sde (version 2.0.18)

gmm: Generalized method of moments estimator

Description

Implementation of the estimator of the generalized method of moments by Hansen.

Usage

gmm(X, u, dim, guess, lower, upper, maxiter=30, tol1=1e-3, 
   tol2=1e-3)

Value

x

a list with parameter estimates, the value of Q1 at the minimum, and the Hessian

Arguments

X

a ts object containing a sample path of an sde.

u

a function of x, y, and theta and DELTA; see details.

dim

dimension of parameter space; see details.

guess

initial value of the parameters; see details.

lower

lower bounds for the parameters; see details.

upper

upper bounds for the parameters; see details.

tol1

tolerance for parameters; see details.

tol2

tolerance for Q1; see details.

maxiter

maximum number of iterations at the second stage; see details.

Author

Stefano Maria Iacus

Details

The function gmm minimizes at the first stage the function Q(theta) = t(Gn(theta)) * Gn(theta) with respect to theta, where Gn(theta) = mean(u(X[i+1], X[i], theta)). Then a matrix of weights W is obtained by inverting an estimate of the long-run covariance and the quadratic function Q1(theta) = t(Gn(theta)) * W * Gn(theta) with starting value theta1 (the solution at the first stage). The second stage is iterated until the first of these conditions verifies: (1) that the number of iterations reaches maxiter; (2) that the Euclidean distance between theta1 and theta2 < tol1; (3) that Q1 < tol2.

The function u must be a function of (u,y,theta,DELTA) and should return a vector of the same length as the dimension of the parameter space. The sanity checks are left to the user.

References

Hansen, L.P. (1982) Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 50(4), 1029-1054.

Examples

if (FALSE) {
alpha <- 0.5
beta <- 0.2
sigma <- sqrt(0.05)
true <- c(alpha, beta, sigma)
names(true) <- c("alpha", "beta", "sigma")

x0 <- rsCIR(1,theta=true)
set.seed(123)
sde.sim(X0=x0,model="CIR",theta=true,N=500000,delta=0.001) -> X
X <- window(X, deltat=0.1)
DELTA = deltat(X)
n <- length(X) 
X <- window(X, start=n*DELTA*0.5)
plot(X)

u <- function(x, y, theta, DELTA){
  c.mean <- theta[1]/theta[2] + 
             (y-theta[1]/theta[2])*exp(-theta[2]*DELTA)
  c.var <- ((y*theta[3]^2 * 
     (exp(-theta[2]*DELTA)-exp(-2*theta[2]*DELTA))/theta[2] +  
	  theta[1]*theta[3]^2*
	  (1-exp(-2*theta[2]*DELTA))/(2*theta[2]^2)))  
  cbind(x-c.mean,y*(x-c.mean), c.var-(x-c.mean)^2, 
        y*(c.var-(x-c.mean)^2))  
}

CIR.lik <- function(theta1,theta2,theta3) {
 n <- length(X)
 dt <- deltat(X)
 -sum(dcCIR(x=X[2:n], Dt=dt, x0=X[1:(n-1)], 
   theta=c(theta1,theta2,theta3), log=TRUE))
}

fit <- mle(CIR.lik, start=list(theta1=.1,  theta2=.1,theta3=.3), 
    method="L-BFGS-B",lower=c(0.001,0.001,0.001), upper=c(1,1,1))
# maximum likelihood estimates
coef(fit)


gmm(X,u, guess=as.numeric(coef(fit)), lower=c(0,0,0), 
    upper=c(1,1,1))

true
}