rho {gmm} | R Documentation |
It computes the objective function of GEL, its first and second analytical derivatives. It is called by \ref{gel}.
rho(x,lamb,derive=0,type=c("EL","ET","CUE"),drop=TRUE)
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 |
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.
'rho' returns a scalar if "derive=0", a qtime 1 vector if "derive" = 1 and a qtimes q matrix if derive = 2.
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.
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)