a group posts heavily and near one when it posts once. We take these ratios from three sources. For specialists and attention-driven posters we use the experience field of Fallahgoul [34 ] . Self-reported professional users are 4.72 percent of users and 8.73 percent of messages (a ratio of 0.54). Novice users are 4.92 percent of users and 5.10 percent of messages (a ratio of 0.96). For bot accounts we use the ground-truth annotation of Gilani et al. [42 ] . Labeled bot accounts are 43.14 percent of accounts and 51.85 percent of statuses (a ratio of 0.83). We see the same in the per-account counts, with 305 tweets for bot accounts against 231 for human accounts.
After we normalize the bot ratio to one, the relative account-to-tweet-share ratios for specialists, bot accounts, and attention-driven posters are 0.65, 1.00 and 1.16, respectively. A feed can have higher noise even when a few informed users are posting. For validation, we use the account-to-tweet ratios from outside and use them to compare the direction of the difference in contributor composition behind foreign stocks’ higher nonfundamental variance (Φ j \Phi_j Φ j λ I \lambda_I λ I λ I \lambda_I λ I λ I \lambda_I λ I § 5 ). A common value of λ I \lambda_I λ I π j \pi_j π j
Proofs of the Propositions
Throughout, we fix a stock f f f j j j γ = r / ( r − υ ) \gamma= r/(r-\upsilon) γ = r / ( r − υ ) τ d = Σ j d − 1 \tau_d = \Sigma_{jd}^{-1} τ d = Σ j d − 1 K K K τ p \tau_p τ p τ s \tau_s τ s T = τ d + τ p + K + τ s T = \tau_d + \tau_p + K + \tau_s T = τ d + τ p + K + τ s Σ = 1 / T \Sigma= 1/T Σ = 1/ T τ 0 ≡ τ d + τ p + K \tau_0 \equiv\tau_d + \tau_p + K τ 0 ≡ τ d + τ p + K π S \pi_S π S π B \pi_B π B π = π S + π B \pi= \pi_S + \pi_B π = π S + π B λ I > 1 \lambda_I > 1 λ I > 1 ω S \omega_S ω S ω A \omega_A ω A ω B \omega_B ω B σ b , j 2 \sigma_{b,j}^{2} σ b , j 2 σ β , j 2 \sigma_{\beta,j}^{2} σ β , j 2 Φ j = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 \Phi_j = \omega_{A,j}^{2}\sigma_{b,j}^{2} + \omega_{B,j}^{2}\sigma_{\beta,j}^{2} Φ j = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 r r r
Proof of Proposition 1
Given the feed structure, on a day with A t A_t A t π S A t \pi_S A_t π S A t π B A t \pi_B A_t π B A t λ I \lambda_I λ I ( 1 − π S − π B ) A t (1-\pi_S-\pi_B)A_t ( 1 − π S − π B ) A t
N t = A t [ ( π S + π B ) λ I + ( 1 − π S − π B ) ] = A t [ π λ I + ( 1 − π ) ] , N_t = A_t\left[(\pi_S+\pi_B)\lambda_I+(1-\pi_S-\pi_B)\right] = A_t\left[\pi\lambda_I+(1-\pi)\right], N t = A t [ ( π S + π B ) λ I + ( 1 − π S − π B ) ] = A t [ π λ I + ( 1 − π ) ] , with π = π S + π B \pi=\pi_S+\pi_B π = π S + π B
D = A t N t = 1 π λ I + ( 1 − π ) = 1 1 + π ( λ I − 1 ) . (16) D = \frac{A_t}{N_t} = \frac{1}{\pi\lambda_I+(1-\pi)} = \frac{1}{1+\pi(\lambda_I-1)}. \tag*{(16)} D = N t A t = π λ I + ( 1 − π ) 1 = 1 + π ( λ I − 1 ) 1 . (16) which is deterministic given ( π , λ I ) (\pi,\lambda_I) ( π , λ I ) π \pi π λ I > 1 \lambda_I>1 λ I > 1 D ∈ ( 1 / λ I , 1 ] D\in(1/\lambda_I,1] D ∈ ( 1/ λ I , 1 ] π \pi π [ 0 , 1 ) [0,1) [ 0 , 1 ) 1 / D = 1 + π ( λ I − 1 ) 1/D=1+\pi(\lambda_I-1) 1/ D = 1 + π ( λ I − 1 ) π = ( 1 − D ) / [ D ( λ I − 1 ) ] \pi=(1-D)/[D(\lambda_I-1)] π = ( 1 − D ) / [ D ( λ I − 1 )] (2) , with π = 0 \pi=0 π = 0 D = 1 D=1 D = 1 D ≤ 1 / λ I D\leq1/\lambda_I D ≤ 1/ λ I π ≥ 1 \pi\geq1 π ≥ 1 π = 1 \pi=1 π = 1 λ S \lambda_S λ S λ B \lambda_B λ B D = [ 1 + π S ( λ S − 1 ) + π B ( λ B − 1 ) ] − 1 D=[1+\pi_S(\lambda_S-1)+\pi_B(\lambda_B-1)]^{-1} D = [ 1 + π S ( λ S − 1 ) + π B ( λ B − 1 ) ] − 1 π S ( λ S − 1 ) + π B ( λ B − 1 ) \pi_S(\lambda_S-1)+\pi_B(\lambda_B-1) π S ( λ S − 1 ) + π B ( λ B − 1 ) π S \pi_S π S π B \pi_B π B ω S = π S λ I / [ π λ I + 1 − π ] \omega_S=\pi_S\lambda_I/[\pi\lambda_I+1-\pi] ω S = π S λ I / [ π λ I + 1 − π ] ω B = π B λ I / [ π λ I + 1 − π ] \omega_B=\pi_B\lambda_I/[\pi\lambda_I+1-\pi] ω B = π B λ I / [ π λ I + 1 − π ] ω A = ( 1 − π ) / [ π λ I + 1 − π ] \omega_A=(1-\pi)/[\pi\lambda_I+1-\pi] ω A = ( 1 − π ) / [ π λ I + 1 − π ] ω = ω S + ω B = π λ I / [ π λ I + 1 − π ] \omega=\omega_S+\omega_B=\pi\lambda_I/[\pi\lambda_I+1-\pi] ω = ω S + ω B = π λ I / [ π λ I + 1 − π ] D D D 1 / A t 1/A_t 1/ A t A t A_t A t 41 ^{41} 41 D ‾ \overline{D} D
C.2. Proof of Proposition 2: We first record an averaging fact used throughout. The mean of n n n σ e 2 \sigma_e^2 σ e 2 σ e 2 / n \sigma_e^2/n σ e 2 / n n σ e 2 n\sigma_e^2 n σ e 2 n n n 1 / n 2 1/n^2 1/ n 2
Equation (3) gives the day-t t t
s ˉ t = ( 1 − ω B , j ) ϵ f 1 + ω A , j b t + ω B , j β t + u ˉ t , (17) \bar{s}_t = (1-\omega_{\mathrm{B},j})\epsilon_{f1}+\omega_{\mathrm{A},j}b_t+\omega_{\mathrm{B},j}\beta_t+\bar{u}_t, \tag*{(17)} s ˉ t = ( 1 − ω B , j ) ϵ f 1 + ω A , j b t + ω B , j β t + u ˉ t , (17) where u ˉ t \bar{u}_t u ˉ t σ e 2 \sigma_e^2 σ e 2 s ˉ t \bar{s}_t s ˉ t N t N_t N t u ˉ t \bar{u}_t u ˉ t N t N_t N t Var ( u ˉ t ) = σ e 2 / N t \operatorname{Var}(\bar{u}_t)=\sigma_e^2/N_t Var ( u ˉ t ) = σ e 2 / N t
Take the variance of (17) for a single day, conditional on the innovation ϵ f 1 \epsilon_{f1} ϵ f 1 b t b_t b t β t \beta_t β t u ˉ t \bar{u}_t u ˉ t
Var ( s ˉ t ∣ ϵ f 1 ) = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 + σ e 2 N t = Φ j + σ e 2 N t , (18) \operatorname{Var}(\bar{s}_t \mid\epsilon_{f1})=\omega_{\mathrm{A},j}^2\sigma_{b,j}^2+\omega_{\mathrm{B},j}^2\sigma_{\beta,j}^2+\frac{\sigma_e^2}{N_t}=\Phi_j+\frac{\sigma_e^2}{N_t}, \tag*{(18)} Var ( s ˉ t ∣ ϵ f 1 ) = ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 + N t σ e 2 = Φ j + N t σ e 2 , (18) where Φ j ≡ ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2 \Phi_j\equiv\omega_{\mathrm{A},j}^2\sigma_{b,j}^2+\omega_{\mathrm{B},j}^2\sigma_{\beta,j}^2 Φ j ≡ ω A , j 2 σ b , j 2 + ω B , j 2 σ β , j 2
The vendor’s series adds a weighting refinement to the sampling term. The series S t S_t S t s ˉ t = S t / N t \bar{s}_t=S_t/N_t s ˉ t = S t / N t N t N_t N t w m w_m w m σ e 2 ∑ m w m 2 / ( ∑ m w m ) 2 \sigma_e^2\sum_m w_m^2/(\sum_m w_m)^2 σ e 2 ∑ m w m 2 / ( ∑ m w m ) 2 c 1 , t σ e 2 / N t c_{1,t}\sigma_e^2/N_t c 1 , t σ e 2 / N t c 1 , t ≡ N t ∑ m w m 2 / ( ∑ m w m ) 2 ≥ 1 c_{1,t}\equiv N_t\sum_m w_m^2/(\sum_m w_m)^2\geq1 c 1 , t ≡ N t ∑ m w m 2 / ( ∑ m w m ) 2 ≥ 1 42 c 1 , t c_{1,t} c 1 , t c w c_w c w c w ( 1 / N t ) c_w(1/N_t) c w ( 1/ N t ) c 1 , t / N t c_{1,t}/N_t c 1 , t / N t c w c_w c w c w c_w c w σ e 2 \sigma_e^2 σ e 2 c w σ e 2 c_w\sigma_e^2 c w σ e 2 v t ≡ Φ j + c w σ e 2 / N t v_t\equiv\Phi_j+c_w\sigma_e^2/N_t v t ≡ Φ j + c w σ e 2 / N t v ˉ f q = T q − 1 ∑ t ∈ q v t = Φ j + c w σ e 2 ( 1 / N t ) \bar{v}_{fq}=T_q^{-1}\sum_{t\in q}v_t=\Phi_j+c_w\sigma_e^2(1/N_t) v ˉ f q = T q − 1 ∑ t ∈ q v t = Φ j + c w σ e 2 ( 1/ N t )
The measured VAQSVf q 2 ^2_{fq} f q 2 v ˉ f q \bar{v}_{fq} v ˉ f q d t ≡ s ˉ t − ( 1 − ω B , j ) ϵ f 1 d_t \equiv\bar{s}_t - (1-\omega_{\mathrm{B},j})\epsilon_{f1} d t ≡ s ˉ t − ( 1 − ω B , j ) ϵ f 1 t t t ϵ f 1 \epsilon_{f1} ϵ f 1 d t d_t d t Var ( d t ) = v t \operatorname{Var}(d_t)=v_t Var ( d t ) = v t
E [ V A Q S V f q 2 ∣ ϵ f 1 , { N t } ] = 1 T q ∑ t ∈ q v t − 2 T q ( T q − 1 ) ∑ t < t ′ Cov ( d t , d t ′ ) = v ˉ f q − 2 T q ( T q − 1 ) ∑ t < t ′ Cov ( d t , d t ′ ) . (19) \begin{aligned}
E\left[\mathrm{VAQSV}_{fq}^{2}\mid\epsilon_{f1},\{N_t\}\right] &= \frac{1}{T_q}\sum_{t\in q}v_t-\frac{2}{T_q(T_q-1)}\sum_{t<t'}\operatorname{Cov}(d_t,d_{t'})\\
&=\bar{v}_{fq}-\frac{2}{T_q(T_q-1)}\sum_{t<t'}\operatorname{Cov}(d_t,d_{t'}). \tag*{(19)}
\end{aligned} E [ VAQSV f q 2 ∣ ϵ f 1 , { N t } ] = T q 1 t ∈ q ∑ v t − T q ( T q − 1 ) 2 t < t ′ ∑ Cov ( d t , d t ′ ) = v ˉ f q − T q ( T q − 1 ) 2 t < t ′ ∑ Cov ( d t , d t ′ ) . (19) The within-quarter standard deviation is computed with the T q − 1 T_q-1 T q − 1 43 v ˉ f q = Φ j + c w σ e 2 ( 1 / N t ) \bar{v}_{fq}=\Phi_j+c_w\sigma_e^2(1/N_t) v ˉ f q = Φ j + c w σ e 2 ( 1/ N t ) (4) . Estimating (4) across firm-quarters by ordinary least squares of V A Q S V f q 2 \mathrm{VAQSV}_{fq}^{2} VAQSV f q 2 ( 1 / N t ) (1/N_t) ( 1/ N t ) Φ j \Phi_j Φ j c w σ e 2 c_w\sigma_e^2 c w σ e 2
A one-parameter version of Φ j \Phi_j Φ j σ β , j = 0 \sigma_{\beta,j}=0 σ β , j = 0 Φ j \Phi_j Φ j Φ j = ω A , j 2 σ b , j 2 \Phi_j=\omega_{\mathrm{A},j}^{2}\sigma_{b,j}^{2} Φ j = ω A , j 2 σ b , j 2 ω ≡ ω S , j + ω B , j \omega\equiv\omega_{\mathrm{S},j}+\omega_{\mathrm{B},j} ω ≡ ω S , j + ω B , j ω A , j = 1 − ω \omega_{\mathrm{A},j}=1-\omega ω A , j = 1 − ω Φ j = ( 1 − ω ) 2 σ b , j 2 \Phi_j=(1-\omega)^2\sigma_{b,j}^{2} Φ j = ( 1 − ω ) 2 σ b , j 2 § 3.2 . The two-term composite is the general object it restricts. The falsification battery probes departures from the attribution (see § 6 ).
The precision in (5) comes from the quarterly average of the daily series. Averaging (17) over the quarter’s T q T_q T q s = ( 1 − ω B , j ) ϵ f 1 + u s=(1-\omega_{\mathrm{B},j})\epsilon_{f1}+u s = ( 1 − ω B , j ) ϵ f 1 + u u = T q − 1 ∑ t ∈ q d t u=T_q^{-1}\sum_{t\in q}d_t u = T q − 1 ∑ t ∈ q d t
Var ( u ) = 1 T q 2 [ ∑ t ∈ q Var ( d t ) + ∑ t ≠ t ′ Cov ( d t , d t ′ ) ] = 1 T q [ v ˉ f q + 2 T q ∑ t < t ′ Cov ( d t , d t ′ ) ] . (20) \operatorname{Var}(u)=\frac{1}{T_q^2}\left[\sum_{t\in q}\operatorname{Var}(d_t)+\sum_{t\ne t'}\operatorname{Cov}(d_t,d_{t'})\right]=\frac{1}{T_q}\left[\bar{v}_{fq}+\frac{2}{T_q}\sum_{t<t'}\operatorname{Cov}(d_t,d_{t'})\right]. \tag*{(20)} Var ( u ) = T q 2 1 t ∈ q ∑ Var ( d t ) + t = t ′ ∑ Cov ( d t , d t ′ ) = T q 1 [ v ˉ f q + T q 2 t < t ′ ∑ Cov ( d t , d t ′ ) ] . (20) Define the dependence multiplier κ 0 , f q \kappa_{0,fq} κ 0 , f q v ˉ f q \bar{v}_{fq} v ˉ f q
Var ( u ) = κ 0 , f q v ˉ f q T q , κ 0 , f q = 1 + 2 T q v ˉ f q ∑ t < t ′ Cov ( d t , d t ′ ) ≥ 1 , (21) \operatorname{Var}(u)=\kappa_{0,fq}\frac{\bar{v}_{fq}}{T_q},\qquad\kappa_{0,fq}=1+\frac{2}{T_q\bar{v}_{fq}}\sum_{t<t'}\operatorname{Cov}(d_t,d_{t'})\ge1, \tag*{(21)} Var ( u ) = κ 0 , f q T q v ˉ f q , κ 0 , f q = 1 + T q v ˉ f q 2 t < t ′ ∑ Cov ( d t , d t ′ ) ≥ 1 , (21) which equals one under the independence of A1 and exceeds one under positive serial correlation.44 ϵ f 1 \epsilon_{f1} ϵ f 1 s s s ( 1 − ω B , j ) − 1 (1-\omega_{B,j})^{-1} ( 1 − ω B , j ) − 1
τ s = ( 1 − ω B , j ) 2 Var ( u ) = ( 1 − ω B , j ) 2 T q κ 0 , f q v ˉ f q . (22) \tau_s=\frac{(1-\omega_{B,j})^2}{\operatorname{Var}(u)}=\frac{(1-\omega_{B,j})^2T_q}{\kappa_{0,fq}\bar{v}_{fq}}. \tag*{(22)} τ s = Var ( u ) ( 1 − ω B , j ) 2 = κ 0 , f q v ˉ f q ( 1 − ω B , j ) 2 T q . (22) The population variance v ˉ f q \bar{v}_{fq} v ˉ f q (19) with (21) gives
E [ V A Q S V f q 2 ∣ ϵ f 1 , { N t } ] = v ˉ f q T q − κ 0 , f q T q − 1 , (23) E\left[\mathrm{VAQSV}_{fq}^{2}\mid\epsilon_{f1},\{N_t\}\right]=\bar{v}_{fq}\frac{T_q-\kappa_{0,fq}}{T_q-1}, \tag*{(23)} E [ VAQSV f q 2 ∣ ϵ f 1 , { N t } ] = v ˉ f q T q − 1 T q − κ 0 , f q , (23) so the realized statistic is unbiased for v ˉ f q \bar{v}_{fq} v ˉ f q κ 0 , f q = 1 \kappa_{0,fq}=1 κ 0 , f q = 1 κ 0 , f q \kappa_{0,fq} κ 0 , f q κ \kappa κ (5) , τ s = T q / ( κ V A Q S V f q 2 ) \tau_s=T_q/(\kappa\mathrm{VAQSV}_{fq}^{2}) τ s = T q / ( κ VAQSV f q 2 ) κ \kappa κ κ 0 , f q \kappa_{0,fq} κ 0 , f q (23) ), the bot-dilution factor ( 1 − ω B , j ) − 2 (1-\omega_{B,j})^{-2} ( 1 − ω B , j ) − 2 (17) ). These are not separately identified, and C.3 takes up which of them can differ across groups. □
The conversion constant and the bot share across groups
We compare the feed’s precision for domestic and foreign stocks and for the three terciles of informed-trading-intensity. The conversion factor (κ \kappa κ κ 0 , f q \kappa_{0,fq} κ 0 , f q ( 1 − ω B , j ) − 2 (1-\omega_{B,j})^{-2} ( 1 − ω B , j ) − 2 (3) ). κ \kappa κ ω B , j \omega_{\mathrm{B},j} ω B , j κ \kappa κ § 6 ).
We conduct sensitivity analysis to put a bound on the extent to which a systematic domestic-foreign difference in the bot share could reach while holding the other two components (the scale and the serial-dependence multiplier) common across the groups. At a common κ \kappa κ 1 − ω B , F 1-\omega_{\mathrm{B},F} 1 − ω B , F κ F \kappa_F κ F κ \kappa κ κ \kappa κ
Proof of Proposition 3
C.4. Proof of Proposition 3: Posterior beliefs. Conjecture the price function ((6) ). Investor i i i η i = ϵ + e i \eta^i=\epsilon+e^i η i = ϵ + e i k i k^i k i e i ∼ N ( 0 , 1 / k i ) e^i\sim N(0,1/k^i) e i ∼ N ( 0 , 1/ k i ) s = ϵ + u s=\epsilon+u s = ϵ + u u ∼ N ( 0 , τ s − 1 ) u\sim N(0,\tau_s^{-1}) u ∼ N ( 0 , τ s − 1 ) ( P , s ) (P, s) ( P , s ) ( ζ , s ) (\zeta, s) ( ζ , s )
ζ ≡ r P − A − G s B − G = ϵ + C B − G x ~ , (24) \zeta\equiv\frac{rP-A-Gs}{B-G}=\epsilon+\frac{C}{B-G}\tilde{x}, \tag*{(24)} ζ ≡ B − G r P − A − G s = ϵ + B − G C x ~ , (24) a signal about ϵ \epsilon ϵ τ p = [ ( B − G ) / C ] 2 Σ j x c − 1 \tau_p=[(B-G)/C]^2\Sigma_{jxc}^{-1} τ p = [( B − G ) / C ] 2 Σ j x c − 1 45 s s s s s s Σ i = ( τ d + τ p + k i + τ s ) − 1 \Sigma^i=(\tau_d+\tau_p+k^i+\tau_s)^{-1} Σ i = ( τ d + τ p + k i + τ s ) − 1 E i [ ϵ ] = Σ i ( τ p ζ + k i η i + τ s s ) E^i[\epsilon]=\Sigma^i(\tau_p\zeta+k^i\eta^i+\tau_ss) E i [ ϵ ] = Σ i ( τ p ζ + k i η i + τ s s )
Demands and market clearing. With V = γ ( v d 0 + ϵ ˉ + ϵ ) V=\gamma(vd_0+\bar{\epsilon}+\epsilon) V = γ ( v d 0 + ϵ ˉ + ϵ ) q i = ( E i [ V ] − r P ) / ( ρ ˉ γ 2 Σ i ) q^i=(E^i[V]-rP)/(\bar{\rho}\gamma^2\Sigma^i) q i = ( E i [ V ] − r P ) / ( ρ ˉ γ 2 Σ i ) 46 ∫ q i d i = x ˉ + x ~ \int q^i\,di=\bar{x}+\tilde{x} ∫ q i d i = x ˉ + x ~ ρ ˉ γ 2 \bar{\rho}\gamma^2 ρ ˉ γ 2 ∫ ( Σ i ) − 1 d i = Σ ˉ − 1 \int(\Sigma^i)^{-1}d_i=\bar{\Sigma}^{-1} ∫ ( Σ i ) − 1 d i = Σ ˉ − 1 ∫ k i η i d i = K ϵ \int k^i\eta^i\,di=K\epsilon ∫ k i η i d i = K ϵ
γ ( v d 0 + ϵ ˉ ) Σ ˉ − 1 + γ ( τ p ζ + K ϵ + τ s s ) − r P Σ ˉ − 1 = ρ ˉ γ 2 ( x ˉ + x ~ ) , (25) \gamma(vd_0+\bar{\epsilon})\bar{\Sigma}^{-1}+\gamma(\tau_p\zeta+K\epsilon+\tau_ss)-rP\bar{\Sigma}^{-1}=\bar{\rho}\gamma^2(\bar{x}+\tilde{x}), \tag*{(25)} γ ( v d 0 + ϵ ˉ ) Σ ˉ − 1 + γ ( τ p ζ + K ϵ + τ s s ) − r P Σ ˉ − 1 = ρ ˉ γ 2 ( x ˉ + x ~ ) , (25) so that
r P = γ ( v d 0 + ϵ ˉ ) + γ Σ ˉ ( τ p ζ + K ϵ + τ s s ) − ρ ˉ γ 2 Σ ˉ ( x ˉ + x ~ ) . (26) rP=\gamma(vd_0+\bar{\epsilon})+\gamma\bar{\Sigma}(\tau_p\zeta+K\epsilon+\tau_ss)-\bar{\rho}\gamma^2\bar{\Sigma}(\bar{x}+\tilde{x}). \tag*{(26)} r P = γ ( v d 0 + ϵ ˉ ) + γ Σ ˉ ( τ p ζ + K ϵ + τ s s ) − ρ ˉ γ 2 Σ ˉ ( x ˉ + x ~ ) . (26) Substituting ζ = ϵ + C B − G x ~ \zeta=\epsilon+\frac{C}{B-G}\tilde{x} ζ = ϵ + B − G C x ~ s = ϵ + u s=\epsilon+u s = ϵ + u
A = γ ( v d 0 + ϵ ˉ ) − ρ ˉ γ 2 Σ ˉ x ˉ , (27) A=\gamma(vd_0+\bar{\epsilon})-\bar{\rho}\gamma^2\bar{\Sigma}\bar{x}, \tag*{(27)} A = γ ( v d 0 + ϵ ˉ ) − ρ ˉ γ 2 Σ ˉ x ˉ , (27) B = γ Σ ˉ ( τ p + K + τ s ) = γ ( 1 − Σ ˉ τ d ) = γ ( 1 − Σ ˉ / Σ j d ) , (28) B=\gamma\bar{\Sigma}(\tau_p+K+\tau_s)=\gamma(1-\bar{\Sigma}\tau_d)=\gamma(1-\bar{\Sigma}/\Sigma_{jd}), \tag*{(28)} B = γ Σ ˉ ( τ p + K + τ s ) = γ ( 1 − Σ ˉ τ d ) = γ ( 1 − Σ ˉ / Σ j d ) , (28) G = γ Σ ˉ τ s , (29) G=\gamma\bar{\Sigma}\tau_s, \tag*{(29)} G = γ Σ ˉ τ s , (29) C = γ Σ ˉ τ p C B − G − ρ ˉ γ 2 Σ ˉ . (30) C=\gamma\bar{\Sigma}\tau_p\frac{C}{B-G}-\bar{\rho}\gamma^2\bar{\Sigma}. \tag*{(30)} C = γ Σ ˉ τ p B − G C − ρ ˉ γ 2 Σ ˉ . (30) From (28) and (29) , B − G = γ Σ ˉ ( τ p + K ) > 0 B-G=\gamma\bar{\Sigma}(\tau_p+K)>0 B − G = γ Σ ˉ ( τ p + K ) > 0 γ Σ ˉ τ p / ( B − G ) = τ p / ( τ p + K ) \gamma\bar{\Sigma}\tau_p/(B-G)=\tau_p/(\tau_p+K) γ Σ ˉ τ p / ( B − G ) = τ p / ( τ p + K ) (30) becomes C K / ( τ p + K ) = − ρ ˉ γ 2 Σ ˉ CK/(\tau_p+K)=-\bar{\rho}\gamma^2\bar{\Sigma} C K / ( τ p + K ) = − ρ ˉ γ 2 Σ ˉ
C = − ρ ˉ γ 2 Σ ˉ τ p + K K < 0. (31) C=-\bar{\rho}\gamma^2\bar{\Sigma}\frac{\tau_p+K}{K}<0. \tag*{(31)} C = − ρ ˉ γ 2 Σ ˉ K τ p + K < 0. (31) Then ( B − G ) / C = − K / ( ρ ˉ γ ) (B-G)/C=-K/(\bar{\rho}\gamma) ( B − G ) / C = − K / ( ρ ˉ γ )
τ p = ( B − G C ) 2 Σ ˉ j x − 1 = K 2 γ 2 ρ ˉ 2 Σ ˉ j x , (32) \tau_p=\left(\frac{B-G}{C}\right)^2\bar{\Sigma}_{jx}^{-1}=\frac{K^2}{\gamma^2\bar{\rho}^2\bar{\Sigma}_{jx}}, \tag*{(32)} τ p = ( C B − G ) 2 Σ ˉ j x − 1 = γ 2 ρ ˉ 2 Σ ˉ j x K 2 , (32) a closed form free of τ s \tau_s τ s τ p \tau_p τ p Σ ˉ = ( τ d + τ p + K + τ s ) − 1 \bar{\Sigma}=(\tau_d+\tau_p+K+\tau_s)^{-1} Σ ˉ = ( τ d + τ p + K + τ s ) − 1 (27) through (31) deliver all coefficients uniquely within the linear class, establishing existence and the stated independence of τ p \tau_p τ p τ s \tau_s τ s
Moments of the quarterly excess return. With R = V − r P R=V-rP R = V − r P (27) , R = ( γ − B ) ϵ − G u − C x ~ + ρ ˉ γ 2 Σ ˉ x ˉ R=(\gamma-B)\epsilon-Gu-C\tilde{x}+\bar{\rho}\gamma^2\bar{\Sigma}\bar{x} R = ( γ − B ) ϵ − G u − C x ~ + ρ ˉ γ 2 Σ ˉ x ˉ γ − B = γ Σ ˉ τ d \gamma-B=\gamma\bar{\Sigma}\tau_d γ − B = γ Σ ˉ τ d E [ R ] = ρ ˉ γ 2 Σ ˉ x ˉ E[R]=\bar{\rho}\gamma^2\bar{\Sigma}\bar{x} E [ R ] = ρ ˉ γ 2 Σ ˉ x ˉ
Var [ R ] = γ 2 Σ ˉ 2 τ d 2 Σ j d + γ 2 Σ ˉ 2 τ s 2 Σ j s − 1 + C 2 Σ j x = γ 2 Σ ˉ 2 ( τ d + τ s ) + C 2 Σ j x . (33) \operatorname{Var}[R]=\gamma^2\bar{\Sigma}^2\tau_d^2\Sigma_{jd}+\gamma^2\bar{\Sigma}^2\tau_s^2\Sigma_{js}^{-1}+C^2\Sigma_{jx}=\gamma^2\bar{\Sigma}^2(\tau_d+\tau_s)+C^2\Sigma_{jx}. \tag*{(33)} Var [ R ] = γ 2 Σ ˉ 2 τ d 2 Σ j d + γ 2 Σ ˉ 2 τ s 2 Σ j s − 1 + C 2 Σ j x = γ 2 Σ ˉ 2 ( τ d + τ s ) + C 2 Σ j x . (33) Using (31) and Σ j x = K 2 / ( γ 2 ρ ˉ 2 τ p ) \Sigma_{jx}=K^2/(\gamma^2\bar{\rho}^2\tau_p) Σ j x = K 2 / ( γ 2 ρ ˉ 2 τ p ) (32) , C 2 Σ j x = γ 2 Σ ˉ 2 ( τ p + K ) 2 / τ p C^2\Sigma_{jx}=\gamma^2\bar{\Sigma}^2(\tau_p+K)^2/\tau_p C 2 Σ j x = γ 2 Σ ˉ 2 ( τ p + K ) 2 / τ p (9) .
Monotonicity in τ s \tau_s τ s Write Var [ R ] = γ 2 N ( τ s ) / T ( τ s ) 2 \operatorname{Var}[R]=\gamma^2N(\tau_s)/T(\tau_s)^2 Var [ R ] = γ 2 N ( τ s ) / T ( τ s ) 2 N = τ d + τ s + ( τ p + K ) 2 / τ p N=\tau_d+\tau_s+(\tau_p+K)^2/\tau_p N = τ d + τ s + ( τ p + K ) 2 / τ p T = τ 0 + τ s T=\tau_0+\tau_s T = τ 0 + τ s τ s \tau_s τ s (32) . Then ∂ Var [ R ] / ∂ τ s = γ 2 ( T − 2 N ) / T 3 \partial\operatorname{Var}[R]/\partial\tau_s=\gamma^2(T-2N)/T^3 ∂ Var [ R ] / ∂ τ s = γ 2 ( T − 2 N ) / T 3 ( τ p + K ) 2 / τ p ≥ τ p + K (\tau_p+K)^2/\tau_p\ge\tau_p+K ( τ p + K ) 2 / τ p ≥ τ p + K 47
T − 2 N = ( τ p + K ) − τ d − τ s − 2 ( τ p + K ) 2 τ p ≤ − ( τ d + τ s + τ p + K ) = − T < 0 , (34) T-2N=(\tau_p+K)-\tau_d-\tau_s-2\frac{(\tau_p+K)^2}{\tau_p}\le-(\tau_d+\tau_s+\tau_p+K)=-T<0, \tag*{(34)} T − 2 N = ( τ p + K ) − τ d − τ s − 2 τ p ( τ p + K ) 2 ≤ − ( τ d + τ s + τ p + K ) = − T < 0 , (34) so Var [ R ] \operatorname{Var}[R] Var [ R ] τ s \tau_s τ s □ \square □
Proof of Proposition 4
The asset’s earnings stream from the pricing date q q q E f , q + s = υ s d f 1 ∗ E_{f,q+s}=\upsilon^sd_{f1}^* E f , q + s = υ s d f 1 ∗ s ≥ 0 s\ge0 s ≥ 0 V f 1 ∗ = γ d f 1 ∗ V_{f1}^*=\gamma d_{f1}^* V f 1 ∗ = γ d f 1 ∗ ∑ s ≥ 0 ( υ / r ) s = r / ( r − υ ) \sum_{s\ge0}(\upsilon/r)^s=r/(r-\upsilon) ∑ s ≥ 0 ( υ / r ) s = r / ( r − υ ) υ s ϵ \upsilon^s\epsilon υ s ϵ ( B ε + G u + C x ~ ) / r (B\varepsilon+ Gu + C\tilde{x})/r ( B ε + G u + C x ~ ) / r d 0 d_0 d 0
Cov ( E f , q + s , P f 1 ∗ ) = υ s B r Σ j d = υ s 1 − Σ ‾ / Σ j d r − υ Σ j d , (35) \operatorname{Cov}(E_{f,q+s}, P_{f1}^{*}) = \upsilon^{s} \frac{B}{r}\Sigma_{jd} = \upsilon^{s}\frac{1-\overline{\Sigma}/\Sigma_{jd}}{r-\upsilon}\Sigma_{jd}, \tag*{(35)} Cov ( E f , q + s , P f 1 ∗ ) = υ s r B Σ j d = υ s r − υ 1 − Σ / Σ j d Σ j d , (35) using B / r = ( 1 − Σ ‾ / Σ j d ) / ( r − υ ) B/r=(1-\overline{\Sigma}/\Sigma_{jd})/(r-\upsilon) B / r = ( 1 − Σ / Σ j d ) / ( r − υ ) (28) . Dividing by SD ( P f 1 ∗ ) \operatorname{SD}(P_{f1}^{*}) SD ( P f 1 ∗ ) (10) . For the information component, 1 − Σ ‾ / Σ j d = ( τ p + K + τ s ) / T 48 1-\overline{\Sigma}/\Sigma_{jd}=(\tau_p+K+\tau_s)/T^{48} 1 − Σ / Σ j d = ( τ p + K + τ s ) / T 48
∂ ∂ τ s τ p + K + τ s T = T − ( τ p + K + τ s ) T 2 = τ d T 2 > 0 , (36) \frac{\partial}{\partial\tau_s}\frac{\tau_p+K+\tau_s}{T}=\frac{T-(\tau_p+K+\tau_s)}{T^2}=\frac{\tau_d}{T^2}>0, \tag*{(36)} ∂ τ s ∂ T τ p + K + τ s = T 2 T − ( τ p + K + τ s ) = T 2 τ d > 0 , (36) so the information component is strictly increasing in τ s \tau_s τ s
Proof of Proposition 5
For X ∼ N ( μ , σ 2 ) X\sim N(\mu,\sigma^2) X ∼ N ( μ , σ 2 ) z = ( x − μ ) / σ z=(x-\mu)/\sigma z = ( x − μ ) / σ
∫ a ∞ z ϕ ( z ) d z = ϕ ( a ) , \int_a^\infty z\phi(z)\,\mathrm{d}z=\phi(a), ∫ a ∞ z ϕ ( z ) d z = ϕ ( a ) , E [ max ( X , 0 ) ] = μ Φ ( μ σ ) + σ ϕ ( μ σ ) , E [ min ( X , 0 ) ] = μ Φ ( − μ σ ) − σ ϕ ( μ σ ) , (37) E[\max(X,0)]=\mu\Phi\left(\frac{\mu}{\sigma}\right)+\sigma\phi\left(\frac{\mu}{\sigma}\right), \qquad E[\min(X,0)]=\mu\Phi\left(-\frac{\mu}{\sigma}\right)-\sigma\phi\left(\frac{\mu}{\sigma}\right), \tag*{(37)} E [ max ( X , 0 )] = μ Φ ( σ μ ) + σ ϕ ( σ μ ) , E [ min ( X , 0 )] = μ Φ ( − σ μ ) − σ ϕ ( σ μ ) , (37) whose sum is μ \mu μ T d a y s T_{\mathrm{days}} T days E [ G a i n ] − E [ ∣ L o s s ∣ ] = T d a y s μ d E[\mathrm{Gain}]-E[|\mathrm{Loss}|]=T_{\mathrm{days}}\mu_d E [ Gain ] − E [ ∣ Loss ∣ ] = T days μ d (37) with respect to σ \sigma σ ϕ ′ ( z ) = − z ϕ ( z ) \phi'(z)=-z\phi(z) ϕ ′ ( z ) = − z ϕ ( z )
∂ E [ G a i n ] ∂ σ d = ∂ E [ ∣ L o s s ∣ ] ∂ σ d = T d a y s ϕ ( μ d σ d ) > 0 , (38) \frac{\partial E[\mathrm{Gain}]}{\partial\sigma_d}=\frac{\partial E[|\mathrm{Loss}|]}{\partial\sigma_d}=T_{\mathrm{days}}\phi\left(\frac{\mu_d}{\sigma_d}\right)>0, \tag*{(38)} ∂ σ d ∂ E [ Gain ] = ∂ σ d ∂ E [ ∣ Loss ∣ ] = T days ϕ ( σ d μ d ) > 0 , (38) so both tails rise with daily volatility at the same rate. Differentiating (37) with respect to μ \mu μ ∂ E [ G a i n ] / ∂ μ d = T d a y s Φ ( μ d / σ d ) \partial E[\mathrm{Gain}]/\partial\mu_d=T_{\mathrm{days}}\Phi(\mu_d/\sigma_d) ∂ E [ Gain ] / ∂ μ d = T days Φ ( μ d / σ d ) ∂ E [ ∣ L o s s ∣ ] / ∂ μ d = − T d a y s Φ ( − μ d / σ d ) \partial E[|\mathrm{Loss}|]/\partial\mu_d=-T_{\mathrm{days}}\Phi(-\mu_d/\sigma_d) ∂ E [ ∣ Loss ∣ ] / ∂ μ d = − T days Φ ( − μ d / σ d ) μ d \mu_d μ d T d a y s σ d 2 = Var [ R ] T_{\mathrm{days}}\sigma_d^2=\operatorname{Var}[R] T days σ d 2 = Var [ R ] Var [ R ] \operatorname{Var}[R] Var [ R ] τ s \tau_s τ s μ d \mu_d μ d E [ G a i n ] E[\mathrm{Gain}] E [ Gain ] E [ ∣ L o s s ∣ ] E[|\mathrm{Loss}|] E [ ∣ Loss ∣ ] τ s \tau_s τ s
Proof of Proposition 6
Under A3(ii), the part of the optimistic price impact that survives the quarter is δ G max ( u , 0 ) \delta G\max(u,0) δ G max ( u , 0 ) V f 1 ∗ V^{*}_{f1} V f 1 ∗
O P = δ G E [ max ( u , 0 ) ] = δ G σ u 2 π = δ γ Σ ‾ τ s τ s − 1 / 2 2 π = δ γ 2 π τ s τ 0 + τ s . (39) \mathrm{OP}=\delta G E[\max(u,0)]=\delta G\frac{\sigma_u}{\sqrt{2\pi}}=\frac{\delta\gamma\sqrt{\overline{\Sigma}}\tau_s\tau_s^{-1/2}}{\sqrt{2\pi}}=\frac{\delta\gamma}{\sqrt{2\pi}}\frac{\sqrt{\tau_s}}{\tau_0+\tau_s}. \tag*{(39)} OP = δ GE [ max ( u , 0 )] = δ G 2 π σ u = 2 π δ γ Σ τ s τ s − 1/2 = 2 π δ γ τ 0 + τ s τ s . (39) The first equality takes the mean of the surviving component, with the constants δ \delta δ G G G (37) at μ = 0 \mu=0 μ = 0 μ Φ ( μ / σ ) \mu\Phi(\mu/\sigma) μ Φ ( μ / σ ) σ ϕ ( μ / σ ) \sigma\phi(\mu/\sigma) σ ϕ ( μ / σ ) σ u ϕ ( 0 ) = σ u / 2 π \sigma_u\phi(0)=\sigma_u/\sqrt{2\pi} σ u ϕ ( 0 ) = σ u / 2 π ϕ ( 0 ) = 1 / 2 π \phi(0)=1/\sqrt{2\pi} ϕ ( 0 ) = 1/ 2 π σ u = τ s − 1 / 2 \sigma_u=\tau_s^{-1/2} σ u = τ s − 1/2 u u u G = γ Σ ‾ τ s G=\gamma\sqrt{\overline{\Sigma}}\tau_s G = γ Σ τ s (7) . The fourth uses Σ ‾ = 1 / ( τ 0 + τ s ) \overline{\Sigma}=1/(\tau_0+\tau_s) Σ = 1/ ( τ 0 + τ s ) τ s τ s − 1 / 2 = τ s \tau_s\tau_s^{-1/2}=\sqrt{\tau_s} τ s τ s − 1/2 = τ s (11) , and μ d r i f t = − O P \mu^{drift}=-\mathrm{OP} μ d r i f t = − OP O P = ( δ γ / 2 π ) f ( τ s ) \mathrm{OP}=(\delta\gamma/\sqrt{2\pi})f(\tau_s) OP = ( δ γ / 2 π ) f ( τ s ) f ( τ s ) = τ s / ( τ 0 + τ s ) f(\tau_s)=\sqrt{\tau_s}/(\tau_0+\tau_s) f ( τ s ) = τ s / ( τ 0 + τ s )
f ′ ( τ s ) = τ 0 − τ s 2 τ s ( τ 0 + τ s ) 2 . (40) f'(\tau_s)=\frac{\tau_0-\tau_s}{2\sqrt{\tau_s}(\tau_0+\tau_s)^2}. \tag*{(40)} f ′ ( τ s ) = 2 τ s ( τ 0 + τ s ) 2 τ 0 − τ s . (40) The derivative is positive for τ s < τ 0 \tau_s<\tau_0 τ s < τ 0 τ s = τ 0 \tau_s=\tau_0 τ s = τ 0 τ s > τ 0 \tau_s>\tau_0 τ s > τ 0 f f f τ s → 0 \tau_s\to0 τ s → 0 τ s → ∞ \tau_s\to\infty τ s → ∞ τ s ∗ = τ 0 \tau_s^*=\tau_0 τ s ∗ = τ 0 (12) .
In measured-noise units, substitute τ s = T q / ( κ w 2 ) \tau_s=T_q/(\kappa w^2) τ s = T q / ( κ w 2 ) (5) with w = V A Q S V f q w=\mathrm{VAQSV}_{fq} w = VAQSV f q a ≡ T q / κ a\equiv T_q/\kappa a ≡ T q / κ
O P ( w ) = δ γ a 2 π w τ 0 w 2 + a , ∂ O P ∂ w = δ γ a 2 π a − τ 0 w 2 ( τ 0 w 2 + a ) 2 . (41) \mathrm{OP}(w)=\frac{\delta\gamma\sqrt{a}}{\sqrt{2\pi}}\frac{w}{\tau_0w^2+a},\qquad
\frac{\partial\mathrm{OP}}{\partial w}=\frac{\delta\gamma\sqrt{a}}{\sqrt{2\pi}}\frac{a-\tau_0w^2}{(\tau_0w^2+a)^2}. \tag*{(41)} OP ( w ) = 2 π δ γ a τ 0 w 2 + a w , ∂ w ∂ OP = 2 π δ γ a ( τ 0 w 2 + a ) 2 a − τ 0 w 2 . (41) The peak in w w w w ∗ = a / τ 0 = T q / ( κ τ 0 ) w^*=\sqrt{a/\tau_0}=\sqrt{T_q/(\kappa\tau_0)} w ∗ = a / τ 0 = T q / ( κ τ 0 ) τ 0 \tau_0 τ 0 T q T_q T q κ \kappa κ K j K_j K j τ 0 = τ d + K + K 2 / ( γ 2 ρ ˉ 2 Σ j x ) \tau_0=\tau_d+K+K^2/(\gamma^2\bar{\rho}^{\,2}\Sigma_{jx}) τ 0 = τ d + K + K 2 / ( γ 2 ρ ˉ 2 Σ j x ) K K K (32) .49 w ∗ w^* w ∗ T q / κ \sqrt{T_q/\kappa} T q / κ T q T_q T q κ \kappa κ δ G u 1 { u > 0 } \delta G u 1\{u>0\} δ G u 1 { u > 0 }
The symmetric-mechanism alternative and the two predictions
§ 3.4 states the two predictions and the symmetric mechanism that is the alternative to the model. In each quarter the noise u f 1 u_{f1} u f 1 q + 1 q+1 q + 1 q + 1 q+1 q + 1 q q q q + 1 q+1 q + 1
§ 3.4 states the first prediction, about shape, and why a symmetric mechanism produces no interior peak, and the second prediction, about sign. The optimistic case is a positive realization of the quarter q q q u f 1 u_{f1} u f 1 G j u f 1 G_j u_{f1} G j u f 1 (6) . This impact is a part of the quarter q q q V f 1 ∗ V^*_{f1} V f 1 ∗ q q q Gain f q \operatorname{Gain}_{fq} Gain f q ∣ Loss f q ∣ |\operatorname{Loss}_{fq}| ∣ Loss f q ∣ ∣ Loss f q ∣ − Gain f q |\operatorname{Loss}_{fq}|-\operatorname{Gain}_{fq} ∣ Loss f q ∣ − Gain f q q q q G j u f 1 G_j u_{f1} G j u f 1 q q q q q q A 3 ( i i ) A3(ii) A 3 ( ii ) δ \delta δ
At the end of quarter ( q ) (q) ( q ) V f 1 ∗ V^*_{f1} V f 1 ∗ δ G j u f 1 \delta G_j u_{f1} δ G j u f 1 V f 1 ∗ V^*_{f1} V f 1 ∗ ( q + 1 ) (q+1) ( q + 1 ) ( q + 1 ) (q+1) ( q + 1 ) δ G j u f 1 \delta G_j u_{f1} δ G j u f 1 T d a y s T_{days} T d a y s A 2 A2 A 2 Gain f , q + 1 \operatorname{Gain}_{f,q+1} Gain f , q + 1 ∣ Loss f , q + 1 ∣ \lvert\operatorname{Loss}_{f,q+1}\rvert ∣ Loss f , q + 1 ∣ ∣ Loss f , q + 1 ∣ − Gain f , q + 1 \lvert\operatorname{Loss}_{f,q+1}\rvert-\operatorname{Gain}_{f,q+1} ∣ Loss f , q + 1 ∣ − Gain f , q + 1
Quarter q + 1 q+1 q + 1 q + 1 q+1 q + 1 Gain f , q + 1 \operatorname{Gain}_{f,q+1} Gain f , q + 1 ∣ Loss f , q + 1 ∣ \lvert\operatorname{Loss}_{f,q+1}\rvert ∣ Loss f , q + 1 ∣ q q q q + 1 q+1 q + 1 q q q q q q q + 1 q+1 q + 1 q + 1 q+1 q + 1 V A Q S V f q VAQSV_{fq} V A QS V f q
In the tests, the share of positive-sentiment days among quarter q q q q q q § 6 ).
The daily dealer market and the liquidity vector
The informed-trading intensity K j K_j K j 50 K j K_j K j 51 [43 , 55 ] . The dealer’s exposure is the variance of the informed investor’s estimate of the payoff-relevant value v = γ j ϵ f 1 v = \gamma_j \epsilon_{f1} v = γ j ϵ f 1
σ I , j 2 = Var ( E [ v ∣ I i n f ] ∣ I d e a l e r ) = γ j 2 K j ( τ d + τ s ) ( τ d + τ s + K j ) . (42) \sigma_{I,j}^{2} = \operatorname{Var}\left(E[v \mid\mathcal{I}_{\mathrm{inf}}] \mid\mathcal{I}_{\mathrm{dealer}}\right) = \gamma_j^{2}\frac{K_j}{(\tau_d+\tau_s)(\tau_d+\tau_s+K_j)}. \tag*{(42)} σ I , j 2 = Var ( E [ v ∣ I inf ] ∣ I dealer ) = γ j 2 ( τ d + τ s ) ( τ d + τ s + K j ) K j . (42) The exposure rises with the private precision K j K_j K j τ s \tau_s τ s [55 ] , the informed investor’s optimal position and the dealer’s break-even condition tie the spread and the market-impact cost to this exposure and to the amount of noise trading, measured by its standard deviation σ x , j \sigma_{x,j} σ x , j σ x , j 2 = Σ j x \sigma_{x,j}^{2} = \Sigma_{jx} σ x , j 2 = Σ j x
Proposition 7 (The liquidity vector. ) With noise-trading standard deviation σ x , j \sigma_{x,j} σ x , j λ j = σ I , j / ( 2 σ x , j ) \lambda_j = \sigma_{I,j}/(2\sigma_{x,j}) λ j = σ I , j / ( 2 σ x , j ) impact j \operatorname{impact}_j impact j σ I , j \sigma_{I,j} σ I , j
spread j = a s σ I , j , impact j = a λ σ I , j σ x , j , turnover j = a T σ x , j , (43) \operatorname{spread}_j = a_s \sigma_{I,j}, \qquad\operatorname{impact}_j = a_\lambda\frac{\sigma_{I,j}}{\sigma_{x,j}}, \qquad\operatorname{turnover}_j = a_T \sigma_{x,j}, \tag*{(43)} spread j = a s σ I , j , impact j = a λ σ x , j σ I , j , turnover j = a T σ x , j , (43) The positive constants a s a_s a s a λ a_\lambda a λ a T a_T a T a T = 1 a_T = 1 a T = 1 σ x , j \sigma_{x,j} σ x , j a s a_s a s a λ a_\lambda a λ § 5 ). Hence the spread increases in K j K_j K j τ s \tau_s τ s Σ j x \Sigma_{jx} Σ j x
price-impact ratio jointly invert to ( σ I , j , σ x , j ) (\sigma_{I,j}, \sigma_{x,j}) ( σ I , j , σ x , j )
K j = σ I , j 2 ( τ d + τ s ) 2 γ j 2 − σ I , j 2 ( τ d + τ s ) (44) K_j = \frac{\sigma_{I,j}^{2}(\tau_d+\tau_s)^2}{\gamma_j^2-\sigma_{I,j}^{2}(\tau_d+\tau_s)} \tag*{(44)} K j = γ j 2 − σ I , j 2 ( τ d + τ s ) σ I , j 2 ( τ d + τ s ) 2 (44) recovers informed-trading intensity from the liquidity vector, with turnover over-identifying σ x , j \sigma_{x,j} σ x , j
We refer to the order flow from traders who trade for reasons unrelated to the firm’s value, such as liquidity needs, as noise trading. The spread and the price-impact ratio identify the two unknowns, the informed investor’s advantage ( σ I , j ) (\sigma_{I,j}) ( σ I , j ) ( σ x , j ) (\sigma_{x,j}) ( σ x , j ) σ I , j \sigma_{I,j} σ I , j σ I , j \sigma_{I,j} σ I , j σ x , j \sigma_{x,j} σ x , j σ I , j \sigma_{I,j} σ I , j K j K_j K j (44) , A.1, A.2). The identification carries one caveat. The spread’s independence from the amount of noise trading is a property of the Kyle batch market. Thus, we treat the recovery as model-based and report over-identifying checks (A.2).
C.10 derives the dealer’s exposure (equation (42) ) with its comparative statics in K j K_j K j τ s \tau_s τ s I d e a l e r \mathcal{I}_{dealer} I d e a l er ϵ f 1 \epsilon_{f1} ϵ f 1 I d e a l e r \mathcal{I}_{dealer} I d e a l er I i n f \mathcal{I}_{inf} I in f τ p \tau_p τ p τ p \tau_p τ p σ I , j \sigma_{I,j} σ I , j K j K_j K j τ s \tau_s τ s j j j σ x , j \sigma_{x,j} σ x , j σ ^ x , j = Σ j x \widehat{\sigma}_{x,j}=\Sigma_{jx} σ x , j = Σ j x
The level of K ^ \widehat{K} K K ^ \widehat{K} K τ p = K j 2 / ( γ j 2 ρ ˉ 2 Σ j x ) \tau_p = K_j^2/(\gamma_j^2\bar{\rho}^2\Sigma_{jx}) τ p = K j 2 / ( γ j 2 ρ ˉ 2 Σ j x ) (8) depends on K j K_j K j γ j \gamma_j γ j Σ j x \Sigma_{jx} Σ j x K ^ \widehat{K} K K j K_j K j
Proof of Proposition 7
Proof of Proposition 7. Throughout this proof the group index j j j γ \gamma γ K K K τ s \tau_s τ s σ I \sigma_I σ I λ \lambda λ j j j γ j \gamma_j γ j K j K_j K j τ s \tau_s τ s σ I , j \sigma_{I,j} σ I , j λ j \lambda_j λ j Informational disadvantage. In the daily dealer market the dealer conditions on the prior and the public signal, so the dealer’s conditional variance of the asset value is γ 2 / ( τ d + τ s ) \gamma^2/(\tau_d+\tau_s) γ 2 / ( τ d + τ s ) K K K γ 2 / ( τ d + τ s + K ) \gamma^2/(\tau_d+\tau_s+K) γ 2 / ( τ d + τ s + K ) 52
σ I 2 = Var ( E [ v ∣ I i n f ] ∣ I d e a l e r ) = γ 2 τ d + τ s − γ 2 τ d + τ s + K = γ 2 K ( τ d + τ s ) ( τ d + τ s + K ) . (45) \sigma_I^2 = \operatorname{Var}(E[v \mid I_{\mathrm{inf}}] \mid I_{\mathrm{dealer}}) = \frac{\gamma^2}{\tau_d+\tau_s} - \frac{\gamma^2}{\tau_d+\tau_s+K} = \frac{\gamma^2 K}{(\tau_d+\tau_s)(\tau_d+\tau_s+K)}. \tag*{(45)} σ I 2 = Var ( E [ v ∣ I inf ] ∣ I dealer ) = τ d + τ s γ 2 − τ d + τ s + K γ 2 = ( τ d + τ s ) ( τ d + τ s + K ) γ 2 K . (45) equation (42) , strictly increasing in K K K γ 2 / ( τ d + τ s + K ) 2 > 0 \gamma^2/(\tau_d+\tau_s+K)^2 > 0 γ 2 / ( τ d + τ s + K ) 2 > 0 τ s \tau_s τ s τ p u b \tau_{\mathrm{pub}} τ pub τ d + τ s \tau_d+\tau_s τ d + τ s τ d + τ s + τ p u b \tau_d+\tau_s+\tau_{\mathrm{pub}} τ d + τ s + τ pub K K K τ s \tau_s τ s K ^ \widehat{K} K τ ^ p \widehat{\tau}_p τ p K ^ \widehat{K} K K K K τ d + τ ^ s \sqrt{\tau_d+\widehat{\tau}_s} τ d + τ s K ^ \widehat{K} K § 5 ).
Kyle equilibrium. Let m ≡ E [ v ∣ I i n f ] m \equiv E[v \mid I_{\mathrm{inf}}] m ≡ E [ v ∣ I inf ] Var ( m ∣ I d e a l e r ) = σ I 2 \operatorname{Var}(m \mid I_{\mathrm{dealer}}) = \sigma_I^2 Var ( m ∣ I dealer ) = σ I 2 x n ∼ N ( 0 , σ x 2 ) x_n \sim N(0,\sigma_x^2) x n ∼ N ( 0 , σ x 2 ) p = μ D + λ q p = \mu_D+\lambda q p = μ D + λ q ( m − μ D − λ q ) q (m-\mu_D-\lambda q)q ( m − μ D − λ q ) q q i n f = ( m − μ D ) / ( 2 λ ) ≡ ψ ( m − μ D ) q_{\mathrm{inf}}=(m-\mu_D)/(2\lambda)\equiv\psi(m-\mu_D) q inf = ( m − μ D ) / ( 2 λ ) ≡ ψ ( m − μ D ) ψ = 1 / ( 2 λ ) \psi=1/(2\lambda) ψ = 1/ ( 2 λ ) λ = Cov ( v , q ) / Var ( q ) = ψ σ I 2 / ( ψ 2 σ I 2 + σ x 2 ) \lambda= \operatorname{Cov}(v,q)/\operatorname{Var}(q) = \psi\sigma_I^2/(\psi^2\sigma_I^2+\sigma_x^2) λ = Cov ( v , q ) / Var ( q ) = ψ σ I 2 / ( ψ 2 σ I 2 + σ x 2 ) ψ = 1 / ( 2 λ ) \psi= 1/(2\lambda) ψ = 1/ ( 2 λ ) λ 2 = σ I 2 / ( 4 σ x 2 ) \lambda^2 = \sigma_I^2/(4\sigma_x^2) λ 2 = σ I 2 / ( 4 σ x 2 )
λ = σ I 2 σ x , ψ = σ x σ I , (46) \lambda= \frac{\sigma_I}{2\sigma_x},\qquad\psi= \frac{\sigma_x}{\sigma_I}, \tag*{(46)} λ = 2 σ x σ I , ψ = σ I σ x , (46) the standard solution of Kyle [55 ] . The total order quantity q = ψ ( m − μ D ) + x n q = \psi(m-\mu_D)+x_n q = ψ ( m − μ D ) + x n ψ 2 σ I 2 + σ x 2 = 2 σ x 2 \psi^2\sigma_I^2+\sigma_x^2=2\sigma_x^2 ψ 2 σ I 2 + σ x 2 = 2 σ x 2 E ∣ q ∣ = 2 / π ⋅ 2 σ x = 2 σ x / π E|q|=\sqrt{2/\pi}\cdot\sqrt{2}\sigma_x=2\sigma_x/\sqrt{\pi} E ∣ q ∣ = 2/ π ⋅ 2 σ x = 2 σ x / π 53
Liquidity vector. The expected effective half-spread, the adverse-selection cost per unit traded under the regret-free linear quotes, is
λ E ∣ q ∣ = σ I 2 σ x ⋅ 2 σ x π = σ I π , (47) \lambda E|q|=\frac{\sigma_I}{2\sigma_x}\cdot\frac{2\sigma_x}{\sqrt{\pi}}=\frac{\sigma_I}{\sqrt{\pi}}, \tag*{(47)} λ E ∣ q ∣ = 2 σ x σ I ⋅ π 2 σ x = π σ I , (47) proportional to σ I \sigma_I σ I σ x \sigma_x σ x a s σ I a_s\sigma_I a s σ I a s a_s a s 1 / π 1/\sqrt{\pi} 1/ π E ∣ Δ p ∣ / E [ vol ] ∝ λ E|\Delta p|/E[\operatorname{vol}]\propto\lambda E ∣Δ p ∣/ E [ vol ] ∝ λ = a λ σ I / σ x =a_\lambda\sigma_I/\sigma_x = a λ σ I / σ x E ∣ q ∣ ∝ σ x E|q|\propto\sigma_x E ∣ q ∣ ∝ σ x = a T σ x =a_T\sigma_x = a T σ x [57 ] , provides an auxiliary trading-cost indicator used as an over-identifying check on the same pair. These are equations (43) . The signs in the proposition follow from the monotonicity of σ I 2 \sigma_I^2 σ I 2 ∂ spread / ∂ Σ j x = 0 \partial\operatorname{spread}/\partial\Sigma_{jx}=0 ∂ spread / ∂ Σ j x = 0
Inversion. Given ( σ ^ I , σ ^ x ) (\widehat{\sigma}_I,\widehat{\sigma}_x) ( σ I , σ x ) σ I 2 ( τ d + τ s ) ( τ d + τ s + K ) = γ 2 K \sigma_I^2(\tau_d+\tau_s)(\tau_d+\tau_s+K)=\gamma^2K σ I 2 ( τ d + τ s ) ( τ d + τ s + K ) = γ 2 K K [ γ 2 − σ I 2 ( τ d + τ s ) ] = σ I 2 ( τ d + τ s ) 2 K[\gamma^2-\sigma_I^2(\tau_d+\tau_s)]=\sigma_I^2(\tau_d+\tau_s)^2 K [ γ 2 − σ I 2 ( τ d + τ s )] = σ I 2 ( τ d + τ s ) 2 (44) . The denominator is positive whenever the data are generated by the model, since σ I 2 = γ 2 K / [ ( τ d + τ s ) ( τ d + τ s + K ) ] < γ 2 / ( τ d + τ s ) \sigma_I^2=\gamma^2K/[(\tau_d+\tau_s)(\tau_d+\tau_s+K)]<\gamma^2/(\tau_d+\tau_s) σ I 2 = γ 2 K / [( τ d + τ s ) ( τ d + τ s + K )] < γ 2 / ( τ d + τ s ) Σ j x \Sigma_{jx} Σ j x σ ^ x \widehat{\sigma}_x σ x ( a λ , a T ) (a_\lambda,a_T) ( a λ , a T ) τ p = K 2 / ( γ 2 ρ ˉ 2 Σ j x ) \tau_p=K^2/(\gamma^2\bar{\rho}^2\Sigma_{jx}) τ p = K 2 / ( γ 2 ρ ˉ 2 Σ j x )
Time-varying uncertainty
This subsection deals with the state dependence of the drift channel (the condition under which next-quarter losses respond more to measured noise). That response depends on how much investors know from outside the feed. The prior is the part of that knowledge that moves with the state of the economy. Thus, this subsection lets the prior weaken in bad times.
Assumption A4 (time-varying prior precision). τ d , q = τ d ⋅ h ( F U q ) \tau_{d,q}=\tau_d \cdot h(\mathrm{FU}_q) τ d , q = τ d ⋅ h ( FU q ) h ′ < 0 h' < 0 h ′ < 0 F U q \mathrm{FU}_q FU q [60 ] .54
The assumption allows the strength of prior beliefs to move with the state of the economy. The index F U q \mathrm{FU}_q FU q q q q τ d , q \tau_{d,q} τ d , q q q q τ d \tau_d τ d h ( F U q ) h(\mathrm{FU}_q) h ( FU q ) § 3.1 ). When financial uncertainty is high, that history tells less about the next dividend, which makes investors then place less confidence in their prior view. Hence, the prior precision τ d , q \tau_{d,q} τ d , q
Proposition 8 (Uncertainty amplification. ) For measured noise below the turning point, the slope of expected overpricing, ∂ O P / ∂ w \partial\mathrm{OP}/\partial w ∂ OP / ∂ w τ 0 \tau_0 τ 0 τ d , q \tau_{d,q} τ d , q τ 0 \tau_0 τ 0 w ∗ w^{*} w ∗
This proposition makes the drift channel (Proposition 6 ) state dependent. Expected overpricing is the average amount by which optimism holds the quarter-end price above the firm’s value V f 1 ∗ V_{f1}^{*} V f 1 ∗ (11) ). Measured noise (w w w (5) ). The object of the proposition is the derivative of expected overpricing with respect to measured noise (∂ O P / ∂ w \partial\mathrm{OP}/\partial w ∂ OP / ∂ w w ∗ w^{*} w ∗ (12) ). This derivative depends on the outside knowledge (what investors know from sources other than the feed), whose precision (τ 0 \tau_0 τ 0 (12) ). The precision of the public signal (τ s \tau_s τ s τ 0 \tau_0 τ 0 § 3.1 ). The three do not vary across investors. The private-signal precision enters as the aggregate K j K_j K j (7) ). When outside knowledge is weak, the signal noise then produces a large impact on price (equation (7) ). A larger price impact of an optimistic error (a positive realization of the signal noise u f 1 u_{f1} u f 1 (11) ). Thus, the derivative is larger when outside knowledge is weaker. Of the three precisions in τ 0 \tau_0 τ 0 τ 0 \tau_0 τ 0 Proposition 6 ). Thus, the next-quarter losses respond more to measured noise in high-uncertainty quarters. The turning point also shifts. The peak of expected overpricing lies where the feed’s precision equals the precision of outside knowledge (τ 0 \tau_0 τ 0 (12) ). Weaker outside knowledge places the peak at a noisier feed, which in measured units is a higher turning point (C.12). The result rationalizes the empirical pattern in García [40 ] that the return-predictive content of sentiment concentrates in bad times. Proposition 8 is tested by adding an interaction of measured noise with the financial-uncertainty index to the regression of next-quarter losses on current-quarter measured noise (row R7, Table IV).
Proof of Proposition 8
C.12. Proof of Proposition 8: Under A4, a rise in financial uncertainty lowers τ d , q \tau_{d,q} τ d , q τ 0 = τ d , q + τ p + K \tau_0 = \tau_{d,q} + \tau_p + K τ 0 = τ d , q + τ p + K τ p \tau_p τ p (32) does not involve τ d , q \tau_{d,q} τ d , q w ∗ = a / τ 0 w^* = \sqrt{a/\tau_0} w ∗ = a / τ 0 (41) is strictly decreasing in τ 0 \tau_0 τ 0 τ 0 \tau_0 τ 0 ∂ O P / ∂ w \partial\mathrm{OP}/\partial w ∂ OP / ∂ w (41) and differentiate it with respect to τ 0 \tau_0 τ 0 w w w c ≡ δ γ a / 2 π c \equiv\delta\gamma\sqrt{a}/\sqrt{2\pi} c ≡ δ γ a / 2 π (41) ,
∂ ∂ τ 0 ( ∂ O P ∂ w ) = c − w 2 ( τ 0 w 2 + a ) 2 − ( a − τ 0 w 2 ) ⋅ 2 ( τ 0 w 2 + a ) w 2 ( τ 0 w 2 + a ) 4 = − c w 2 ( 3 a − τ 0 w 2 ) ( τ 0 w 2 + a ) 3 . (48) \frac{\partial}{\partial\tau_0}\left(\frac{\partial\mathrm{OP}}{\partial w}\right)
= c\frac{-w^2(\tau_0w^2+a)^2-(a-\tau_0w^2)\cdot2(\tau_0w^2+a)w^2}{(\tau_0w^2+a)^4}
= -c\frac{w^2(3a-\tau_0w^2)}{(\tau_0w^2+a)^3}. \tag*{(48)} ∂ τ 0 ∂ ( ∂ w ∂ OP ) = c ( τ 0 w 2 + a ) 4 − w 2 ( τ 0 w 2 + a ) 2 − ( a − τ 0 w 2 ) ⋅ 2 ( τ 0 w 2 + a ) w 2 = − c ( τ 0 w 2 + a ) 3 w 2 ( 3 a − τ 0 w 2 ) . (48) For w < w ∗ w < w^* w < w ∗ τ 0 w 2 < a \tau_0w^2 < a τ 0 w 2 < a 3 a − τ 0 w 2 > 2 a > 0 3a-\tau_0w^2 > 2a > 0 3 a − τ 0 w 2 > 2 a > 0 τ 0 \tau_0 τ 0 τ 0 \tau_0 τ 0
Partial observation of the public signal
Partial observation of the public signal
In this subsection, we derive the claim in the footnote to H1. We take common observation as the baseline, since the feed’s aggregate sentiment is a freely observable public series while producing firm-specific knowledge is the costly activity (Van Nieuwenburgh and Veldkamp, 2009). Fix a group and suppress its index. A mass θ ∈ ( 0 , 1 ] \theta\in(0,1] θ ∈ ( 0 , 1 ] s s s s s s ζ \zeta ζ ζ \zeta ζ s s s τ s \tau_s τ s Σ i = ( τ d + τ p + k i + τ s ) − 1 \Sigma^i=(\tau_d+\tau_p+k^i+\tau_s)^{-1} Σ i = ( τ d + τ p + k i + τ s ) − 1 τ p ζ + k i η i + τ s s \tau_p\zeta+k^i\eta^i+\tau_s s τ p ζ + k i η i + τ s s ( τ d + τ p + k i ) − 1 (\tau_d+\tau_p+k^i)^{-1} ( τ d + τ p + k i ) − 1 τ s s \tau_s s τ s s
τ p ζ + K ϵ + θ τ s s . \tau_p\zeta+K\epsilon+\theta\tau_s s. τ p ζ + K ϵ + θ τ s s . The private terms aggregate to K ϵ K\epsilon K ϵ θ \theta θ θ τ s s \theta\tau_s s θ τ s s
T ˉ = τ d + τ p + K + θ τ s . \bar{T}=\tau_d+\tau_p+K+\theta\tau_s. T ˉ = τ d + τ p + K + θ τ s . Every appearance of τ s \tau_s τ s τ p = K 2 / ( γ 2 ρ ˉ 2 ∑ j x j ) \tau_p=K^2/(\gamma^2\bar{\rho}^2\sum_j x_j) τ p = K 2 / ( γ 2 ρ ˉ 2 ∑ j x j ) τ s \tau_s τ s (8) ). Thus, the market-clearing condition coincides with that of a full-observation economy with signal precision, θ τ s \theta\tau_s θ τ s (7) then follow with τ s \tau_s τ s θ τ s \theta\tau_s θ τ s θ τ s \theta\tau_s θ τ s
This equivalence is only about the price function. The realized return has the sentiment shock, so the return variance and the expected overpricing under partial observation are not those of the θ τ s \theta\tau_s θ τ s
We now make the footnote’s bound precise. Write the price-relevant precision of group j j j θ j τ s , j \theta_j\tau_{s,j} θ j τ s , j θ F ≤ θ D \theta_F \le\theta_D θ F ≤ θ D θ D τ s , D / ( θ F τ s , F ) ≥ τ s , D / τ s , F \theta_D\tau_{s,D}/(\theta_F\tau_{s,F}) \ge\tau_{s,D}/\tau_{s,F} θ D τ s , D / ( θ F τ s , F ) ≥ τ s , D / τ s , F τ s , D / τ s , F \tau_{s,D}/\tau_{s,F} τ s , D / τ s , F θ < 1 \theta< 1 θ < 1 θ ( τ s , D − τ s , F ) \theta(\tau_{s,D}-\tau_{s,F}) θ ( τ s , D − τ s , F ) τ s , D − τ s , F \tau_{s,D}-\tau_{s,F} τ s , D − τ s , F τ s , D > τ s , F \tau_{s,D} > \tau_{s,F} τ s , D > τ s , F
Supporting Derivations
Value of the dividend claim
Value of the dividend claim: We derive two environment statements. The first is the equivalence between the dividend law and its one-step recursion. The second is the value of the claim.
For the first, fix s ≥ 2 s \ge2 s ≥ 2 d f s ∗ = v j s − 1 d f 1 ∗ = v j ( v j s − 2 d f 1 ∗ ) = v j d f , s − 1 ∗ d^*_{fs} = v_j^{s-1}d^*_{f1} = v_j(v_j^{s-2}d^*_{f1}) = v_jd^*_{f,s-1} d f s ∗ = v j s − 1 d f 1 ∗ = v j ( v j s − 2 d f 1 ∗ ) = v j d f , s − 1 ∗ d f , s + 1 ∗ = v j d f s ∗ d^*_{f,s+1} = v_jd^*_{fs} d f , s + 1 ∗ = v j d f s ∗ d f s ∗ = v j s − 1 d f 1 ∗ d^*_{fs} = v_j^{s-1}d^*_{f1} d f s ∗ = v j s − 1 d f 1 ∗ s s s d f 1 ∗ d^*_{f1} d f 1 ∗
For the second, the position pays the period-1 value of the stream, V f 1 ∗ = ∑ s ≥ 1 d f s ∗ / r s − 1 V^*_{f1} = \sum_{s\ge1} d^*_{fs}/r^{s-1} V f 1 ∗ = ∑ s ≥ 1 d f s ∗ / r s − 1 s s s s − 1 s-1 s − 1 r s − 1 r^{s-1} r s − 1 q ≡ v j / r q \equiv v_j/r q ≡ v j / r V f 1 ∗ = d f 1 ∗ ∑ k ≥ 0 q k V^*_{f1} = d^*_{f1}\sum_{k\ge0}q^k V f 1 ∗ = d f 1 ∗ ∑ k ≥ 0 q k S N = 1 + q + ⋯ + q N S_N = 1+q+\cdots+q^N S N = 1 + q + ⋯ + q N S N − q S N = 1 − q N + 1 S_N-qS_N = 1-q^{N+1} S N − q S N = 1 − q N + 1 S N = ( 1 − q N + 1 ) / ( 1 − q ) S_N = (1-q^{N+1})/(1-q) S N = ( 1 − q N + 1 ) / ( 1 − q ) v j < r v_j < r v j < r q q q q N + 1 q^{N+1} q N + 1 1 / ( 1 − q ) = r / ( r − v j ) = γ j 1/(1-q) = r/(r-v_j) = \gamma_j 1/ ( 1 − q ) = r / ( r − v j ) = γ j V f 1 ∗ = γ j d f 1 ∗ V^*_{f1} = \gamma_jd^*_{f1} V f 1 ∗ = γ j d f 1 ∗ v j / r v_j/r v j / r
Additional Related Literature
Hong and Stein [49 ] survey the literature on heterogeneous beliefs, where divergence of opinions raises prices under short-sale constraints because optimists set the price [62 ] . Diether et al. [29 ] show how the analyst-forecast dispersion predicts lower subsequent returns. Cookson and Niessner [19 ] show that disagreement driven by different information is more tightly related to trading volume than disagreement driven by different investment approaches. Cookson et al. [19 ] show that partisan identity shapes how investors read the same signals. The last three papers measure the variance of stated beliefs. Our dispersion measure instead helps capture the composition of the contributor pool. Dispersion is the ratio of the number of unique posting accounts to the number of tweets. A low ratio indicates that a few accounts are posting repeatedly about a stock. A high ratio indicates many accounts each posting rarely about a stock. Accounts that post repeatedly about a firm are specialists (Van Nieuwerburgh and Veldkamp [64 ] ) or automated accounts, while accounts which post rarely are attention-driven contributors (Barber and Odean [11 ] ). In the model, the tweets of accounts that post rarely carry the firm-specific news together with the day’s fad (a transient component that carries none of that news), while the tweets of specialists carry the firm-specific news without the fad. Thus, at a given fad variance, a larger share of tweets from accounts that post rarely implies a larger nonfundamental component of the feed (§ 3.2 ). The presence of a fad in such tweets is further supported by field evidence that people’s beliefs and decisions reflect overconfidence, limited attention, persuasion and emotion (DellaVigna [28 ] ).
Our paper also relates to the literature on cross-listing. Karolyi [53 ] surveys the evidence and reports favorable share-price reactions in the first month after listing, an overall improvement in liquidity, and a reduction in domestic market risk that can lower the cost of equity capital. Foerster and Karolyi [36 ] find that firms cross-listing as American Depositary Receipts earn abnormal returns before and at listing and lose part of them in the year after. The pattern is related to an expansion of these firms’ shareholder base and to the capital raised at listing. It supports the market segmentation hypothesis and the investor recognition hypothesis of Merton [61 ] . Eun and Sabherwal [33 ] find that for Canadian stocks listed in both Toronto and the United States, prices in the two markets are cointegrated and mutually adjusting, with the home market contributing most of the price discovery on average. Gagnon and Karolyi [39 ] find that deviations from price parity between U.S. cross-listed shares and their home-market shares are small on average but volatile and are related to proxies for holding costs that can impede arbitrage. Our producer-knowledge mismatch hypothesis rests on the premise that the home-market investors of a foreign firm tend to possess an informational advantage about it (H1). Since ADR prices are linked to home-market prices through the depositary structure, the finding that domestic-foreign difference in the slope of next-quarter losses on sentiment noise is strongest for ADRs admits a mechanical explanation (§ 6 ).
