The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
157,150 characters
Where Has the Foreign Signal Gone?
\addtocontents{toc}{\protect\setcounter{tocdepth}{-10}}
\onehalfspacing
\begin{frontmatter}
\title{Where Has the Foreign Signal Gone?\tnoteref{t1}}
\author[]{Piyush Akimitsu\corref{cor1}}
\author[]{Ulrich Hounyo\corref{cor1}}
\address{ SUNY at Albany}
\cortext[cor1]{Akimitsu: [email removed], Hounyo: [email removed]}
\tnotetext[t1]{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 Montr\'eal 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).}
\begin{abstract}\setstretch{1.5}
We compare the relation between sentiment noise in the Twitter feed and losses for stocks of domestic and foreign firms. The mean quarterly domestic-foreign difference in the slope of next-quarter losses on measured noise is −1.40 (Newey-West t = −2.99). This slope is negative for 78 and 48 percent of the quarters for domestic and foreign stocks, respectively (block-bootstrap p = 0.016). Thus, this relationship is temporally more stable for domestic stocks. We use a noisy rational expectations model to explain these patterns through the composition of the contributor-pool. The feeds for stocks of foreign firms have higher repeat-author concentration and a larger model-implied nonfundamental component than those for domestic firms. We interpret these differences through the producer-knowledge mismatch hypothesis. For foreign firms, the English-language contributor pool underrepresents the home-market investors who tend to possess informational advantages about them. Thus, the foreign signal is not gone but diluted.
\end{abstract}
\end{frontmatter}
\thispagestyle{plain}
\vspace*{\fill}
\noindent{\footnotesize\textit{Keywords:} Social Media Sentiment, Asset Pricing, Information Frictions, Home Bias\\[3pt]\textit{JEL:} C22, C53, G12\par}
\clearpage
\section{Introduction}\label{sec:intro}
There has been an increasing trend in using social media sentiment for predicting stock returns. However, the predictive power of such sentiment depends on how much of it is information and how much of it is noise. This distinction matters because it shapes how reliable the sentiment signal is. A large literature documents the link between sentiment and returns, incorporates sentiment measures into trading strategies, and refines the text-processing methods behind them \citep{antweiler2004talk, tetlock2007giving, chen2014wisdom, bartov2018twitter}.
A sentiment index constructed from the social media feed consists of two components. The first is the beliefs of investors who hold information about the firm. The second is fads and automated content that carry no information about the firm. These two components have opposite implications. The informed component moves prices closer to fundamental value. The uninformed component can move prices away from fundamental value and create temporary mispricing.
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 GMM. Specifically, for the composition of the contributor pool, we estimate the concentration 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). The results show that both the concentration of the repeat authors and the scale of the fad component are larger for foreign stocks than for domestic stocks. Our hypothesis is that the foreign feed is less informative than the domestic feed. The main 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 by 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 it is the mismatch between contributors who write the tweet and people who hold information about the firm. 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 \citep{twitter2019annual}.\footnote{U.S. users accounted for approximately 20 percent of monetizable daily active users as of 2019 \citep{twitter2019annual}. \citet{paydarzarnaghi2023twitter} analyze over 9 million stock-specific tweets and show that the U.S. demographic user groups drive equity price reactions substantially more than any other group.} The U.S. investors lean more towards domestic stocks which represents information concentration where the U.S. investors are rationally deepening their knowledge about domestic firms rather than diversifying \citep{van2009information, mondria2010home, sims2003rational}. The home market investors of a firm with headquarters in a foreign country tend to possess an informational advantage about such a firm \citep{dvorak2005domestic, bae2008local}. The advantage is visible in prices, since home markets lead price discovery for cross-listed stocks on average \citep{eun2003cross}.\footnote{The informational advantage of the home market investors about these foreign firms applies to both American Depositary Receipts and directly listed foreign shares, which form our sub-sample of foreign stocks. More than half of sub-sample of such foreign stocks belongs to non-English-speaking countries. We call a country English-speaking when English is a first language there, following \citet{crystal2003english}.} Under our hypothesis, these home-market investors of foreign firms are underrepresented on English-language finance Twitter. This implies that the signal for foreign firms should have a weaker information component and more noise than domestic stocks. Investors cannot separate the signal from its error. Therefore, the prices absorb the noise carried by the signal. Since a foreign signal is noisier, the prices of foreign stocks are noisier too. If arbitrage is limited, it lets those price effects persist \citep{shleifer1997limits}.
This matters for three main reasons. First, investors who trade using Twitter sentiment get a signal whose reliability varies with firm geography. Weighting the two groups equally conceals that variation. 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 (informed specialists, attention-driven contributors posting 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 the investors here. The recognition they gain lowers their cost of raising the capital \citep{merton1987simple, karolyi1998companies}. The prices reflect the noise in a stock's feed and the market is unable to distinguish between signal and noise as they are absorbed into the prices. A noisier feed could move the prices of the foreign listings for reasons unrelated to the true firm value. The aggregate sentiment index doesn't show how much of that sentiment is noise. Such a distortion in prices which is unseen also stays uncorrected.
We formalize the mechanism in a noisy rational expectations model with three ingredients beyond the standard signal structure. The first ingredient is repeat authorship. Some accounts are specialists or automated accounts that post about the same stock more than once on a day, while attention-driven contributors post once. Measured sentiment aggregates tweets rather than accounts. The weight of the repeat authors in measured sentiment is their share of tweets, which exceeds their share of accounts. The model identifies the account share of repeat authors (specialists and automated accounts) from the daily posting dispersion (count of distinct posting accounts divided by the count of their tweets). This account share and the posting intensity of the repeat authors give the tweet share of the repeat authors. A repeat author posts more than one tweet per account. An attention-driven contributor posts one tweet per account. As the share of repeat authors rises, the number of accounts per tweet is diluted and the dispersion falls. Another ingredient in the model is a short-sale constraint as in \citet{miller1977risk}. Since short selling is costly, a pessimist who holds no shares and cannot borrow doesn't sell, while a pessimist who holds shares can still sell. Moreover, arbitrageurs who can borrow can still short. When the signal understates the firm's value, the price is low which attracts the investors. The underpricing is corrected within the quarter through trading. When the signal overstates the firm's value, the pricing correction requires selling. Since the pessimists who hold no shares and can't borrow can't sell and don't buy at a price above their valuation, the market for the stock is dominated by the optimists whose valuations are reflected in the price more than those of pessimists. Therefore, the overpricing is only partly corrected within the quarter.
The model predicts that the surviving overpricing reverses in the following quarter as an excess of losses over gains. Figure~\ref{fig:timeline} shows the timing of these events and of the empirical design. The third ingredient is a dealer market in the spirit of \citet{kyle1985continuous}. Some investors hold private information about the firm and others do not. The dealer receives orders from both types of investors and cannot tell them apart. Since any order might carry information, the dealer charges a spread and moves the price after each trade as compensation for adverse selection. The size of that price move per unit traded is the market-impact cost. The spread and the market-impact cost are the model's two liquidity objects. In the data, they correspond to the relative quoted spread and the \citet{amihud2002illiquidity} illiquidity ratio. The model recovers an informed-trading-intensity rank for each stock from this spread. This informed-trading-intensity is higher when more of a stock's investors hold private information. Given the spread, the Amihud ratio shows how much trading comes from the investors without private information. Then domestic-foreign comparisons are made within terciles of that intensity ranking. The model delivers closed-form moments for sentiment dispersion and volatility, return volatility, gains, losses, liquidity, and price informativeness. Each of these moments varies across domestic and foreign stocks and across terciles of that rank.
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.\footnote{The vendor changed its sampling methodology in 2017. So, the pre-change feed is not comparable with the post-change feed. Hence, the primary sample begins in 2018 and earlier quarters are not used in the estimation.} Domestic and foreign stocks are sorted separately into terciles of the model-based informed-trading-intensity rank. Domestic-foreign comparisons are then made within each tercile of that rank. Moreover, the platform-wide changes in feed volume move measured noise for every stock. Hence, in the reduced-form regressions the domestic-foreign comparisons are made quarter by quarter. This quarter-by-quarter comparison keeps those common movements out of those regressions.
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,\footnote{The noise measure is the within-quarter volatility of daily per-tweet sentiment.} the mean quarterly slope is more negative for domestic stocks than for foreign stocks by 1.40 (Newey-West $t=-2.99$). This relation keeps its sign in nearly 4 out of 5 quarters for domestic stocks, against about 1 out of 2 for foreign stocks. In economic terms, a one standard deviation increase in sentiment noise increases the next-quarter losses by 0.64 percentage points of excess quarterly returns for domestic stocks. The domestic-foreign difference in that response is 1.39 percentage points, which is about two thirds of the sample's mean quarterly excess return. If scaled to the yearly level, the domestic response and the domestic-foreign difference response are about 2.6 and 5.6 percentage points of annual excess return---a descriptive scaling rather than a portfolio return. This domestic-foreign contrast also appears as negative quarterly slope---for 78\% of quarters for domestic stocks as opposed to 48\% of the quarters for foreign stocks---in the regressions of next-quarter losses on sentiment noise. This difference of 30 percentage points has a block-bootstrap $p$ value of 0.016 (battery row R14, Table~\ref{tab:battery}). If this pattern is broken down by terciles, this domestic-foreign gap in the share of quarters with a negative slope is significant in the lowest informed trading intensity tercile.\footnote{In the lowest intensity tercile the share of quarters with a negative slope is 0.852 for domestic stocks against 0.556 for foreign, a difference of 0.296 with a block-bootstrap $p$ value of 0.004.} The gap is positive but not significant in the middle tercile and is close to zero in the highest tercile where the informed trading intensity is heaviest. Here informed trading may hold prices near their fundamental values, so that the feed's noise moves prices little for either group and the gap between them closes down. Additionally, the mean slope difference between domestic and foreign is strongest for ADRs at $-2.00$ (Newey-West $t=-3.72$). It is to be noted that an ADR's U.S. price is tied to its home-market share price by arbitrage, the depositary link, which itself could weaken the loss-sentiment noise relation. Therefore, we look at the directly listed foreign shares, which do not have this link. The estimate of the slope difference keeps its sign at $-0.74$ but is imprecise (Newey-West $t=-1.29$).\footnote{The listing-type, headquarters-language, and country splits are in Table~\ref{tab:reliability_splits} (\ref{app:robustness}). The listing-type slope differences are in \ref{app:rob_r14d}.}
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 domestic, and the foreign feed also carries a larger fad component than domestic. The estimated share of repeat authors is higher for foreign stocks than for domestic stocks in all terciles of informed trading intensity. Thus, the foreign-domestic gap in repeat author share is the true gap and doesn't merely reflect the foreign-domestic gap in informed trading intensity. The foreign-domestic gap in repeat author share is the widest among stocks with the most informed trading intensity. In the topmost tercile, the repeat author share for foreign and domestic are 0.170 and 0.145, respectively (Table~\ref{tab:theta_hat}). The fad component, 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.\footnote{A joint Wald test rejects equality of the four parameters for foreign and domestic stocks, with $p$ values of $7.2\times 10^{-10}$ under firm weighting, $1.9\times 10^{-14}$ under quarter weighting, and $1.9\times 10^{-5}$ under a two-way firm-and-quarter covariance (Table~\ref{tab:h1_contrasts}). The excess for foreign stocks holds in all bootstrap replications for the fad scale and the top-tercile share.} The repeat-author share and the fad scale are obtained from structural estimation. The third measure is observable signal-quality index ($T_q/\mathrm{VAQSV}^2$) obtained directly from the observed daily feed readings. The higher the feed precision, the higher this index. The median of this index for foreign stocks is lower than that for domestic stocks in all intensity terciles---by 17 percent in the low tercile, 16 percent in the middle tercile and 7 percent in the top tercile (Table~\ref{tab:r6_dataterm}). Each of the domestic-foreign gaps is modest on its own.
The third result is that a higher value of a model-motivated signal-noise index is followed by a larger loss-gain differential. The structural model shows this asymmetry and the mispricing correction mechanism (Section~\ref{sec:model_gl}). The reduced form regressions show that the loss-gain differential indeed does widen with that index. However, the estimated short-sale friction parameter lies at its zero lower bound. That means the costly short selling channel that would have caused the overpricing to survive is silent. In a quarter with a larger share of positive-sentiment days, the price rises above value, and under that channel the overpricing would survive and correct into a larger loss the next quarter. That stronger loss response does not appear (battery row R8, Table~\ref{tab:battery}). Instead, we have the model-motivated signal-noise index (Section~\ref{sec:rf_correction}) which varies with stock-quarter and the next-quarter loss-gain excess rises with that index in a reduced-form projection. Within each group, it basically ranks stock-quarters by the amount of overpricing the feed's noise would create. It combines two things---how informative a stock's feed is in a quarter and how much the market already knows about the stock from everything other than the feed, the outside knowledge about the stock. The model estimates this outside information as one value for each of the six groups, domestic and foreign at each of three informed-trading-intensity terciles. Outside information about a stock can vary from quarter to quarter depending on which tercile the stock lies in each quarter. With quarter fixed effects and geography-and-tercile group fixed effects, the excess of next-quarter losses over gains rises with the index under the announcement alignment ($t = 9.63$, with standard errors double clustered by firm and quarter).\footnote{Because the signal-noise index has no natural scale, to make the coefficient of this regression interpretable, the text reports the direction of the relation and the $t$ statistic rather than the coefficient itself. Measuring the loss and gain in the next calendar quarter, instead of in the quarter of the next earnings announcement, the index still predicts a larger loss-gain difference ($t = 9.33$, with standard errors double clustered by firm and quarter). Table~\ref{tab:b3_drift} reports the coefficient under both the announcement and the calendar alignment.}\textsuperscript{,}\footnote{An alternative specification removes the geography-and-tercile group fixed effects and keeps the quarter fixed effects, turning the coefficient negative ($t = -4.18$, with standard errors double clustered by firm and quarter). The reversal is a composition effect, since the groups differ in their averages and a regression without the group fixed effects is driven by those differences across groups rather than by the relation within them. Section~\ref{sec:rf_correction} and Table~\ref{tab:b3_drift} report the coefficient with and without the group fixed effects.}
The model gives some additional predictions which are tested using a battery of reduced-form checks.\footnote{We re-estimate the loss-on-noise regression (battery row R1, Table~\ref{tab:battery}) across thirty-one subsamples and control specifications in Table~\ref{tab:additional_robustness}.} The noisier the sentiment, the larger the losses. The raw and the volume-adjusted measures of quarterly sentiment volatility, in a regression of next-quarter losses on both and tweet volume, have coefficients of $-0.317$ and $-0.612$ ($t = -6.02$ and $-2.11$, with standard errors double clustered by firm and quarter), so the loss prediction is not driven by tweet volume, to which the raw measure is tied and the volume-adjusted one is not (\ref{app:battery}). In a regression of next-quarter losses on noise and the stock's realized return volatility (the within-quarter standard deviation of its daily excess returns), the coefficients on the noise are insignificant and return volatility has a coefficient of $-14.70$ ($t = -40.91$, with standard errors double clustered by firm and quarter). Thus, noisier feeds predict larger losses because they come with more volatile returns, not because they push down the average next-quarter return. This is consistent with the short-sale friction at its zero lower bound. Next-quarter gains (the sum of a stock's positive daily excess returns over the quarter) rise with lagged dispersion (mean ratio of distinct posting accounts to tweets within the quarter). The response concentrates in the lowest intensity tercile, at 0.128 ($t = 3.15$, with standard errors double clustered by firm and quarter).
A limitation of the structural estimates is that each stock's signal precision is recovered only as its observable signal-quality index divided by one conversion constant common to all stocks, whose size absorbs the scale of the vendor's score, serial dependence across days, and the dilution of the signal by automated content, which the data do not separate (Section~\ref{sec:structural}). The estimate of this conversion constant is about 400, with a standard error of 34 (Table~\ref{tab:theta_hat}). In the domestic-foreign comparisons of signal precision, the constant cancels and the comparisons depend on the observable index alone. Second, about 84\% of the weighted distance between model predicted and data computed moments comes from the four liquidity moments, 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 rest of the paper proceeds as follows. Section 2 reviews the literature. Section 3 presents the model. Section 4 describes the data. Section 5 reports the structural estimation. Section 6 reports the reduced-form evidence. Section 7 concludes.
\section{Related Literature}\label{sec:lit}
Prior work in this area shows that investors overweigh domestic and nearby securities \citep{french1991investor}, while investment funds hold more domestic than foreign stocks \citep{lambert2026home}. This geographical proximity is more than just familiarity. It also gives real informational advantages to the investors \citep{kang1997home, ahearne2004information, coval1999home, coval2001geography}. Some other related work shows that domestic investors outperform foreign investors in the same securities \citep{dvorak2005domestic}, and that local analysts forecast more accurately in less transparent environments \citep{bae2008local}. Additionally, the investors rationally deepen initial local advantages and \citet{van2009information} provide a theoretical basis for that. In addition to geography, language shapes the attention which foreign firms receive. \citep{boulland2017news} 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 \citep{merton1987simple}. Investors with finite capacity direct their attention to the signals with the highest informational return \citep{sims2003rational, peng2006investor}. This leads to endogenous home bias \citep{mondria2010home, van2009information}. Retail investors are net buyers of stocks that get more attention \citep{barber2008all}. 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 \citep{andre2026mental}. English-language finance 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 \citep{antweiler2004talk}. Pessimistic coverage in a Wall Street Journal column predicts downward price pressure which reverses later \citep{tetlock2007giving}, while the predictability from newspaper content is concentrated in recessions \citep{garcia2013sentiment}. This holds for social media as well. Seeking Alpha articles and their reader commentaries predict returns and earnings surprises \citep{chen2014wisdom}. The aggregate opinion in tweets posted just before an earnings announcement predicts the firm's earnings and its announcement return \citep{bartov2018twitter}. Similarly, a comparison between global and local news sentiment shows that the former moves returns more, and more persistently, than the latter \citep{fraiberger2021media}. 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. \citet{gomezcram2026prediction} 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. 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. \citet{cookson2024social} trace the differences in the sentiment signal of various platforms to their user population composition and platform design. \citet{huang2024local} 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 \citet{farboodi2022data} decomposition of the \citet{bai2016informative} measure of price informativeness. In our model the precision of the sentiment signal enters that decomposition (Section~\ref{sec:model_eq}). \citet{dugast2018data} show that cheaper low-precision signals can reduce price informativeness by lowering the demand for more precise signals.\footnote{The related work on heterogeneous beliefs and on cross-listing is discussed in \ref{app:lit_extra}.}
\section{A Model of Sentiment, Information, and Liquidity}\label{sec:model}
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 \citet{hellwig1980aggregation}, \citet{admati1985noisy}, and \citet{farboodi2022data}: In such an equilibrium, prices aggregate dispersed information.\footnote{In a noisy rational expectations model investors learn from the price itself. In the supply of shares, there is a random component, without which the price would let every investor infer the private information of the others. With that random component, the price reveals that information only in part. Thus, private information still retains value. The \textit{precision} of a signal is the inverse of the variance of its noise. An investor's belief after observing the private signal, the price and the public signal has a precision equal to the precision of the belief held before observing them in addition to the individual precision of each of the three signals.} 2. The short-sale constraint in the tradition of \citet{miller1977risk}: 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~\ref{prop:drift}). 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~\ref{prop:vol}). 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 \citet{kyle1985continuous} and \citet{glosten1985bid}: 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 (Section~\ref{sec:model_micro}). 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 ($S_t$), and the tweet count ($N_t$). $S_t/N_t$ gives the day's sentiment per tweet. In the model, the score of each tweet contains noise specific to that tweet. An average over many tweets reduces such noise but does not eliminate it. The more tweets a day has, the less of this noise remains. This remaining noise is the sampling part of the day-to-day variation in sentiment per tweet. The fad and the automated content behave differently. The fad is a transient component of the feed for the day, so each attention-driven tweet posted that day has it. The automated content is also a component of the feed for the day, so each bot tweet posted on that day has it. Averaging the sentiment scores of the day's tweets does not reduce either of these components. The fad's contribution to a day's average sentiment score equals the fad multiplied by the fraction of the day's tweets that are attention-driven (if a tenth of the day's tweets are attention-driven then the average score carries a tenth of the day's fad, irrespective of the number of tweets). This is true also for the automated content and the fraction of the day's tweets that come from bots. These two components are the second part, the nonfundamental part, of the day-to-day variation in sentiment per tweet. The data can tell the two parts apart because the sampling part depends on the tweet count while the nonfundamental part does not (Section~\ref{sec:model_sent}). 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.\footnote{This index is computed for each stock-quarter. The structural estimation aggregates it to the group (domestic or foreign stocks) level for each informed-trading-intensity tercile. The headline reduced-form 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.} The composition of contributor types is a secondary question.\footnote{This question is addressed by the calibration check in \ref{app:composition_check} and the reduced-form battery in Section~\ref{sec:reducedform}. The model objects and their empirical counterparts are in Table~\ref{tab:notation_map}.}
\subsection{The economic environment}\label{sec:model_env}
Time is a sequence of quarters. We describe one quarter and suppress the quarter index except where needed. Within the description, periods $s = 0, 1, 2, \ldots$ count quarters from the present, so $s = 0$ is the current quarter. There is a continuum of stocks $f$. Each stock belongs to a stock group $j$, a set of stocks that share the same parameters. A stock group is defined on stocks, not on the contributors who post about them. Geography is one such grouping, splitting stocks by whether the firm is headquartered at home or abroad. The empirical work also groups stocks by the model-based informed-trading-intensity rank. The dividend that stock $f$ pays in period $s$ is $d^{*}_{fs}$. Starred symbols denote quantities in the model's value units throughout.\footnote{Assumption~\ref{assump:daily} below specifies the single positive scale that converts value units to returns.} The current dividend $d^{*}_{f0}$ is known. A share of stock $f$ is a claim to the dividend stream
\begin{equation}
d^{*}_{f1} = \upsilon_j\, d^{*}_{f0} + \bar{\epsilon}_{f1} + \epsilon_{f1}, \qquad \epsilon_{f1} \sim N(0,\Sigma_{jd}), \qquad d^{*}_{fs} = \upsilon_j^{\,s-1} d^{*}_{f1} \ \ (s \geq 2),
\end{equation}
where $\upsilon_j$ is the gross one-period persistence factor of cash flows. Let $r$ denote the gross riskless rate, one plus the net rate. We maintain $0 \leq \upsilon_j < r$ throughout, so the dividend claim has finite value. The first equality says next period's dividend equals $\upsilon_j$ times the current dividend plus two shocks. The last equality is the solved form of the recursion $d^{*}_{f,s+1} = \upsilon_j\, d^{*}_{fs}$ for $s \geq 1$ (\ref{app:proof_value}). Therefore, each dividend after period 1 is $\upsilon_j$ times the one before it. The law carries no error term, so there are no further shocks. The shock $\bar{\epsilon}_{f1}$ is a common component shared with other firms. The shock $\epsilon_{f1}$ is the firm-specific innovation, the one object investors are uncertain about. Its variance is $\Sigma_{jd}$. As in \citet{farboodi2022data}, all investors observe the common component, so all learning concerns $\epsilon_{f1}$. Trade occurs once within the quarter, at a price $P^{*}_{f1}$ set before the innovation becomes public. With no residual uncertainty after period 1, the position pays the period-1 value of the dividend stream, $V^{*}_{f1} = \sum_{s \geq 1} d^{*}_{fs}/r^{\,s-1}$, when the innovation becomes public. Substituting the dividend persistence $d^{*}_{fs} = \upsilon_j^{\,s-1} d^{*}_{f1}$ into this present-value sum gives a geometric series in $\upsilon_j/r$, which sums to $V^{*}_{f1} = \gamma_j\, d^{*}_{f1}$, where $\gamma_j \equiv r/(r-\upsilon_j)$ is the present-value multiplier of a growing perpetuity (\ref{app:proof_value}). The supply of shares is $\bar{x}_j + \tilde{x}_f$, the sum of a known mean component $\bar{x}_j$ and a random component $\tilde{x}_f \sim N(0,\Sigma_{jx})$. The random component of the quantity supplied of shares represents trades unrelated to the firm's value \citep{delong1990noise}. 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~\ref{prop:price}). 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~\ref{fig:timeline} shows the timing of the four events, the feed, the trading, the announcement, and the correction. The feed and the trading fall in quarter $q$. The announcement and the correction fall in quarter $q+1$.
\begin{figure}[!htbp]
\centering
\caption{Timing of the mechanism and the empirical design}\label{fig:timeline}
\medskip
\resizebox{\textwidth}{!}{\begin{tikzpicture}[
box/.style={draw, rounded corners=2pt, align=left, inner sep=5pt, text width=0.163\textwidth, minimum height=3.7cm, font=\scriptsize, execute at begin node={\hyphenpenalty=10000\relax}},
mbox/.style={draw, dashed, rounded corners=2pt, align=left, inner sep=5pt, text width=0.56\textwidth, font=\scriptsize, execute at begin node={\hyphenpenalty=10000\relax}},
arr/.style={->, thick, shorten >=2pt, shorten <=2pt}]
\node[box] (b1) at (0,0) {The feed is created: The contributors post about the stock on each day of quarter $q$. The quarter's daily readings form the public signal whose noise is given by the within-quarter volatility of daily sentiment per tweet, VAQSV.};
\node[box, right=0.42cm of b1] (b2) {The quarterly trade: Investors use the prior, their private signals, the price and the public signal. The market clears at the price $P^{*}_{f1}$, which incorporates the public signal with its noise.};
\node[box, right=0.42cm of b2] (b3) {The asymmetry: The pessimistic error in the signal is corrected within the quarter because of buying. The optimistic error is corrected only in part since short selling is closed to some investors. The rest of the optimistic error survives as overpricing.};
\node[box, right=0.72cm of b3] (b4) {The announcement: In quarter $q+1$, the firm reports its earnings from quarter-$q$. The innovation $\epsilon_{f1}$ becomes public.};
\node[box, right=0.42cm of b4] (b5) {The correction: The price falls to the true value revealed. The surviving overpricing is removed. As a results, the losses exceed gains in magnitude.};
\draw[arr] (b1) -- (b2);
\draw[arr] (b2) -- (b3);
\draw[arr] (b3) -- (b4);
\draw[arr] (b4) -- (b5);
\draw[decorate, decoration={brace, amplitude=6pt}]
([yshift=0.25cm]b3.north east -| b1.north west) -- ([yshift=0.25cm]b3.north east)
node[midway, yshift=0.42cm, font=\footnotesize]{Quarter $q$};
\draw[decorate, decoration={brace, amplitude=6pt}]
([yshift=0.25cm]b4.north west) -- ([yshift=0.25cm]b5.north east)
node[midway, yshift=0.42cm, font=\footnotesize]{Quarter $q+1$};
\node[mbox, anchor=north] (m) at ([yshift=-1.45cm]b2.south) {On each trading day of quarter $q$, a dealer takes the other side of informed and uninformed orders and sets the bid-ask spread and the market-impact cost. These two trading costs are the measurement layer from which informed-trading intensity is recovered.};
\draw[dotted, thick] (b2.south) -- (m.north);
\end{tikzpicture}}
\begin{minipage}{0.95\textwidth}\footnotesize\medskip
\textit{Notes:} A firm announces its earnings for quarter $q$ after the quarter ends, with a median lag of 35 days and 40 days for domestic and foreign firms, respectively, so the announcement typically falls in quarter $q+1$. The next-quarter gains and losses are measured over the next quarter in which the announcement falls. In the model, the investors trade once in quarter $q$, at a single price which is set before the earnings announcement. The arrows show the order of events. The dashed box shows the daily dealer market, which runs on each trading day of the quarter. The informed-trading intensity is recovered spread and the market-impact cost. The solid boxes show the mechanism, right from the feed to the overpricing correction.
\par\medskip\noindent Alt text: There are five boxes---three for quarter $q$ and the rest for quarter $q+1$. Box 1: The feed forms in quarter $q$. Box 2: Investors trade at a price that incorporates the public signal and its noise. Box 3: Pessimistic noise is bought away within the quarter, while part of the optimistic noise survives because short selling is costly. Box 4: The firm announces its earnings. Box 5: The surviving overpricing is removed, so losses exceed gains.
The dashed box under box two shows the daily dealer market, where the spread and the price impact are set and from which informed-trading intensity is measured.
\end{minipage}
\end{figure}
A continuum of investors $i$ of unit mass with mean-variance preferences and common risk-aversion coefficient $\bar{\rho}$ chooses share demands $q^i_f$ to maximize $E[W^i\,|\,\mathcal{I}^i] - \tfrac{\bar{\rho}}{2}\mathrm{Var}[W^i\,|\,\mathcal{I}^i]$. Here $\mathcal{I}^i$ is investor $i$'s information set, $W^i_0$ is initial wealth. End-of-period wealth is $W^i = rW^i_0 + \sum_f q^i_f (V^{*}_{f1} - rP^{*}_{f1})$. Investor $i$'s information set for stock $f$ contains four items. The first is the prior on next period's dividend. The cash-flow history and the common component fix its center. Its one uncertain part is the firm-specific innovation $\epsilon_{f1}$, so the prior precision is $\tau_d \equiv \Sigma_{jd}^{-1}$. The second is a private signal of precision $k^i$. A private signal is a noisy observation of the innovation that investor $i$ alone receives, for example from proprietary research or soft information. Its realized value is $\eta^i_{f1} \equiv \epsilon_{f1} + e^i_{f1}$, where the idiosyncratic noise $e^i_{f1} \sim N(0, 1/k^i)$ is independent across stocks and investors.
A higher $k^i$ implies less idiosyncratic noise. Thus, the precision $k^i$ measures how well informed investor $i$ is. The third is the equilibrium price $P^{*}_{f1}$. The private signals enter the price through their average. Hence, observing the price gives an investor additional information about the innovation. The precision of that additional information is $\tau_p$ (equation~\eqref{eq:taup}). The fourth is a public signal extracted from the firm's Twitter feed. Section~\ref{sec:model_sent} 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 $k^i_j$ denote investor $i$'s private-signal precision for the stocks in group $j$. The aggregate private precision in group $j$ is then given by: $K_j = \int_0^1 k^i_j\, di$, which is the integral of that precision over the unit mass of investors. $K_j$ is the total private-signal precision the price aggregates from those investors. It measures how much private information about the firm's innovation the market as a whole holds. It is not a count of investors. The aggregate is high when many investors are informed and each informed investor knows a great deal. The model does not need these two margins separately. $K_j$ is the informed-trading intensity of group $j$.\footnote{It is the model's counterpart of the probability of informed trading which \citet{easley1996liquidity} estimate from trade data. Reduced-form studies often use the share of a stock held by institutional investors for the same object.} The empirical analysis recovers $K_j$ from the trading costs set in a dealer market that operates on each trading day of the quarter (Section~\ref{sec:model_micro}).\footnote{In the data, the liquidity related variables are built from daily measures and averaged within the quarter, so the dealer market in the model is at the daily frequency too.} Each of the four information sources mentioned above has a precision. The prior has a precision of $\tau_d$. The price has a precision of $\tau_p$. The private signals have a precision of $K_j$. The public signal has a precision of $\tau_s$. Thus, the total precision is $T_j \equiv \tau_d + \tau_p + K_j + \tau_s$. The posterior variance of the innovation is $\overline{\Sigma}_j = 1/T_j$. The prior precision $\tau_d$, the price-signal precision $\tau_p$, and the informed-trading intensity $K_j$ take one value for each stock group $j$. The precision of the public signal $\tau_s$ is obtained for each stock-quarter (Section~\ref{sec:model_sent}). Where a group value of $\tau_s$ is needed, the estimation uses the within-group median. We keep the group subscript on $K_j$, $T_j$, and $\overline{\Sigma}_j$. We suppress it on $\tau_d$, $\tau_p$, and $\tau_s$ except when a contrast across groups is needed. In the estimation, the precisions take one value for each geography at each informed-trading-intensity tercile, six stock groups in all (Section~\ref{sec:model_micro}).
\subsection{Social media as a public signal}\label{sec:model_sent}
The Twitter feed is public, available through the vendor's aggregate metrics, and carries the public signal in the model. Proposition~\ref{prop:vaqsv} shows how it is constructed from the quarter's daily readings. The estimates specifically characterize the Twitter feed rather than public information in general.\footnote{This is a statement about the scope. We neither claim that the feed captures all public information relevant to pricing nor that contributor composition on Twitter matches that on other platforms. Information on other venues lies outside the scope of the model. Some of the prior work in this area supports treating a single venue's sentiment as an informative public signal. For instance, \citet{Sprenger2013} find that the tweets of users who give better investment advice are retweeted more, a mechanism through which the platform aggregates information. \citet{garcia2013sentiment} and \citet{bartov2018twitter} link media and Twitter sentiment to returns and fundamentals.}
The signal is $s_{f1} = \epsilon_{f1} + u_{f1}$, where the public noise $u_{f1}$ is normal with mean zero and precision $\tau_s$. Investors condition on this signal.\footnote{The human contributors who post on the feed are investors themselves. The information they post is also the one they trade on. The predictor is the noise in the quarter $q$'s feed, and the outcomes are the gains and losses of quarter $q+1$, which rules out reverse causality. Noise that is corrected inside quarter $q$ also moves its return. We do not use that same-quarter relation for identification.} Private information doesn't show up on the feed. It reaches investors through other channels, such as proprietary research and soft information, and enters the private precision $K_j$ (Section~\ref{sec:model_env}).\footnote{Investors have finite attention. A less attentive investor extracts less from the feed and remains more uncertain about the innovation \citep{sims2003rational, peng2006investor}. In the model, the price depends on the feed's precision multiplied by the share of investors who read the feed. A more precise feed read by a few investors and a less precise feed read by many investors can move the price in the same way. Therefore, sentiment and returns alone cannot tell these two scenarios apart. The precision $\tau_s$ that we recover from the feed does not contain that share. If fewer investors read the foreign feed than the domestic feed, the gap in price-relevant precision is larger than the measured gap (footnote to \ref{hyp:mismatch}). Investor attention affects how much they trade on a signal. For instance, \citet{jiang2022attention} find that anomaly returns are higher after days with high investor attention.}
Three vendor metrics summarize the feed at the firm-quarter level. The volume-adjusted quarterly sentiment volatility $\mathrm{VAQSV}_{fq}$ is the quarter's sentiment volatility measured per tweet. A higher value maps to a less precise public signal.\footnote{The feed reports, for each stock-day, an exponentially time-weighted sentiment total $S_t$ over per-tweet scores in $[-1,1]$ and the tweet count $N_t$, with $q$ indexing quarters and $t$ days within the quarter. These are the stock-level Twitter sentiment metrics of Context Analytics, formerly Social Market Analytics, obtained through Dewey Data \citep{contextanalytics2022data}.} Quarterly sentiment volatility is the within-quarter standard deviation of the daily total, $\mathrm{QSV}_{fq} = \mathrm{SD}_{t \in q}(S_t)$. $\mathrm{VAQSV}_{fq}= \mathrm{SD}_{t \in q}(S_t/N_t)$ is the volume-adjusted quarterly sentiment volatility. Quarterly sentiment volatility, $\mathrm{QSV}_{fq} = \mathrm{SD}_{t \in q}(S_t)$, applies the same operator to the daily total.\footnote{These two volatility measures have correlations of $0.32$ and $0.06$ for the full sample and the primary sample (2018Q1 onward), respectively.} As the day's tweet volume increases, so does $S_t$, which is why the volatility is measured on the per-tweet average $S_t/N_t$. The vendor's total $S_t$ weights recent tweets more than earlier ones. In the model, each per-tweet sentiment score has its own noise, $e$. The noise in a weighted average of $N_t$ such sentiment scores has a larger variance than the noise in a simple average of $N_t$ such sentiment scores, by a factor that depends on those weights. Equation~\eqref{eq:vaqsv} absorbs that factor into the constant $c_w \geq 1$. Further, $T_q$ denotes the number of days in the quarter with at least one scored tweet, weekends included, while $T_{days}$ denotes the number of trading days in the quarter with a return. The quarterly gain and loss measures are obtained by summing over $T_{days}$. For each stock and day, the vendor data gives the dispersion, $D_t = A_t/N_t$, where $A_t$ is the number of distinct accounts that post that day. Average dispersion $\overline{D}_{fq}$ is the mean of $D_t$ over $T_q$. A lower value for dispersion indicates that the feed is generated by fewer accounts. Because the formula never inspects the sentiment scores, $D_t$ measures contributor concentration rather than disagreement of opinion. Additionally, we have the average inverse tweet count, $\overline{(1/N_t)}_{fq} = \tfrac{1}{T_q}\sum_{t \in q} 1/N_t$, which is high when the daily tweet volume is low. A high value means more sampling noise in the quarter's signal.\footnote{There are two vendor methodology changes which should be noted. First, there was a dictionary update in March 2016 which changed $S_t$ across quarters. Our within-quarter moments are robust to these level shifts of $S_t$ across quarters. Second, in 2017, there was a switch from cashtag filtering to a topic-model classifier affecting the selection of tweets which would become part of the feed. To deal with this, our primary estimation sample starts with 2018Q1 (the post change period). We exclude the period prior to 2018Q1 from the estimation.}
For each stock, posts come from three contributor types. 1. The specialists $(\mathbb{S})$: 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 \citep{van2009information}. 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 $\tau_s$. The specialist's private signal (a separate noisy reading of the same innovation with precision $k^i$) stays with the specialist and enters the price through trading alone. 2. Attention-driven contributors $(\mathbb{A})$: Accounts which post a single tweet about a firm on a day \citep{barber2008all}. 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 $(\mathbb{B})$: 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.\footnote{Twitter's disclosures put false or spam accounts below five percent of its active users, while an independent audit from September 2022 attributed about twenty to thirty percent of U.S. content on the platform to bots. These industry measurements along with their classification error are given in \ref{app:bot_prevalence}.} The vendor's sentiment metrics are aggregated over tweets instead of accounts or contributor types. Hence, each contributor type enters these metrics through its share of tweets, and bots account for a substantial share of the tweets behind them.
$A_t$ accounts post about the stock on day $t$. Among the accounts posting about a stock in group $j$, the share of specialists is $\pi_{\mathbb{S},j}$, the share of bots is $\pi_{\mathbb{B},j}$ and the share of attention-driven contributors is $1-\pi_{\mathbb{S},j}-\pi_{\mathbb{B},j}$. Each type $g$ posts a fixed number of tweets per account per day, its posting intensity $\lambda_g$, with $\lambda_{\mathbb{S}} > 1$, $\lambda_{\mathbb{B}} > 1$ and $\lambda_{\mathbb{A}} = 1$. The two repeat intensities may differ.\footnote{Since the dispersion in the data identifies only the combined share of the two repeat types, the data cannot tell the two rates apart (Proposition~\ref{prop:dispersion}). Hence, the estimation uses a common repeat intensity $\lambda_I$ for both and fixes it at $1.5$ tweets per account per day (Section~\ref{sec:structural}).} Each tweet has a sentiment score. The sentiment scores are $\epsilon_{f1} + e$, $\beta_t + e$ and $\epsilon_{f1} + b_t + e$ for a specialist tweet, a bot tweet, and an attention-driven tweet, respectively. $b_t$ is the fad on day $t$, $\beta_t$ is the bot content on day $t$ and $e$ is the noise of the individual tweet. The firm-specific innovation, $\epsilon_{f1}$, is the part of the firm's next dividend that no investor observes without noise. Each of the model's signals, be it the private signal, the price, or the public signal, is a noisy reading of it (Section~\ref{sec:model_env}). Both a specialist tweet and an attention-driven tweet contain $\epsilon_{f1}$ in their scores. This forms the fundamental part of the score since it is news about the dividend. The fad $b_t$ and the bot content $\beta_t$ don't contain $\epsilon_{f1}$. Hence, they form the nonfundamental parts of a score. Assumption~\ref{assump:dist} states the distributions of these terms.
\refstepcounter{assump}\label{assump:dist}\textbf{Assumption A\arabic{assump}\ (distribution).} The fad $b_t \sim N(0,\sigma_{b,j}^2)$ and the bot process $\beta_t \sim N(0,\sigma_{\beta,j}^2)$ are i.i.d.\ across days. The per-tweet noise $e \sim N(0,\sigma_e^2)$ is i.i.d.\ across tweets and days. The processes $b_t$ and $\beta_t$ and the noise $e$ are mutually independent and independent of $\epsilon_{f1}$.
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 $A_t$ posting accounts, specialists and bot accounts contribute $(\pi_{\mathbb{S},j} + \pi_{\mathbb{B},j})\lambda_I A_t$ tweets and the attention-driven type contributes $(1 - \pi_{\mathbb{S},j} - \pi_{\mathbb{B},j})A_t$ tweets. Therefore, tweets per account equal $\pi_j \lambda_I + (1 - \pi_j)$, with $\pi_j \equiv \pi_{\mathbb{S},j} + \pi_{\mathbb{B},j}$ being the share of accounts that post more than once. We call these specialists and bot accounts repeat authors.
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:dispersion}
~\textit{(Dispersion determines the repeat-author share for a given posting intensity.) Given the feed structure, daily dispersion in group $j$ is $D = \left[\pi_j \lambda_I + (1-\pi_j)\right]^{-1}$ with $\pi_j = \pi_{\mathbb{S},j} + \pi_{\mathbb{B},j}$, strictly decreasing in $\pi_j$, and the inversion}
\begin{equation}\label{eq:pi_inversion}
\pi_j \;=\; \frac{1 - D}{D\,(\lambda_I - 1)}
\end{equation}
\textit{recovers the total repeat-author share from observed dispersion for a given $\lambda_I > 1$, with $\pi_j = 0$ at $D = 1$. Dispersion determines $\pi_j$ as a whole. It does not separate specialists from bot accounts.} \textit{(Proof in \ref{app:proof_dispersion}.)}
Proposition~\ref{prop:vaqsv} 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 $\bar{s}_t$ is the average of the sentiment scores of the day's $N_t$ tweets. We build it up from the three contributor types. Dispersion gives the day's tweet count as $N_t = A_t/D$. The tweet share of a type is its tweets divided by $N_t$, $\omega_{\mathbb{S},j} \equiv \pi_{\mathbb{S},j}\lambda_I D$, $\omega_{\mathbb{B},j} \equiv \pi_{\mathbb{B},j}\lambda_I D$, and $\omega_{\mathbb{A},j} \equiv (1-\pi_j)D$. The three tweet shares sum to $D\,[\pi_j \lambda_I + (1-\pi_j)]$, which equals one by Proposition~\ref{prop:dispersion}. Averaging the sentiment scores within each type gives $\epsilon_{f1} + \bar{e}^{\mathbb{S}}_t$ for specialist tweets, $\epsilon_{f1} + b_t + \bar{e}^{\mathbb{A}}_t$ for attention-driven tweets, and $\beta_t + \bar{e}^{\mathbb{B}}_t$ for bot tweets, where $\bar{e}^{g}_t$ is the average of the per-tweet noises of the $\omega_{g,j} N_t$ tweets of type $g$ and has variance $\sigma_e^2/(\omega_{g,j} N_t)$. The day's per-tweet sentiment equals the three type averages weighted by the tweet shares,
\begin{equation}\label{eq:sbar_body}
\bar{s}_t = \omega_{\mathbb{S},j}\,(\epsilon_{f1} + \bar{e}^{\mathbb{S}}_t) + \omega_{\mathbb{A},j}\,(\epsilon_{f1} + b_t + \bar{e}^{\mathbb{A}}_t) + \omega_{\mathbb{B},j}\,(\beta_t + \bar{e}^{\mathbb{B}}_t) = (1 - \omega_{\mathbb{B},j})\,\epsilon_{f1} + \omega_{\mathbb{A},j}\, b_t + \omega_{\mathbb{B},j}\,\beta_t + \bar{u}_t.
\end{equation}
The loading on the innovation is $\omega_{\mathbb{S},j} + \omega_{\mathbb{A},j} = 1 - \omega_{\mathbb{B},j}$, the sum of the tweet shares of specialists and attention-driven contributors, the two types whose tweets carry the innovation. The term $\bar{u}_t = \omega_{\mathbb{S},j}\bar{e}^{\mathbb{S}}_t + \omega_{\mathbb{A},j}\bar{e}^{\mathbb{A}}_t + \omega_{\mathbb{B},j}\bar{e}^{\mathbb{B}}_t$ is the sampling noise. It has mean zero and variance $\sigma_e^2/N_t$ (\ref{app:proof_vaqsv}). The ratio $S_t/N_t$ of the vendor's sentiment total to its tweet count is the empirical counterpart of $\bar{s}_t$.
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:vaqsv}
~\textit{(Volume-adjusted quarterly sentiment volatility and the precision of the public signal.) Under \ref{assump:dist}, and when the factor $c_{1,t}$ defined below is the same on every day of the quarter, the conditional mean of the squared volume-adjusted quarterly sentiment volatility of a stock in group $j$, given the quarter's firm-specific innovation and the daily tweet counts, is}
\begin{equation}\label{eq:vaqsv}
E\!\left[\,\mathrm{VAQSV}_{fq}^2 \;\middle|\; \epsilon_{f1},\,\{N_t\}_{t\in q}\,\right] \;=\; \underbrace{\omega_{\mathbb{A},j}^2\,\sigma_{b,j}^2 \;+\; \omega_{\mathbb{B},j}^2\,\sigma_{\beta,j}^2}_{\;\Phi_j\;\equiv\;\text{nonfundamental}} \;+\; c_w\, \sigma_e^2\, \overline{\left(1/N_t\right)} \;\equiv\; \bar{v}_{fq},
\end{equation}
\textit{where $\Phi_j$ is the nonfundamental variance. The daily sampling variance of the per-tweet average $S_t/N_t$ is $c_{1,t}\sigma_e^2/N_t$, $c_{1,t}=1$ under equal weights and $c_{1,t}>1$ under the vendor's recency weighting, so that $c_w \geq 1$. Only the sampling-noise coefficient ($c_w\sigma_e^2$) on the average inverse tweet count ($\overline{(1/N_t)}$) enters equation~\eqref{eq:vaqsv}. $\bar{v}_{fq}$ is the average over the quarter's $T_q$ days of the daily conditional variance of the daily per-tweet sentiment. $\mathrm{VAQSV}_{fq}^2$ is unbiased for $\bar{v}_{fq}$ under \ref{assump:dist} and understates it under positive serial dependence across days (\ref{app:proof_vaqsv}). The investors condition on the public signal $s_{f1} = \epsilon_{f1} + u_{f1}$ obtained by averaging $\bar{s}_t$ over the quarter's $T_q$ days and dividing by $1-\omega_{\mathbb{B},j}$. The public noise $u_{f1}$ is the quarterly average of $\omega_{\mathbb{A},j} b_t + \omega_{\mathbb{B},j}\beta_t + \bar{u}_t$ divided by $1-\omega_{\mathbb{B},j}$. It is normally distributed with mean zero as per \ref{assump:dist}. The precision of that signal is}
\begin{equation}\label{eq:tau_s}
\tau_s \;=\; \frac{T_q}{\kappa \cdot \mathrm{VAQSV}_{fq}^2}, \qquad \kappa > 0,
\end{equation}
\textit{where $\kappa$ is the conversion constant that maps the observable signal-quality index $T_q/\mathrm{VAQSV}_{fq}^2$ into the latent precision $\tau_s$. The signal-quality index is the precision if the daily readings were independent, the day's sentiment moved one-for-one with the innovation and the vendor's score were in innovation's units. The data departs from all three of these conditions. The daily readings are serially dependent. Bot tweets don't carry any of the innovation. The day's sentiment therefore moves with the innovation by $1-\omega_{\mathbb{B},j}$ instead of one-for-one. A fixed scale converts the vendor's score in sentiment units to the innovation's cash-flow units. $\kappa$ absorbs these three departures.\footnote{The estimation treats $\kappa$ as one parameter common across firm-quarters (Section~\ref{sec:structural}). The data identify it as a whole and do not separate the three (\ref{app:proof_vaqsv}).} The bot tweet share raises it by lowering the weight on the innovation to $1-\omega_{\mathbb{B},j}$. The squared scale that links the vendor's per-tweet score to the cash-flow innovation enters it as well. It enters because equation~\eqref{eq:sbar_body} normalizes the specialist loading to one.\footnote{The estimation treats $\kappa$ as one parameter common across firm-quarters (Section~\ref{sec:structural}). The data identify it as a whole and do not separate the three components (\ref{app:proof_vaqsv}).}} \textit{(Proofs of equations \eqref{eq:vaqsv} and \eqref{eq:tau_s} in \ref{app:proof_vaqsv}.)}
The group-level precision of the public signal is $\tau_{s,j}$ where $j$ is the stock group introduced in Section~\ref{sec:model_env}. This precision of public signal differs across groups because the tweet shares ($\omega_{g,j}$) and the fad and bot variances ($\sigma_{b,j}^2$ and $\sigma_{\beta,j}^2$, respectively) that enter $\Phi_j$ are group specific (equations~\eqref{eq:vaqsv} and~\eqref{eq:tau_s}).\footnote{We write $\tau_s$ for the precision of the public signal. Where a domestic-foreign contrast is made at a fixed informed-trading-intensity tercile, the two stock groups compared are the domestic and the foreign group at that tercile (Section~\ref{sec:model_micro}).} The observable signal-quality index ($T_q/\mathrm{VAQSV}_{fq}^2$) that gives $\tau_{s,j}$ (equation~\eqref{eq:tau_s}) varies at the stock-quarter level. The estimation aggregates it to the group level (Section~\ref{sec:structural}). The linear regression in Equation~\eqref{eq:vaqsv} recovers the nonfundamental variance ($\Phi_j$) as the intercept and the sampling-noise coefficient ($c_w\sigma_e^2$) as the slope. The structural GMM estimation (Section~\ref{sec:structural}) uses the equations~\eqref{eq:vaqsv} and~\eqref{eq:tau_s} under two conventions. The first one sets the bot-content variance $\sigma_{\beta,j}^2$ to zero. That means that the nonfundamental variance $\Phi_j$ comes solely from the fad. The second one treats $\kappa$ as a parameter common to all firm-quarters.\footnote{Both the conventions and the identification of $\lambda_I$ and $\sigma_{b,j}$ are in \ref{app:vaqsv_estimation}.}
\subsection{Equilibrium prices and informativeness}\label{sec:model_eq}
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 \citet{hellwig1980aggregation} and \citet{admati1985noisy}. 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.
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:price}
~\textit{(Equilibrium price.) There is a linear rational expectations equilibrium in which the price $P^{*}_{f1}$ satisfies}
\begin{equation}\label{eq:price}
rP^{*}_{f1} \;=\; A_f \;+\; B_j\,\epsilon_{f1} \;+\; G_j\,u_{f1} \;+\; C_j\,\tilde{x}_f,
\end{equation}
\textit{with}
\begin{equation}\label{eq:coefs}
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 = -\,\bar{\rho}\,\gamma_j^2\, \overline{\Sigma}_j\, \frac{\tau_p + K_j}{K_j},
\end{equation}
\textit{and price-signal precision}
\begin{equation}\label{eq:taup}
\tau_p \;=\; \frac{K_j^2}{\gamma_j^2\, \bar{\rho}^2\, \Sigma_{jx}},
\end{equation}
\textit{which is independent of $\tau_s$.} \textit{(Proof in \ref{app:proof_price}.)}
Risk-averse investors pay the price $P^{*}_{f1}$ today for a persistent yet uncertain dividend stream, discounted at the gross rate $r$ (Equation~\eqref{eq:price} shows its components). A one-unit increase in the innovation ($\epsilon_{f1}$) raises the period-1 dividend ($d^{*}_{f1}$) by one. The firm's value is $V^{*}_{f1} = \gamma_j\, d^{*}_{f1}$, with $\gamma_j = r/(r-\upsilon_j)$ the present-value multiplier. The increase therefore raises $V^{*}_{f1}$ by $\gamma_j$. This multiplier is larger when the innovation is more persistent (when $\upsilon_j$ is larger). The market holds residual uncertainty about the innovation. Hence, the price reflects the innovation's value only to the degree to which the market learns about it. Investors cannot separate the feed's signal from its error. Hence, the price carries the feed's noise along with its signal. Part of the share supply is random and comes from investors who trade for reasons unrelated to the firm's value. This randomness keeps the price only partly revealing. The price is itself a signal, but it adds information only through the private signals that trading aggregates into it. Its response to the public signal repeats what investors already observe.\footnote{All equilibrium coefficients are derived from first principles (\ref{app:model_proofs}). At $\tau_s = 0$, $B_j$, the total precision $T_j$, and the informativeness decomposition in Proposition~\ref{prop:pinf} reduce to those in \citet{farboodi2022data}. The supply-noise coefficient $C_j$ takes the price-impact form of \citet{hellwig1980aggregation} and \citet{admati1985noisy}.} The quarterly excess return is $R \equiv V^{*}_{f1} - rP^{*}_{f1}$. Its first two moments follow from the equilibrium price (\ref{app:proof_price}). Its mean $E[R] = \bar{\rho}\gamma_j^2 \overline{\Sigma}_j \bar{x}_j$ is the premium that compensates investors for bearing the average supply $\bar{x}_j$. Its variance,
\begin{equation}\label{eq:varR}
\mathrm{Var}[R] \;=\; \gamma_j^2\, \overline{\Sigma}_j^{2} \left[\, \tau_d + \tau_s + \frac{(\tau_p + K_j)^2}{\tau_p} \,\right],
\end{equation}
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 (Section~\ref{sec:model_gl}).
The second question for this subsection is how sentiment affects what the price reveals about the firm's future earnings. Following \citet{bai2016informative} and \citet{farboodi2022data}, stock-level informativeness at horizon $s$ is the covariance of horizon-$s$ earnings with the price, scaled by the price's standard deviation.\footnote{Horizon-$s$ earnings are the dividend $s$ periods after the pricing date, equal to $\upsilon_j^s\, d^{*}_{f1}$ by the dividend law (Section~\ref{sec:model_env}).}
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:pinf}
~\textit{(Price informativeness.) Stock-level price informativeness at horizon $s$ decomposes as}
\begin{equation}\label{eq:pinf}
\mathrm{PINF}_{j,s} \;=\; \underbrace{\frac{\Sigma_{jd}}{\mathrm{SD}(P^{*}_{f1})}}_{\text{volatility}} \;\cdot\; \underbrace{\frac{\upsilon_j^{\,s}}{r - \upsilon_j}}_{\text{growth}} \;\cdot\; \underbrace{\left(1 - \frac{\overline{\Sigma}_j}{\Sigma_{jd}}\right)}_{\text{information}}, \qquad 1 - \frac{\overline{\Sigma}_j}{\Sigma_{jd}} \;=\; \frac{\tau_p + K_j + \tau_s}{\tau_d + \tau_p + K_j + \tau_s},
\end{equation}
\textit{so the information component is strictly increasing in $\tau_s$.} \textit{(Proof in \ref{app:proof_pinf}.)}
The decomposition of price informativeness has three factors. The volatility factor ($\Sigma_{jd}/\mathrm{SD}(P^{*}_{f1})$) is the innovation variance relative to the volatility of the price. A noisier price lowers measured informativeness through this factor. The growth factor ($\upsilon_j^s/(r-\upsilon_j)$) is the ratio of how strongly the innovation appears in horizon-$s$ earnings ($\upsilon_j^s$) to the rate that discounts the persistent earnings stream to present value ($r-\upsilon_j$). Today's news matters for distant earnings because of this persistence. The information factor $1-\overline{\Sigma}_j/\Sigma_{jd}$ is the fraction of the innovation variance that the price, the private signals, and the feed together remove, equal to $(\tau_p+K_j+\tau_s)/T_j$. The signal precision $\tau_s$ enters this factor directly (equation~\eqref{eq:pinf}). It reaches the volatility factor only through the price's standard deviation. In \citet{farboodi2022data}, 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~\eqref{eq:tau_s} measures the signal precision $\tau_s$ from the observable signal-quality index $T_q/\mathrm{VAQSV}_{fq}^2$ up to the conversion constant $\kappa$. The model predicts that informativeness, net of the volatility and growth components, rises with $\tau_s$.
\subsection{Short-sale constraints and the asymmetry between gains and losses}\label{sec:model_gl}
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 $r_{ft}$, $\mathrm{Gain}_{fq} = \sum_{t \in q} \max(r_{ft}, 0)$ and $\mathrm{Loss}_{fq} = \sum_{t \in q} \min(r_{ft}, 0)$. Connecting these outcomes to the quarterly equilibrium requires a statement about how the quarterly excess return accrues over days.
\refstepcounter{assump}\label{assump:daily}\textbf{Assumption A\arabic{assump}\ (daily accrual).} The quarter contains $T_{days}$ trading days. The daily excess returns $r_{ft}$ are i.i.d.\ $N(\mu_d, \sigma_d^2)$ within the quarter. The daily parameters satisfy $T_{days}\mu_d = E[R] + \mu^{drift}$ and $T_{days}\sigma_d^2 = \mathrm{Var}[R]$, where $\mu^{drift}$ is the correction drift (Proposition~\ref{prop:drift}), a predictable component of the mean daily excess return that reverses the previous quarter's overpricing.\footnote{Returns are measured per dollar of the asset's unconditional value. A single positive scale, written $s_P$, converts the model's value units into return units and is estimated as a shared parameter (see Section~\ref{sec:structural}). A positive scale preserves the sign and the monotonicity of all predictions. Under \ref{assump:daily}, the quarterly and daily volatilities are linked by $\sigma_R = T_{days}^{1/2}\,\sigma_d$, where $\sigma_R \equiv \mathrm{Var}[R]^{1/2}$ is the standard deviation of the quarterly excess return $R$.}
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 \ref{assump:daily}, the expected tails are
\[
\begin{aligned}
E[\mathrm{Gain}] &= T_{days}\left[\mu_d \Phi(\mu_d/\sigma_d) + \sigma_d\,\phi(\mu_d/\sigma_d)\right],\\
E[|\mathrm{Loss}|] &= T_{days}\left[-\mu_d \Phi(-\mu_d/\sigma_d) + \sigma_d\,\phi(\mu_d/\sigma_d)\right],
\end{aligned}
\]
where $\Phi$ and $\phi$ are the standard normal distribution function and density (Proof in \ref{app:proof_vol}). Subtracting the two equations gives $E[\mathrm{Gain}] - E[|\mathrm{Loss}|] = T_{days}\,\mu_d$. The increase in daily volatility ($\sigma_d$) increases $E[\mathrm{Gain}]$ and $E[|\mathrm{Loss}|]$ equally and thus cancels in the difference. The gain-loss gap then depends on the mean daily excess return ($\mu_d$) alone. This separation between the mean and the volatility channels is the underlying mechanism for the identification in the empirical exercise. In the data, the daily volatility is the within-quarter standard deviation of daily excess returns. In a regression with the quarter's losses $\mathrm{Loss}_{fq}$ as the outcome, controlling for the daily volatility absorbs the volatility channel. Any remaining effect operates through the mean daily excess return, whose quarterly sum ($T_{days}\mu_d$) contains the correction drift ($\mu^{drift}$) (\ref{assump:daily}, Proposition~\ref{prop:drift}). The quarterly sum also contains the group-level risk premium ($E[R]$), which the group intercepts absorb. The correction drift is then identified from the variation within groups (see Section~\ref{sec:reducedform}).
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:vol}
~\textit{(Volatility channel.) At fixed mean daily excess return $\mu_d$, $E[\mathrm{Gain}]$ and $E[|\mathrm{Loss}|]$ are increasing in the daily volatility $\sigma_d$. Under \ref{assump:daily}, $T_{days}\sigma_d^2 = \mathrm{Var}[R]$, and $\mathrm{Var}[R]$ is strictly decreasing in $\tau_s$ by equation~\eqref{eq:varR}, so a noisier public signal raises both $E[\mathrm{Gain}]$ and $E[|\mathrm{Loss}|]$ symmetrically. The drift channel is Proposition~\ref{prop:drift}. (Proof in \ref{app:proof_vol}.)}
The proposition shows separation of the volatility channel from the mean channel. A noisier feed raises return volatility (equation~\ref{eq:varR}) 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.
\refstepcounter{assump}\label{assump:ss}\textbf{Assumption A\arabic{assump}\ (short-sale constraints).} \textit{(i)} Negative realizations of the signal noise $u_{f1}$ are corrected within the quarter, meaning their price impact is removed by trading before the quarter ends. \textit{(ii)} A fraction $\delta \in [0,1]$ of the price impact of positive realizations of $u_{f1}$ survives the quarter.
A positive realization of the signal noise $u_{f1}$ holds the price above the firm's value (equation~\ref{eq:price}), an optimistic error. A negative realization holds it below, a pessimistic error. The noise $u_{f1}$ is normally distributed with mean zero. Optimistic and pessimistic errors of any given size are equally likely. Without a friction, trading would correct both errors within the quarter. The signal noise would then affect gains and losses only through the volatility channel, which leaves their gap unchanged (Proposition~\ref{prop:vol}). A loss-gain gap therefore requires a friction that blocks the correction on one side. That friction is the short-sale constraint, as in \citet{miller1977risk}. Selling short requires borrowing the stock. Mandates against short sales prohibit it for some investors \citep{almazan2004constrain}, while borrowing costs make it expensive for others \citep{davolio2002market}. Purchasing is not prohibited for any investor. This corrects the pessimistic error. An optimistic error can be corrected only by a sale. The selling pressure comes from the pessimists who hold shares and from the arbitrageurs who can borrow and short. The arbitrageurs have limited capital and limited horizons, the limits to arbitrage \citep{shleifer1997limits}. A pessimist who holds no shares and cannot borrow has no sale to make. Abstaining from a purchase does not push the price down. Thus, part of the overpricing stays in the price. The surviving fraction is $\delta$.\footnote{This pattern has empirical support. Stocks with greater analyst disagreement earn lower subsequent returns. Prices reflect the optimistic valuations when the constrained pessimists do not trade \citep{diether2002differences}.}
The surviving overpricing is corrected at the next earnings date, when the firm's value becomes public. No buyer will then pay more than it. Thus, the price settles at that value. That fall is a negative expected drift in the next quarter's returns. The daily excess returns scatter around a mean that carries this drift (\ref{assump:daily}). 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 $G_j u_{f1}$ by \eqref{eq:price}. Under \ref{assump:ss}, the part that survives is $\delta G_j \max(u_{f1}, 0)$, the price impact of the positive realizations scaled by $\delta$. Expected overpricing is the mean of this surviving part. The expectation $E[\max(u_{f1},0)]$ has a closed form, obtained in the same way as the expected tails with the mean set to zero (\ref{app:proof_drift}).
\refstepcounter{prop}
\textbf{Proposition \arabic{prop}}\label{prop:drift}
~\textit{(Overpricing and its correction.) Under \ref{assump:ss}, expected overpricing is}
\begin{equation}\label{eq:op}
\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},
\end{equation}
\textit{and under \ref{assump:daily} this is corrected as next-quarter drift $\mu^{drift} = -\mathrm{OP}_j$. $\mathrm{OP}_j$ vanishes as $\tau_s \to 0$ and as $\tau_s \to \infty$ and attains an interior maximum at}
\begin{equation}\label{eq:peak}
\tau_s^{*} \;=\; \tau_0 \;\equiv\; \tau_d + \tau_p + K_j .
\end{equation}
\textit{In measured-noise units $w \equiv \mathrm{VAQSV}$, the maximum is at $w^{*} = \sqrt{T_q/(\kappa\,\tau_0)}$. Holding $T_q$, $\kappa$, and the remaining group primitives fixed, the turning point is strictly decreasing in $K_j$.} \textit{(Proof in \ref{app:proof_drift}.)}
Expected overpricing in \eqref{eq:op} is the product of two opposing forces, scaled by the constant $\delta$. The first force is the weight the price places on the public signal, $G_j = \gamma_j\,\tau_s/(\tau_0 + \tau_s)$, from \eqref{eq:coefs} with $\overline{\Sigma}_j = 1/T_j$. This weight rises with the signal precision $\tau_s$. The second force is the mean positive noise $E[\max(u_{f1},0)]$, which equals $\sigma_u/\sqrt{2\pi}$ with $\sigma_u$ the standard deviation of $u_{f1}$. That standard deviation is $1/\sqrt{\tau_s}$, since $u_{f1}$ has precision $\tau_s$, so the second force falls as the first rises. When the feed is nearly uninformative the market ignores it, so even large noise cannot enter the price and there is nothing to correct. When the feed is nearly perfect the market gives it nearly full weight, but it carries almost no noise. Expected overpricing peaks in between, at the point where the precision of the public signal equals the combined precision of all other information, $\tau_0 = \tau_d + \tau_p + K_j$ in \eqref{eq:peak}. At that point the feed supplies exactly half of the market's total precision. The total is $T_j = \tau_0 + \tau_s$. The feed contributes the $\tau_s$ part. At $\tau_s = \tau_0$ the feed's share is $\tau_s/T_j = \tau_0/(2\tau_0) = 1/2$. Below $\tau_0$, an increase in $\tau_s$ raises the weight faster than it shrinks the noise, so expected overpricing rises. Above $\tau_0$, an increase in $\tau_s$ shrinks the noise faster than it raises the weight, so expected overpricing falls. Accordingly, the derivative of expected overpricing in $\tau_s$ is positive below $\tau_0$, zero at $\tau_0$, and negative above it.
Under \ref{assump:ss}(ii), a fraction $\delta$ of the optimistic price impact survives quarter $q$, with expected size $\mathrm{OP}_j$ in \eqref{eq:op}, and is corrected during quarter $q+1$. Accordingly, the drift specifications in Section~\ref{sec:reducedform} relate the quarter $q+1$ correction to quarter $q$ signal noise. Pessimistic noise is corrected within quarter $q$. Its price impact falls on quarter $q$ returns, not on the quarter $q+1$ correction that the drift specifications measure.
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, $|\mathrm{Loss}_{f,q+1}| - \mathrm{Gain}_{f,q+1}$, to current-quarter measured noise first rises and then falls. The peak arrives at lower measured noise for stocks with more informed trading (Proposition~\ref{prop:drift}). For comparisons at a common $T_q$, the ordering of these peaks across groups does not depend on the conversion constant $\kappa$. The second is about sign. The correction comes only from optimistic errors. The next-quarter response is therefore stronger after quarters in which a larger share of the sentiment days were positive.
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. 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~\ref{prop:vol}). It produces neither an interior peak nor a correction that depends on the sign of the noise. Section~\ref{sec:reducedform} tests both predictions against it.\footnote{\ref{app:symmetric} gives the full argument, including how an optimistic error that survives quarter $q$ shows up as gains in quarter $q$ and as losses in quarter $q+1$.}\textsuperscript{,}\footnote{The state dependence of the drift channel is derived in \ref{sec:model_state}.}
\subsection{The daily dealer market and the ordinal intensity index}\label{sec:model_micro}
The level of $\widehat{K}$ for foreign and domestic stocks is not comparable because $\widehat{K}$ is recovered from the spread in the daily dealer market where neither the dealer nor the informed investor conditions on the quarterly price signal, as a model simplification (\ref{app:dealer_detail}). The precision of that omitted signal ($\tau_p = K_j^2/(\gamma_j^2\bar{\rho}^2\Sigma_{jx})$ from equation~\eqref{eq:taup}) depends on $K_j$ and on the group parameters $\gamma_j$ and $\Sigma_{jx}$. Thus, the precision differs between domestic and foreign stocks. A group with more informed trading has a more informative price. Omitting that quarterly price signal loses more information for that group. Thus, this omission shifts the level of $\widehat{K}$ by an amount that differs across groups while leaving the comparative statics of the spread in $K_j$ unchanged. The order of stocks within a group is therefore preserved.
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.\footnote{Controlling for $\widehat{K}$ in a regression would allow comparison of the domestic and foreign stocks at the same value of $\widehat{K}$. These stock groups differ in informed-trading intensity. The regression would attribute that difference to the foreign indicator instead of removing it. As a robustness check, the three liquidity measures of Proposition~\ref{prop:micro} (the lagged quoted spread, the Amihud illiquidity ratio, and turnover) are nonetheless added as controls. The domestic-foreign difference in the loss-on-noise slope survives, at $-0.86$ (Newey-West $t=-2.21$, \ref{app:rob_r14d}).} The empirical work sorts stocks into terciles of informed-trading intensity formed inside each geography-quarter.\footnote{The headline regression (battery row R1, Table~\ref{tab:battery}) is one pooled regression. Its slope of next-quarter losses on lagged measured noise differs by geography and by intensity tercile. This gives six slopes, one for each geography in each of the three terciles (equation~\eqref{eq:r1cells}). The domestic-foreign difference at a tercile is the difference between the domestic and the foreign slope in that tercile. The structural estimation reports the corresponding domestic-foreign contrasts of the repeat-author share by tercile in Table~\ref{tab:h1_contrasts}.} Thus, the empirical work uses $\widehat{K}$ only as an ordinal intensity index. For each stock-quarter, we form a ranking statistic (the previous quarter's quoted spread multiplied by $\sqrt{\tau_d + \widehat{\tau}_s}$, with $\tau_d$ the prior precision estimated for each geography and $\widehat{\tau}_s$ also from the previous quarter). Then within each geography-quarter, we rank stocks on this statistic and cut them into three terciles. Then we estimate $\widehat{K}$ for each geography and tercile, six values in all. The ranking cannot use $\widehat{K}$ itself, which is a single number per geography and tercile. $\widehat{K}$ is estimated only after the terciles are created. Unlike $\widehat{K}$, the ranking does not depend on the estimated scale constants of equation~\eqref{eq:micro}. The recovered $\widehat{K}$ increases across the three terciles (\ref{app:proof_micro}). The domestic-foreign comparison conditions on the intensity tercile from the model and the foreign indicator from the data.
\subsection{Geography and the producer-knowledge mismatch hypothesis}\label{sec:model_geo}
Geography (domestic and foreign) enters the model as a grouping of stocks (Section~\ref{sec:model_env}). 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~\ref{eq:sbar_body} and~\ref{eq:vaqsv}). Nothing else that differs between the two geographies enters the nonfundamental variance (equation~\ref{eq:vaqsv}). A domestic-foreign difference in it therefore reflects a difference in the pool composition. The hypothesis states the direction of that difference.
\refstepcounter{hyp}\label{hyp:mismatch}\textbf{Hypothesis H\arabic{hyp}\ (producer-knowledge mismatch).} \textit{(a)} On English-language Twitter, foreign stocks have a larger nonfundamental variance ($\Phi_j$) than domestic stocks. \textit{(b)} Under a common conversion factor ($\kappa$), the observable signal-quality index ($T_q/\mathrm{VAQSV}_{fq}^2$) is lower for foreign stocks than for domestic stocks.
The hypothesis compares the informativeness of the feed for foreign and domestic stocks. The nonfundamental variance ($\Phi_j$) is the part of the feed's day-to-day variation that comes from the human fads and the automated content together (equation~\ref{eq:vaqsv}). It carries no information about the firm-specific innovation and does not change with the tweet count. When the measurement equation (equation~\ref{eq:vaqsv}) is fitted as a linear regression (Section~\ref{sec:model_sent}), $\Phi_j$ is the intercept. The signal precision ($\tau_s$) measures how sharply the quarter's feed narrows uncertainty about the firm-specific innovation (equation~\ref{eq:tau_s}). A larger nonfundamental variance makes the day's per-tweet sentiment average ($S_t/N_t$) more volatile across the days of the quarter. A more volatile average lowers the observable signal-quality index (holding the quarter's number of scored days and its tweet volume fixed) and delivers a less precise signal at a common conversion factor $\kappa$ (equations~\ref{eq:vaqsv} and~\ref{eq:tau_s}). Thus, a larger nonfundamental variance (the object of part (a) of \ref{hyp:mismatch}) 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 $\tau_s$ for the following reason---$\tau_s$ equals the index divided by the conversion factor $\kappa$ (equation~\ref{eq:tau_s}). The conversion constant $\kappa$ is one constant shared by all stocks (Section~\ref{sec:structural}). Hence, the ordering of the precision of the public signal $\tau_s$ across stock groups is the ordering of the signal-quality index and does not depend on the estimate of $\kappa$. With a common $\kappa$, the ordering of $\tau_s$ across groups is the ordering of the signal-quality index (Section~\ref{sec:model_sent}). Both parts of \ref{hyp:mismatch} compare objects computed from the vendor's daily sentiment total and tweet count alone (Section~\ref{sec:model_sent}).
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 \ref{hyp:mismatch} relates to the nonfundamental variance as a whole, not split by contributor types. The feed has three contributor types (Section~\ref{sec:model_sent}). 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~\ref{eq:vaqsv}). Although automated accounts can be identified in ground-truth samples \citep{gilani2017bots}, human fads cannot be. The data vendor's metrics are aggregated over tweets. Hence, these metrics have no account related identities (Section~\ref{sec:model_sent}). 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 (Section~\ref{sec:reducedform}). 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~\ref{tab:battery}). 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~\ref{tab:battery}).\footnote{Both specifications have quarter fixed effects with standard errors double clustered by firm and quarter.}
For part (a) of \ref{hyp:mismatch}, the evidence is the foreign coefficient in the measurement equation (equation~\ref{eq:vaqsv}). 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 (Section~\ref{sec:model_sent}). The foreign coefficient is the difference in the intercept $\Phi_j$ between foreign and domestic stocks. We report it with and without log capitalization held fixed (Section~\ref{sec:structural}, Table~\ref{tab:size_contrasts}). Since a larger nonfundamental variance lowers the observable signal-quality index at a given number of scored days and tweet volume (equation~\ref{eq:vaqsv}), 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 ($T_q/\mathrm{VAQSV}_{fq}^2$) for foreign and for domestic stocks (Table~\ref{tab:r6_dataterm}).\footnote{Both parts of the hypothesis pertain to the contributors, not to the investors who read the feed. Nonetheless, the number of investors who read the feed matters for prices. The model assumes that every investor observes the public signal. If only a fraction $\theta_j$ of investors observe it, the price responds to the signal as if its precision were $\theta_j\tau_s$ rather than $\tau_s$ (\ref{app:theta_lemma}). Home-market investors of a foreign firm, who tend to possess an informational advantage about that firm, may not read its English-language feed. Hence, a smaller $\theta_j$ for foreign stocks is plausible. Two consequences follow for the hypothesis. First, $\theta_j$ does not enter the nonfundamental variance or the signal-quality index, which are computed from the vendor's daily sentiment total and tweet count. Thus, parts (a) and (b) are unaffected. Second, let $\theta_D$ and $\theta_F$ denote the fractions for domestic and for foreign stocks, and $\tau_{s,D}$ and $\tau_{s,F}$ the corresponding precisions of the public signal. If $\theta_F \le \theta_D$, then the ratio of the domestic to the foreign signal-quality index understates the ratio of the precisions the price responds to, $\theta_D\tau_{s,D}/(\theta_F\tau_{s,F})$ (\ref{app:theta_lemma}). A smaller $\theta_F$ can then only widen the gap that part (b) states.}
\section{Data}\label{sec:data}
\subsection{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, \citealp{contextanalytics2022data}) 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\%.\footnote{Table~\ref{tab:composition_country} in \ref{app:robustness} reports the sample composition by headquarters country.}\textsuperscript{,}\footnote{Calibrated constants and dependence conventions are in \ref{app:conventions}.}
\section{Structural Estimation}\label{sec:structural}
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. Section~\ref{sec:reducedform} reports the validation exercises. Table~\ref{tab:exercise_map} lists all the exercises with their classification.
\subsection{Estimation design}
We estimate the model in one step by the generalized method of moments on ninety-one closed-form moments.\footnote{Since the model moments have closed forms, a simulated method of moments would add simulation error without any gain. Maximum likelihood would require a joint density for every data object behind the moments and would score each daily return against the normal distribution of Assumption~\ref{assump:daily}. The method of moments uses this normality through expected gain and expected loss and leaves the other moment blocks distribution free.} The ninety-one moments are the model-implied values of the data moments (\ref{sec:model_to_estimation}), which cover the liquidity measures, the firm's cash flows, price informativeness, dispersion, squared measured noise, and the quarter's gains and losses. Using all of them has the same two advantages as in the microstructure inversion. The first advantage is that the measurement error in any one moment does not pass into the estimates because each parameter is estimated from every moment block jointly. The second advantage is that it allows testing of the relations which the model imposes upon the moments. A parameter vector that reproduces all 91 moments can exist only if the data satisfies each one of the relations which the model imposes upon the moments. Those relations hold for the population moments. The data moments are sample estimates of the population moments and, in a finite sample, carry sampling error. Hence, some amount of misfit remains even when the model is correctly specified. A test of over-identifying restrictions checks whether the misfit that remains at the estimates is larger than what the sampling error alone would produce. It shows that the misfit is indeed larger. The test (forty-six degrees of freedom for forty-five free parameters) rejects the null that all ninety-one relations hold (\ref{app:fit_detail}). Thus, the model is misspecified in at least one relation, which matters for the choice of weighting matrix. Under correct specification the probability limit of the estimate is the true parameter vector, irrespective of the weighting matrix. Under misspecification it is the parameter vector that minimizes the weighted distance between the model moments and the data moments, which depends on the weighting matrix. Thus, the weighting must be fixed in advance rather than chosen from the fit \citep{hall2003large}. 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 (Section~\ref{sec:model_micro}), 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.\footnote{The measurement conventions behind each moment block and the starting values are shown in \ref{app:estdetail}. \ref{app:estdetail} also shows the saturation criterion, which excludes any candidate estimate at which the model's gain and loss moments underflow to zero.}
\subsection{Headline estimates and the joint Wald test}
Table~\ref{tab:theta_hat} 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 (\ref{app:basin_detail}). The repeat-author shares are identified from contributor dispersion (Proposition~\ref{prop:dispersion}) and the fad scale from the nonfundamental variance (equation~\ref{eq:vaqsv}). 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 ($t=-4.34$). The nonfundamental variance for foreign stocks remains higher than that of domestic stocks ($t=6.12$) (Table~\ref{tab:size_contrasts}). To translate the dispersion into repeat-author shares (Proposition~\ref{prop:dispersion}) 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 (\ref{app:headline_detail}). 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 \ref{app:composition_check}. 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, \ref{app:vaqsv_estimation}) 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 ($p = 7.2\times 10^{-10}$), 70.38 with standard errors clustered by quarter ($p = 1.9\times 10^{-14}$) and 27.05 with standard errors double clustered by firm and quarter ($p = 1.9\times 10^{-5}$) (Table~\ref{tab:h1_contrasts}). To make sure that the value 1.5 does not drive these findings, we repeat the estimation at eight different values of the posting intensity from 1.2 to 1.85. The feed for foreign stocks maintains the higher share of repeat-authors and fad scale at every such value of posting intensity.\footnote{For each of these eight values of posting intensity, the joint Wald test rejects the null that the three shares of repeat-authors and the fad scale are equal across geography, when standard errors are clustered by firm, when standard errors are clustered by quarter and when the standard errors are double clustered by firm and quarter. The largest value among these $p$ values is $1.0\times 10^{-4}$ (Table~\ref{tab:lambda_profile}). This shows that the result that the foreign feed has the higher repeat-author shares and the higher fad scale is statistically significant at every value of the posting intensity in this range and robust to clustering variations. \ref{app:headline_detail} reports the identification diagnostics.}
\begin{table}[!htbp]\centering
\begin{small}
\caption{Headline structural parameter estimates with asymptotic sandwich standard errors and firm-block bootstrap intervals}\label{tab:theta_hat}
\begin{tabular}{lcc >{\raggedright\arraybackslash}p{6.2cm}}
\hline\hline
Parameter & Estimate & Sandwich SE & Bootstrap summary \\
\hline
$\pi$, Domestic T1 & 0.1744 & 0.0020 & 95\% interval: [0.1712, 0.1775] \\
$\pi$, Domestic T2 & 0.1573 & 0.0019 & 95\% interval: [0.1542, 0.1604] \\
$\pi$, Domestic T3 & 0.1447 & 0.0018 & 95\% interval: [0.1417, 0.1472] \\
$\pi$, Foreign T1 & 0.1820 & 0.0050 & 95\% interval: [0.1744, 0.1895] \\
$\pi$, Foreign T2 & 0.1637 & 0.0045 & 95\% interval: [0.1566, 0.1710] \\
$\pi$, Foreign T3 & 0.1698 & 0.0047 & 95\% interval: [0.1616, 0.1769] \\
$\sigma_b$, domestic & 0.02732 & 0.00031 & 95\% interval: [0.02686, 0.02775] \\
$\sigma_b$, foreign & 0.02927 & 0.00041 & 95\% interval: [0.02858, 0.02985] \\
$\sigma_e^2$ & 0.0001642 & 1.70e-05 & 95\% interval: [0.0001377, 0.0001878] \\
$s_P$ & 1.564 & 0.0574 & 95\% interval: [1.479, 1.721] \\
$\kappa$ & 394.2 & 34.2 & 95\% interval: [294, 490] \\
\hline
\addlinespace
$\delta$, at its zero lower bound & 0.0003728 & -- & Same value in each of the 399 replications \\
\hline\hline
\end{tabular}
\begin{minipage}{0.95\textwidth}\footnotesize\medskip
\textit{Notes:} 45 parameters are estimated using a one-step GMM with 91 moments while the posting intensity is set at 1.5 (Section~\ref{sec:structural}). 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 (Section~\ref{sec:structural}). 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 ($\delta = 1/(1+e^{-z})$ with $z$ unrestricted) is strictly between 0 and 1. It approaches 0 only as $z$ decreases without bound. At the reported value ($\delta = 0.0003728, z = -7.89$), the GMM objective still increases in $\delta$. Thus, the estimate lies at its zero lower bound. The reported value is where the optimizer stopped along the transform. Each of the 399 replications starts from the point estimate and stops at the same value. Since the delta method does not apply to an estimate at the edge of its parameter space, no standard error is reported for $\delta$. Treating the posting intensity as free in the covariance (at the same point estimates) multiplies the standard errors of the repeat-author shares by 7.9 to 18.7. Dispersion identifies the shares only jointly with the posting intensity (Section~\ref{sec:structural}).
\end{minipage}
\end{small}
\end{table}
All four domestic-foreign contrasts in Table~\ref{tab:h1_contrasts} 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.
\begin{table}[!htbp]\centering
\begin{small}
\caption{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}\label{tab:h1_contrasts}
\begin{footnotesize}\setlength{\tabcolsep}{4pt}
\begin{tabular}{lcccc}
\hline\hline
Object & Estimate & Bootstrap median & \shortstack{95\% interval\\or $p$ value} & \shortstack{Share of\\replications $<0$} \\
\hline
$\pi$ contrast, tercile 1 & -0.00766 & -0.00778 & [-0.01565, 0.00032] & 0.972 \\
$\pi$ contrast, tercile 2 & -0.00638 & -0.00647 & [-0.01361, 0.00127] & 0.950 \\
$\pi$ contrast, tercile 3 & -0.02512 & -0.02454 & [-0.03260, -0.01723] & 1.000 \\
$\sigma_b$ contrast & -0.00195 & -0.00195 & [-0.00247, -0.00139] & 1.000 \\
Joint Wald, clustered by firm & 48.56 & -- & $p = 7.2\times 10^{-10}$ & -- \\
Joint Wald, clustered by quarter & 70.38 & -- & $p = 1.9\times 10^{-14}$ & -- \\
Joint Wald, double clustered & 27.05 & -- & $p = 1.9\times 10^{-5}$ & -- \\
\hline\hline
\end{tabular}
\end{footnotesize}
\begin{minipage}{0.95\textwidth}\footnotesize\medskip
\textit{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 \citep{cameron2011robust}.
\end{minipage}
\end{small}
\end{table}
The measurement contrasts behind the concentration and fad estimates also survive direct size conditioning (\ref{app:robustness}, Table~\ref{tab:size_contrasts}, battery row R16). Two tests locate the repeat-author margin. Quarters with a narrower contributor pool coincide with more trading, which is consistent with specialists who post repeatedly and also trade, rather than with automated accounts that post without trading (battery row R13b, Table~\ref{tab:battery}). A direct test provides no evidence of a systematic differential automated-content channel (battery row R13a, Table~\ref{tab:battery}). Its dispersion coefficient is $1.6\times 10^{-5}$, with a 95 percent confidence interval from $-2.5\times 10^{-4}$ to $2.8\times 10^{-4}$. A non-rejection of this kind does not establish that the channel is small. The evidence is consistent with a substantial human repeat component, while the specialist and automated decomposition remains unidentified. We place no number on the automated share. The detection accuracy the literature reports cannot support one.\footnote{The break-date analysis and the event-window checks are in \ref{app:battery}.}
\subsection{The conversion constant}
The estimate of the conversion constant ($\kappa$), which links the observable signal-quality index to the precision of the public signal ($\tau_s$), is 394 (SE 34, 95\% CI [294, 490], Table~\ref{tab:theta_hat}). Fixing $\kappa$ at 240 or at 565, outside the confidence interval on either side, and re-estimating the other parameters worsens the fit by only 0.7 and 0.4 percent (Table~\ref{tab:kappa_profile}). $\kappa$ absorbs the three departures of the data from the conditions under which the signal-quality index would equal $\tau_s$, namely serial dependence of the daily readings, the attenuation of the day's sentiment by automated content, and the scale of the vendor's per-tweet sentiment score relative to the firm's cash-flow innovation (Section~\ref{sec:model_sent}). 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 (\ref{app:serial_dependence}). The remainder of $\kappa$ reflects the scale of the vendor's score and the attenuation by automated content, which the data do not separate (\ref{app:proof_vaqsv}). Thus, we don't calibrate an automated-account level. The sampling uncertainty in the estimate of $\kappa$ affects none of the domestic-foreign comparisons of signal quality. Those comparisons are ratios of $\tau_s$ across groups, in which $\kappa$ (common to all groups) cancels and leaves a ratio of the signal-quality index. The slopes of gains and of losses on the lagged signal-quality index are estimated with the index in logarithms, where $\kappa$ enters only as an intercept. Only the level of $\tau_s$ depends on $\kappa$. None of our findings uses that level.\footnote{The profile and a pre-specified check of the shape of the loss and variance responses that the estimates render uninformative are shown in \ref{app:kappa_detail}.}
\subsection{The short-sale friction}
In the model, a fraction ($\delta$) of the overpricing from an optimistic error in the public signal survives the quarter because of the short-sale friction. This surviving fraction of overpricing is corrected in the next quarter and it manifests as an excess of losses over gains (Proposition~\ref{prop:drift}). The estimate of $\delta$ lies at its zero lower bound (Table~\ref{tab:theta_hat}). With $\delta$ at its zero lower bound, the model is able to explain the next-quarter gains and losses without the mechanism involving correction of surviving overpricing. The model moments for the means of next-quarter gains and losses and for their slopes on the lagged signal-quality index depend on the quarter's return variance (the volatility channel) and on the mean daily excess return (the group-level risk premium plus the correction drift, \ref{assump:daily}). The correction drift is proportional to $\delta$. With $\delta$ at its zero lower bound, just the return variance and the risk premium match these moments to their data counterparts.
The zero lower bound doesn't affect the concentration and fad contrasts, which are identified from contributor dispersion and the nonfundamental variance (Section~\ref{sec:structural}), or the reliability gap, which is reduced-form evidence (battery row R14, Table~\ref{tab:battery}).
\subsection{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 (Section~\ref{sec:structural}). At the headline estimate (Table~\ref{tab:theta_hat}), 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 $J$ test rejects the null hypothesis that these 46 restrictions hold at the 5 percent level (\ref{app:fit_detail}). Table~\ref{tab:perblock_j} 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 $t$ statistic (the misfit of a moment divided by the moment's bootstrap standard error). The liquidity block holds 24 moments. Each is the mean of one of the four liquidity measures of \ref{sec:model_to_estimation} (the quoted spread, the high-low spread, the Amihud ratio, and turnover) in one of the six geography-tercile groups (domestic and foreign stocks at each of the three informed-trading-intensity terciles). The weighted squared misfit of these 24 moments makes up 84.8 percent of the objective. The largest $t$ statistic among them is 28.5 (a miss of 28.5 bootstrap standard errors on one moment). Most of the liquidity block's misfit is due to the restriction that a single scale constant ($a_s$) links the model's spread to the quoted spread in all six groups (equation~\eqref{eq:micro}). If we let $a_s$ differ across the six groups (which adds five free parameters) the objective falls from 4,338 to 1,588. The misfit raises two questions. First, whether the sorting of stocks into informed-trading-intensity terciles is contaminated by it. The answer is negative since the terciles are assigned before the joint estimation, on a ranking statistic which is the lagged quoted spread multiplied by the square root of prior precision plus lagged signal precision (\ref{app:estdetail}). The scale constants ($a_s$, $a_{cs}$ and $a_\lambda$ of \ref{sec:model_to_estimation}) do not enter this ranking statistic. Hence, the joint estimation cannot feed back into the tercile assignment. The ranking statistic does rest on the quoted spread, one of the four liquidity measures. Hence, battery row R10 re-assigns the terciles from the high-low spread and under the adverse-selection bounds on the quoted spread and re-runs the headline regression of battery row R1 (\ref{app:robustness}). The domestic-foreign difference in the response of next-quarter losses to lagged measured noise in the lowest intensity tercile (equation~\eqref{eq:r1cells}) keeps its sign under each of these re-assignments. The ranking statistic also carries model content through the multiplication of the lagged quoted spread by the square root of prior precision plus lagged signal precision. Therefore, battery row R14C sorts stocks into terciles on the lagged quoted spread alone and re-computes the reliability gap (the domestic-foreign difference in the share of quarters with a negative slope of losses on lagged measured noise, battery row R14, Table~\ref{tab:battery}) within those spread terciles. The gap is positive in each of the three spread terciles and significant in the middle one (battery row R14C, Table~\ref{tab:spread_sort}). Second, whether the misfit spills into the parameters that carry the findings (the repeat-author shares, $\pi_j$ and the fad scales, $\sigma_{b,j}$). It does not because excluding the liquidity block from the moment system leaves the domestic-foreign contrasts in the repeat-author share by tercile and in the fad scale unchanged up to five decimals. If we let $a_s$ differ across groups, the domestic-foreign contrasts move by less than 12 percent and change no sign (Table~\ref{tab:liq_robust}). Foreign stocks also keep the higher repeat-author share in each tercile and the higher fad scale under the Ledoit-Wolf-shrunk full-matrix weighting (Table~\ref{tab:variants}). The raw liquidity regressions show the same limitation from the reduced-form side. The quoted spread and the Amihud illiquidity ratio fall with lagged noise ($-0.034$, $t=-5.41$, and $-6.30$, $t=-3.88$, battery row R5, Table~\ref{tab:battery}). Both signs run against the adverse-selection prediction that spreads widen as the public signal grows noisier (\ref{app:dealer_detail}). Per-day turnover does not absorb the pattern. Price informativeness is measured at four horizons (the covariance of the price with earnings one to four quarters ahead, Proposition~\ref{prop:pinf}). For foreign stocks in the second and third informed-trading-intensity terciles, the price informativeness is negative at each of the four horizons. For foreign stocks in the first tercile, the price informativeness is positive at two horizons and negative at the other two (Table~\ref{tab:r6_dataterm}). 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 (\ref{app:fit_detail}). If we remove the informativeness block from the moment system, it moves the repeat-author shares and the fad scales by less than $10^{-5}$ (Table~\ref{tab:variants}). Thus, the concentration and fad contrasts do not depend on this block.\footnote{\ref{app:fit_detail} reports the decomposition of the objective by block, the informativeness data term, and the variance-bridge diagnostics.}
\begin{table}[!htbp]\centering
\caption{Concentration and fad contrasts under liquidity-block robustness}\label{tab:liq_robust}
\begin{tabular}{@{}lccc@{}}
\hline\hline
Domestic-foreign contrast & Headline & \shortstack{Liquidity block\\excluded} & \shortstack{Common spread\\scale relaxed} \\
\hline
$\pi$ contrast, tercile 1 & $-0.00766$ & $-0.00766$ & $-0.00678$ \\
$\pi$ contrast, tercile 2 & $-0.00638$ & $-0.00638$ & $-0.00701$ \\
$\pi$ contrast, tercile 3 & $-0.02512$ & $-0.02512$ & $-0.02594$ \\
$\sigma_b$ contrast & $-0.00195$ & $-0.00195$ & $-0.00198$ \\
\hline\hline
\end{tabular}
\begin{minipage}{0.95\textwidth}\footnotesize\medskip
\textit{Notes:} The first three rows show the domestic-foreign contrasts in the repeat-author share ($\pi$) for each informed-trading-intensity tercile in that order. The last row is the domestic-foreign contrast in the fad scale ($\sigma_b$). When a contrast is negative, foreign stocks have the higher value. The headline column is the same as the estimate column of Table~\ref{tab:h1_contrasts}. The middle column shows the contrasts re-estimated with the 24 liquidity moments (the quoted spread, the high-low spread, the Amihud ratio, and turnover for each of the six geography-tercile groups) excluded from the moment system, with the free parameters re-estimated on the remaining 67 moments. They equal the headline contrasts up to five decimal places. The parameters that the liquidity moments identify (the three scale constants, $a_s$, $a_{cs}$, $a_\lambda$, and the six noise-trading standard deviations, $\sigma_{x,j}$), and the conversion constant $\kappa$, are held at their headline values. In the last column, the single scale constant $a_s$ that links the model's spread to the quoted spread in all six groups (equation~\eqref{eq:micro}) is replaced by one $a_s$ per group and all parameters are re-estimated. Both variants start from the headline estimate with the posting intensity set at 1.5. The variants that remove the informativeness block or use the Ledoit-Wolf-shrunk full-matrix weighting are in Table~\ref{tab:variants}.
\end{minipage}
\end{table}
\section{Reduced-Form Evidence and Falsification}\label{sec:reducedform}
The parameter estimates in Section~\ref{sec:structural} 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~\ref{sec:data}.\footnote{Since the standard errors are clustered by quarter and firm for a panel which spans only 27 quarters, the cluster-robust standard errors can understate sampling variation and the $t$ tests can over-reject. Therefore, for five of the battery's headline coefficients, we also mark the $p$ values from the wild-cluster bootstrap with Rademacher weights \citep{cameron2008bootstrap} (which remains valid with few clusters) with stars (Table~\ref{tab:battery}). Each of the five values is below 0.05.} Wherever the lagged sentiment enters the regressions, the lag is formed after the sample is cut at 2018Q1, which ensures that no sentiment value from before the vendor's methodology change enters any regression. The specification used is,
\begin{equation}\label{eq:rfspec}
Y_{f,q+h} \;=\; \beta\, X_{fq} \;+\; \Gamma' Z_{fq} \;+\; \alpha_q \;+\; \eta_{fq},
\end{equation}
where $X_{fq}$ is the row's sentiment regressor measured in quarter $q$, $Y_{f,q+h}$ is the row's outcome measured $h$ quarters later, the vector $Z_{fq}$ collects the row's stated controls and conditioning, and $\alpha_q$ are quarter fixed effects. The horizon is $h=1$ for every row in which sentiment enters lagged (the prediction rows and the two liquidity rows, battery row R5, Table~\ref{tab:battery}). The horizon is $h=0$ for the two rows in which the outcome and the regressor are measured in the same quarter (battery rows R13a and R13b, Table~\ref{tab:battery}). For the rows that use the announcement-aligned outcome, the outcome quarter is the calendar quarter of the earnings announcement that follows the quarter for which sentiment is measured (the next quarter for most firm-quarters, Table~\ref{tab:seams}).\footnote{Two rows depart from this specification. Battery row R14 (Table~\ref{tab:battery}) runs one cross-sectional regression of next-quarter losses on lagged measured noise per quarter and geography, and tests the domestic-foreign difference in the share of quarters with a negative slope by a circular block bootstrap over quarters. Battery row R12 (Table~\ref{tab:battery}) is not a regression but a rank correlation between two stock rankings, described in \ref{app:battery}. Each row's specification is in Table~\ref{tab:battery_specs}.}
The slope of next-quarter total losses on lagged measured noise differs across the six geography-tercile cells (battery row R1, Table~\ref{tab:battery}). The regressor is interacted with the indicators for geography and for intensity tercile, and each cell has its own intercept,
\begin{equation}\label{eq:r1cells}
Y_{f,q+1} \;=\; \sum_{g\in\{D,F\}}\;\sum_{k=1}^{3}\left(\beta_{gk}\,X_{fq} + \delta_{gk}\right)\mathbf{1}\{f\in gk\} \;+\; \Gamma' Z_{fq} \;+\; \alpha_q \;+\; \eta_{fq},
\end{equation}
where cell $gk$ collects the stock-quarters of geography $g$ in intensity tercile $k$ and $\delta_{gk}$ is its intercept, with one cell intercept normalized. The terciles are formed within each geography-quarter. Thus, cell membership is reassigned every quarter. The vector $Z_{fq}$ consists of the controls only (the FF3 betas, lagged MAX, log capitalization, and buzz). The implied slope $\beta_{gk}$ is the response of next-quarter total losses to lagged measured noise in cell $gk$. The domestic-foreign difference at intensity tercile $k$ is given by $\beta_{Dk} - \beta_{Fk}$.
\begin{table}[!htbp]\centering
\caption{Reduced-form battery headline results}\label{tab:battery}
\begin{tabularx}{\textwidth}{@{}>{\RaggedRight}X>{\RaggedRight}p{0.30\textwidth}>{\RaggedRight\arraybackslash}p{0.20\textwidth}@{}}
\hline\hline
Row & Headline estimate & Statistic \\
\hline
\multicolumn{3}{@{}l}{\textit{Panel A. Main reduced-form evidence}} \\
\addlinespace
Reliability by geography (R14) & Mean quarterly domestic-foreign slope difference, $-1.40$. Domestic share of negative-beta quarters 0.778, foreign share 0.481 & Newey-West $t = -2.99$, share difference $p = 0.016$ \\
Volatility and losses, unconditional levels (R1) & Implied slopes negative in all six cells & Largest $|t| = 1.60$ \\
Dispersion and gains by tercile (R4) & $+0.128^{***}$ in the low tercile & $t = 3.15$ \\
Volume-contamination horse race (R9) & Raw signal $-0.317^{***}$, adjusted $-0.612^{**}$ & $t = -6.02$ and $-2.11$ \\
Consolidated joint regression, losses (R11) & Noise $-1.64^{***}$, dispersion $+1.10^{***}$ & $t = -4.88$ and $+12.7$ \\
\addlinespace
\multicolumn{3}{@{}l}{\textit{Panel B. Mechanism and diagnostic evidence}} \\
\addlinespace
Fad-filtered noise diagnostic (R3FAD) & Squared term negative in all three terciles & Largest $|t| = 0.31$ \\
Drift-isolated level, announcement aligned (R2) & $+0.211$ & $t = 1.15$ \\
State dependence, announcement aligned (R7) & $-23.8$ & $t = -1.98$ \\
Correction after optimism, announcement aligned (R8) & $+0.214$ & $t = 0.21$ \\
Spread on lagged noise (R5) & $-0.034$ & $t = -5.41$ \\
Price impact on lagged noise (R5) & $-6.30$ & $t = -3.88$ \\
Attribution sign test (R13a) & $+1.6\times 10^{-5}$ & $t = 0.12$ \\
Trading coincidence (R13b) & $+8{,}121$ trades & $t = 10.5$ \\
Similarity-index bridge (R12) & Rank correlation $-0.379$ & Diagonal share $0.227$ \\
\hline\hline
\end{tabularx}
\begin{minipage}{0.95\textwidth}\footnotesize\medskip
\textit{Notes:} Each row's specification is in Table~\ref{tab:battery_specs}. Stars mark the wild-cluster bootstrap $p$ value for the five coefficients of rows R4, R9, and R11, with three stars for $p$ below 0.01, two for $p$ below 0.05 and one for $p$ below 0.10.
\end{minipage}
\end{table}
\subsection{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 (battery row R14, Table~\ref{tab:battery}). 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 $t=-2.99$).\footnote{27 of the 28 quarters of the primary sample are used as the first quarter has no in-sample predecessor to supply the lagged regressor. The Newey-West standard error of the mean slope difference allows autocorrelation in the quarterly slope differences up to four quarters. The difference in shares is tested with a circular block bootstrap over quarters, with blocks of four consecutive quarters and 999 draws (the same resampled quarters for domestic and foreign). The mean slope difference is negative and significant at 5 percent when any single sample year is left out (\ref{app:rob_r14e}).} The slope is negative in about four of five quarters for domestic stocks as opposed to one in two for foreign stocks (a domestic-foreign difference in shares of quarters with a negative slope of 0.296, block-bootstrap $p=0.016$). A one-standard-deviation increase in measured noise (0.0099 in the estimation sample) deepens next-quarter losses by 0.64 percentage points of quarterly excess return for domestic stocks. The mean domestic-foreign slope difference of $-1.40$ times the standard deviation of 0.0099 is $-0.0139$ in the loss variable (a next-quarter loss deeper by 1.39 percentage points for domestic stocks relative to foreign stocks). The mean domestic-foreign slope difference survives in sign after we add a richer control set consisting of the lagged quoted spread, the Amihud illiquidity ratio, and turnover to the size and factor-beta controls ($-0.86$, Newey-West $t=-2.21$, \ref{app:rob_r14d}). Figure~\ref{fig:r14_reliability} 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.
The reliability gap could reflect a domestic-foreign difference in informed-trading intensity rather than a difference in the feeds (Section~\ref{sec:model_micro}). 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 (share for domestic stocks is 0.852 and for foreign stocks is 0.556, block-bootstrap $p=0.004$). The difference for tercile 2 is 0.148 and is not significant. In tercile 3, the difference closes to $-0.037$ and is not significant. Hence, informed-trading intensity does not explain the gap away, but the gap is confined to tercile 1. Across terciles 1 to 3, the share of quarters with a negative slope is 0.852, 0.889, and 0.741 for domestic stocks (23, 24, and 20 of 27 quarters), and 0.556, 0.741, and 0.778 for foreign stocks (15, 20, and 21 of 27 quarters). The differences between adjacent terciles within a geography are not tested. 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~\ref{tab:h1_contrasts} is widest in tercile 3. The model ties 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 ($\tau_0$ in equation~\eqref{eq:peak}) is the sum of the prior precision ($\tau_d$), the price-signal precision ($\tau_p$), and the informed-trading intensity ($K_j$), and it takes one value for each geography at each tercile. Investors give less weight to the feed for a stock about which their outside knowledge is stronger. Hence, the price responds less to the feed ($G_j$ in equation~\eqref{eq:coefs}). The feed's noise then matters less for the stock's next-quarter losses (Section~\ref{sec:model_micro}). 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 $\tau_0$ varies across the terciles. Holding the other group primitives fixed, the precision of outside knowledge rises with the informed-trading intensity (equation~\eqref{eq:peak}). The ranking statistic of Section~\ref{sec:model_micro} 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.
We classify the foreign stocks using three categories---by listing type (ADRs and directly listed shares), by headquarters language (English and non-English), and by country (Canada and other foreign). Only the separation by listing type shows a significant difference. The share of quarters with a negative slope is 0.296 for ADRs against 0.593 for directly listed foreign shares (a difference of $-0.296$, block-bootstrap $p=0.022$). The price of an ADR in the U.S. is linked to its home-market share price through the depositary link, which can weaken the relation between the feed's noise and next-quarter losses for reasons unrelated to information.\footnote{The headquarters-language and country differences are $-0.074$ each and are not significant. The split cells and differences are in Table~\ref{tab:reliability_splits} (\ref{app:reliability_splits}). The listing-type mean slopes are in \ref{app:rob_r14d}.}\textsuperscript{,}\footnote{There are other alternatives to informed-trading intensity as the conditioning variable, such as analyst coverage, institutional ownership, and media coverage. Analyst coverage requires IBES data, institutional ownership requires 13F holdings records and the media coverage requires a news archive. Since the licensed access to this data is not available, we cannot test these alternatives directly.} The reliability gap survives within informed-trading-intensity tercile 1 and with log capitalization controlled in every cross-section. These two results leave the alternatives less room, but we do not test them.
\begin{figure}[!htbp]\centering
\caption{Quarterly reliability of the sentiment-loss relation by geography}\label{fig:r14_reliability}
\includegraphics[width=0.85\textwidth]{fig_r14_reliability.pdf}
\begin{minipage}{0.85\textwidth}\footnotesize\medskip
\textit{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 ($-1.40$, Newey-West $t=-2.99$) at four lags (battery row R14, Table~\ref{tab:battery}).
\par\medskip\noindent Alt text: The two panels cover 2018 to 2024. The top panel plots the quarterly loss-on-noise slope for domestic stocks as a solid line and for foreign stocks as a dashed line. The domestic line is negative in more quarters. The bottom panel shows bars for the domestic-foreign difference by quarter. The dashed line shows the mean of $-1.40$ (Newey-West $t=-2.99$).
\end{minipage}
\end{figure}
\subsection{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 lagged noise in the headline loss regression is negative in all six geography-tercile cells. However, no cell slope is individually significant (battery row R1, Table~\ref{tab:battery}). The slope of next-quarter losses on lagged noise, in a pooled specification with dispersion and the full set of control variables, is negative and significant (battery row R11, Table~\ref{tab:battery}). 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 (battery row R2, Table~\ref{tab:battery}). Thus, the response of next-quarter losses to current-quarter measured noise works through return volatility instead of through a lower mean return. A less precise public signal leads to more swings in price, which show up as larger losses. The coefficients on measured noise and dispersion have opposite signs in the joint regression (battery row R11, Table~\ref{tab:battery}). Thus, while a noisier signal 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 breadth of the feed's contributors. The paper treats them as distinct channels.
\subsection{The correction channel in reduced form}\label{sec:rf_correction}
In the model, the expected overpricing that survives a quarter is proportional to $\sqrt{\tau_s}/(\tau_0+\tau_s)$ (equation~\eqref{eq:op}), which varies with the precision of the public signal (equation~\eqref{eq:peak}). For each stock-quarter we evaluate this factor with $\tau_0$ at the headline estimate of the group's precisions ($\tau_d$, $\tau_p$ and $K_j$) and with $\tau_s$ at the stock's estimated precision of the public signal in the previous quarter (which is equal to the previous quarter's signal-quality index divided by the headline estimate of the conversion constant, equation~\eqref{eq:tau_s}). 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,
\begin{equation}\label{eq:driftproj}
\bigl|\mathrm{Loss}_{f,q+1}\bigr| - \mathrm{Gain}_{f,q+1} \;=\; b_3\,\mathrm{Index}_{fq} \;+\; \Gamma' Z_{fq} \;+\; \alpha_q \;+\; \gamma_g \;+\; \eta_{fq},
\end{equation}
where $\mathrm{Index}_{fq}$ is the lagged model-motivated signal-noise index, $Z_{fq}$ consists of the factor betas, the lagged lottery control and log capitalization. $\alpha_q$ are quarter fixed effects and $\gamma_g$ are geography-tercile group effects which confine the comparison to stock-quarters within the same geography-tercile group. The excess of losses over gains in the quarter of the next earnings announcement rises with the index ($t=9.63$). If the group effects are removed, the slope turns negative, since the comparison then also runs across groups and mixes the differences in group averages with the relation within groups (\ref{app:drift_detail}). 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~\ref{prop:drift}). 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~\ref{tab:r3_turning}). The peak level for the top tercile is too imprecise and is left out of the ordering.\footnote{\ref{app:drift_detail} reports the projection with and without the group effects. Table~\ref{tab:r3_turning} reports the peak levels.} Since the structural friction parameter lies at its zero lower bound (Table~\ref{tab:theta_hat}) and the correction after optimistic quarters is not detectable (battery row R8, Table~\ref{tab:battery}), the rise of the loss-gain excess with the index has the sign which the overpricing correction mechanism predicts but the fitted model does not produce. Thus, the rise is reported as a reduced-form pattern instead of a structural identification of an overpricing correction caused by short-sale constraints.
\subsection{Gains and the breadth of the contributor pool}\label{sec:rf_gains}
The results also show that the next-quarter gains rise with the breadth of the contributor pool, which is measured by lagged dispersion (the number of distinct posting accounts per tweet), mainly where informed trading intensity is light. The slope of next-quarter gains on lagged dispersion is positive and significant in informed-trading-intensity tercile 1 (battery row R4, Table~\ref{tab:battery}), while in terciles 2 and 3 it is statistically indistinguishable from zero ($t = 0.11$ and $0.34$). Holding the other group primitives fixed, the model implies that the price responds more to the feed where informed trading is lighter (equation~\eqref{eq:coefs}). 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. This pattern survives after we control for the firm's cash-flow innovation variance. However, within informed-trading-intensity tercile 1, the pattern is strongest for the firms with the most predictable cash flows (\ref{app:rob_r4_innov}, Table~\ref{tab:r4_innovvar}). 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 lagged dispersion should be the weakest among the most predictable firms. Neither of the model's return channels (the volatility channel of Proposition~\ref{prop:vol} and the correction of surviving overpricing of Proposition~\ref{prop:drift}) produces that effect. A wider contributor pool predicts larger next-quarter gains (battery row R4, Table~\ref{tab:battery}) and smaller next-quarter losses in the joint regression (battery row R11, Table~\ref{tab:battery}). 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~\ref{prop:vol}). The correction channel cannot produce it either, since it is closed by the zero lower bound on the short-sale friction. We therefore report the relation between the breadth of the contributor pool and next-quarter gains as a reduced-form regularity that the model does not account for.
The tercile orderings in Sections~\ref{sec:rf_correction} and \ref{sec:rf_gains} are subject to the same limitation. The estimated informed-trading intensities ($K_j$) are also smallest in tercile 3 for both geographies. Thus, the tercile orderings in the three subsections 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 $\tau_0$ and $K_j$ across the groups.
\section{Conclusion}\label{sec: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.
\bibliographystyle{aer}
\bibliography{Sentiment_stocks}
\clearpage
\addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
\thispagestyle{empty}
\begin{center}
\vspace*{0.04\textheight}
{\LARGE\bfseries Internet Appendix\par}
\bigskip
{\Large to\par}
\bigskip
{\Large\bfseries Where Has the Foreign Signal Gone?\par}
\end{center}
\vspace{1em}
{\small\setstretch{1.15}
\makeatletter\def\tmptocnumberline#1{\hb@xt@ 12em{#1\hfill}}\makeatother
\tableofcontents
}
\clearpage