aspg-34-4474-1791482495 20261008180135 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Prospects for Applied Mathematics and Data Analysis PAMDA 2836-4449 2025 5 2 Ika Agustin Department of Mathematics, University of Jember, Jember, East Java, Indonesia Harmonic Regression–Stochastic Residual Decomposition of Atmospheric Carbon Dioxide Concentration: Spectral Characterization and Forecast-Error Structure Ika Hesti Agustin1,* 1 Department of Mathematics, University of Jember, Jember, East Java, Indonesia Email: ikahesti.fmipa@unej.ac.id Received: May 31, 2025 Revised: August 09, 2025 Accepted: October 14, 2025 ⋆ Corresponding author For a series yt =gt +εt combining a curving trend with a strong seasonal cycle, this paper develops and proves properties of an explicit alternative to seasonal differencing: gt =β0+β1t+β2t2+ΣKk =1[ak sin(2πkt/m)+bk cos(2πkt/m)] and εt ∼ ARMA(p,q)×(P,Q)m, stationary and invertible. Four results are proved: near-orthogonality of the harmonic regressors, with Var( ˆ ak) ≈ 2σ2 ε /n; spectral concentration of the periodogram at ωk = 2πk/m; stationarity of the fitted residual via its characteristic roots, guaranteeing aWold representation εt = Σj ψjat−j with Σj ψ2j < ∞; and a three-way decomposition MSE(h) = Bias(h)2+Var( ˆ gT+h)+σ2 a Σh−1 j=0 ψ2j, whose noise term is shown to converge under stationarity but to diverge linearly, as in the exact random-walk case, under integration. Every result is verified numerically. Applied to the Mauna Loa CO2 record (h = 12,24,36,60 months against a linear-trend and a directly differenced SARIMA(1,1,1)(1,1,1)12 benchmark), gt explains R2 = 0.998 of variance with a significant, HAC-robust quadratic coefficient; the SARIMA benchmark attains marginally lower error at every horizon, a gap a Diebold–Mariano test does not find significant (p = 0.275 and 0.862), even though the two models’ forecast variances are confirmed, against their own state-space output, to grow through different mechanisms – bounded for the proposed model, unbounded for SARIMA. The proposed model’s own error decomposition further shows parameter-estimation variance uniformly negligible, so its non-stochastic error is attributable almost entirely to trend-misspecification bias. 2025 01 08 4474 https://creativecommons.org/licenses/by/4.0/ 10.54216/PAMDA.050201 https://www.americaspg.com/journal/34/article/4474 https://www.americaspg.com/storage/articles/manuscripts/01KZPT568GBKYVVJ3PFWEYS58N.pdf