CarmaNoise function

Estimation for the underlying Levy in a carma model

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

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

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.

Returns

• incr.Levy: a numeric object contains the estimated increments.

Author(s)

The YUIMA Project Team

Note

The function qmle uses the function CarmaNoise for estimation of underlying Levy in the carma model.

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

## Not run:

#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)
## End(Not run)