gmm function

Generalized method of moments estimator

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

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

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.

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.

Returns

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

Author(s)

Stefano Maria Iacus

Examples

## Not run:

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
## End(Not run)