A note on calculating autocovariances of periodic ARMA models

Autocovariance calculation procedures for P ARMA models are generally carried out using the periodic Yule Walker equations (e.g. Bentarzi and Aknouche, 2005). This approach has been considered earlier by Li and Hui (1988) for calculating P ARMA autocovariances, where the (p+1)-start-up autocovariances, for all seasons, were given through a matrix equation Aγ = y, which is solved for γ (γ being the (p + 1)S × 1-vector of the start-up autocovariances). The latter equation is however analytically and computationally intractable since the matrix A is not given explicitly but formed through an appropriate algorithm. Adopting the same approach, Shao and Lund (2004) showed that the r-start-up (r = max(p, q) + 1) autocovariances may be obtained by solving a linear system ΓUγ = κ for γ, where Γ and U are matrices of dimensions rS × (p + 1)rS and (p + 1)rS × rS, respectively. While these matrices are given explicitly, the method remains relatively cumbersome since it requires an increasing bookkeeping due to the matrix product. This note proposes an improved computation procedure for calculating the P ARMA autocovariances based on the latter approach.

The proposed method computes the (p + 1)-start-up autocovariances based on a linear system with a corresponding matrix given explicitly, whose analytical form exhibits a circular structure, naturally assorted with the model periodicity.

Consider a causal P ARMA model of orders (p, q) and period

where φ

0 = -1 and {ε t , t ∈ Z} is a periodic white noise process, i.e., a sequence of uncorrelated random variables with mean zero and variance

h = E (y v+nS y v+nS-h ) be the autocovariance function at season v and lag h ∈ Z. Then, it is well known (Li and Hui, 1988;Shao and Lund, 2004) that multiplying (1) by y v+nS-h and tacking expectation, the γ

where the normalized cross-autocovariances (ψ

k ), coefficients of the unique causal representation of the P ARMA process {y t , t ∈ Z}, are given by (see e.g. Lund and Basawa, 2000;Shao and Lund, 2004)

with ψ

stands for the indicator function). Equation (2) needs to be started from the knowledge of γ

Once these start-up values are given, the γ (v) h for h > p may be obtained recursively from (2) while invoking (3). For γ (v) h with negative lags, we may use the well-known relation

. The main result of this note is to formulate a linear system for computing the p + 1 necessary starting autocovariances. Let γ = (γ

be the S(p + 1)-vector of such values and ζ be the S(p + 1)-vector whose entries

, for 1 ≤ v ≤ S and 0 ≤ h ≤ p, are the right-hand sides of (2).

Define the (p + 1)-square matrices ϕ

, for h = 0, …, p

and the S(p + 1)-square matrix

where 0 m×n denotes the null matrix of dimension m × n. Then, the starting autocovariance vector γ is the unique solution of the following linear system

whenever model ( 1) is causal. Note that, in view of (4), the infinite sum in (5) contains only p non zero terms. It may be possible to reduce the complexity of forming Φ using its circular property. Indeed, equation ( 5) may be used to only evaluate the first bloc Φ

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