Volume 5 • Issue 2 • PP: 01–08 • 2025
Harmonic Regression–Stochastic Residual Decomposition of Atmospheric Carbon Dioxide Concentration: Spectral Characterization and Forecast-Error Structure
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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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.
Keywords
References
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