This paper develops a Stein-like shrinkage estimator for heterogeneous panel data models with cross-sectional heteroskedasticity and contemporaneous error dependence, where the cross-section dimension is fixed and the time dimension increases. The estimator combines unrestricted and restricted feasible generalized least squares (FGLS) estimators, with a weight determined by a Wald statistic assessing the imposed linear restrictions. Under independence over time and suitable regularity conditions, we derive analytical expressions for the asymptotic bias, mean squared error matrix, and risk of the standard shrinkage estimator. We establish sufficient conditions for asymptotic risk dominance over the unrestricted FGLS estimator and characterize the limiting relative asymptotic risk under sequential limits with increasing regressor and restriction dimensions. Monte Carlo simulations examine the finite-sample performance of positive-part shrinkage under homogeneity and exclusion restrictions. An application to forecasting output growth in 19 countries illustrates modest improvements in forecast accuracy.
Macroeconomic panels display two forms of cross-sectional dependence usually treated as rival specifications: unobserved common factors with heterogeneous loadings, which motivate common correlated effects estimators, and spatial spillovers, which motivate spatial autoregressive models. This article specifies and evaluates a sequential pretest: common factors are filtered through cross-sectional averages, the residuals are tested for spatial dependence with Moran’s I, and the estimator is chosen accordingly. A Monte Carlo experiment with 96 cells and 1000 replications each varies the weight matrix, spatial parameter, number of factors and panel dimensions. The annual Moran test on filtered residuals is correctly sized, but a rule requiring rejection in a quarter of the periods is conservative by construction. Detection does not translate into lower risk: with common factors and moderate spatial dependence, a more powerful rule raises the mean squared error of the post-test estimator, even when calibrated by permutation, while a joint spatial–factor estimator keeps a low risk and close to nominal coverage if the weight matrix is correctly specified. Applied to 18 Latin American economies (2000–2021), the pretest selects the factor-based estimator and the mean group joint estimator, unlike the pooled one, finds no significant spatial lag.
This study investigates whether rising income inequality fuels credit-financed household consumption in the United States, and how economic policy uncertainty affects credit, spending, and inequality. We estimate a vector autoregression with exogenous variables (VARX) in first differences using annual U.S. data for the 1972–2025 period, drawn from the U.S. Census Bureau and the Federal Reserve Economic Data. Inequality is measured by the Gini index and the 90th-/10th- and 50th-/10th-percentile income ratios. We find no evidence that rising inequality increases household credit growth. Instead, inequality affects spending directly, as a wider 90th-/10th-percentile gap raises expenditure growth. Policy uncertainty consistently reduces credit growth and widens the gap between middle and low incomes. Inequality is persistent: the 90th-/10th-percentile ratio partly corrects after a shock but retains about three-quarters of it and drifts upward, and its short-run changes are driven by consumption rather than credit. The study contributes aggregate time-series evidence that challenges the inequality–debt hypothesis, extends the credit-channel effect of policy uncertainty to household credit, and shows that its distributional cost falls on those at the bottom of the income distribution. The findings point to policy priorities in affordable credit access, income support, and macroprudential regulation.
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.
