Reliable modelling of carbon prices is critical for climate policy design, investment appraisal, and risk management issues in emissions trading systems. In this paper, a seasonal Ornstein–Uhlenbeck (OU) jump-diffusion model is developed for the monthly EU Allowance (EUA) prices. The long-run mean is a causal 12-month trailing moving average plus a Fourier seasonal component without lookahead bias. The likelihood employs exact Poisson probabilities and is modeled by a compound Poisson process with Laplace jumps, capturing discontinuous movements. Maximum likelihood estimation of parameters is performed on EUA prices between January 2013 and December 2025. The model is evaluated in-sample (2013–2023) and out-of-sample (2024–2025) using RMSE, MAE, directional accuracy, and Diebold–Mariano tests, and benchmarked against GARCH (1,1), GJR-GARCH, and EGARCH, the natural models for return-scale volatility clustering, complementary to the OU family’s focus on price-level dynamics. AIC and BIC favor the jump-augmented specification in-sample, but this preference does not carry over out-of-sample: point forecasts only slightly outperform a random walk, Diebold–Mariano tests cannot rule out equal accuracy over the short evaluation window, and the jump model’s directional accuracy (48%) falls below both the seasonal-only specification (61%) and chance. Four further checks—joint maximum-likelihood estimation of the jump intensity λ, block-bootstrap parameter uncertainty, an extended 48-month out-of-sample window, and density-forecast diagnostics (log-score, CRPS, PIT)—are treated as central results rather than as robustness add-ons. They show that λ is weakly identified when estimated jointly, that several parameters carry wide confidence intervals, and that the in-sample preference of AIC and BIC does not translate into better out-of-sample point or density forecasts. These identification and forecasting findings, rather than the jump specification’s in-sample fit, are the paper’s principal contribution, and limitations are discussed explicitly.
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.
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.
The nonlinear ARDL (NARDL) model by Shin et al. tests for asymmetric adjustment by decomposing a forcing variable into cumulated positive and negative changes. We study this linear-in-parameters partial-sum design when the forcing variable is stationary, as is commonly assumed for composite pressure indices. The two partial sums then share one stochastic trend, carry opposite deterministic trends, and are cointegrated by construction; the long-run symmetry test concerns the coefficient on cumulative absolute variation. If the dependent variable’s drift is not explained by the other level regressors, the estimated asymmetry converges to the unexplained drift divided by half the mean absolute change in the forcing variable, and the test rejects with a probability approaching one. Without a levels relationship, the Wald statistic has a nonstandard limit. Monte Carlo experiments confirm both results. Where a valid levels relationship exists, a linear trend removes the gross distortion and a restricted bootstrap with the trend restores size. For the Global Supply Chain Pressure Index and U.S. long-distance less-than-truckload freight prices over 1998M1–2026M7, the conventional asymmetry estimate (0.0111) is close to its drift benchmark (0.0121), and symmetry is rejected at every lag order without a trend and at none with one. Sign-split innovation models do not support rockets-and-feathers asymmetry, although their estimates are imprecise.
Installed renewable capacity is the leading indicator of energy-transition efforts, yet its conversion into realized generation is rarely estimated directly. This study estimates that rate in a panel of 118 developing and emerging economies over 2001–2023, treating proportional conversion, a unit coefficient, as a testable benchmark. In the preferred two-way fixed-effects specification, non-hydro renewable generation per person rises with one-year-lagged capacity per person, both in log(1 + x) form, at a rate of 1.042 (standard error, SE = 0.029), statistically indistinguishable from proportional conversion (test of unit coefficient, p = 0.148); the pure log–log elasticity among positive observations is lower, 0.845 (SE = 0.039). The estimate is stable under country-specific trends (0.924, SE = 0.046) and fixed technology shares (1.025, SE = 0.029), and a distributed lag in exact first differences shows that generation responds mainly to the current and previous year’s capacity changes, with a cumulative response of 1.070 (SE = 0.044). Higher lagged capacity is associated with a 3.630 percentage-point rise in the renewable output share (SE = 0.813) and a 3.258 percentage-point fall in the fossil share (SE = 0.771). Governance interactions are small and fragile under multiple-testing correction, and Poisson and pure log–log checks address the units-sensitivity of log(1 + x) estimates. Conditional on lag structure and functional form, deployment statistics are informative about realized generation and fossil substitution, whereas broader institutional and development gains remain unconfirmed.
