acfARp {forecastSNSTS}

Compute autocovariances of an AR(p) process

Description

This functions returns the autocovariances Cov(X_{t-k}, X_t) of a stationary time series (Y_t) that fulfills the following equation:

Y_t = \sum_{j=1}^p a_j Y_{t-j} + \sigma \varepsilon_{t},

where \sigma > 0, \varepsilon_t is white noise and a_1, \ldots, a_p are real numbers satisfying that the roots z_0 of the polynomial 1 - \sum_{j=1}^p a_j z^j lie strictly outside the unit circle.

Usage

acfARp(a = NULL, sigma, k)

Arguments

a

vector (a_1, \ldots, a_p) of coefficients; default NULL, corresponding to p = 0, white noise with variance \sigma^2,

sigma

standard deviation of \varepsilon_t; default 1,

k

lag for which to compute the autocovariances.

Value

Returns autocovariance at lag k of the AR(p) process.

Examples

## Taken from Section 6 in Dahlhaus (1997, AoS)
a1 <- function(u) {1.8 * cos(1.5 - cos(4*pi*u))}
a2 <- function(u) {-0.81}
# local autocovariance for u === 1/2: lag 1
acfARp(a = c(a1(1/2), a2(1/2)), sigma = 1, k = 1)
# local autocovariance for u === 1/2: lag -2
acfARp(a = c(a1(1/2), a2(1/2)), sigma = 1, k = -1)
# local autocovariance for u === 1/2: the variance
acfARp(a = c(a1(1/2), a2(1/2)), sigma = 1, k = 0)