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.
