A Diagnostic Evaluation of Seasonal Ornstein–Uhlenbeck Jump Models for Carbon Prices: In-Sample Fit vs. Out-of-Sample Performance

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.