Difference-in-differences (DID) is one of the most popular tools used to evaluate causal effects of policy interventions. This paper extends the DID methodology to accommodate interval outcomes, which are often encountered in empirical studies using survey or administrative data. We point out that a naive application or extension of the conventional parallel trends assumption may yield uninformative or counterintuitive results, and present a suitable identification strategy, called parallel shifts, which exhibits desirable properties. Practical attractiveness of the proposed method is illustrated by revisiting an influential minimum wage study by Card and Krueger (1994).

When customers must visit a seller to learn the valuation of its product, sellers potentially benefit from charging a lower price on the first visit and a higher price when a buyer returns. Armstrong and Zhou (2016) show that such price discrimination can arise in equilibrium when buyers learn a seller's pricing policy only upon visiting. We depart from this assumption by supposing that sellers commit to observable pricing policies that guide consumer search and buyers can choose whom to visit first. We show that no seller engages in price discrimination in equilibrium.

Standard decision theory seeks conditions under which a preference relation can be compressed into a single real-valued function. However, when preferences are incomplete or intransitive, a single function fails to capture the agent's evaluative structure. Recent literature on multi-utility representations suggests that such preferences are better represented by families of functions. This paper provides a canonical and intrinsic geometric characterization of this family. We construct the \textit{ledger group} $U(P)$, a partially ordered group that faithfully encodes the native structure of the agent's preferences in terms of trade-offs. We show that the set of all admissible utility functions is precisely the \textit{dual cone} $U^*$ of this structure. This perspective shifts the focus of utility theory from the existence of a specific map to the geometry of the measurement space itself. We demonstrate the power of this framework by explicitly reconstructing the standard multi-attribute utility representation as the intersection of the abstract dual cone with a subspace of continuous functionals, and showing the impossibility of this for a set of lexicographic preferences.

There has been significant progress in Bayesian inference based on sparsity-inducing (e.g., spike-and-slab and horseshoe-type) priors for high-dimensional regression models. The resulting posteriors, however, in general do not possess desirable frequentist properties, and the credible sets thus cannot serve as valid confidence sets even asymptotically. We introduce a novel debiasing approach that corrects the bias for the entire Bayesian posterior distribution. We establish a new Bernstein-von Mises theorem that guarantees the frequentist validity of the debiased posterior. We demonstrate the practical performance of our proposal through Monte Carlo simulations and two empirical applications in economics.

This paper develops the Theory of Strategic Evolution, a general model for systems in which the population of players, strategies, and institutional rules evolve together. The theory extends replicator dynamics to settings with endogenous players, multi level selection, innovation, constitutional change, and meta governance. The central mathematical object is a Poiesis stack: a hierarchy of strategic layers linked by cross level gain matrices. Under small gain conditions, the system admits a global Lyapunov function and satisfies selection, tracking, and stochastic stability results at every finite depth. We prove that the class is closed under block extension, innovation events, heterogeneous utilities, continuous strategy spaces, and constitutional evolution. The closure theorem shows that no new dynamics arise at higher levels and that unrestricted self modification cannot preserve Lyapunov structure. The theory unifies results from evolutionary game theory, institutional design, innovation dynamics, and constitutional political economy, providing a general mathematical model of long run strategic adaptation.

We consider an adaptive experiment for treatment choice and design a minimax and Bayes optimal adaptive experiment with respect to regret. Given binary treatments, the experimenter's goal is to choose the treatment with the highest expected outcome through an adaptive experiment, in order to maximize welfare. We consider adaptive experiments that consist of two phases, the treatment allocation phase and the treatment choice phase. The experiment starts with the treatment allocation phase, where the experimenter allocates treatments to experimental subjects to gather observations. During this phase, the experimenter can adaptively update the allocation probabilities using the observations obtained in the experiment. After the allocation phase, the experimenter proceeds to the treatment choice phase, where one of the treatments is selected as the best. For this adaptive experimental procedure, we propose an adaptive experiment that splits the treatment allocation phase into two stages, where we first estimate the standard deviations and then allocate each treatment proportionally to its standard deviation. We show that this experiment, often referred to as Neyman allocation, is minimax and Bayes optimal in the sense that its regret upper bounds exactly match the lower bounds that we derive. To show this optimality, we derive minimax and Bayes lower bounds for the regret using change-of-measure arguments. Then, we evaluate the corresponding upper bounds using the central limit theorem and large deviation bounds.

\citet{farrell2021deep} establish non-asymptotic high-probability bounds for general deep feedforward neural network (with rectified linear unit activation function) estimators, with \citet[Theorem 1]{farrell2021deep} achieving a suboptimal convergence rate for fully connected feedforward networks. The authors suggest that improved approximation of fully connected networks could yield sharper versions of \citet[Theorem 1]{farrell2021deep} without altering the theoretical framework. By deriving approximation bounds specifically for a narrower fully connected deep neural network, this note demonstrates that \citet[Theorem 1]{farrell2021deep} can be improved to achieve an optimal rate (up to a logarithmic factor). Furthermore, this note briefly shows that deep neural network estimators can mitigate the curse of dimensionality for functions with compositional structure and functions defined on manifolds.

I relax the standard assumptions of transitivity and partition structure in economic models of information to formalize vague knowledge: non-transitive indistinguishability over states. I show that vague knowledge, while failing to partition the state space, remains informative by distinguishing some states from others. Moreover, it can only be faithfully expressed through vague communication with blurred boundaries. My results provide microfoundations for the prevalence of natural language communication and qualitative reasoning in the real world, where knowledge is often vague.

This paper investigates both short and long-run interaction between BIST-100 index and CDS prices over January 2008 to May 2015 using ARDL technique. The paper documents several findings. First, ARDL analysis shows that 1 TL increase in CDS shrinks BIST-100 index by 22.5 TL in short-run and 85.5 TL in long-run. Second, 1000 TL increase in BIST index price causes 25 TL and 44 TL reducation in Turkey's CDS prices in short- and long-run respectively. Third, a percentage increase in interest rate shrinks BIST index by 359 TL and a percentage increase in inflation rate scales CDS prices up to 13.34 TL both in long-run. In case of short-run, these impacts are limited with 231 TL and 5.73 TL respectively. Fourth, a kurush increase in TL/USD exchange rate leads 24.5 TL (short-run) and 78 TL (long-run) reductions in BIST, while it augments CDS prices by 2.5 TL (short-run) and 3 TL (long-run) respectively. Fifth, each negative political events decreases BIST by 237 TL in short-run and 538 TL in long-run, while it increases CDS prices by 33 TL in short-run and 89 TL in long-run. These findings imply the highly dollar indebted capital structure of Turkish firms, and overly sensitivity of financial markets to the uncertainties in political sphere. Finally, the paper provides evidence for that BIST and CDS with control variables drift too far apart, and converge to a long-run equilibrium at a moderate monthly speed.

We develop a theoretical framework for sample splitting in A/B testing environments, where data for each test are partitioned into two splits to measure methodological performance when the true impacts of tests are unobserved. We show that sample-split estimators are generally biased for full-sample performance but consistently estimate sample-split analogues of it. We derive their asymptotic distributions, construct valid confidence intervals, and characterize the bias-variance trade-offs underlying sample-split design choices. We validate our theoretical results through simulations and provide implementation guidance for A/B testing products seeking to evaluate new estimators and decision rules.

We propose a new formulation of the maximum score estimator that uses compositions of rectified linear unit (ReLU) functions, instead of indicator functions as in Manski (1975,1985), to encode the sign alignment restrictions. Since the ReLU function is Lipschitz, our new ReLU-based maximum score criterion function is substantially easier to optimize using standard gradient-based optimization pacakges. We also show that our ReLU-based maximum score (RMS) estimator can be generalized to an umbrella framework defined by multi-index single-crossing (MISC) conditions, while the original maximum score estimator cannot be applied. We establish the $n^{-s/(2s+1)}$ convergence rate and asymptotic normality for the RMS estimator under order-$s$ Holder smoothness. In addition, we propose an alternative estimator using a further reformulation of RMS as a special layer in a deep neural network (DNN) architecture, which allows the estimation procedure to be implemented via state-of-the-art software and hardware for DNN.

Timely assessment of current conditions is essential especially for small, open economies such as Singapore, where external shocks transmit rapidly to domestic activity. We develop a real-time nowcasting framework for quarterly GDP growth using a high-dimensional panel of approximately 70 indicators, encompassing economic and financial indicators over 1990Q1-2023Q2. The analysis covers penalized regressions, dimensionality-reduction methods, ensemble learning algorithms, and neural architectures, benchmarked against a Random Walk, an AR(3), and a Dynamic Factor Model. The pipeline preserves temporal ordering through an expanding-window walk-forward design with Bayesian hyperparameter optimization, and uses moving block-bootstrap procedures both to construct prediction intervals and to obtain confidence bands for feature-importance measures. It adopts model-specific and XAI-based explainability tools. A Model Confidence Set procedure identifies statistically superior learners, which are then combined through simple, weighted, and exponentially weighted schemes; the resulting time-varying weights provide an interpretable representation of model contributions. Predictive ability is assessed via Giacomini-White tests. Empirical results show that penalized regressions, dimensionality-reduction models, and GRU networks consistently outperform all benchmarks, with RMSFE reductions of roughly 40-60%; aggregation delivers further gains. Feature-attribution methods highlight industrial production, external trade, and labor-market indicators as dominant drivers of Singapore's short-run growth dynamics.

Foundation models, and in particular large language models, can generate highly informative responses, prompting growing interest in using these ''synthetic'' outputs as data in empirical research and decision-making. This paper introduces the idea of a foundation prior, which shows that model-generated outputs are not as real observations, but draws from the foundation prior induced prior predictive distribution. As such synthetic data reflects both the model's learned patterns and the user's subjective priors, expectations, and biases. We model the subjectivity of the generative process by making explicit the dependence of synthetic outputs on the user's anticipated data distribution, the prompt-engineering process, and the trust placed in the foundation model. We derive the foundation prior as an exponential-tilted, generalized Bayesian update of the user's primitive prior, where a trust parameter governs the weight assigned to synthetic data. We then show how synthetic data and the associated foundation prior can be incorporated into standard statistical and econometric workflows, and discuss their use in applications such as refining complex models, informing latent constructs, guiding experimental design, and augmenting random-coefficient and partially linear specifications. By treating generative outputs as structured, explicitly subjective priors rather than as empirical observations, the framework offers a principled way to harness foundation models in empirical work while avoiding the conflation of synthetic ''facts'' with real data.

We investigate the extent to which groups with elevated mortality rates ex ante might opt out of guaranteed national pensions in favour of demographically aligned plans, which we label equitable longevity risk sharing (ELRiS) pools, even if this involves accepting some idiosyncratic risk. Technically, this paper develops a stochastic model of retirement income within an ELRiS structure that is calibrated to equate the discounted expected utility of a guaranteed national pension. The practical motivation for developing this alternative is that working members of First Nations peoples of Canada: (1) experience much higher mortality rates than average over their entire life cycle, and (2) some are actually allowed by current legislation to opt out of the Canada Pension Plan (CPP). We then demonstrate that under reasonable economic preferences and parameters, a sub-group with a 10-year life expectancy gap relative to the population could attain equivalent lifetime utility by contributing a mere two-thirds to the plan, even if they were pooled with only 30 members. For a longevity gap of 20 years, such as between an Indigenous male versus a non-Indigenous female, the contribution rate falls to less than a third. The difference between the statutory and mandatory contribution rates to a guaranteed national pension and those needed within these self-sustaining pools is an implicit subsidy from Indigenous to non-Indigenous. From a policy perspective, this paper aspires to jump-start a conversation that sparks a change in a status quo, which is obviously unfair and inequitable.

We curate the DeXposure dataset, the first large-scale dataset for inter-protocol credit exposure in decentralized financial networks, covering global markets of 43.7 million entries across 4.3 thousand protocols, 602 blockchains, and 24.3 thousand tokens, from 2020 to 2025. A new measure, value-linked credit exposure between protocols, is defined as the inferred financial dependency relationships derived from changes in Total Value Locked (TVL). We develop a token-to-protocol model using DefiLlama metadata to infer inter-protocol credit exposure from the token's stock dynamics, as reported by the protocols. Based on the curated dataset, we develop three benchmarks for machine learning research with financial applications: (1) graph clustering for global network measurement, tracking the structural evolution of credit exposure networks, (2) vector autoregression for sector-level credit exposure dynamics during major shocks (Terra and FTX), and (3) temporal graph neural networks for dynamic link prediction on temporal graphs. From the analysis, we observe (1) a rapid growth of network volume, (2) a trend of concentration to key protocols, (3) a decline of network density (the ratio of actual connections to possible connections), and (4) distinct shock propagation across sectors, such as lending platforms, trading exchanges, and asset management protocols. The DeXposure dataset and code have been released publicly. We envision they will help with research and practice in machine learning as well as financial risk monitoring, policy analysis, DeFi market modeling, amongst others. The dataset also contributes to machine learning research by offering benchmarks for graph clustering, vector autoregression, and temporal graph analysis.

Understanding household behaviour is essential for modelling macroeconomic dynamics and designing effective policy. While heterogeneous agent models offer a more realistic alternative to representative agent frameworks, their implementation poses significant computational challenges, particularly in continuous time. The Aiyagari-Bewley-Huggett (ABH) framework, recast as a system of partial differential equations, typically relies on grid-based solvers that suffer from the curse of dimensionality, high computational cost, and numerical inaccuracies. This paper introduces the ABH-PINN solver, an approach based on Physics-Informed Neural Networks (PINNs), which embeds the Hamilton-Jacobi-Bellman and Kolmogorov Forward equations directly into the neural network training objective. By replacing grid-based approximation with mesh-free, differentiable function learning, the ABH-PINN solver benefits from the advantages of PINNs of improved scalability, smoother solutions, and computational efficiency. Preliminary results show that the PINN-based approach is able to obtain economically valid results matching the established finite-difference solvers.

Crypto enthusiasts claim that buying and holding crypto assets yields high returns, often citing Bitcoin's past performance to promote other tokens and fuel fear of missing out. However, understanding the real risk-return trade-off and what factors affect future crypto returns is crucial as crypto becomes increasingly accessible to retail investors through major brokerages. We examine the HODL strategy through two independent analyses. First, we implement 480 million Monte Carlo simulations across 378 non-stablecoin crypto assets, net of trading fees and the opportunity cost of 1-month Treasury bills, and find strong evidence of survivorship bias and extreme downside concentration. At the 2-3 year horizon, the median excess return is -28.4 percent, the 1 percent conditional value at risk indicates that tail scenarios wipe out principal after all costs, and only the top quartile achieves very large gains, with a mean excess return of 1,326.7 percent. These results challenge the HODL narrative: across a broad set of assets, simple buy-and-hold loads extreme downside risk onto most investors, and the miracles mostly belong to the luckiest quarter. Second, using a Bayesian multi-horizon local projection framework, we find that endogenous predictors based on realized risk-return metrics have economically negligible and unstable effects, while macro-finance factors, especially the 24-week exponential moving average of the Fear and Greed Index, display persistent long-horizon impacts and high cross-basket stability. Where significant, a one-standard-deviation sentiment shock reduces forward top-quartile mean excess returns by 15-22 percentage points and median returns by 6-10 percentage points over 1-3 year horizons, suggesting that macro-sentiment conditions, rather than realized return histories, are the dominant indicators for future outcomes.