Many explanations for the same fact

Asset-pricing research consistently produces many different explanations for the same empirical facts. As a rule of thumb, you should expect asset-pricing researchers to wildly overachieve. Behavioral researchers can typically point to several psychological biases which might explain the same anomaly. e.g., it is possible to argue that the excess trading puzzle is due to a preference for gambling, the disposition effect, and social interactions among other things. In cross-sectional asset-pricing, there is an entire zoo of different explanations for the size and value effects, most of which “have little in common economically with each other.” In a 2017 review article, John Cochrane lists ten different explanations for the equity premium puzzle.

Unfortunately, the existence of so many different explanations for the same few facts is a real problem for the field. With so many to choose from, which explanation should a researcher use when evaluating counterfactuals or doing policy analysis? While each explanation is consistent with observed market data, different explanations will generally make different predictions out-of-sample in novel market environments. e.g., even if firm size and liquidity were perfectly correlated in all past data, researchers would still care which was the correct explanation for the size effect because they’d want to know what to expect if they ever encountered a bunch of large illiquid stocks.

So what is it about the asset-pricing research process that consistently produces multiple explanations for the same set of facts? That’s the topic of this post.

The Research Process

To answer this question, I need a working model of what it means to do asset-pricing research. What does the process entail? What counts as a new empirical fact? When asset-pricing researchers encounter one, how do they go about constructing a model to explain it? What primitives do they start with? What’s the general form of the model they eventually arrive at? How do different models differ from one another?

Asset-pricing models study the behavior of investors who can take actions today and want things tomorrow. I will use $a$ to denote an action that investors can take today. e.g., think about allocating a portfolio, deciding how much to consume, or making a capital improvement. I will use $w$ to denote a future outcome that investors want. e.g., think about consumption, wealth, or eternal happiness in the afterlife.

An asset-pricing model is a constrained optimization problem of the form

(1) $\begin{equation*} \begin{split} {\textstyle \max_a} &\phantom{=} \Exp[ \, \mathrm{Utility}(w) \, ] \\ \text{s.t.} &\phantom{=} 0 \geq \mathrm{Constraints}(a) \end{split} \end{equation*}$

The $\max_a(\cdot)$ operator captures the idea that investors strategically choose which actions to take today, $a$ . They realize that their choice of actions today, $a$ , affects how much of the thing they want they get to enjoy tomorrow, $w = \mathrm{W}(a)$ . The function $\mathrm{Utility}(w)$ denotes the utility that investors get from having a given amount of $w$ tomorrow. The expectation operator $\Exp[w] = \int \, w \cdot \mathrm{pdf}(w|a) \cdot \mathrm{d}w$ represents their conditional beliefs about the likelihood of $w$ tomorrow given their choice of actions today with $\mathrm{pdf}(w|a)$ denoting their subjective probability distribution function. This is the distribution that investors have in their heads, and because it is a subjective distribution, it might not be objectively correct. Investors also realize that they face various constraints, which I write as the requirement $0 \geq \mathrm{Constraints}(a)$ . Not all actions are possible today.

All asset-pricing models agree on this basic setup. Different asset-pricing models just make different claims about the functional forms of investor preferences, $\mathrm{Utility}(w)$ ; beliefs, $\mathrm{pdf}(w|a)$ ; and, constraints, $\mathrm{Constraints}(a)$ . Models based on habit formation, recursive utility, hyperbolic discounting, and loss aversion all monkey around with investor preferences. When a model says that investors have rational expectations, that they extrapolate, or that they seek sparsity, it is making a choice about the functional form of investors’ beliefs. The limits-to-arbitrage literature gives a taxonomy of investor constraints (short sale, margin, etc).

Researchers test whether an asset-pricing model’s specific choices for $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ fit the data by examining the model’s first-order conditions (FOCs). e.g., the stochastic discount factor (SDF) implied by an asset-pricing model comes from the FOC with respect to consumption. See Chapter 1 verse 1 of the Book of John. The function $\mathrm{f}(x) = 1 - x^2$ is an upside down parabola, which has a maximum value of $\mathrm{f}(0)=1$ . You could double check that $x=0$ maximizes $\mathrm{f}(x)=1-x^2$ by verifying that $\mathrm{f}'(0) = 0$ . Likewise, you can double check whether investors’ actions are optimal by verifying that

(2) $\begin{equation*} {\textstyle \frac{\mathrm{d}\phantom{a}}{\mathrm{d}a}} \big\{ \, \Exp[ \mathrm{Utility}(w) ] - \lambda \cdot \mathrm{Constraints}(a) \, \big\} = 0 \end{equation*}$

If an asset-pricing model’s FOCs aren’t satisfied in the data, then the model’s choice of functional forms must be missing something about investor behavior. The new parameter $\lambda$ in the equation above is a Lagrange multiplier, which captures how much a particular constraint distorts investors’ optimal choice of actions.

Here’s the asset-pricing research process in action. At time $t$ , the literature contains a bunch of models and data on various empirical settings (different countries, time periods, asset classes, etc). Each asset-pricing model makes different assumptions about $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ . But they all take the general form outlined in Equation (1). Asset-pricing researchers check whether the FOCs implied by each existing model hold in the various empirical settings observed at time $t$ . If no model adequately explains market data in some important empirical setting, then we call it a new empirical fact. Researchers then work backwards from the non-zero values of the FOCs in Equation (2) to try and guess the correct functional forms for $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ . If successful, we add a new asset-pricing model to the literature and the process repeats at time $(t+1)$ , perhaps with data on a few more empirical settings.

Ill-Posed Inverse Problem

The whole goal of the asset-pricing research process is to guess the correct functional forms for $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ to plug into Equation (1). Researchers do this by working backwards from violations of known models’ FOCs in Equation (2). If researchers guess correctly, they will find that the resulting model’s FOCs fit the observed market data in all empirical settings. But this excellent empirical fit is just a signature verifying that the model is correct. The point of guessing the true asset-pricing model is not to maximize empirical fit. With enough free parameters, any model can fit the data arbitrarily well.

Researchers care about guessing the true model (i.e., the optimization problem that investors are actually trying to solve) because the true model will allow them to predict how investors will behave in novel market environments that they haven’t encountered before. Put differently, knowing the correct functional forms for $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ allows researchers to evaluate counterfactual scenarios and do policy analysis. Knowing the correct asset-pricing model allows you to answer questions like, ‘What should the expected return of a large illiquid stock be?’, even if you have never encountered such a stock in the past.

This is an inverse problem. Asset-pricing research doesn’t involve finding the maximum of some known widely-agreed-upon optimization program. It involves staring at a bunch of data that is assumed to maximize some optimization problem and trying to figure out the details of that unknown problem. Put another way, asset-pricing researchers aren’t in the business of finding the maximum of a known function, $\mathrm{f}(x) = 1 - x^2$ . They observe $(x, \, y) = (0, \, 1)$ and assume this data is the maximum of some unknown function, $\mathrm{f}(x) = y$ . Then they try to guess the details of this unknown function.

The study of inverse optimization problems represents an entire mathematical field of inquiry. But there’s one detail in particular that’s relevant here: inverse problems are typically ill-posed. Put simply, many different functions can share the same derivative values. Functions with quite different global behavior can have the same slope locally. e.g., $\mathrm{f}(x) = 1 - x^2$ , $\mathrm{g}(x) = e^{-x^2}$ , and $\mathrm{h}(x) = \cos(\pi/2 \cdot x)$ all achieve a maximum value of $1$ at $x=0$ . These curves are all roughly the same when $|x| < 1/2$ . Yet, they have very different behavior globally: $\mathrm{f}(4) = -15$ , $\mathrm{g}(4) \approx 0$ , and $\mathrm{h}(4) = 1$ .

Neighborhood of x=0

Global behavior

Asset-pricing researchers are solving an inverse problem. They are trying to reverse engineer an entire optimization problem by studying finite data about its first-order conditions. And inverse problems are typically ill-posed. This is why they consistently produce multiple explanations for the same facts. We should expect many different optimization problems to produce equivalent FOCs in the observed market data. Moreover, we should expect this to be true even though the optimization problems, which look equivalent locally, will generally display quite different global behavior in novel as-yet-unseen market environments. And these global differences are what matter when evaluating counterfactuals and doing policy analysis.

A Potential Solution

If asset-pricing researchers generate multiple explanations for the same fact because they’re solving an ill-posed inverse problem, then how might we make this problem well-posed? The mathematical literature on inverse optimization problems makes one suggestion: severely limit the class of functions you are willing to consider. If you see derivative data in the neighborhood of $x=0$ that was generated by either $\mathrm{f}(x) = 1 - x^2$ , $\mathrm{g}(x) = e^{-x^2}$ , or $\mathrm{h}(x) = \cos(\pi/2 \cdot x)$ , you must find some grounds for ruling out two of the three options. e.g., if you were only willing to consider polynomial solutions, then $\mathrm{g}(x) = e^{-x^2}$ and $\mathrm{h}(x) = \cos(\pi/2 \cdot x)$ would be off limits. Your only remaining choice would be $\mathrm{f}(x) = 1 - x^2$ , making the inverse problem well-posed.

Unfortunately, this isn’t a particularly promising route for asset-pricing researchers. There aren’t good economic grounds for ruling out functional forms for $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ that yield similar FOCs in the observed data. In fact, some of the most cited papers in asset pricing involve dreaming new and exotic functional forms for these objects. e.g., think about models involving recursive preferences, which argue that investors’ utility takes the functional form:

(3) $\begin{equation*} \mathrm{Utility}_t(w_t) = \Big\{ \, (1 - \delta) \cdot w_t^{1 - 1/\psi} + \delta \cdot \big(\Exp_t[\mathrm{Utility}_{t+1}(w_{t+1})]^{1-\gamma}\big)^{\frac{1-1/\psi}{1-\gamma}} \, \Big\}^{\frac{1}{1 - 1/\psi}} \end{equation*}$

There’s simply no way this heifer would have been accepted into the asset-pricing cannon if researchers had decided to severely restrict the kinds of utility functions they were willing to consider sometime back in the early 1980s. If you think the discovery of recursive utility was progress, then that’s a problem.

Luckily, this isn’t the only way forward. Asset-pricing models do more than just make predictions about which FOCs should hold in the observed market data. They also say why these FOCs should be satisfied: because investors are optimizing with a specific tradeoff in mind. There is economic content in the $\max_a(\cdot)$ operator as well as in the FOCs this operator produces. An asset-pricing model doesn’t say that a set of FOCs should just happen to hold in the data. It says that these FOCs should hold because investors are optimizing the tradeoff embodied by the model’s choice of $\mathrm{Utility}(w)$ , $\mathrm{pdf}(w|a)$ , and $\mathrm{Constraints}(a)$ . So in most cases it should be possible to ask investors whether they are thinking about this tradeoff.

This represents a practical step we could take towards converting the asset-pricing research process into a well-posed inverse problem. It is a straightforward way of ruling out lots of spurious solutions. e.g., Cross-sectional differences in expected returns can’t be explained by investors demanding a risk premium for holding assets with exposure to X if real-world investors show no desire to hedge their exposure to X when given the chance. Likewise, if the excess trading puzzle is due to a widespread preference for gambling among retail investors, then it should be easy to find retail investors who express a preference for gambling. Yes, it is possible for markets to move in ways that no individual investor understands. But asset-pricing models don’t explain those kinds of market fluctuations. Asset-pricing models make predictions about how investors strategically respond to market fluctuations they do understand, so it should be possible to ask them about these fluctuations and the logic behind their responses.

The Research Process

Ill-Posed Inverse Problem

A Potential Solution

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