We explore investor beliefs, sentiment and disagreement (dispersion of views across investors), on stock market returns during the COVID-19 pandemic using a large number of messages of investors on a social media investing platform, StockTwits. Stock-Twits is a microblogging platform in which investors can post their views as a Tweeter type format, e.g., 140 characters. 1 Perhaps more importantly, they also can label their messages with a sentiment which can be bullish, bearish, or leave it unspecified, neutral. Since extracting sentiment from text data is a challenging task, this is an important feature. 2 Besides, they can use "cashtags", e.g., $AAPL, to link their message to a particular firm. 3 Using a survey conducted on wealthy retail investors who are clients of Vanguard, Giglio et al. (2020) provide a data-driven analysis of how investor expectations about economic growth and stock market returns changed during the February-March 2020 stock market crash induced by the COVID-19 pandemic. They find that, among others, investor beliefs before and during the COVID-19 crisis in February-March 2020 1 StockTwits was founded in 2008 as a social networking platform for investors on the financial markets.
It is similar to Tweeter with more additional options that are designed for investors to express their views. Cookson and Niessner (2019) provide an in-depth analysis of the StockTwits data for measuring disagreement and exploring its source. To have reliable and high quality data, they conduct their analysis of messages posted between January 2013 and September 2014. Since then, the number of users, as well as messages, haven exponentially increased. For example, the number of messages posted between December 2019 and March 2020 is about a quarter of their messages. Although we have access to the whole dataset of StockTwits, however, we limit our analysis on posted messages during the COVID-19 pandemic. 2 A major problem in textual analysis is related to the correct interpretation of context in which certain words are used. In other words, it is difficult to evaluate what truly is a positive, neutral, and negative statement. For example, it is not clear how to deal with sarcasm, see Rosenthal et al. (2019) , among others. 3 Detailed information about why do people post messages? and why do their messages represent their view? can be found in Cookson and Niessner (2019) and therein references.
2 shows a sharp increased disagreement about the stock market outcomes and future of the economy. This paper complements their findings in several important ways by analyzing the messages on StockTwits.
The contribution of this paper is three-fold. First, thanks to the availability of investors' messages at the high-frequency level, e.g., minute, augmented with their sentiment and cashtags, we can measure disagreement at a high-frequency level. 4 Comparing the disagreement and its evolution which are extracted from StockTwits with their counterparts in Giglio et al. (2020) is an interesting exercise, given the heterogeneity in the sources of data.
Second, it is important to measure the disagreement around the COVID-19 pandemic, however, understanding the source of disagreement is more vital for both the design of the ongoing economic policy response and in further advancing economic theories. It is not clear what is the source of disagreement in Giglio et al. (2020) : information or interpretation of information. The StockTwits data can be used to explore this direction of disagreement as in Cookson and Niessner (2019) .
Third, the rich and multimodal nature of the StockTwits data allows us to take a magnifier on sentiment/disagreement and explore how does it vary across sectors, as well as investors: homogeneous or heterogeneous. As we have seen while the price of most companies has been dropped during the COVID-19 pandemic, some pharmaceutical and technological companies had a big jump in their price. For example, a tech 4 The number of posted messages per minute is on the scale of hundreds.
3 company such as Zoom had more than 20% jump in its price. Hence, these companies might have different sources of disagreement with stocks that dropped. This aspect of StockTwits data can help to understand this direction of disagreement.
To explore the investor beliefs, sentiment and disagreement, on stock market returns during the COVID-19 pandemic, we conduct a comprehensive analysis of all posted messages on the StockTwits platform between November 30, 2020, and March 31, 2020. In total, we have 3,676,169 messages of 179,468 unique users mentioning 10,715 unique tickers. We establish the following empirical facts about sentiment and disagreement of investors that post messages on the StockTwits platform.
Daily time series of the sentiment and disagreement is not a stationary process. This is true across and within all investors with any investment philosophy, horizon, and experience. The same results are valid across sectors. The immediate implication is that before any investigation/analysis it is helpful to deal with non-stationary via either transformation, rolling window, or differencing.
There is a V−shape (Λ−shape ) in sentiment (disagreement) between February 19, 2020 and March 31, 2020. Specifically, there is a sharp decrease (increase) in sentiment (disagreement) until March 23, 2020, and then they reverse back until the end of our sample. This result is consistent with findings of Giglio et al. (2020) that the average investor turned more pessimistic about the short-run performance of both stock markets and the economy.
The pattern of sentiment and disagreement is homogeneous for investors across and within investment philosophy, horizons, and experiences. Daily time series of sentiment and disagreement support this finding. Furthermore, the average and standard deviation of sentiment and/or disagreement are ranging from 0.6 to 0.8 during the COVID-19 pandemic.
The pattern of sentiment and disagreement is heterogeneous across sectors. Daily time series of sentiment and disagreement across sectors behave quite differently across sectors which is in contrast with the behavior of sentiment and disagreement across investors. For instance, the daily time series of sentiment for the healthcare sector is almost flat while it has a downward trend for the financial sector.
The financial sector is the most pessimistic while healthcare is the most optimistic during the COVID-19 pandemic. The median of sentiment for the financial sector is 0.257 while this statistic for the healthcare sector is 0.782. Moreover, the disagreement for the financial sector is 0.901 which is pretty high while this statistic for disagreement of healthcare is 0.581.
To explore investor beliefs, sentiment and disagreement, on stock market returns during COVID-19 pandemic, we limit our sample on messages posted between November 30, 2019 and March 31, 2020. 5 The format of downloaded messages is JSON. 6 We use the Python 3.7 for our analysis.
In total, we have 3,676,169 messages of 179,468 unique users mentioning 10,715 5 The reason for this range is the following. It is believed that the virus has started spreading in Wuhan in the beginning of December 2019. Also, we are only able to update our dataset once per month, at the end of each month. The last month in our dataset at the conducing this paper was March 2020. 6 See Listing 1 for a detailed information about JSON format and an example of it from StockTwits. unique tickers. For each message/idea, we observe a unique message and user identifier among much other information, see Listing 1. Panel A of Table 1 presents summary statistics for some characteristics of raw StockTwits Data, without any filtration.
The average number of messages per stock-date is 23.99, with as many as 26,424 messages for some stocks on some days. The average number of people a user follows is 23.92 where this stat for the number of followers is 121.37. The median number of ideas (likes) a user has is 37 (22), with as many as 2,126,704 (809,351) ideas (likes).
Textual analysis of a dataset by Natural Language Processing (NLP) techniques requires an initial assessment. We do need an NLP technique for extracting the sentiment of unspecified messages where they are neither bullish nor bearish. Panel B of Table 1 shows the frequency distribution of messages' basic features such as the number of words, characters, stopdwrods, and cashtags, as well as, the average length of words. $SPY is the cashtage with the highest number of frequencies which is 368,086 (10% of total cashtags).
The top graph of Figure 
StockTwits's users can fill out their profiles with information about themselves as investors. Specifically, they are able to specify their investment approach, investment horizon (holding period), and experience level. Table 2 , we present the breakdown of users by investment approach, holding period, and experience. This table reveals several important messages. First, more than 80% of users do not specify their approach, investment horizon, as well as, experience level. This is consist with the dataset of Cookson and Niessner (2019) . The number of users in their dataset drop to 12,029 from 107,808 users after removing those people that do not report their investment approach, holding period, and experience in their profile information.
Second, on StockTwits, the most common approach is technical, representing 5.92% of users and about 9.7% of messages. Growth and momentum investors represent the next two most common investment philosophies (3.44% and 3.42% of investors, respectively), followed by fundamental and value investors (2.48% and 1.86% 7 of investors, respectively), representing 3.57% and 2.6% of messages, respectively.
Global macro investors make up only 0.78% of overall investors and 0.56% of messages. 7
Third, on StockTwits, swing trader has the largest number of users for investment horizon, representing 6.85% of users and 11.78% messages. Long term investor and day trader are the next two common investment preferences for holding period (4.51% and 3.61% of investors, respectively), followed by position traders with 2.98% and 4.95% of investors and messages, respectively.
Finally, users with intermediate experience are the most common investors, representing 8.46% of users and 15.79% of messages. Novice and professional users share the same percentage of users by 4.92% and 4.72% of investors, respectively. However, professional investors tend to post messages more often than novice investors. The percentage of posted messages for professional investors is 8.73% while this number is 5.10% for novice investors.
Eliciting sentiment from a text document such as a tweet, message, or review of a product on Amazon, among others, is a challenging but important task. This task means identifying whether a piece of text, e.g., tweet, express positive, negative, or neutral sentiment. 8 Thanks to an outstanding feature of StockTwits, allowing users 7 The same systematic analysis as been done in Cookson and Niessner (2019) can be conducted to examine whether the StockTwits investment approaches reliably categorize users into truly different investment philosophies. 8 Rosenthal et al. (2019) describe some details about sentiment analysis for tweets. Gentzkow et al.
(2019) provide a gentle introduction to the use of text as an input to economic research. 8 to label their messages, identifying the sentiment of its messages is not a problem.
This critical feature enormously facilitates the analysis of StockTwits's messages. Although the field of textual analysis has been provided with many advanced algorithms, however, they can not be as accurate as of the person that posts the message.
To carry our analysis, we drop the messages that left their sentiment as null. We also eliminate messages with multiple Cashtages from our analysis. 9 We follow the same approach as Antweiler and Frank (2004) and Cookson and Niessner (2019) in constructing a sentiment measure from bullish and bearish messages. Specifically, we first label each bearish message as −1 and each bullish message as 1. We then take the arithmetic average of these classifications at the group1 × day × group2 level:
where N Bullish itg and N bearish itg are number of bullish and bearish messages per group1, day, and group2, respectively. Group1 can be either all firms, sectors, industries, or a specific firm, sector, or industry. Group2 can either be all investors or investors with a given investment philosophy, experience, or holding period (investment horizon) level.
This measure of sentiment has important features. Among others, measuring the sentiment at the sector or industry level can be potentially very useful. For example, during the COVID-19 pandemic knowing the sentiment at the level of industry can be fruitful for the designing ongoing economic policy response. Unfortunately, this direction of a sentiment measure is not explored in Cookson and Niessner (2019) . 9 More then half of the messages on our dataset are unclassified, neither bullish nor bearish. Although, by excluding unclassified messages, we lose about half of our dataset, however, the number of messages that remain is still substantial. Possibly, it would be interesting if we label unclassified messages from classifies ones as it is done in Cookson and Niessner (2019) . We leave it for further investigation. Disagreement and sentiment go hand-in-hand and they cannot be considered separately from each other. We calculate the disagreement similar to Cookson and Niessner (2019) ( Antweiler and Frank (2004) ) where disagreement is measured by computing the standard deviation of expressed sentiment across group1 × day × group2 (messages) as following
where group1 can be either all firms, sectors, industries, or a specific firm, sector, or industry and group2 can either be all investors or investors with a given investment philosophy, experience, or holding period (investment horizon) level. 10
The rich and multimodal features of the StockTwits data enable us to construct a time series of sentiment at the daily frequency. In this section, we discuss possible dynamics for modeling the time series of average sentiment and its decomposition.
We calculate the average sentiment measure, AvgSentiment itg , for day t from messages posted between the market close of day t − 1 to the market close of day t. 11 Panel (a) of Figure 3 presents the daily time series of sentiment where group1 is all sectors, group2 users that group by their investment approach and t represents a day. As we can see, the time series is noisy. A closer visual inspection on it reveals, this time se-10 Detailed information about this measure can be found in Antweiler and Frank (2004) and Cookson and Niessner (2019). 11 As in Cookson and Niessner (2019) , for this analysis, we compute the average sentiment measure by assigning each message an equal weight. However, our dataset allows us to scale the sentiment of each message by the number of likes or number of followers its user has. We leave this robustness check for further investigation.
ries is not stationary, that is the mean, variance, and covariance are time-dependent.
In this paper, we are interested in the trend of sentiment and disagreement process.
Therefore, we need to transform the data somehow to identify its short-term trend.
Investors often use moving average approach to detect a signal for taking long or short positions, trend-following approach. The sentiment at day t equals the averages of past seven days of AvgSentiment itg . We use simple moving average (SMA) which is unweighted mean of previous days. Given the length of our sample, we use the seven days moving average -one week. However, weighted/exponential moving average with different lags can be considered for this analysis. Specifically,
The rolling window can be justified as follows. It is rational to assume that the sentiment of a user is unlikely to change instantaneously, contrary, its evolution is likely gradual. Statistically speaking, the process of the sentiment is unlikely to be stationary. Hence, using the past seven observations for extracting the sentiment can be justified. 12 Right side of Panel (a) in Figure 3 shows the daily time series of sentiment with the rolling window on past seven days. As we can see, the daily time series of sentiment after applying the rolling window is less noisy, perhaps, more importantly, its short-term trend is clear.
In general, depending on the nature of the trend and seasonality, there are two main approaches, additive or multiplicative, wherein, each observation in the series can be manifested as either a sum or a product of the components, respectively. For example, an additive model can have a decomposable time series alike Harvey and Peters (1990) . 13 The model has three main components: trend, seasonality, and holidays:
where g t is the trend function, s t represents periodic changes, e.g., weekly and monthly, h t represents the role of holidays, and t is any idiosyncratic changes which are not accommodated by the model. This specification is similar to a generalized additive model of Hastie and Tibshirani (1987) . Figure 3 presents the disagreement correlation matrix with (right) and without (left) rolling window where group2 can be either investment approach, experience, or holding period. 14 A close inspection reveals two important messages.
First, the rolling window does not affect the structure of the disagreement correlation matrix. The graphs on the left and right side of Panel (b) are very similar. Second, the disagreement is strongly correlated together across investors with any approach, experience, and investment horizon.
In this section, we explore the evolution of sentiment and disagreement across and within all investors and sectors. 13 There is a useful Python and R package for empirical implementation of this model, see Taylor and Letham (2018) . 14 To save space, we have not reported the daily time series as Panel (a) for investors' experience and their investment horizon. Their time series have a similar pattern as their counterpart, investment approach.
Summary. We find that the sentiment (disagreement) has a sharp decrease (increase) between February 19, 2020, and March 23, 2020, across all investors with any investment philosophy, horizon, and experience, where a historical market hight followed by a record drop. The pattern of sentiment and disagreement is homogeneous for investors with different investment philosophies, horizons, and experiences. Furthermore, The differential interpretation of information has likely a small contribution as a source of disagreement during the COVID-19 pandemic. Table 3 presents summary statistics on average sentiment broken down by investment philosophy (approach), horizon (holding period), and experience. As investors tend to post bullish messages more frequently than bearish messages, it makes sense that the average sentiment across investors with any investment philosophy, horizon, and experience is closer to 1 than −1. To save space, we do not report daily time series for average sentiment and disagreement within investors' philosophy, experience, and investment horizon. 15 We observe the following messages from these time series. Within investors' philosophy, all investors, except growth, share the same opinion on sentiment and disagreement.
This reconfirms what we found in Panel A of Table 3 . As for investors' experience:
(i) that professional investors are more pessimistic than novice and intermediate in- (ii) day trader investors are more pessimistic than other investors within this group.
Consequently, disagreement is much higher, from the middle of January it is around 1 which is pretty high.
On February 19, 2020, the US stock market was at a historical high and followed 15 These results are available upon request. by a record drop on March 23, 2020. We refer to these dates the good and bad state of the economy, respectively. By the end of March, the market has recovered some of its losses. We refer to this state, which is the last observation in our sample, as the "recovered" state of the economy. It is interesting to investigate the disagreement correlation matrix within investors with different investment philosophies, horizons, and experiences at different stages of the economy. The outcome of such an investigation potentially can help to identify the source of disagreement in these states of the economy. Consequently, it can provide good guidance for the designing ongoing economic policy response to the COVID-19 pandemic. It has been documented that disagreement, volume, and volatility are highly correlated, especially during a bear market, see Bollerslev et al. (2018) , among others.
However, it is not clear what is the source of disagreement: information or interpretation of information (investment philosophy). Cookson and Niessner (2019) find that disagreement is evenly split between both sources of disagreement. Surprisingly, a conclusion from Figure 4 is that the differential interpretation of information is not likely a source of disagreement during the COVID-19 pandemic. The main argument to support this claim is the high correlation among investment philosophies.
Summary. By exploiting the evolution of average sentiment and disagreement across and within sectors, in contrast to our finding across investors, we find that the pattern of sentiment and disagreement is heterogeneous across sectors. Moreover, the financial sector is the most pessimistic while healthcare is the most optimistic during the COVID-19 pandemic. 17 The average sentiment for the financial sector is even entering within negative territory during March. tor. 18 The Healthcare sector has the highest average number of messages per day, 3, 910 messages, followed by financial and technology sectors with 4, 125 and 1, 997 messages, respectively. This is consistent with our original sample, except the finan- A visualization re-confirm our findings from Table 4 . Moreover, this figure allows us to see their evolution which is not possible from {"object":"Message", 2 "action":"create", 3 "data":{"id":185405776,"body":"In the last years, $PLAB&#39;s CEO bought shares at market price only once. Six months after the purchase, the stock was up by +45.24%. Check out https://www.ceo-buys.com to learn more.","created _ at":"2019-12-01T00:00:00Z","user":{"id":1719889," username":"CEOBuys","name":"CEO-Buys.com","avatar _ url":"https:// avatar _ url _ ssl":"https://avatars.stocktwits.com/production/1719889/ identity":"User","classification":[],"followers":1140,"following":0," ideas":29818,"watchlist _ stocks _ count":0,"like _ count":0,"plus _ tier":""
,"premium _ room":"","subscribers _ count":120,"subscribed _ to _ count":0," following _ stocks":0,"location":"","bio":null,"website _ url":null,"
trading _ strategy":{"assets _ frequently _ traded":["Equities"],"approach"
:"Momentum","holding _ period":"Swing Trader","experience":"
Professional"}},"source":{"id":4467,"title":"CEO Buys","url":"https:
//www.ceo-buys.com"},"symbols":[{"id":3146,"symbol":"PLAB","title":" Photronics Inc.","aliases":[],"is _ following":false,"watchlist _ count": 683,"exchange":"NASDAQ","sector":"Technology","industry":"
Semiconductor -Integrated Circuits","logo _ url":"http://logos.xignite .com/NASDAQGS/00021796.gif","trending":false,"trending _ score":-0.4598 07}],"prices":[{"id":3146,"symbol":"PLAB","price":"11.76"}]," mentioned _ users":[],"entities":{"sentiment":null},"sentiment":{" sentiment _ score":0.7582}}, Detailed information about this format can be found in https://www.json.org/ json-en.html, among others. Each JSON file has four main categories: object, action, data, and time. Except for the time of the creation for a tweet, all of the information is inside the data section. Each tweet and user has a unique id. To view the structure of the above file, simply just copy and paste it to http://jsonviewer.stack.hu. Panel (a): this figure exhibits the daily time series of sentiment and disagreement of users based on their approach. A user can fill out his/her profile by approach, experience, and holding period (investment horizon). We first label each bearish message as −1 and each bullish message as 1. We then take the arithmetic average of these classifications at the group1 × day × group2 level:
Group1 can be either all firms, sectors, industries, or specific firm, sector, or industry. Group2 can either be all investors or investors with a given investment philosophy, experience, or holding period (investment horizon) level. Furthermore, to follow the sentiment's trend, we use rolling windows of the past seven days AvgSentiment itg for measuring the sentiment at each day. Disagreement is calculated
itg . Panel (b): these two heatmaps present the disagreement correlation matrix of users by investment philosophy, experience, and holding period. This figure exhibits the daily time series of sentiment and disagreement of users within their approach. A posted messages on StockTwits can include a cashtage, i.e., $ticker, to specify the messages is related to which firm. By mentioning a ticker the posted message can belong to one of these sectors: basic materials, financial, consumer goods, healthcare, industrial goods, technology, utilities, service, and conglomerates.There are four options for this section: day trader, position trader, long term trader, and swing trader. We first label each bearish message as −1 and each bullish message as 1. We then take the arithmetic average of these classifications at the group1 × day × group2 level: AvgSentiment itg = 