rho {gmm}R Documentation

Objective function of Generalized Empirical Likelihood (GEL)

Description

It computes the objective function of GEL, its first and second analytical derivatives. It is called by \ref{gel}.

Usage

rho(x,lamb,derive=0,type=c("EL","ET","CUE"),drop=TRUE)

Arguments

x A ntimes q matrix with typical element (t,i), g_i(theta,x_t)
lamb A q times 1 vector of lagrange multipliers
derive 0 for the objective function, 1 for the first derivative with respect to λ and 2 for the second derivative with respect to λ.
type "EL" for empirical likelihood, "ET" for exponential tilting and "CUE" for continuous updated estimator.
drop Because the solution may not be in the domain of rho(v) forall t in small sample, we can drop those observations to avoid the return of NaN

Details

The objective function is frac{1}{n}sum_{t=1}^n rho(<g(theta,x_t),λ>), where rho(v)=log{(1-v)} for empirical likelihood, -e^v for exponential tilting and (-v-0.5v^2) for continuous updated estimator.

Value

'rho' returns a scalar if "derive=0", a qtime 1 vector if "derive" = 1 and a qtimes q matrix if derive = 2.

References

Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219-255.

Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. Journal of Business and Economic Statistics, 14 262-280.

Examples

g <- function(tet,x)
        {
        n <- nrow(x)
        u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)])
        f <- cbind(u,u*x[4:(n-3)],u*x[3:(n-4)],u*x[2:(n-5)],u*x[1:(n-6)])
        return(f)
        }
n = 500
phi<-c(.2,.7)
thet <- 0.2
sd <- .2
x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1)
gt <- g(thet,x)
lamb <- rep(0,5)
rho(gt,lamb,type='EL',derive=0)

[Package gmm version 1.0-4 Index]