The Basic Recipe For Rationalizing Errors In Belief

Behavioral-finance models are often written down so that, although each individual trader holds incorrect beliefs, market events nevertheless unfold in such a way that traders can rationalize their own errors. e.g., consider the model in Scheinkman and Xiong (2003). In this model, each individual trader knows that every other trader is over-confident, and he knows that every other trader thinks that he himself is over-confident. He just doesn’t think that they’re correct. He’s pig-headedly insists that he’s the only unbiased trader. And yet, in spite of this error, the model is set up so that he can interpret the realized price path in his own internally consistent way. Each trader thinks the price distortion caused by his own over-confidence is actually coming from the value of the option to resell at a later date to some other over-confident trader.

There’s a good reason why researchers write down models this way. The idea is to write down a model that’s exactly one-step away from a rational benchmark. That way, any new predictions made by the model can be attributed to the behavioral bias. In this post, I first outline the basic recipe for rationalizing traders’ errors in beliefs. Then, I point out something slightly paradoxical about this recipe—namely, it requires fine-tuning the model parameters. And, while a researcher can to do this fine-tuning in a theoretical model, it’s not clear who can turn the appropriate knobs in the real world. These models are like stage magic. And, while we can learn about which cognitive biases people suffer from by studying a good magician’s sleight of hand, most missing coins wind up between the couch cushions rather than in The Amazing Randi‘s pocket.

Errors In Belief

Here’s a simple framework for digesting errors in beliefs. To start with, consider a market where a trader has correct beliefs. i.e., suppose that a trader receives a noiseless signal:

$\begin{equation*} \mathit{Signal} = \mathit{News} \qquad \text{where} \qquad \mathit{News} \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{N}(0, \, 1) \end{equation*}$

Then, given the trader’s optimal demand in response to this noiseless signal, suppose that the structural relationship between the trader’s noiseless signal and realized returns is given by:

$\begin{equation*} \mathit{Return} = \beta^{\star} \cdot \mathit{Signal} + \varepsilon^{\star} \qquad \text{where} \qquad \varepsilon^{\star} \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{N}(0, \, 1) \end{equation*}$

We typically think that $\beta^\star \in (0, \, 1)$ with larger values of $\beta^\star$ indicating more informative prices. I’m using the term “structural relationship” for $\beta^{\star} \overset{\scriptscriptstyle \mathrm{def}}{=} {\textstyle \frac{\partial \phantom{s}}{\partial s}} \mathrm{E}^{\star}[ \, \mathit{Return} \, | \, \mathrm{do}(\mathit{Signal} = s) \, ]$ because this parameter reflects the expected change in returns due to an exogenous shift in the trader’s signal. Note that this structural relationship could reflect other traders’ errors in belief, as was the case in Scheinkman and Xiong (2003).

But, in reality, suppose that the trader is over-confident about the precision of his signal. While he thinks it’s noiseless, his signal actually contains noise:

$\begin{equation*} \mathit{Signal} = \alpha \cdot \mathit{Noise} + \sqrt{1 - \alpha^2} \cdot \mathit{News} \qquad \text{where} \qquad \mathit{Noise} \overset{\scriptscriptstyle \text{iid}}{\sim} \mathrm{N}(0, \, 1) \end{equation*}$

And, the parameter $\alpha \in [0, \, 1]$ governs the relative contribution of noise to the trader’s signal: $\alpha = 0$ corresponds to correct beliefs; whereas, $\alpha = 1$ corresponds to a signal that is pure noise. Then, given the trader’s optimal demand, suppose that the structural relationship between the trader’s noisy signal and realized returns is actually given by:

$\begin{equation*} \mathit{Return} = \beta \cdot \mathit{Signal} + \varepsilon \qquad \text{where} \qquad \varepsilon \sim \mathrm{N}(0, \, 1) \end{equation*}$

Notice that, in reality, idiosyncratic-return shocks are no longer drawn IID. Let $\rho \overset{\scriptscriptstyle \mathrm{def}}{=} \mathrm{Cor}[\mathit{News}, \, \varepsilon]$ denote the correlation between the news about fundamentals in the trader’s signal and idiosyncratic-return shocks. e.g., in a model of disagreement, you might think about $\rho < 0$ due to the existence of another trader whose disagreement stems from negatively correlated signals or negatively correlated mistakes.

The Basic Recipe

Suppose that the trader, who doesn’t realizing that he’s getting a noisy signal, is still carefully monitoring price informativeness. i.e., he’s carefully monitoring the relationship between his signal and realized returns. Here’s what it would take for this trader to rationalize his error in beliefs. Notice that the covariance of the trader’s signal and market returns is given by:

$\begin{align*} \mathrm{Cov}[\mathit{Return}, \, \mathit{Signal}] &= \mathrm{Cov}[\beta \cdot \mathit{Signal} + \varepsilon, \, \mathit{Signal}] \\ &= \mathrm{Cov}[\beta \cdot \mathit{Signal}, \,\mathit{Signal}] + \mathrm{Cov}[\varepsilon, \, \mathit{Signal}] \\ &= \beta + \mathrm{Cov}\big[ \, \varepsilon, \, \alpha \cdot \mathit{Noise} + \sqrt{1-\alpha^2} \cdot \mathit{News} \, \big] \\ &= \beta + \rho \cdot \sqrt{1-\alpha^2} \end{align*}$

So, since the variance of his signal is $\mathrm{Var}[\mathit{Signal}] = 1$ , if the trader regresses realized returns on his signal, he’ll find a slope coefficient of

$\begin{equation*} \hat{\beta}^{\text{OLS}} \overset{\scriptscriptstyle \mathrm{def}}{=} {\textstyle \frac{\mathrm{Cov}[\mathit{Return}, \, \mathit{Signal}]}{\mathrm{Var}[\mathit{Signal}]}} = \beta + \rho \cdot \sqrt{1-\alpha^2} \end{equation*}$

Thus, if a researcher chooses the values of $\rho$ and $\alpha$ so that $\beta^{\star} = \hat{\beta}^{\text{OLS}} = \beta + \rho \cdot \sqrt{1-\alpha^2}$ , then the trader will see data that’s consistent with his erroneous belief about his signal being noiseless.

It’s important to emphasize that, when a researcher chooses $\alpha$ and $\rho$ so that $\beta^{\star} - \beta = \rho \cdot \sqrt{1-\alpha^2}$ , he’s not giving the trader correct beliefs, though. Although price informativeness will look correct to the trader, his error in beliefs will still cause returns to respond to pure noise. The covariance of noise and returns will be:

$\begin{align*} \mathrm{Cov}[\mathit{Return}, \, \mathit{Noise}] &= \mathrm{Cov}[\beta \cdot \mathit{Signal} + \varepsilon, \, \mathit{Noise}] \\ &= \mathrm{Cov}\big[ \, \beta \cdot \big( \, \alpha \cdot \mathit{Noise} + \sqrt{1-\alpha^2} \cdot \mathit{News} \, \big), \, \mathit{Noise} \, \big] + \mathrm{Cov}[\varepsilon, \, \mathit{Noise}] \\ &= \beta \cdot \alpha \end{align*}$

So, returns will react to pure noise whenever $\alpha > 0$ . And, in principle, the trader could notice this fact if he cared to inspect $\mathrm{Cov}[\mathit{Return}, \, \mathit{Noise}]$ rather than just $\mathrm{Cov}[\mathit{Return}, \, \mathit{Signal}]$ .

Like Stage Magic

That’s how you write down a model where biased traders can rationalize their own errors in belief. The basic recipe is simple enough. Just introduce a hidden correlation into the information structure of the model (i.e., the parameter $\rho$ ) and then fine-tune this correlation so that it cancels out the effects of the trader’s behavioral bias (the parameter $\alpha$ ). It’s really pretty when this sort of cancellation takes place. Models that manage to use this basic recipe, such as Scheinkman and Xiong (2003), are really beautiful. But, this approach raises an obvious question: in the real world, why should we expect $\alpha$ and $\rho$ to take on the precise values needed to hide a trader’s error? Where does the required fine-tuning come from? What’s the underlying mechanism at work?

These models are like stage magic. They’re expertly scripted illusions that demonstrate how behavioral biases can go undetected… even by traders who are actively trying to detect them. And, this is not a slight. This is really informative in the same way that going to a good magic show is really informative. It teaches you something useful about the limits of human perception, about how your attention can be managed, about how you can be deceived. But, you don’t leave magic shows thinking that the next deck of cards you open will contain 52 copies of the $6\clubsuit$ because you happened to be thinking of that card when you opened the box. No one expects everyday situations to operate by the rules of stage magic. Most of the time, there’s no magician to carefully script the illusion. And, the same logic applies to financial markets. It’s useful to know that you can fine-tune parameters to hide an error, but we shouldn’t assume that markets typically operate with the parameters dialed in this way. Why should we? Who exactly would be the one turning the knobs?

Errors In Belief

The Basic Recipe

Like Stage Magic

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