Econometrics

Background: The claim that a power law governs the distribution of value in a digital economy is usually asserted about sectors, while the evidence for heavy tails concerns firms. The two are different objects, and the claim is rarely tested at the level at which it is made. Methods: We test it at three levels using two censuses: every US employer firm from 1978 to 2023 in ten published size classes and 71 industries in the national accounts from 1997 to 2025. Distributions are fitted to interval-censored counts by maximum likelihood and compared by Vuong’s test; the tail is located by a threshold sweep; concentration from grouped data is bounded rather than estimated. Results: A log-normal is preferred over a Pareto in all forty-six years, yet underpredicts the number of firms with 10,000 employees or more by a factor of twenty-one, so the distribution is a mixture whose tail alone is consistent with a power law. When refitted with more than twenty employees, the exponent is 1.056, within three thousandths of a published microdata estimate obtained by a different method. Between sectors, the claim fails on two tests: industry scale carries no information about the share of value an industry retains, with correlations of 0.009 (Pearson) and $-0.047$ (Spearman), and the rise in the large-firm employment share is not compositional, since employment moved toward less concentrated sectors throughout. Conclusions: The power law operates on firms within industries, in the tail, which is where every instrument used here stops resolving differences. Rejecting a sectoral power law does not refute firm-level winner-take-all; it places that phenomenon below the resolution of the public statistics in which the debate is usually conducted.

We propose a joint covariate balancing estimator for the average treatment effect on the treated (ATT) in a difference-in-differences (DID) design when the outcome evolution, rather than a specific outcome level, is missing at random (MAR). The proposed estimator combines treatment and response into one indicator and calibrates the two group-based density ratios directly. Point estimation therefore requires only two nuisance working models, and inference adds two linear projections, giving four in total, whereas a factorized implementation of the efficient influence function requires five: one treatment probability, two response probabilities, and two outcome-evolution models. The estimator is automatically self-normalized and attains the semiparametric efficiency bound for the coarsened data model when all working models are correctly specified. Furthermore, we show that the proposed estimator is doubly robust for both consistency and inference. The simulation results support componentwise robustness and confirm the efficiency result when all nuisance models are correctly specified, while highlighting substantial small-sample limitations under low response probabilities. An all-correct design further shows that the efficiency ordering relative to an estimator using the always-observed outcome level can reverse.

Cognitive biases influence investor decision-making and how media outlets frame and report news. This study investigates whether cognitive biases embedded in news data can serve as early warning indicators of shifts in market sentiment and systemic risk, and whether emotion adds analytical value. News data from three South African outlets were analyzed for the period from January 2025 to May 2026, alongside major events that materially affected systemic risk. The study establishes the relationship between cognitive biases and the economic cycle, defines key cognitive bias proxy measures covering herding, availability, overconfidence, and loss aversion, and connects these measures to sentiment, emotion, and systemic risk. Large language models are used to extract sentiment and emotion from the news data. A systemic risk index is derived from South African equity, bond, foreign exchange, and credit variables. It captures financial stress and is more responsive to shocks than macroeconomic-based indices. The study shows that cognitive bias variables help explain changes in sentiment and emotional intensity, which often precede structural breaks in market variables. Spillover risk analysis shows that exogenous shocks coincide with more negatively toned news articles and a sharp increase in systemic risk.