Generalized Auto-Correlation Function of Higher Order Arma Processes: Application to Pandemic Data

Abstract

The Autocorrelation Function (ACF) of a time series process reveals the 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 ACF of processes as far as ARMA(3,0). The procedure has shown a clear connection among the autocovariances at consecutive lags of the respective process as well as between particular lags of consecutive orders of the process. It is also observed that the Yule-Walker relation emerges after lag (q + 2) for processes higher than ARMA(2,1). This means that there is the need for the computation of individual γ(k) for k ≤ (q + 2). The derived approach is applied to daily new Covid-19 cases for three countries with stationary series, and are found to have different ARMA processes. The results are compared with those based on ”ARIMAfit” function in R. In each case, the results of the two methods are found to be the same with damp exponential decay, an indication that the pandemic would cease eventually in these countries. The results provide useful relations that may be utilized as diagnostic tests for determining whether a given data follows a specified process.