# ARMAacf: Compute Theoretical ACF for an ARMA Process

## Description

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

## Usage

 `1` ```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?

## Details

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

## Value

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

## 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`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```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)) ```