LL {JLLprod}R Documentation

Homothetic Production Function: Main Estimator

Description

This function implements the Lewbel & Linton's (2003) estimator. In general it estimates the model Y=r(x,z)+e, imposing the following structure r(x,z)=E[Y|X=x,Z=z]=h[g(x,z)], and g(bx,bz)=b*g(x,z). The unknown function g is assumed to be smooth and h is assumed to be a strictly monotonic smooth function.

Usage

LL(xx, zz, yy, xxo, zzo, Vmin=NULL, Vmax = NULL,
   k, j, h = NULL, kernel = NULL)

Arguments

xx Numerical: Nx1 vector.
zz Numerical: Nx1 vector.
yy Numerical: Nx1 vector.
xxo Scalar: Normalization in xx direction.
zzo Scalar: Normalization in zz direction.
Vmin Scalar: Minimum value for Vk's, where Vk's are elements in the interval [Vmin,Vmax]. Default value is -1.
Vmax Scalar: Maximum value for Vk's, where Vk's are elements in the interval [Vmin,Vmax]. Default value is 3.
k Scalar: See Lewbel & Linton (2003). There is NO default, you must provide a number, i.e. 80
j Scalar: See Lewbel & Linton (2003). There is NO default, you must provide a number, i.e. 100
h Numerical: 2x1 vector of bandwidths, [hxx,hzz], used in the estimation. Default is the Silverman's rule of thumb in each direction.
kernel Kernel function in all steps. Default is `gauss'.

Details

User may choose a variety of kernel functions. For example `uniform', `triangular', `quartic', `epanech', `triweight' or `gauss', see Yatchew (2003), pp 33. Another choice may be `order34', `order56' or `order78', which are third, fifth and seventh (gauss based) order kernel functions, see Pagan and Ullah (1999), pp 55.

Vmax should be chosen wisely. The user should make sure that Vmax*xxo belongs to the interior of the observed support of xx and similarly Vmax*zzo belongs to the interior of the observed support of zz.

Value

r N x 1 vector: Unrestricted Nonparametric r (see above) evaluated at data points, i.e. r(xxi,zzi).
g N x 1 vector: Nonparametric component g (see above) evaluated at data points, i.e. g(xxi,zzi).
h N x 1 vector: Nonparametric component h (see above) evaluated at data points, i.e. h[g(xxi,zzi)].
hd N x 1 vector: Nonparametric first derivative of h (see above) evaluated at data points, i.e. h'[g(xxi,zzi)].

Warning

Simple and fast kernel regression is used in each step for computational time gain. However, it could take several minutes to complete for sample sizes bigger than 300 observations.

k & j should be chosen accordingly with Vmin and Vmax. Try keeping them below 100.

Author(s)

David Tomás Jacho-Chávez

References

Lewbel, A., and Linton, O.B. (2003) Nonparametric Estimation of Homothetic and Homothetically Separable Functions. Unpublished manuscript.

Yatchew, A. (2003) Semiparametric Regression for the Applied Econometrician. Cambridge University Press.

Pagan, A. and Ullah, A. (1999) Nonparametric Econometrics. Cambridge Universtiy Press.

See Also

JLL, LLef , locpoly, Blocc

Examples


data(ecu)
##This part simply does some data sorting & trimming
xlnK <- ecu$lnk
xlnL <- ecu$lnl
xlnY <- ecu$lny
xqKL <- quantile(exp(xlnK)/exp(xlnL),  probs=c(2.5,97.5)/100)
yx <- cbind(xlnY,xlnK,xlnL)
tlnklnl <- yx[((exp(yx[,2])/exp(yx[,3]))>=xqKL[1]) 
              & ((exp(yx[,2])/exp(yx[,3]))<=xqKL[2]),]
Y<-tlnklnl[,1]
K<-exp(tlnklnl[,2])/median(exp(tlnklnl[,2]))
L<-exp(tlnklnl[,3])/median(exp(tlnklnl[,3]))

LLb<-LL(xx=K,zz=L,yy=Y,xxo=median(K),zzo=median(L),k=80,j=100)

#win.graph()
nf <- layout(matrix(c(1,2,1,2),2,2, byrow=TRUE),respect=TRUE)
plot(log(K)-log(L),log(LLb$g)-log(L),pch=3,xlab="ln(K/L)"
     ,ylab="ln(g(K/L,1))",main="Homogeneous Component g")
plot(log(LLb$g),LLb$h,xlab="ln(g)",pch=3,ylab="h(g)"
     ,main="Nonhomogeneous Component h",ylim=c(min(min(LLb$h)
     ,min(LLb$r)),max(max(LLb$h),max(LLb$r))))
points(log(LLb$g),LLb$r,type="p",pch=1,col="blue",lwd=2)
legend(-0.5,15.5,c("Nonparametric","Kernel Regression")
       ,merge=TRUE,lty=c(1,-1),pch=c(3,1),lwd=c(1,2)
       ,col=c("black","blue"),cex=0.95)

[Package JLLprod version 1.0.1 Index]