stats (version 3.6.2)

# ARMAacf: Compute Theoretical ACF for an ARMA Process

## Description

Compute the theoretical autocorrelation function or partial autocorrelation function for an ARMA process.

## Usage

`ARMAacf(ar = numeric(), ma = numeric(), lag.max = r, pacf = FALSE)`

## Arguments

ar

numeric vector of AR coefficients

ma

numeric vector of MA coefficients

lag.max

integer. Maximum lag required. Defaults to `max(p, q+1)`, where `p, q` are the numbers of AR and MA terms respectively.

pacf

logical. Should the partial autocorrelations be returned?

## Value

A vector of (partial) autocorrelations, named by the lags.

## Details

The methods used follow Brockwell & Davis (1991, section 3.3). Their equations (3.3.8) are solved for the autocovariances at lags \(0, \dots, \max(p, q+1)\), and the remaining autocorrelations are given by a recursive filter.

## References

Brockwell, P. J. and Davis, R. A. (1991) Time Series: Theory and Methods, Second Edition. Springer.

`arima`, `ARMAtoMA`, `acf2AR` for inverting part of `ARMAacf`; further `filter`.

## Examples

```# NOT RUN {
ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10)

## Example from Brockwell & Davis (1991, pp.92-4)
## answer: 2^(-n) * (32/3 + 8 * n) /(32/3)
n <- 1:10
a.n <- 2^(-n) * (32/3 + 8 * n) /(32/3)
(A.n <- ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10))
stopifnot(all.equal(unname(A.n), c(1, a.n)))

ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10, pacf = TRUE)
zapsmall(ARMAacf(c(1.0, -0.25), lag.max = 10, pacf = TRUE))

## Cov-Matrix of length-7 sub-sample of AR(1) example:
toeplitz(ARMAacf(0.8, lag.max = 7))
# }
```