setCarma function

Continuous Autoregressive Moving Average (p, q) Model

'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.

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, ...)

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.

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 .

Returns

• model: an object of yuima.carma-class.

Author(s)

The YUIMA Project Team

Note

There may be missing information in the model description. Please contribute with suggestions and fixings.

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)