yuima (version 1.15.27)

CarmaNoise: Estimation for the underlying Levy in a carma model

Description

Retrieve the increment of the underlying Levy for the carma(p,q) process using the approach developed in Brockwell et al.(2011)

Usage

CarmaNoise(yuima, param, data=NULL, NoNeg.Noise=FALSE)

Value

incr.Levy

a numeric object contains the estimated increments.

Arguments

yuima

a yuima object or an object of yuima.carma-class.

param

list of parameters for the carma.

data

an object of class yuima.data-class contains the observations available at uniformly spaced time. If data=NULL, the default, the 'CarmaNoise' uses the data in an object of yuima.data-class.

NoNeg.Noise

Estimate a non-negative Levy-Driven Carma process. By default NoNeg.Noise=FALSE.

Author

The YUIMA Project Team

References

Brockwell, P., Davis, A. R. and Yang. Y. (2011) Estimation for Non-Negative Levy-Driven CARMA Process, Journal of Business And Economic Statistics, 29 - 2, 250-259.

Examples

if (FALSE) {
#Ex.1: Carma(p=3, q=0) process driven by a brownian motion.

mod0<-setCarma(p=3,q=0)

# We fix the autoregressive and moving average parameters
# to ensure the existence of a second order stationary solution for the process.

true.parm0 <-list(a1=4,a2=4.75,a3=1.5,b0=1)

# We simulate a trajectory of the Carma model.

numb.sim<-1000
samp0<-setSampling(Terminal=100,n=numb.sim)
set.seed(100)
incr.W<-matrix(rnorm(n=numb.sim,mean=0,sd=sqrt(100/numb.sim)),1,numb.sim)

sim0<-simulate(mod0,
               true.parameter=true.parm0,
               sampling=samp0, increment.W=incr.W)

#Applying the CarmaNoise

system.time(
  inc.Levy0<-CarmaNoise(sim0,true.parm0)
)

# We compare the orginal with the estimated noise increments 

par(mfrow=c(1,2))
plot(t(incr.W)[1:998],type="l", ylab="",xlab="time")
title(main="True Brownian Motion",font.main="1")
plot(inc.Levy0,type="l", main="Filtered Brownian Motion",font.main="1",ylab="",xlab="time")

# Ex.2: carma(2,1) driven  by a compound poisson
# where jump size is normally distributed and
# the lambda is equal to 1.

mod1<-setCarma(p=2,               
               q=1,
               measure=list(intensity="Lamb",df=list("dnorm(z, 0, 1)")),
               measure.type="CP") 

true.parm1 <-list(a1=1.39631, a2=0.05029,
                  b0=1,b1=2,
                  Lamb=1)

# We generate a sample path.

samp1<-setSampling(Terminal=100,n=200)
set.seed(123)
sim1<-simulate(mod1,
               true.parameter=true.parm1,
               sampling=samp1)

# We estimate the parameter using qmle.
carmaopt1 <- qmle(sim1, start=true.parm1)
summary(carmaopt1)
# Internally qmle uses CarmaNoise. The result is in 
plot(carmaopt1)

# Ex.3: Carma(p=2,q=1) with scale and location parameters 
# driven by a Compound Poisson
# with jump size normally distributed.
mod2<-setCarma(p=2,                
               q=1,
               loc.par="mu",
               scale.par="sig",
               measure=list(intensity="Lamb",df=list("dnorm(z, 0, 1)")),
               measure.type="CP") 

true.parm2 <-list(a1=1.39631,
                  a2=0.05029,
                  b0=1,
                  b1=2,
                  Lamb=1,
                  mu=0.5,
                  sig=0.23)
# We simulate the sample path 
set.seed(123)
sim2<-simulate(mod2,
               true.parameter=true.parm2,
               sampling=samp1)

# We estimate the Carma and we plot the underlying noise.

carmaopt2 <- qmle(sim2, start=true.parm2)
summary(carmaopt2)

# Increments estimated by CarmaNoise
plot(carmaopt2)
}