gmm (version 1.0-2)

rho: 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 gel.

Usage

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

Arguments

x

A $n\times q$ 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 $\lambda$ and 2 for the second derivative with respect to $\lambda$.

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

Value

• 'rho' returns a scalar if "derive=0", a $q\time 1$ vector if "derive" = 1 and a $q\times q$ matrix if derive = 2.

Details

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

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)