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Modeling Heterogeneous Beliefs

Updated: Nov 24, 2022

Extracting disagreement from text is a hard semantic problem. The major empirical challenge in studying beliefs about individual stocks is the difficulty in obtaining a reasonably large time-series and cross-sectional sample of belief statements. This section describes a model in which investors’ beliefs are formulated using Markov chains with finite state space. The model nests from a version of a model of Harrison and Kreps (1978) where a dividend-yielding asset is traded by two types of investors:optimists and pessimists.

We then need belief observations to understand how real world investors think regarding stocks, i.e., giving a current state, what is their views of transiting probability for the next state? As a demonstration, we first obtain every posts from Guba forum for every stocks which is active during the period 2015-01-01 to 2017-12-31. We then use NLP to classify the tone of all posts and regroup them into three categories: positive, neutral and negative. At a given day, a company will have three figures that in turn, pos, neu, neg, representing the total count for positive, neutral and negative posts at that day. In theory, an ordinary investors shall take into all posts of the three categories in an unbiased manner, however, optimists (pessimists) are biased in the sense that they tend to interpret more posts as ”good (bad) side” than ordinary investors do. This tendency nests on a well known psychosocial phenomena called ”selective exposure” (Hart et al., 2009). Using more detailed data, a recent study, Cookson et al. (2020) show that the selective exposure lingers even to those most confirmatory information during financial decision-making.

Above figure shows a series of demo predictions. The blue line represents the real weekly return of PingAn Bank (000001.SZ) while the orange line is the predicted ret based on post sentiment from Guba.

Figure 1 plots unconditional beliefs derived from Guba posts. Apparently, the blue line on the top represents what probability optimists think that the PingAn bank will pay high dividend over time. Since optimists are permanent, they always think that PingAn bank pays higher dividend than pessimists do. The average probability of having a good state between these two types of investors are 58% vs. 31%.

Finally, the last figure shows equilibrium prices based on Guba’s post sentiments. Recall that the difference in beliefs are driven by subjective views on how often a good state are expected between two groups of investors and how many dividend are expected to be paid between two sates.

The exact equilibrium price of an asset will of course requires most updated and accurate estimates on expected dividend payoffs. However, it is easy to realize that in this build, the impact of expected dividend payoffs should be qualitatively the same in later regression analysis as the core idea of our model is that we abstract sentiment data from social media to form heterogeneous transition matrices. The bottom line is we should always delivering good explanatory power while keeping the model (input) as compact as possible!

PS: A working paper based on this project with more technical details will be available on SSRN soon.


Cookson, J. A., Engelberg, J., Mullins, W., 2020. Echo chambers. SSRN Electronic Journal.

Harrison, J. M., Kreps, D. M., 1978. Speculative investor behavior in a stock market with heterogeneous expectations. The Quarterly Journal of Economics 92, 323.

Hart, W., Albarracín, D., Eagly, A. H., Brechan, I., Lindberg, M. J., Merrill, L., 2009. Feeling validated versus being correct: A meta-analysis of selective exposure to information. Psychological Bulletin 135, 555–588.


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