Sequential equilibrium is one of the most fundamental refinements of Nash equilibrium for games in extensive form. However, it is not defined for extensive-form games in which a player can choose among a continuum of actions. We define a class of infinite extensive form games in which information behaves continuously as a function of past actions and define a natural notion of sequential equilibrium for this class. Sequential equilibria exist in this class and refine Nash equilibria. In standard finite extensive-form games, our definition selects the same strategy profiles as the traditional notion of sequential equilibrium.

Between 1949 and 1980, every U.S. state mandated public schools to provide educational services for disabled students. This is one of the largest education reforms in U.S. history, but little is known about its impacts. Given scarce data in this period, I compile survey and administrative datasets and set up a difference-in-difference design using variation in the mandates' timing. I show that the mandates increased both services for disabled students and preschool enrollments. In adulthood, disabled individuals below school age at a mandate's implementation became about 20% less likely to have no education, attained up to 0.23 more years of education, and were more likely to have worked. Although this policy could have taken away resources from non-disabled students, in fact, education and employment also increased for non-disabled individuals. These effects align with evidence that the mandates increased spending per student by up to 15%. Families were also impacted: the mandates increased employment among mothers of disabled children and the probability that disabled individuals became household heads. Over the long term, the mandates paid for themselves by generating government revenues in excess of their cost. These results provide new evidence on the large, broad impacts of expanding access to education for disabled students.

An economy, large or small, has traditionally been defined in terms of an explicit set of agents and an assignment of characteristics to each agent. But when individual agents are negligible, most economically relevant properties of an economy can be defined in terms of the distribution of characteristics alone. Agents need not be specified. It has been frequently asserted that the distributional description of an economy is too sparse for core analysis. Notions of coalitions and blocking require the individualistic description of agents. This paper shows that this is not so. The presence of blocking coalitions can be directly identified in terms of distributions alone. Indeed, we give a purely distributional proof of the classical core-equivalence theorem that delivers the core-equivalence theorem for individualistic economies as a corollary. Our methods have applications outside of general equilibrium theory. They apply to large matching markets and to analogs of the Shapley-value for atomless economies.

Heinsohn and Steiger's "Eigentum, Zins und Geld" (1996) proposes the property premium as the foundational determinant of interest, replacing time preference. This paper examines whether the replacement succeeds. It does not. The two arguments against time preference, the savings-inelasticity claim after Hahn and the portfolio-shift claim after Keynes, both fail on standard microeconomic grounds. With time preference intact, the property premium sits within the standard decomposition of the interest rate. In ordinary collateralized credit it coincides with the risk premium. Only when the lender is a money-issuing bank with a real redemption obligation does a third term enter the decomposition that standard asset-pricing theory does not articulate. That third term is Heinsohn and Steiger's genuine contribution. The paper discusses its apparent disappearance or disguised operation after 2008, and the circularity of a property anchor measured in money.

Prediction markets are widely treated as forecasting devices that reveal collective expectations about uncertain futures. This article argues that under specifiable conditions they also function as coordination mechanisms: public probabilities that organize the behavior of voters, donors, journalists, traders, and institutions in ways that can be self-fulfilling or self-defeating. Most existing work asks whether prediction markets forecast accurately; this paper asks whether accurate forecasting is even the right criterion for a market that has become a public coordination device. Drawing on transaction-level evidence from the 2024 U.S. presidential election, we show that the social force of a market signal depends less on its size than on its persistence, the breadth of responding trader types, and cross-platform consensus. We introduce a Signal Credibility Index (SCI) -- combining the variance ratio VR(6), a two-sidedness diagnostic, and a trader-concentration adjustment -- as a microstructure-grounded criterion for predicting when price moves acquire behavioral traction. Applied to three major 2024 political shocks, the framework reveals that superficially similar events generated qualitatively distinct signal types with different implications for elite coordination. A cross-platform comparison establishes a systematic decoupling of social authority from epistemic robustness: the most visible market produced the least accurate forecasts. The framework carries direct implications for regulating prediction markets as democratic information infrastructure.

We propose using a discounted version of a convex combination of the log-likelihood with the corresponding expected log-likelihood such that when they are maximized they yield a filter, predictor and smoother for time series. This paper then focuses on working out the implications of this in the case of the canonical exponential family. The results are simple exact filters, predictors and smoothers with linear recursions. A theory for these models is developed and the models are illustrated on simulated and real data.

Using only retrospective data, we study the problem of predicting treatment effects for the same treatment/policy implemented in a different location or time period. We propose a distributionally robust estimator that minimizes the worst-case mean squared error for the prediction of treatment effect over a class of distributions defined by a Wasserstein neighborhood around the source distribution. Because the joint distribution of potential outcomes is unidentified, the problem is inherently one of partial identification. We characterize the sharp upper and lower bounds of the minimax optimizer by exploiting the Fréchet class of distributions consistent with the marginal distributions of potential outcomes. The resulting predictor preserves the sign of the average treatment effect under the source distribution but is shrunk toward zero, with the degree of shrinkage depending on the extent of treatment effect heterogeneity. We establish consistency and asymptotic normality of the bound estimators, develop a two-step inference procedure, and discuss the choice of the robustness parameter.

Many modern areas have not learned their lessons and often hope for the wisdom of later generations, resulting in them only possessing modern technology and difficult to iterate ancient civilizations. At present, there is no way to tell how we should learn from history and promote the gradual upgrading of civilization. Therefore, we must tell the history of civilization's progress and the means of governance, learn from experience to improve the comprehensive strength and survival ability of civilization, and achieve an optimal solution for the tempering brought by conflicts and the reduction of internal conflicts. Firstly, we must follow the footsteps of history and explore the reasons for the long-term stability of each country in conflict, including providing economic benefits to the people and means of suppressing them; then, use mathematical methods to demonstrate how we can achieve the optimal solution at the current stage. After analysis, we can conclude that the civilization transformed from human plowing to horse plowing can easily suppress the resistance of the people and provide them with the ability to resist; The selection of rulers should consider multiple institutional aspects, such as exams, elections, and drawing lots; Economic development follows a lognormal distribution and can be adjusted by population mean and standard deviation. Using a lognormal distribution with the maximum value to divide equity can adjust the wealth gap.

We consider bootstrap inference in predictive (or Granger-causality) regressions when the parameter of interest may lie on the boundary of the parameter space, here defined by means of a smooth inequality constraint. For instance, this situation occurs when the definition of the parameter space allows for the cases of either no predictability or sign-restricted predictability. We show that in this context constrained estimation gives rise to bootstrap statistics whose limit distribution is, in general, random, and thus distinct from the limit null distribution of the original statistics of interest. This is due to both (i) the possible location of the true parameter vector on the boundary of the parameter space, and (ii) the possible non-stationarity of the posited predicting (resp. Granger-causing) variable. We discuss a modification of the standard fixed-regressor wild bootstrap scheme where the bootstrap parameter space is shifted by a data-dependent function in order to eliminate the portion of limiting bootstrap randomness attributable to the boundary, and prove validity of the associated bootstrap inference under non-stationarity of the predicting variable as the only remaining source of limiting bootstrap randomness. Our approach, which is initially presented in a simple location model, has bearing on inference in parameter-on-the-boundary situations beyond the predictive regression problem.

This study presents empirical evidence to support the validity of new definitions in financial markets. The author develops a new method to determine investors' risk attitudes in financial markets. The risk attitudes of investors in US financial markets from 1889-1978 are analyzed and the results indicate that equity investors who invested in the composite S&P 500 index were risk-averse in 1977. Conversely, risk-free asset investors who invested in US Treasury bills were found to exhibit not enough risk-loving behavior, which can be considered a type of risk-averse behavior. These findings suggest that the new definitions in financial markets accurately reflect the behavior of investors and should be considered in investment strategies.

In this study, we propose a test for the coefficient randomness in autoregressive models where the autoregressive coefficient is local to unity, which is empirically relevant given the results of earlier studies. Under this specification, we theoretically analyze the effect of the correlation between the random coefficient and disturbance on tests' properties, which remains largely unexplored in the literature. Our analysis reveals that the correlation crucially affects the power of tests for coefficient randomness and that tests proposed by earlier studies can perform poorly when the degree of the correlation is moderate to large. The test we propose in this paper is designed to have a power function robust to the correlation. Because the asymptotic null distribution of our test statistic depends on the correlation $ψ$ between the disturbance and its square as earlier tests do, we also propose a modified version of the test statistic such that its asymptotic null distribution is free from the nuisance parameter $ψ$. The modified test is shown to have better power properties than existing ones in large and finite samples.

We establish theoretical results about the low frequency contamination (i.e., long memory effects) induced by general nonstationarity for estimates such as the sample autocovariance and the periodogram, and deduce consequences for heteroskedasticity and autocorrelation robust (HAR) inference. We present explicit expressions for the asymptotic bias of these estimates. We distinguish cases where this contamination only occurs as a small-sample problem and cases where the contamination continues to hold asymptotically. We show theoretically that nonparametric smoothing over time is robust to low frequency contamination. Our results provide new insights on the debate between consistent versus inconsistent long-run variance (LRV) estimation. Existing LRV estimators tend to be in inflated when the data are nonstationary. This results in HAR tests that can be undersized and exhibit dramatic power losses. Our theory indicates that long bandwidths or fixed-b HAR tests suffer more from low frequency contamination relative to HAR tests based on HAC estimators, whereas recently introduced double kernel HAC estimators do not super from this problem. Finally, we present second-order Edgeworth expansions under nonstationarity about the distribution of HAC and DK-HAC estimators and about the corresponding t-test in the linear regression model.

We propose a new simple procedure called Population-Mean-Based Aggregation (PMBA) that enables a principal to "aggregate" information about an unknown state of the world from agents without understanding the information structure among them. PMBA only requires agents to communicate their beliefs about the state, and some agents to communicate their expectations of the population average belief. In a large population, for any finite number of possible states, and under weak assumptions on the information structure, allowing individual agents' beliefs to be misspecified, we show that PMBA infers the true state (in probability or almost surely under the stated conditions). We show how PMBA can be reinterpreted as a linear regression procedure, and how it can be used to aggregate information from a finite number of agents, allowing us to reuse existing results on inference in linear models. We conduct a novel experiment to show that the real-world performance of our procedure exceeds that of existing methods.

The start of a human's life can be characterized by two lotteries: that of your genes (nature) and the family you were born into (nurture). These set in motion a trajectory, from birth onward, in health and human capital. Leveraging three longitudinal social-science data sets, we systematically analyze the relationship between an individual's genotype, the socioeconomic status (SES) of the families they grew up in, and their realized traits in adulthood. We proxy an individual's genetic predisposition by polygenic indexes (PGIs) and family SES by a latent factor of parental education and father's (former) occupational status. We then investigate how PGIs, parental SES, and their interaction contribute to later-life outcomes across a range of forty-five socioeconomic, anthropometric, health, behavioral, and personality traits. We find strong genetic and socioeconomic associations with these phenotypes, but no evidence of sizable gene-environment interactions.

We develop a nonparametric approach to identify and estimate consumer preferences and unobserved heterogeneity under nonlinear price schedules. Leveraging variation across multiple price schedules, we show that both the utility function and the distribution of preference types can be nonparametrically identified. The quantile function of unobserved types becomes solution of a functional equation, and we derive conditions ensuring identification. We propose an iterative approach for estimation, in which the regularization bias decays exponentially in the number of iterations while the variance grows only polynomially, yielding a near-parametric convergence rate. We propose a valid bootstrap procedure for finite-sample inference and extend the framework to accommodate potential endogeneity of prices and additional observed heterogeneity. Monte Carlo simulations and an empirical application to data from a European mail carrier demonstrate how we can recover the utility functions and preference distributions in finite samples.

A forward-looking agent observes signals of a state that follows a Gaussian AR(1) process. He balances the cost of having imprecise beliefs with the cost of acquiring more precise signals. I characterize his optimal information acquisition policy, and analyze how his steady-state beliefs and costs depend on persistence (the AR(1) parameter) and patience (the agent's discount factor). Higher persistence has a non-monotone effect on belief precision and raises overall costs. Higher patience makes beliefs more precise and lowers overall costs.

We study welfare analysis for policy changes when supply and demand behavior are only partially known. We augment the robust approach pioneered by Kang and Vasserman (2025) by incorporating the supply side. We posit intervals of feasible pass-through and conduct (market-power) parameters, then apply them to two equilibrium snapshots to characterize the extremal supply-side terms entering welfare. We show that the supply-side bounds are attained by inverse pass-through functions that take only the two endpoint values of the specified interval, separated by a single price cutoff. Combining these supply-side extrema with demand-side shape restrictions, we produce simple bounds for consumer surplus, producer surplus, total surplus, and deadweight loss.

We characterize the family of utility functions satisfying linear fractional relative risk aversion (LFRRA) in terms of the Gauss hypergeometric functions. We apply this family, which nests various utility functions used in different strands of literature, to monopolistic competition and obtain the profit-maximizing price by generalizing the Lambert W function. We let firm-level data decide whether the RRA in each sector or in the aggregate economy is increasing, decreasing, or constant, which in turn determines whether markups are decreasing, increasing, or constant with respect to marginal costs.

Without the assumption of complete, shared awareness, it is necessary to consider communication between agents who may entertain different representations of the world. A syntactic (language-based) approach provides powerful tools to address this problem. In this paper, we define translation operators between two languages which provide a `best approximation' for the meaning of propositions in the target language subject to its expressive power. We show that, in general, the translation operators preserve some, but not all, logical operations. We derive necessary and sufficient conditions for the existence of a joint state space and a joint language, in which the subjective state spaces of each agent, and their individual languages, may be embedded. This approach allows us to compare languages with respect to their expressiveness and thus, with respect to the properties of the associated state space.

We model a market for data where an incumbent and a challenger compete for data from a producer. The incumbent has access to an exclusive data producer, and it uses this exclusive access, together with economies of scope in the aggregation of the data, as a strategy against the potential entry by the challenger. We assess the incumbent incentives to either deter or accommodate the entry of the challenger. We show that the incumbent will accommodate when the exclusive access is costly and when the economies of scope are low, and it will blockade or deter otherwise. The results would justify an access regulation that incentivizes the entry of the challenger, e.g., by increasing production costs for the exclusive data.

This paper studies how communication across experts prior to aggregation by a decision-maker affects the efficiency of forecast combination. When experts exchange information before reporting their forecasts, their signals become correlated through the communication network, altering aggregation efficiency even when forecasts are unbiased. The analysis introduces a statistic that characterizes how network structure shapes aggregation efficiency and shows that degree heterogeneity plays a central role. Among connected networks, regular networks attain the minimal level of aggregation distortion, while star networks generate the largest distortions within sparse connected structures. Random network benchmarks show that aggregation efficiency approaches the regular-network benchmark when expected degree either vanishes or becomes large as network size increases, whereas networks with constant expected degree generate intermediate distortions. These results provide a theoretical foundation for understanding how communication across experts affects forecast combination and establish a connection between the forecast combination literature and models of social learning in networks.