yuima (version 1.15.27)

setCarma: Continuous Autoregressive Moving Average (p, q) model

Description

'setCarma' describes the following model:

Vt = c0 + sigma (b0 Xt(0) + ... + b(q) Xt(q))

dXt(0) = Xt(1) dt

...

dXt(p-2) = Xt(p-1) dt

dXt(p-1) = (-a(p) Xt(0) - ... - a(1) Xt(p-1))dt + (gamma(0) + gamma(1) Xt(0) + ... + gamma(p) Xt(p-1))dZt

The continuous ARMA process using the state-space representation as in Brockwell (2000) is obtained by choosing:

gamma(0) = 1, gamma(1) = gamma(2) = ... = gamma(p) = 0.

Please refer to the vignettes and the examples or the yuima documentation for details.

Usage

setCarma(p,q,loc.par=NULL,scale.par=NULL,ar.par="a",ma.par="b",
lin.par=NULL,Carma.var="v",Latent.var="x",XinExpr=FALSE, Cogarch=FALSE, ...)

Value

model

an object of yuima.carma-class.

Arguments

p

a non-negative integer that indicates the number of the autoregressive coefficients.

q

a non-negative integer that indicates the number of the moving average coefficients.

loc.par

location coefficient. The default value loc.par=NULL implies that c0=0.

scale.par

scale coefficient. The default value scale.par=NULL implies that sigma=1.

ar.par

a character-string that is the label of the autoregressive coefficients. The default Value is ar.par="a".

ma.par

a character-string that is the label of the moving average coefficients. The default Value is ma.par="b".

Carma.var

a character-string that is the label of the observed process. Defaults to "v".

Latent.var

a character-string that is the label of the unobserved process. Defaults to "x".

lin.par

a character-string that is the label of the linear coefficients. If lin.par=NULL, the default, the 'setCarma' builds the CARMA(p, q) model defined as in Brockwell (2000).

XinExpr

a logical variable. The default value XinExpr=FALSE implies that the starting condition for Latent.var is zero. If XinExpr=TRUE, each component of Latent.var has a parameter as a initial value.

Cogarch

a logical variable. The default value Cogarch=FALSE implies that the parameters are specified according to Brockwell (2000).

...

Arguments to be passed to 'setCarma', such as the slots of yuima.model-class

measure

Levy measure of jump variables.

measure.type

type specification for Levy measure.

xinit

a vector of expressions identyfying the starting conditions for CARMA model.

Author

The YUIMA Project Team

Details

Please refer to the vignettes and the examples or to the yuimadocs package.

An object of yuima.carma-class contains:

info:

It is an object of carma.info-class which is a list of arguments that identifies the carma(p,q) model

and the same slots in an object of yuima.model-class .

References

Brockwell, P. (2000) Continuous-time ARMA processes, Stochastic Processes: Theory and Methods. Handbook of Statistics, 19, (C. R. Rao and D. N. Shandhag, eds.) 249-276. North-Holland, Amsterdam.

Examples

# Ex 1. (Continuous ARMA process driven by a Brownian Motion)
# To describe the state-space representation of a CARMA(p=3,q=1) model:
# Vt=c0+alpha0*X0t+alpha1*X1t
# dX0t = X1t*dt 
# dX1t = X2t*dt
# dX2t = (-beta3*X0t-beta2*X1t-beta1*X2t)dt+dWt
# we set
mod1<-setCarma(p=3, 
               q=1, 
               loc.par="c0")
# Look at the model structure by
str(mod1)

# Ex 2. (General setCarma model driven by a Brownian Motion)
# To describe the model defined as:
# Vt=c0+alpha0*X0t+alpha1*X1t
# dX0t = X1t*dt 
# dX1t = X2t*dt
# dX2t = (-beta3*X0t-beta2*X1t-beta1*X2t)dt+(c0+alpha0*X0t)dWt
# we set 
mod2 <- setCarma(p=3,
                 q=1,
                 loc.par="c0",
                 ma.par="alpha",
                 ar.par="beta",
                 lin.par="alpha")
# Look at the model structure by
str(mod2)

# Ex 3. (Continuous Arma model driven by a Levy process)
# To specify the CARMA(p=3,q=1) model driven by a Compound Poisson process defined as:
# Vt=c0+alpha0*X0t+alpha1*X1t
# dX0t = X1t*dt 
# dX1t = X2t*dt
# dX2t = (-beta3*X0t-beta2*X1t-beta1*X2t)dt+dzt
# we set the Levy measure as in setModel
mod3 <- setCarma(p=3,
                 q=1,
                 loc.par="c0",
                 measure=list(intensity="1",df=list("dnorm(z, 0, 1)")),
                 measure.type="CP")
# Look at the model structure by
str(mod3)

# Ex 4. (General setCarma  model driven by a Levy process)
# Vt=c0+alpha0*X0t+alpha1*X1t
# dX0t = X1t*dt 
# dX1t = X2t*dt
# dX2t = (-beta3*X1t-beta2*X2t-beta1*X3t)dt+(c0+alpha0*X0t)dzt
mod4 <- setCarma(p=3,
                 q=1,
                 loc.par="c0",
                 ma.par="alpha",
                 ar.par="beta",
                 lin.par="alpha",
                 measure=list(intensity="1",df=list("dnorm(z, 0, 1)")),
                 measure.type="CP")
# Look at the model structure by
str(mod4)