portes (version 1.08)

varima.sim: Simulate Data From Nonseasonal ARIMA(p,d,q) or VARIMA(p,d,q) Models

Description

Simulate time series from Integrated AutoRegressive Moving Average, ARIMA(p,d,q), or Vector Integrated AutoRegressive Moving Average, VARIMA(p,d,q), where $d$ is a nonnegative difference integer in the ARIMA case and it is a vector of $k$ differenced components $d_1,...,d_k$ in the VARIMA case. The simulated process may have a deterministic terms, drift constant and time trend, with non-zero mean. The innovations may have finite or infinite variance.

Usage

varima.sim(phi=NULL,theta=NULL,d=NA,sigma,n,constant=NA,trend=NA, 
          demean=NA,StableParameters=NA,Trunc.Series=NA)

Arguments

phi

a numeric or an array of AR or an array of VAR parameters with order $p$.

theta

a numeric or an array of MA or an array of VMA parameters with order $q$.

d

an integer or a vector representing the order of the difference.

sigma

variance of white noise series. It must be entered as matrix in case of bivariate or multivariate time series.

n

length of the series.

constant

a numeric vector represents the intercept in the deterministic equation.

trend

a numeric vector represents the slop in the deterministic equation.

demean

a numeric vector represents the mean of the series.

StableParameters

the four parameters, ALPHA, BETA, GAMMA, and DELTA, as described in the function rstable. This argument is needed to generate data from ARIMA or

Trunc.Series

truncation lag is used to truncate the infinite MA or VMA Process. IF it is NA, then Trunc.Series = min(100,$n/3$).

Value

• Simulated data from ARIMA(p,d,q) or VARIMA(p,d,q) process that may have a drift and deterministic time trend terms.

Details

This function is used to simulate a nonseasonal ARIMA or VARIMA model of order $(p,d,q)$ $$\Phi(B)D(B)(Z_{t}-\mu) = a + b \times t + \Theta(B)e_{t},$$ where $a, b,$ and $\mu$ correspond to the arguments constant, trend, and demean respectively. The series $e_{t}$ represents the innovation with mean zero and finite or infinite variance. $\Phi(B)$ and $\Theta(B)$ are the VAR VMA coefficient matrices respectively and $B$ is the backshift time operator. $D(B)=diag[(1-B)^{d_{1}},\ldots,(1-B)^{d_{k}}]$ is a diagonal matrix. This states that each individual series $Z_{i}, i=1,...,k$ is differenced $d_{i}$ times to reduce to a stationary VARMA(p,q) series.

References

Hipel, K.W. and McLeod, A.I. (2005). "Time Series Modelling of Water Resources and Environmental Systems". Reinsel, G. C. (1997). "Elements of Multivariate Time Series Analysis". Springer-Verlag, 2nd edition.

See Also

arima.sim, vma.sim, ImpulseVMA, InvertQ, fitstable, rstable

Examples

########################################################################
# Simulate ARIMA(2,1,0) process with phi = c(1.3, -0.35), Gaussian innovations
# The series is truncated at lag 50
Trunc.Series <- 40
n <- 1000
phi <- c(1.3, -0.35)
theta <- NULL
d <- 1
sigma <- 1
constant <- NA
trend <- NA
demean <- NA
StableParameters <- NA
set.seed(1)
x <- varima.sim(phi,theta,d,sigma,n,constant,trend,demean,StableParameters,Trunc.Series)
coef(arima(x,order=c(2,1,0)))
########################################################################
# Simulate univariate ARMA(2,1) process with length 500, 
# phi = c(1.3, -0.35), theta = 0.1. Drift equation is 8 + 0.05*t
# Stable innovations with: ALPHA = 1.75, BETA = 0, GAMMA = 1, DELTA = 0
n <- 500
phi <-  c(1.3, -0.35)
theta <-  0.1
constant <- 8
trend <- 0.05
demean <- 0
d <- 0
sigma <-  0.7
ALPHA <- 1.75
BETA <- 0
GAMMA <- 1
DELTA <- 0
StableParameters <- c(ALPHA,BETA,GAMMA,DELTA)
Z <- varima.sim(phi,theta,d,sigma,n,constant,trend,demean,StableParameters)
plot(Z)
########################################################################
# Simulate a bivariate VARMA(1,1) process with length 300. 
# phi = array(c(0.5,0.4,0.1,0.5), dim=c(k,k,1)),
# theta = array(c(0,0.25,0,0), dim=c(k,k,1)).
# The process have mean c(10,12),
# Drift equation a + b * t, where a = c(2,5), and b = c(0.01,0.08)
# The variance covariance is sigma = matrix(c(1,0.71,0.71,2),2,2).
# The series is truncated at default value: Trunc.Series=ceiling(100/3)=34 
k <- 2
n <- 300
Trunc.Series <-  50   
phi <-  array(c(0.5,0.4,0.1,0.5),dim=c(k,k,1))
theta <-  array(c(0,0.25,0,0),dim=c(k,k,1))
d <- c(0,0)
sigma <- matrix(c(1,0.71,0.71,2),k,k)
constant <- c(2,5)
trend <- c(0.01,0.08)
demean <- c(10,12)
Z <- varima.sim(phi, theta, d,sigma, n, constant,trend,demean)
plot(Z)
########################################################################
# Simulate a bivariate VARIMA(1,d,1) process with length 300, where d=(1,2). 
# phi = array(c(0.5,0.4,0.1,0.5), dim=c(k,k,1)),
# theta = array(c(0,0.25,0,0), dim=c(k,k,1)).
# The process have mean zero and no deterministic terms.
# The variance covariance is sigma = matrix(c(1,0.71,0.71,2),2,2).
# The series is truncated at default value: Trunc.Series=ceiling(100/3)=34 
k <- 2
n <- 300
Trunc.Series <-  50   
phi <-  array(c(0.5,0.4,0.1,0.5),dim=c(k,k,1))
theta <-  array(c(0,0.25,0,0),dim=c(k,k,1))
d <- c(1,2)
sigma <- matrix(c(1,0.71,0.71,2),k,k)
Z <- varima.sim(phi, theta, d, sigma, n)
plot(Z)
########################################################################
# Simulate a bivariate VAR(1) process with length 600. 
# Stable distribution: ALPHA=(1.3,1.6), BETA=(0,0.2), GAMMA=(1,1), DELTA=(0,0.2)
# The series is truncated at default value: Trunc.Series=min(100,200)=100 
k <- 2
n <- 600
phi <- array(c(-0.2,-0.6,0.3,1.1),dim=c(k,k,1))
theta <- NULL
d <- NA
sigma <- matrix(c(1,0.71,0.71,2),k,k)
ALPHA <- c(1.3,1.6)
BETA <- c(0,0.2)
GAMMA <-c(1,1)
DELTA <-c(0,0.2)
StableParameters <- c(ALPHA,BETA,GAMMA,DELTA)
varima.sim(phi,theta,d,sigma,n,StableParameters=StableParameters)