Sim.DiffProc (version 2.8)

fitsde: Maximum Pseudo-Likelihood Estimation of 1-Dim SDE

Description

The (S3) generic function "fitsde" of estimate drift and diffusion parameters by the method of maximum pseudo-likelihood of the 1-dim stochastic differential equation.

Usage

fitsde(data, ...)
## S3 method for class 'default':
fitsde(data, drift, diffusion, start = list(), pmle = c("euler","kessler", 
   "ozaki", "shoji"), optim.method = "L-BFGS-B",
   lower = NULL, upper = NULL, ...)
   
## S3 method for class 'fitsde':
summary(object, \dots)
## S3 method for class 'fitsde':
coef(object, \dots)
## S3 method for class 'fitsde':
vcov(object, \dots)
## S3 method for class 'fitsde':
logLik(object, \dots)
## S3 method for class 'fitsde':
AIC(object, \dots)
## S3 method for class 'fitsde':
BIC(object, \dots)
## S3 method for class 'fitsde':
confint(object,parm, level=0.95, \dots)

Arguments

data

a univariate time series (ts class).

drift

drift coefficient: an expression of two variables t, x and theta a vector of parameters of sde. See Examples.

diffusion

diffusion coefficient: an expression of two variables t, x and theta a vector of parameters of sde. See Examples.

start

named list of starting values for optimizer. See Examples.

pmle

a character string specifying the method; can be either: "euler" (Euler pseudo-likelihood), "ozaki" (Ozaki pseudo-likelihood), "shoji" (Shoji pseudo-l

optim.method

the method for optim.

lower, upper

bounds on the variables for the "Brent" or "L-BFGS-B" method.

object

an object inheriting from class "fitsde".

parm

a specification of which parameters are to be given confidence intervals, either a vector of names (example parm='theta1'). If missing, all parameters are considered.

level

the confidence level required.

...

further arguments to pass to optim.

Value

• fitsde returns an object inheriting from class "fitsde".

newcommand

\CRANpkg

href

http://CRAN.R-project.org/package=#1

pkg

#1

Details

The function fitsde returns a pseudo-likelihood estimators of the drift and diffusion parameters in 1-dim stochastic differential equation. The optim optimizer is used to find the maximum of the negative log pseudo-likelihood. An approximate covariance matrix for the parameters is obtained by inverting the Hessian matrix at the optimum. The pmle of pseudo-likelihood can be one among:"euler": Euler pseudo-likelihood), "ozaki": Ozaki pseudo-likelihood, "shoji": Shoji pseudo-likelihood, and "kessler": Kessler pseudo-likelihood. For more details see vignette("FitSDE").

References

Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24, 211-229. Iacus, S.M. (2008). Simulation and inference for stochastic differential equations: with R examples. Springer-Verlag, New York. Iacus, S.M. (2009). sde: Simulation and Inference for Stochastic Differential Equations. Rpackage version 2.0.10. Iacus, S.M. and all. (2014). The yuima Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations. Journal of Statistical Software, 57(4). Ozaki, T. (1992). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach. Statistica Sinica, 2, 25-83. Shoji, L., Ozaki, T. (1998). Estimation for nonlinear stochastic differential equations by a local linearization method. Stochastic Analysis and Applications, 16, 733-752. Dacunha, D.C. and Florens, D.Z. (1986). Estimation of the Coefficients of a Diffusion from Discrete Observations. Stochastics. 19, 263--284. Dohnal, G. (1987). On estimating the diffusion coefficient. J. Appl.Prob., 24, 105--114. Genon, V.C. (1990). Maximum constrast estimation for diffusion processes from discrete observation. Statistics, 21, 99--116. Nicolau, J. (2004). Introduction to the estimation of stochastic differential equations based on discrete observations. Autumn School and International Conference, Stochastic Finance. Ait-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. The Journal of Finance, 54, 1361--1395. Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica. 70, 223--262. B.L.S. Prakasa Rao. (1999). Statistical Inference for Diffusion Type Processes. Arnold, London and Oxford University press, New York. Kutoyants, Y.A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.

See Also

dcEuler, dcElerian, dcOzaki, dcShoji, dcKessler and dcSim for approximated conditional law of a diffusion process. gmm estimator of the generalized method of moments by Hansen, and HPloglik these functions are useful to calculate approximated maximum likelihood estimators when the transition density of the process is not known, in package sde. qmle in package yuima calculate quasi-likelihood and ML estimator of least squares estimator. PSM.estimate in package PSM for estimation of linear and non-linear mixed-effects models using stochastic differential equations.

Examples

#####  Example 1:

## Modele GBM (BS)
## dX(t) = theta1 * X(t) * dt + theta2 * x * dW(t)
## Simulation of data
X <- GBM(N =1000,theta=4,sigma=1)
## Estimation: true theta=c(4,1)
fx <- expression(theta[1]*x)
gx <- expression(theta[2]*x)

fres <- fitsde(data=X,drift=fx,diffusion=gx,start = list(theta1=1,theta2=1))
fres			   
summary(fres)
coef(fres)
logLik(fres)
AIC(fres)
BIC(fres)
vcov(fres)
confint(fres,level=0.95)

#####  Example 2:


## Nonlinear mean reversion (Ait-Sahalia) modele
## dX(t) = (theta1 + theta2*x + theta3*x^2) * dt + theta4 * x^theta5 * dW(t) 
## Simulation of the process X(t)
f <- expression(1 - 11*x + 2*x^2)
g <- expression(x^0.5)
res <- snssde1d(drift=f,diffusion=g,M=1,N=1000,Dt=0.001,x0=5)
mydata1 <- res$X

## Estimation
## true param theta= c(1,-11,2,1,0.5)
true <- c(1,-11,2,1,0.5)
pmle <- eval(formals(fitsde.default)$pmle)

fx <- expression(theta[1] + theta[2]*x + theta[3]*x^2)
gx <- expression(theta[4]*x^theta[5])

fres <- lapply(1:4, function(i) fitsde(mydata1,drift=fx,diffusion=gx,
	             pmle=pmle[i],start = list(theta1=1,theta2=1,theta3=1,theta4=1,
                 theta5=1),optim.method = "L-BFGS-B"))
Coef <- data.frame(true,do.call("cbind",lapply(1:4,function(i) coef(fres[[i]]))))
names(Coef) <- c("True",pmle)
Summary <- data.frame(do.call("rbind",lapply(1:4,function(i) logLik(fres[[i]]))),
                      do.call("rbind",lapply(1:4,function(i) AIC(fres[[i]]))),
                      do.call("rbind",lapply(1:4,function(i) BIC(fres[[i]]))),
                      row.names=pmle)
names(Summary) <- c("logLik","AIC","BIC")
Coef	
Summary


#####  Example 3:

## dX(t) = (theta1*x*t+theta2*tan(x)) *dt + theta3*t *dW(t)
## Simulation of data

f <- expression(2*x*t-tan(x))
g <- expression(1.25*t)
sim <- snssde1d(drift=f,diffusion=g,M=1,N=1000,Dt=0.001,x0=10)
mydata2 <- sim$X

## Estimation
## true param theta= c(2,-1,1.25)
true <- c(2,-1,1.25)

fx <- expression(theta[1]*x*t+theta[2]*tan(x))
gx <- expression(theta[3]*t)

fres <- lapply(1:4, function(i) fitsde(mydata2,drift=fx,diffusion=gx,
	             pmle=pmle[i],start = list(theta1=1,theta2=1,theta3=1),
				 optim.method = "L-BFGS-B"))
Coef <- data.frame(true,do.call("cbind",lapply(1:4,function(i) coef(fres[[i]]))))
names(Coef) <- c("True",pmle)
Summary <- data.frame(do.call("rbind",lapply(1:4,function(i) logLik(fres[[i]]))),
                      do.call("rbind",lapply(1:4,function(i) AIC(fres[[i]]))),
                      do.call("rbind",lapply(1:4,function(i) BIC(fres[[i]]))),
                      row.names=pmle)
names(Summary) <- c("logLik","AIC","BIC")
Coef
Summary

#####  Example 4:

## Application to real data
## CKLS modele vs CIR modele 
## CKLS (mod1):  dX(t) = (theta1+theta2* X(t))* dt + theta3 * X(t)^theta4 * dW(t)
## CIR  (mod2):  dX(t) = (theta1+theta2* X(t))* dt + theta3 * sqrt(X(t))  * dW(t)

data(Irates)
rates <- Irates[,"r1"]
rates <- window(rates, start=1964.471, end=1989.333)

fx1 <- expression(theta[1]+theta[2]*x)
gx1 <- expression(theta[3]*x^theta[4])
gx2 <- expression(theta[3]*sqrt(x))

fitmod1 <- fitsde(rates,drift=fx1,diffusion=gx1,pmle="euler",start = list(theta1=1,theta2=1,
                  theta3=1,theta4=1),optim.method = "L-BFGS-B")
fitmod2 <- fitsde(rates,drift=fx1,diffusion=gx2,pmle="euler",start = list(theta1=1,theta2=1,
                  theta3=1),optim.method = "L-BFGS-B")	
summary(fitmod1)
summary(fitmod2)
coef(fitmod1)
coef(fitmod2)
confint(fitmod1,parm=c('theta2','theta3'))
confint(fitmod2,parm=c('theta2','theta3'))
AIC(fitmod1)
AIC(fitmod2)	
## Display
## CKLS Modele
op <- par(mfrow = c(1, 2))
theta <- coef(fitmod1)
N <- length(rates)
res <- snssde1d(drift=fx1,diffusion=gx1,M=200,t0=time(rates)[1],T=time(rates)[N],
                Dt=deltat(rates),x0=rates[1],N)
plot(res,plot.type="single",ylim=c(0,40))
lines(rates,col=2,lwd=2)
legend("topleft",c("real data","CKLS modele"),inset = .01,col=c(2,1),lwd=2,cex=0.8)

## CIR Modele
theta <- coef(fitmod2)
res <- snssde1d(drift=fx1,diffusion=gx2,M=200,t0=time(rates)[1],T=time(rates)[N],
                Dt=deltat(rates),x0=rates[1],N)
plot(res,plot.type="single",ylim=c(0,40))
lines(rates,col=2,lwd=2)
legend("topleft",c("real data","CIR modele"),inset = .01,col=c(2,1),lwd=2,cex=0.8)
par(op)