TARMA.sim: Simulation of a two-regime 'TARMA(p1,p2,q1,q2)' process

In tseriesTARMA: Analysis of Nonlinear Time Series Through Threshold Autoregressive Moving Average Models (TARMA) Models

Simulation of a two-regime TARMA(p1,p2,q1,q2) process

Description

\loadmathjax

Simulates from the following two-regime TARMA(p1,p2,q1,q2) process:

\mjdeqn

X_t = \left\lbrace \beginarrayll \phi_1,0 + \sum_i=1^p_1 \phi_1,i X_t-i + \sum_j=1^q_1 \theta_1,j \varepsilon_t-j + \varepsilon_t, & \quad\mathrmif\quad X_t-d \leq \mathrmthd \\\ &\\\ \phi_2,0 + \sum_i=1^p_2 \phi_2,i X_t-i + \sum_j=1^q_2 \theta_2,j \varepsilon_t-j + \varepsilon_t, & \quad\mathrmif\quad X_t-d > \mathrmthd \endarray \right. X[t] = = \phi[1,0] + \phi[1,1]X[t-1] + ... + \phi[1,p1]X[t-p1] + \theta[1,1]\epsilon[t-1] + ... + \theta[1,q]\epsilon[t-q1] + \epsilon[t] – if X[t-d] <= thd

= \phi[2,0] + \phi[2,1]X[t-1] + ... + \phi[2,p2]X[t-p2] + \theta[2,1]\epsilon[t-1] + ... + \theta[2,q]\epsilon[t-q2] + \epsilon[t] – if X[t-d] > thd

Usage

TARMA.sim(
  n,
  phi1,
  phi2,
  theta1,
  theta2,
  d = 1,
  thd = 0,
  s1 = 1,
  s2 = 1,
  rand.gen = rnorm,
  innov = rand.gen(n, ...),
  n.start = 500,
  xstart,
  start.innov = rand.gen(n.start, ...),
  ...
)

Arguments

n	Length of the series.
phi1	Vector of p1+1 Autoregressive parameters of the lower regime. The first element is the intercept.
phi2	Vector of p2+1 Autoregressive parameters of the upper regime. The first element is the intercept.
theta1	Vector of q1 Moving Average parameters of the lower regime.
theta2	Vector of q2 Moving Average parameters of the upper regime.
d	Delay parameter. Defaults to 1.
thd	Threshold parameter. Defaults to 0.
s1	Innovation variance for the lower regime. Defaults to 1.
s2	Innovation variance for the upper regime. Defaults to 1.
rand.gen	Optional: a function to generate the innovations. Defaults to rnorm.
innov	Optional: a time series of innovations. If not provided, rand.gen is used.
n.start	Length of the burn-in period. Defaults to 500.
xstart	Initial condition as a named list: $ar: AR part length k = max(p1,p2,d), X[k], X[k-1], ... ,X[1]; $ma: MA part length q = ma.ord, e[q], ... , e[1].
start.innov	Optional: a time series of innovations for the burn-in period.
...	Additional arguments for rand.gen.

Details

Note that the parameters are not checked for ergodicity.

Value

A time series object of class ts generated from the above model.

Author(s)

Simone Giannerini, [e-mail]

Greta Goracci, [e-mail]

References

• \insertRef

  Gia21tseriesTARMA

Examples

## a TARMA(1,1,1,1) model
set.seed(123)
x <- TARMA.sim(n=100, phi1=c(0.5,-0.5), phi2=c(0.0,0.8), theta1=-0.5, theta2=0.5, d=1, thd=0.2)

## a TARMA(1,2,1,1) model
x <- TARMA.sim(n=100,phi1=c(0.5,-0.5,0),phi2=c(0,0.5,0.3),theta1=-0.5,theta2=0.5,d=1,thd=0.2)

tseriesTARMA documentation built on Oct. 8, 2024, 5:11 p.m.