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: Harmonic regression Spectral analysis SARIMA Forecast-error decomposition Stationarity Wold

representation Atmospheric carbon dioxide

1. INTRODUCTION

Series with a curving trend and a strong periodic cycle appear

throughout engineering, economics, and the earth sciences,

and two modelling traditions handle them differently. Writing

L for the lag operator (Lyt = yt−1), ∇ = 1−L, and ∇m = 1−

Lm, the Box–Jenkins tradition removes trend and seasonality

by applying ∇d∇Dm

to yt and models the differenced series

with an autoregressive moving-average structure,

ϕ(L)Φ(Lm)∇d∇Dm

yt = θ(L)Θ(Lm)at , (1)

the seasonal autoregressive integrated moving-average

(SARIMA) specification; it is flexible and often accurate,

but encodes trend and seasonality implicitly, through d,D

and the fitted coefficients, rather than as parameters with direct

physical meaning. The alternative sets d = D = 0 and

Prospects for Applied Mathematics and Data Analysis

♦ ISSN: 2836-4449 Vol: 05, No. 02, 2025 ♦ PP. 01–08