CPoint function

Volatility structural change point estimator

CPoint(yuima, param1, param2, print=FALSE, symmetrized=FALSE, plot=FALSE)
qmleL(yuima, t,  ...)
qmleR(yuima, t,  ...)

Arguments

• yuima: a yuima object.
• param1: parameter values before the change point t
• param2: parameter values after the change point t
• plot: plot test statistics? Default is FALSE.
• print: print some debug output. Default is FALSE.
• t: time value. See Details.
• symmetrized: if TRUE uses the symmetrized version of the quasi maximum-likelihood approximation.
• ...: passed to qmle method. See Examples.

Details

CPoint estimates the change point using quasi-maximum likelihood approach.

Function qmleL estimates the parameters in the diffusion matrix using observations up to time t.

Function qmleR estimates the parameters in the diffusion matrix using observations from time t to the end.

Arguments in both qmleL and qmleR follow the same rules as in qmle.

Returns

• ans: a list with change point instant, and paramters before and after the change point.

Author(s)

The YUIMA Project Team

Examples

## Not run:

diff.matrix <- matrix(c("theta1.1*x1","0*x2","0*x1","theta1.2*x2"), 2, 2)

drift.c <- c("1-x1", "3-x2")
drift.matrix <- matrix(drift.c, 2, 1)

ymodel <- setModel(drift=drift.matrix, diffusion=diff.matrix, time.variable="t",
state.variable=c("x1", "x2"), solve.variable=c("x1", "x2"))
n <- 1000

set.seed(123)

t1 <- list(theta1.1=.1, theta1.2=0.2)
t2 <- list(theta1.1=.6, theta1.2=.6)

tau <- 0.4
ysamp1 <- setSampling(n=tau*n, Initial=0, delta=0.01)
yuima1 <- setYuima(model=ymodel, sampling=ysamp1)
yuima1 <- simulate(yuima1, xinit=c(1, 1), true.parameter=t1)

x1 <- yuima1@data@zoo.data[[1]]
x1 <- as.numeric(x1[length(x1)])
x2 <- yuima1@data@zoo.data[[2]]
x2 <- as.numeric(x2[length(x2)])

ysamp2 <- setSampling(Initial=n*tau*0.01, n=n*(1-tau), delta=0.01)
yuima2 <- setYuima(model=ymodel, sampling=ysamp2)

yuima2 <- simulate(yuima2, xinit=c(x1, x2), true.parameter=t2)

yuima <- yuima1
yuima@data@zoo.data[[1]] <- c(yuima1@data@zoo.data[[1]], yuima2@data@zoo.data[[1]][-1])
yuima@data@zoo.data[[2]] <- c(yuima1@data@zoo.data[[2]], yuima2@data@zoo.data[[2]][-1])

plot(yuima)

# estimation of change point for given parameter values
t.est <- CPoint(yuima,param1=t1,param2=t2, plot=TRUE)

low <- list(theta1.1=0, theta1.2=0)

# first state estimate of parameters using small 
# portion of data in the tails
tmp1 <- qmleL(yuima,start=list(theta1.1=0.3,theta1.2=0.5),t=1.5,
        lower=low, method="L-BFGS-B")
tmp1
tmp2 <- qmleR(yuima,start=list(theta1.1=0.3,theta1.2=0.5), t=8.5,
        lower=low, method="L-BFGS-B")
tmp2

# first stage changepoint estimator
t.est2 <- CPoint(yuima,param1=coef(tmp1),param2=coef(tmp2))
t.est2$tau

# second stage estimation of parameters given first stage
# change point estimator
tmp11 <- qmleL(yuima,start=as.list(coef(tmp1)), t=t.est2$tau-0.1,
 lower=low, method="L-BFGS-B")
tmp11

tmp21 <- qmleR(yuima,start=as.list(coef(tmp2)), t=t.est2$tau+0.1,
 lower=low, method="L-BFGS-B")
tmp21

# second stage estimator of the change point
CPoint(yuima,param1=coef(tmp11),param2=coef(tmp21))

## One dimensional example: non linear case
diff.matrix <- matrix("(1+x1^2)^theta1", 1, 1)
drift.c <- c("x1")

ymodel <- setModel(drift=drift.c, diffusion=diff.matrix, time.variable="t",
state.variable=c("x1"), solve.variable=c("x1"))
n <- 500

set.seed(123)

y0 <- 5 # initial value
theta00 <- 1/5
gamma <- 1/4

theta01 <- theta00+n^(-gamma)

t1 <- list(theta1= theta00)
t2 <- list(theta1= theta01)

tau <- 0.4
ysamp1 <- setSampling(n=tau*n, Initial=0, delta=1/n)
yuima1 <- setYuima(model=ymodel, sampling=ysamp1)
yuima1 <- simulate(yuima1, xinit=c(5), true.parameter=t1)
x1 <- yuima1@data@zoo.data[[1]]
x1 <- as.numeric(x1[length(x1)])

ysamp2 <- setSampling(Initial=tau, n=n*(1-tau), delta=1/n)
yuima2 <- setYuima(model=ymodel, sampling=ysamp2)

yuima2 <- simulate(yuima2, xinit=c(x1), true.parameter=t2)

yuima <- yuima1
yuima@data@zoo.data[[1]] <- c(yuima1@data@zoo.data[[1]], yuima2@data@zoo.data[[1]][-1])

plot(yuima)

t.est <- CPoint(yuima,param1=t1,param2=t2)
t.est$tau

low <- list(theta1=0)
upp <- list(theta1=1)

# first state estimate of parameters using small 
# portion of data in the tails
tmp1 <- qmleL(yuima,start=list(theta1=0.5),t=.15,lower=low, upper=upp,method="L-BFGS-B")
tmp1
tmp2 <- qmleR(yuima,start=list(theta1=0.5), t=.85,lower=low, upper=upp,method="L-BFGS-B")
tmp2


# first stage changepoint estimator
t.est2 <- CPoint(yuima,param1=coef(tmp1),param2=coef(tmp2))
t.est2$tau

# second stage estimation of parameters given first stage
# change point estimator
tmp11 <- qmleL(yuima,start=as.list(coef(tmp1)), t=t.est2$tau-0.1,
   lower=low, upper=upp,method="L-BFGS-B")
tmp11

tmp21 <- qmleR(yuima,start=as.list(coef(tmp2)), t=t.est2$tau+0.1,
  lower=low, upper=upp,method="L-BFGS-B")
tmp21

# second stage estimator of the change point
CPoint(yuima,param1=coef(tmp11),param2=coef(tmp21),plot=TRUE)
## End(Not run)