portes (version 2.1-3)

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,innov=NULL,innov.dist=c("normal","t","stable"),  
         df=1,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.

innov

univariate or multivariate innovation series.

innov.dist

innovation distribution from univariate or multivariate normal, t, or stable. Default is normal when innov = NULL is selected.

df

degrees of freedom needed for generating innovations with univariate or multivariate t-distribution.

StableParameters

the four parameters, ALPHA, BETA, GAMMA, and DELTA, as described in the function rStable. This argument is needed when both arguments innov.dist = "stable"<

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 univariate or multivariate series $e_{t}$ represents the innovations series given from the argument innov. If innov = NULL then $e_{t}$ will be generated from a univariate or multivariate normal distribution, t-distribution, or stable distribution of infinite variance specified from the argument innov.dist. $\Phi(B)$ 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 MA(1) where innovation series is provided via argument innov 
########################################################################
set.seed(1234)
n <- 200
phi <-  NULL
theta <-  0.6
d <- NA
sigma <- 1.9
Z <- varima.sim(phi, theta, d, sigma, n,innov=rnorm(n))
plot(Z)
#to save time the other examples are not run
########################################################################
# Simulate ARIMA(2,1,0) process with phi=c(1.3,-0.35), Gaussian innovations
# The series is truncated at lag 50
########################################################################
set.seed(1234)
Trunc.Series <- 40
n <- 1000
phi <- c(1.3, -0.35)
theta <- NULL
d <- 1
sigma <- 1
Z <- varima.sim(phi,theta,d,sigma,n,Trunc.Series=Trunc.Series)
coef(arima(Z,order=c(2,1,0)))
########################################################################
# Simulate MA(1) process with theta = 0.5, t5-distribution innovations
########################################################################
set.seed(1234)
n <- 200
phi <- NULL
theta <- 0.5
Z <- varima.sim(phi, theta, sigma=1, n=n, innov.dist="t", df=5)
plot(Z)
########################################################################
# 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
########################################################################
set.seed(1234)
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
Stable <- c(ALPHA,BETA,GAMMA,DELTA)
Z <- varima.sim(phi,theta,d,sigma,n,constant,trend,demean,
     innov.dist="stable",StableParameters=Stable)
plot(Z)
########################################################################
# Simulate a bivariate white noise series from a multivariate t4-distribution 
########################################################################
set.seed(1234)
Z <- varima.sim(sigma=diag(2),n=200,innov.dist="t",df=4)
plot(Z)
########################################################################
# Simulate a trivariate VARMA(1,1) process with length 300. 
# phi = array(c(0.5,0.4,0.1,0.5,0,0.3,0,0,0.1), dim=c(k,k,1)), where k =3
# theta = array(c(0,0.25,0,0.5,0.1,0.4,0,0.25,0.6), dim=c(k,k,1)).
# innovations are generated from multivariate normal distribution
# The process have mean c(10, 0, 12),
# Drift equation a + b * t, where a = c(2,1,5), and b = c(0.01,0.06,0)
# The series is truncated at default value: Trunc.Series=ceiling(100/3)=34 
########################################################################
set.seed(1234)
k <- 3
n <- 300
Trunc.Series <-  50   
phi <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0,0.1),dim=c(k,k,1))
theta <-  array(c(0,0.25,0,0.5,0.1,0.4,0,0.25,0.6),dim=c(k,k,1))
sigma <- diag(k)
constant <- c(2,1,5)
trend <- c(0.01,0.06,0)
demean <- c(10,0,12)
Z <- varima.sim(phi, theta, d = 0,sigma, n, constant,trend,demean)
plot(Z)
########################################################################
# Simulate a bivariate VARIMA(2,d,1) process with length 300, where d=(1,2). 
# phi = array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2)),
# theta = array(c(0,0.25,0,0), dim=c(k,k,1)).
# innovations are generated from multivariate normal
# 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 
########################################################################
set.seed(1234)
k <- 2
n <- 300
Trunc.Series <-  50   
phi <-  array(c(0.5,0.4,0.1,0.5,0,0.3,0,0),dim=c(k,k,2))
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 
########################################################################
set.seed(1234)
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)
Stable <- c(ALPHA,BETA,GAMMA,DELTA)
Z <- varima.sim(phi,theta,d,sigma,n,innov.dist="stable",StableParameters=Stable)
plot(Z)