Generalized Autocorrelation Function of Stationary Higher Order ARMA Processes: Application to Pandemic Data

Abstract

The Autocorrelation Function (ACF) of a time series process reveals inherent characteristics of the series that may not be visible from the original series. The ACF of the ARMA(p, q) process has been presented in a few studies in understandably rigorous and laborious manner with no explicit form of the function. In this study, the approach of autocovariance generating functions (acvgf) is used to obtain an explicit expression for a series that follows a linear process under condition of distinct real roots of the AR(p) lag operator polynomial. The technique is used to derive ACFs of processes as far as ARMA(2, q) for any value of q and subsequently states results for specific ARMA(3, q) processes. The procedure has shown a clear connection among autocovariances at consecutive lags of the respective process as well as among consecutive orders of the process at particular lags. The derived approach which is applied to daily new Covid-19 cases for countries with stationary series obtains the same results of damp exponential decay in each case as that based on "ARIMAfit" function in R. The results provide useful relations that may be utilized as diagnostic tests for determining whether a given data follows a specified linear process.

Keywords: Autocovariance generating function, linear process, theoretical autocorrelation