bayess (version 1.6)

ARllog: log-likelihood associated with an AR(p) model defined either through its natural coefficients or through the roots of the associated lag-polynomial

Description

This function is related to Chapter 6 on dynamical models. It returns the numerical value of the log-likelihood associated with a time series and an AR(p) model, along with the natural coefficients \(psi\) of the AR(p) model if it is defined via the roots lr and lc of the associated lag-polynomial. The function thus uses either the natural parameterisation of the AR(p) model $$x_t - \mu + \sum_{i=1}^p \psi_i (x_{t-i}-\mu) = \varepsilon_t$$ or the parameterisation via the lag-polynomial roots $$\prod_{i=1}^p (1-\lambda_i B) x_t = \varepsilon_t$$ where \(B^j x_t = x_{t-j}\).

Usage

ARllog(p,dat,pr, pc, lr, lc, mu, sig2, compsi = TRUE, pepsi = c(1, rep(0, p)))

Value

ll

value of the log-likelihood

ps

vector of the \(\psi_i\)'s

Arguments

p

order of the AR\((p)\) model

dat

time series modelled by the AR\((p)\) model

pr

number of real roots

pc

number of non-conjugate complex roots

lr

real roots

lc

complex roots, stored as real part for odd indices and imaginary part for even indices

mu

drift coefficient \(\mu\) such that \((x_t-\mu)_t\) is a standard AR\((p)\) series

sig2

variance of the Gaussian white noise \((\varepsilon_t)_t\)

compsi

boolean variable indicating whether the coefficients \(\psi_i\) need to be retrievedfrom the roots of the lag-polynomial, i.e. if the model is defined by pepsi (when compsi is FALSE) or by lr and lc (when compsi is TRUE).

pepsi

potential p+1 coefficients \(\psi_i\) if compsi is FALSE, with 1 asthe compulsory first value

See Also

MAllog,ARmh

Examples

ARllog(p=3,dat=faithful[,1],pr=3,pc=0,
lr=c(-.1,.5,.2),lc=0,mu=0,sig2=var(faithful[,1]),compsi=FALSE,pepsi=c(1,rep(.1,3)))