HPloglik function

Ait-Sahalia Hermite polynomial expansion approximation of the likelihood

Ait-Sahalia Hermite polynomial expansion and Euler approximation of the likelihood of a process solution of a stochastic differential equation. These functions are useful to calculate approximated maximum likelihood estimators when the transition density of the process is not known.

HPloglik(X, theta, M, F, s, log=TRUE)

Arguments

• X: a ts object containing a sample path of an sde.
• theta: vector of parameters.
• M: list of derivatives; see details.
• F: the transform function; see details.
• s: drift and diffusion coefficient; see details.
• log: logical; if TRUE, the log-likelihood is returned.

Details

The function HPloglik returns the Hermite polynomial approximation of the likelihood of a diffusion process transformed to have a unitary diffusion coefficient. The function F is the transform function, and s is the original diffusion coefficient. The list of functions M contains the transformed drift in M[[1]] and the subsequent six derivatives in x of M[[1]]. The functions F, s, and M have arguments (t,x,theta).

Returns

• x: a number

References

Ait-Sahalia, Y. (1996) Testing Continuous-Time Models of the Spot Interest Rate, Review of Financial Studies, 9(2), 385-426.

Author(s)

Stefano Maria Iacus

Examples

set.seed(123)
d <- expression(-1*x)
s <- expression(2) 
sde.sim(drift=d, sigma=s) -> X

M0 <- function(t, x, theta) -theta[1]*x
M1 <- function(t, x, theta) -theta[1]
M2 <- function(t, x, theta) 0
M3 <- function(t, x, theta) 0
M4 <- function(t, x, theta) 0
M5 <- function(t, x, theta) 0
M6 <- function(t, x, theta) 0
mu <- list(M0, M1, M2, M3, M4, M5, M6)

F <- function(t, x, theta) x/sqrt(theta[2])
S <- function(t, x, theta) sqrt(theta[2])

true.loglik <- function(theta) {
 DELTA <- deltat(X)
 lik <- 0
 for(i in 2:length(X))
  lik <- lik + dnorm(X[i], mean=X[i-1]*exp(-theta[1]*DELTA), 
   sd = sqrt((1-exp(-2*theta[1]*DELTA))*theta[2]/
             (2*theta[1])),TRUE)
 lik  
}

xx <- seq(-3,3,length=100)
sapply(xx, function(x) HPloglik(X,c(x,4),mu,F,S)) -> px
sapply(xx, function(x) true.loglik(c(x,4))) -> py

plot(xx,px,type="l",xlab=expression(beta),ylab="log-likelihood")
lines(xx,py, lty=3) # true