Calculate the autocorrelation of an ARMA(p,q) filter applied to a stationary process.

Calculate the autocorrelation of an ARMA(p,q) filter applied to a stationary process.

arma_acf(acf, phi = numeric(), rho = numeric(), m)

Arguments

acf

Vector of length N + m of autocorrelations of the original stationary process.

phi

Vector of p >= 0 coefficients defining the autoregressive part of the filter (see 'Details').

rho

Vector of q >= 0 coefficients defining the moving-average part of the filter (see 'Details').

m

Order of the MA approximation to the ARMA(p,q) filter.

Value

A vector of length N giving the autocorrelation of the filtered process.

Details

Let X_n denote the observation of the original process at time n, and Y_n denote the corresponding observation of the filtered process. The AR(p,q) filter model is defined as

Y_n = X_n + sum_{i=1}^p phi_i Y_{n-i} + sum_{j=1}^q rho_j X_{n-j}.

The autocorrelation of Y_n can be computed exactly from that of X_n for pure moving-average filters with p = 0. The general ARMA(p,q) filter is first approximated by a moving-average process of order m of the form

Y[n] = X[n] + sum_{j=1}^m psi[j] X[n-j],

where the coefficients psi are determined using the method of Brockwell & Davis (1991) implemented in stats::ARMAtoMA().

References

Brockwell, P.J. and Davis, R.A. "Time Series: Theory and Methods" (1991). Springer, New York. https://doi.org/10.1007/978-1-4899-0004-3.