An agent is a misspecified Bayesian if she updates her belief using Bayes' rule given a subjective, possibly misspecified model of her signals. This paper shows that a belief sequence is consistent with misspecified Bayesian updating if and only if the set of posteriors admits a countable partition such that the prior contains a grain of the conditional average posterior on each cell. The condition imposes essentially no restrictions on posteriors given a full-support prior over a finite state space and reduces to a support inclusion condition on compact state spaces under mild regularity assumptions. However, it rules out posterior beliefs with heavier tails than the prior on unbounded state spaces. In Gaussian environments, it implies that posterior uncertainty cannot exceed prior uncertainty. The results delineate the boundary between updating rules that are observationally equivalent to Bayesian updating under misspecification and genuinely non-Bayesian rules. As an application, the paper shows that diagnostic expectations are consistent with misspecified Bayesianism, whereas some parameterizations of smooth diagnostic expectations are not.

This paper analyses how limits to the complexity of statistical models used by market participants can shape asset prices. We consider an economy in which the stochastic process that governs the evolution of economic variables may not have a simple representation, and yet, agents are only capable of entertaining statistical models with a certain level of complexity. As a result, they may end up with a lower-dimensional approximation that does not fully capture the intertemporal complexity of the true data-generating process. We first characterize the implications of the resulting departure from rational expectations and relate the extent of return and forecast-error predictability at various horizons to the complexity of agents' models and the statistical properties of the underlying process. We then apply our framework to study violations of uncovered interest rate parity in foreign exchange markets. We find that constraints on the complexity of agents' models can generate return predictability patterns that are simultaneously consistent with the well-known forward discount and predictability reversal puzzles.

This paper proposes a framework in which agents are constrained to use simple models to forecast economic variables and characterizes the resulting biases. It considers agents who can only entertain state-space models with no more than d states, where d measures the intertemporal complexity of a model. Agents are boundedly rational in that they can only consider models that are too simple to nest the true process, yet they use the best model among those considered. I show that using simple models adds persistence to forward-looking decisions and increases the comovement among them. I then explain how this insight can bring the predictions of three workhorse macroeconomic models closer to data. In the new-Keynesian model, forward guidance becomes less powerful. In the real business cycle model, consumption responds more sluggishly to productivity shocks. The Diamond–Mortensen–Pissarides model exhibits more internal propagation and more realistic comovement in response to productivity and separation shocks.

This paper presents a theoretical model of an autocrat who controls the media in an attempt to persuade society of his competence. We base our analysis on a Bayesian persuasion framework in which citizens have heterogeneous preferences and beliefs about the autocrat. We characterize the autocrat's information manipulation strategy when society is monolithic and when it is divided. When the preferences and beliefs in society are more diverse, the autocrat engages in less information manipulation. Our findings thus suggest that the diversity of attitudes and opinions can act as a bulwark against information manipulation by hostile actors.

This paper characterizes the conditions under which the observed beliefs of a group of agents are consistent with Bayesian updating. Beliefs are consistent with Bayesianism if they arise from the application of Bayes' rule given some subjective distribution for the state and the signals agents observe between periods. The paper's main finding is that beliefs are consistent with Bayesianism if and only if the mean of the distribution of posteriors is uniformly absolutely continuous with respect to the prior. Furthermore, the paper shows that the existing results on the empirical content of Bayesianism rely on additional restrictions on permissible subjective distributions, such as the requirement that agents have correct beliefs about the distribution of signals.

What are the testable implications of the Bayesian rationality hypothesis? This paper argues that the absolute continuity of posteriors with respect to priors constitutes the entirety of the empirical content of this hypothesis. I consider a decision-maker who chooses a sequence of actions and an econometrician who observes the decision-maker's actions, but not her signals. The econometrician is interested in testing the hypothesis that the decision-maker follows Bayes' rule to update her belief. I show that without a priori knowledge of the set of models considered by the decision-maker, there are almost no observations that would lead the econometrician to conclude that the decision-maker is not Bayesian. The absolute continuity of posteriors with respect to priors remains the only implication of Bayesian rationality, even if the set of actions is sufficiently rich that the decision-maker's actions fully reveal her beliefs, and even if the econometrician observes a large number of ex ante identical agents who observe i.i.d. signals and face the same sequence of decision problems.

This paper analyzes how limits to the complexity of statistical models used by market participants can shape asset prices. We consider an economy in which agents can only entertain models with at most k factors, where k may be distinct from the true number of factors that drive the economy’s fundamentals. We first characterize the implications of the resulting departure from rational expectations for return predictability at various horizons. We then apply our framework to two applications in asset pricing: (i) violation of uncovered interest rate parity at different horizons and (ii) momentum and reversal in equity returns.

I propose an equilibrium search and matching model with permanent worker heterogeneity, asymmetric information, and endogenous separations and study the dynamics of adverse selection in the labor market. The interaction between asymmetric information and endogenous separations leads to a cyclical adverse selection problem that has testable predictions both for the aggregate variables and for individual workers’ outcomes. First, a deterioration in the distribution of ability in the pool of the unemployed leads firms to raise their hiring standards, thus resulting in shifting out of the Beveridge curve. Second, if the separation rate is log-supermodular (log-submodular) in productivity and ability, the pool of the unemployed becomes more (less) adversely selected in downturns. Third, firms rationally discriminate against the long-term unemployed by demanding more unequivocally positive signals of their ability before hiring them. Fourth, this scarring effect is more (less) severe for lower-ability workers and after deeper recessions if the separation rate is log-supermodular (log-submodular). I conclude by providing conditions on the fundamentals of the economy that lead to log-supermodular and log-submodular separation rates.

We study a dynamic game in which short-run players repeatedly play a symmetric, strictly supermodular game whose payoffs depend on a fixed unknown state of nature. Each short-run player inherits the beliefs of his immediate predecessor in addition to observing the actions of the players in his social neighborhood in the previous stage. Because of the strategic complementary between their actions, players have the incentive to coordinate with others and learn from them. We show that in any Markov Bayesian equilibrium of the game, players eventually reach consensus in their actions. They also asymptotically receive similar payoffs despite initial differences in their access to information. We further show that, if the players' payoffs can be represented by a quadratic function, then the private observations are optimally aggregated in the limit for generic specifications of the game. Therefore, players asymptotically coordinate on choosing the best action given the aggregate information available throughout the network. We provide extensions of our results to the case of changing networks and endogenous private signals.

I propose an equilibrium search and matching model with permanent worker heterogeneity, asymmetric information, and endogenous separations and study the dynamics of adverse selection in the labor market. The interaction between asymmetric information and endogenous separations leads to a cyclical adverse selection problem that has testable predictions both for the aggregate variables and for individual workers’ outcomes. First, a deterioration in the distribution of ability in the pool of the unemployed leads firms to raise their hiring standards, thus resulting in shifting out of the Beveridge curve. Second, if the separation rate is log-supermodular (log-submodular) in productivity and ability, the pool of the unemployed becomes more (less) adversely selected in downturns. Third, firms rationally discriminate against the long-term unemployed by demanding more unequivocally positive signals of their ability before hiring them. Fourth, this scarring effect is more (less) severe for lower-ability workers and after deeper recessions if the separation rate is log-supermodular (log-submodular). I conclude by providing conditions on the fundamentals of the economy that lead to log-supermodular and log-submodular separation rates.

In this paper, we study the problem of non-Bayesian learning over social networks by taking an axiomatic approach. As our main behavioral assumption, we postulate that agents follow social learning rules that satisfy imperfect recall, according to which they treat the current beliefs of their neighbors as sufficient statistics for all the information available to them. We establish that as long as imperfect recall represents the only point of departure from Bayesian rationality, agents’ social learning rules take a log-linear form. Our approach also enables us to provide a taxonomy of behavioral assumptions that underpin various non-Bayesian models of learning, including the canonical model of DeGroot. We then show that for a fairly large class of learning rules, the form of bounded rationality represented by imperfect recall is not an impediment to asymptotic learning, as long as agents assign weights of equal orders of magnitude to every independent piece of information. Finally, we show how the dispersion of information among different individuals in the social network determines the rate of learning.

A repeated network game where agents have quadratic utilities that depend on information externalities-an unknown underlying state-as well as payoff externalities-the actions of all other agents in the network-is considered. Agents play Bayesian Nash Equilibrium strategies with respect to their beliefs on the state of the world and the actions of all other nodes in the network. These beliefs are refined over subsequent stages based on the observed actions of neighboring peers. This paper introduces the Quadratic Network Game (QNG) filter that agents can run locally to update their beliefs, select corresponding optimal actions, and eventually learn a sufficient statistic of the network's state. The QNG filter is demonstrated on a Cournot market competition game and a coordination game to implement navigation of an autonomous team.

This work is concerned with the problem of social learning. A network of agents attempt to learn some unknown state of the world which is drawn by nature from a finite set. The sources of information available to the agents are their own private observations as well as the beliefs of their neighboring agents in the social structure in which they interact. The rational approach to the problem of social learning is for each agent to refine her opinion, by using the Bayes' rule to incorporate her neighbors' beliefs and own private signals over time. However, repeated applications of the Bayes' rule in social networks can become computationally intractable, partly due to the fact that each agent needs to use her local data that is increasing over time and make inferences about the global network structure. The inherent complexity of the Bayesian approach has lead to the consideration of behavioral, non-Bayesian updates, where each agent instead of processing all new information in the optimum Bayesian way uses simple rules such as linear or convex combinations and forms updated beliefs. In this paper, it is shown that by replicating the rule that maps the agents' common prior to their Bayesian posterior at the initial time-step for all future steps, one can derive a memoryless Bayesian update that has a log-linear format and can serve as a theoretical justification for some of the non-Bayesian rules suggested in the literature. Convergence and learning under the derived rules are also investigated, and it is shown that while two communicating agents always learn the true state, the beliefs may fail to converge in the general network setting. The proposed approach has the advantage that while preserving some features of the Bayesian inference, they are made tractable.

We consider a repeated game in which a team of agents share a common, but only partially known, task. The team also has the goal to coordinate while completing the task. This creates a trade-off between estimating the task and coordinating with others reminiscent of the kind of trade-off exemplified by the Keynesian beauty contest game. The agents thus can benefit from learning from others. This paper provides a survey of results from [1-4]. We first present a recent result that states repeated play of the game by myopic but Bayesian agents, who observe the actions of their neighbors over a connected network, eventually yield coordination on a single action. Furthermore, the coordinated action is equal to the mean estimate of the common task given individual's information. This indicates that agents in the network have the same mean estimate in the limit despite the differences in the quality of local information. Finally, we state that if the space of signals is a finite set, the coordinated action is equal to the estimate of the common task given full information, that is, agents eventually aggregate the information available throughout the network on the common task optimally.

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This paper focuses on a model of opinion formation over networks with continuously flowing new information and studies the relationship between the network and information structures and agents' ability to reach agreement. At each time period, agents receive private signals in addition to observing the beliefs held by their neighbors in a network. Each agent then updates its belief by aggregating the information available to it in a boundedly rational fashion. Whether this model results in conformity or disparity of beliefs is contingent on connectivity of the network and identifiability of the unknown state. We show that if the network is strongly connected, agents will eventually reach consensus in their beliefs; if in addition the state is globally identifiable, then agents will be able to learn the unknown state. Agents will also reach consensus and learn the state if the network fails to be connected but the state is locally identifiable. In contrast, agents in a non-strongly connected network will almost never reach agreement in their beliefs about a locally unidentifiable state. We also provide a characterization of the rates of convergence in terms of the top Lyapunov exponent of a set of i.i.d. matrices. The proofs use Oseledets' multiplicative ergodic theorem and recent results on stability of Lyapunov regular dynamical systems.

We consider a repeated network game where agents' utilities are quadratic functions of the state of the world and actions of all the agents. The state of the world is represented by a vector on which agents receive private signals with Gaussian noise. We define the solution concept as Bayesian Nash equilibrium and present a recursion to compute equilibrium strategies locally if an equilibrium exists at all stages. We further provide conditions under which a unique equilibrium exists. We conclude with an example of the proposed recursion in a repeated Cournot competition game and discuss properties of convergence such as efficient learning and convergence rate.