Finding an ARMA(p,q) model given its spectral density or its correlogram

TL;DR

This paper presents methods to determine ARMA(p,q) models from their spectral density or correlogram, enabling complete model identification from either spectral or autocorrelation information.

Contribution

It introduces a systematic approach to derive the spectral density and correlogram from an ARMA model, and vice versa, facilitating model identification from spectral or autocorrelation data.

Findings

Provides formulas to compute spectral density from ARMA parameters.

Derives autocorrelation functions from spectral density.

Enables complete model reconstruction from spectral or correlogram data.

Abstract

An ARMA model can be fully determined based on either its spectral density, or its correlogram, i.e. a formula for computing the corresponding k th serial correlation for any integer k. In this article we describe how to find, given one of these three ways of specifying the model, the other two.