JLL {JLLprod}R Documentation

Generalized Homothetic Production Function: Main Estimator

Description

This function implements the Jacho-Chávez, Lewbel & Linton's (2005) 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[M(x,z)], and M(x,z)=G(x)+F(z). The unknown functions G and F are assumed to be smooth and H is assumed to be strictly monotonic.

Usage

JLL(xx, zz, yy, z0, r0, ngrid = NULL, h1 = NULL, h2 = NULL,
    hstar = NULL, k1 = NULL, k2 = NULL, kstar = NULL, p1 = NULL,
    p2 = NULL, pstar = NULL)

Arguments

xx Numerical: Nx1 vector.
zz Numerical: Nx1 vector.
yy Numerical: Nx1 vector.
z0 Scalar: Scale and sign normalization, i.e. F'(z0)=f(z0)=1.
r0 Scalar: Location normalization, i.e. r0=H[0].
ngrid scalar: number of equally spaced grid points in each direction at which the model will be estimated.
h1 Numerical: 2x1 vector in the form c(hxx,hzz). Scalar: same bandwidth in each direction. First step bandwidth. Default is the Silverman's rule of thumb in each direction.
h2 Numerical: 2x1 vector in the form c(hr,hzz). Scalar: same bandwidth in each direction. Second step bandwidth. Default is the Silverman's rule of thumb in each direction.
hstar Scalar: bandwidth used in the estimation of H.
k1 Kernel function in step 1. Default is `gauss'.
k2 Kernel function in step 2. Default is `gauss'.
kstar Kernel function in step 3. Default is `gauss'.
p1 Order of polynomial to be used in step 1. From 0 up to 3.
p2 Order of polynomial to be used in step 2. From 0 up to 3.
pstar Order of polynomial to be used in step 3. From 0 up to 3.

Details

User may choose a variety of kernel functions. For example `uniform', `triangular', `quartic', `epanech', `triweight' or `gauss' when p[l]=1,2 or 3, and l=1,2,star, see Yatchew (2003), pp 33. Another choice may be `order34', `order56' or `order78' when p[ll]=0 for ll=2,star, which are third, fifth and seventh (gauss based) order kernel functions, see Pagan and Ullah (1999), pp 55.

Value

Rhat ngrid x ngrid matrix: Unrestricted nonparametric fit at grid points.
Mhat ngrid x ngrid matrix: Nonparametric component M (see above) evaluated at grid points.
Ghat ngrid x 1 vector: Nonparametric component G (see above) evaluated at grid points.
Fhat ngrid x 1 vector: Nonparametric component F (see above) evaluated at grid points.
x ngrid x 1 vector: grid points in the xx direction.
z ngrid x 1 vector: grid points in the zz direction.
hhat ngrid x 1 vector: Nonparametric component H (see above) evaluated at cbind(x,z).

Warning

Results seem to be more sensitive to the choice of z0 to the choice of r0. Try choosing values of z0 in the interior of the support of zz.

Author(s)

David Tomás Jacho-Chávez

References

Jacho-Chávez, D.T., Lewbel, A., and Linton, O.B. (2005) Identification and Nonparametric Estimation of a Transformed Additively Separable Model. 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.plot, LL, LLef , locpoly

Examples

#A simple simulation
d<- 1;  n=400
xx <- runif(n^d); zz <- runif(n^d);
x <- cbind(xx,zz)
e <- rnorm(n^d,sd=1)
G <- function(x){(1/2)*sin(2*pi*x)}
F <- function(z){-2*(z^2)+2*z-1/3}
M <- function(x,z){G(x)+F(z)}
H <- function(m){1+(16/7)*m}
z0<-1/4; r0<-H(0);
y <- H(M(x[,1],x[,2])) + e
m <- JLL(xx=x[,1],zz=x[,2],yy=y,p1=3,p2=1,pstar=1,h1=0.15,h2=0.7,ngrid=10,
         z0=z0,r0=r0)

#win.graph()
layout(matrix(c(1,2,1,2),2,2, byrow=TRUE),respect=TRUE)
plot(m$x,m$Ghat,type="l",lty=1)
lines(m$x,G(m$x),type="l",lty=2,col="blue")
plot(m$z,m$Fhat,type="l",lty=1)
lines(m$z,F(m$z),type="l",lty=2,col="blue")

[Package JLLprod version 1.0.1 Index]