A note on calculating autocovariances of periodic ARMA models

An analytically simple and tractable formula for the start-up autocovariances of periodic ARMA (PARMA) models is provided.

💡 Research Summary

The paper addresses a long‑standing computational bottleneck in the analysis of periodic autoregressive moving‑average (PARMA) models: the evaluation of their autocovariance function. While PARMA models are indispensable for representing time series with seasonal or cyclic dynamics—such as electricity demand, climate indices, or financial returns—their autocovariances have traditionally required solving large block‑Toeplitz linear systems or performing extensive matrix inversions. The computational cost grows rapidly with the model order (p, q) and the period length S, making real‑time inference and large‑scale applications impractical.

The authors propose a conceptually simple yet mathematically rigorous solution based on the notion of “start‑up autocovariances.” By decomposing the time index t into a period index s (0 ≤ s < S) and an intra‑period lag k, the PARMA process can be viewed as a collection of S ordinary ARMA(p, q) processes, each with its own set of coefficients (\phi_i^{(s)}), (\theta_j^{(s)}) and white‑noise variance (\sigma_{(s)}^2). For each sub‑process the classic Yule‑Walker equations hold, but the key insight is that only the first two autocovariances, (\gamma_0^{(s)}) and (\gamma_1^{(s)}), are required to generate the entire autocovariance sequence via a simple recursion.

The paper derives closed‑form expressions for these start‑up values. By solving the characteristic polynomial of the AR part for each period, the authors obtain the eigenvalues (\lambda_i^{(s)}). Substituting these into the Yule‑Walker framework yields

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