Nonparametric Instrumental Regression

Description

npregiv computes nonparametric estimation of an instrumental regression function phi defined by conditional moment restrictions stemming from a structural econometric model: E [Y - phi (Z,X) | W ] = 0, and involving endogenous variables Y and Z and exogenous variables X and instruments W. The function phi is the solution of an ill-posed inverse problem.

When method="Tikhonov", npregiv uses the approach of Darolles, Fan, Florens and Renault (2011) modified for local polynomial kernel regression of any order (Darolles et al use local constant kernel weighting which corresponds to setting p=0; see below for details). When method="Landweber-Fridman", npregiv uses the approach of Horowitz (2011) again using local polynomial kernel regression (Horowitz uses B-spline weighting).

Usage

npregiv(y,
        z,
        w,
        x = NULL,
        zeval = NULL,
        weval = NULL,
        xeval = NULL,
        p = 1,
        nmulti = 1,
        random.seed = 42,
        optim.maxattempts = 10,
        optim.method = c("Nelder-Mead", "BFGS", "CG"),
        optim.reltol = sqrt(.Machine$double.eps),
        optim.abstol = .Machine$double.eps,
        optim.maxit = 500,
        alpha = NULL,
        alpha.min = 1e-10,
        alpha.max = 1e-01,
        alpha.tol = .Machine$double.eps^0.25,
        iterate.max = 1000,
        iterate.diff.tol = 1.0e-08,
        constant = 0.5,
        method = c("Landweber-Fridman","Tikhonov"),
        penalize.iteration = TRUE,
        smooth.residuals = TRUE,        
        start.from = c("Eyz","EEywz"),
        starting.values  = NULL,
        stop.on.increase = TRUE,
        ...)

Arguments

Details

Tikhonov regularization requires computation of weight matrices of dimension n x n which can be computationally costly in terms of memory requirements and may be unsuitable for large datasets. Landweber-Fridman will be preferred in such settings as it does not require construction and storage of these weight matrices while it also avoids the need for numerical optimization methods to determine alpha.

method="Landweber-Fridman" uses an optimal stopping rule based upon ||E(y|w)-E(phi(z,x)|w)||^2 . However, if insufficient training is conducted the estimates can be overly noisy. To best guard against this eventuality set nmulti to a larger number than the default nmulti=0 for npreg.

When using method="Landweber-Fridman", iteration will terminate when either the change in the value of ||(E(y|w)-E(phi(z,x)|w))/E(y|w)||^2 from iteration to iteration is less than iterate.diff.tol or we hit iterate.max or ||(E(y|w)-E(phi(z,x)|w))/E(y|w)||^2 stops falling in value and starts rising.

Value

npregiv returns a list with components phi and either alpha when method="Tikhonov" or num.iterations, norm.stop and convergence when method="Landweber-Fridman".

Note

This function should be considered to be in ‘beta test’ status until further notice.

Author(s)

Jeffrey S. Racine racinej@mcmaster.ca, Samuele Centorrino samuele.centorrino@univ-tlse1.fr

References

Carrasco, M. and J.P. Florens and E. Renault (2007), “Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization,” In: James J. Heckman and Edward E. Leamer, Editor(s), Handbook of Econometrics, Elsevier, 2007, Volume 6, Part 2, Chapter 77, Pages 5633-5751

Darolles, S. and Y. Fan and J.P. Florens and E. Renault (2011), “Nonparametric instrumental regression,” Econometrica, 79, 1541-1565.

Feve, F. and J.P. Florens (2010), “The practice of non-parametric estimation by solving inverse problems: the example of transformation models,” Econometrics Journal, 13, S1-S27.

Florens, J.P. and J.S. Racine (2012), “Nonparametric instrumental derivatives,” Working Paper.

Fridman, V. M. (1956), “A method of successive approximations for Fredholm integral equations of the first kind,” Uspeskhi, Math. Nauk., 11, 233-334, in Russian.

Horowitz, J.L. (2011), “Applied nonparametric instrumental variables estimation,” Econometrica, 79, 347-394.

Landweber, L. (1951), “An iterative formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, 73, 615-24.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

Li, Q. and J.S. Racine (2004), “Cross-validated Local Linear Nonparametric Regression,” Statistica Sinica, 14, 485-512.

See Also

npregivderiv,npreg

Examples

## Not run: 
## This illustration was made possible by Samuele Centorrino
## <samuele.centorrino@univ-tlse1.fr>

set.seed(42)
n <- 1500

## The DGP is as follows:

## 1) y = phi(z) + u

## 2) E(u|z) != 0 (endogeneity present)

## 3) Suppose there exists an instrument w such that z = f(w) + v and
## E(u|w) = 0

## 4) We generate v, w, and generate u such that u and z are
## correlated. To achieve this we express u as a function of v (i.e. u =
## gamma v + eps)

v <- rnorm(n,mean=0,sd=0.27)
eps <- rnorm(n,mean=0,sd=0.05)
u <- -0.5*v + eps
w <- rnorm(n,mean=0,sd=1)

## In Darolles et al (2011) there exist two DGPs. The first is
## phi(z)=z^2 and the second is phi(z)=exp(-abs(z)) (which is
## discontinuous and has a kink at zero).

fun1 <- function(z) { z^2 }
fun2 <- function(z) { exp(-abs(z)) }

z <- 0.2*w + v

## Generate two y vectors for each function.

y1 <- fun1(z) + u
y2 <- fun2(z) + u

## You set y to be either y1 or y2 (ditto for phi) depending on which
## DGP you are considering:

y <- y1
phi <- fun1

## Sort on z (for plotting)

ivdata <- data.frame(y,z,w)
ivdata <- ivdata[order(ivdata$z),]
rm(y,z,w)
attach(ivdata)

model.iv <- npregiv(y=y,z=z,w=w)
phi.iv <- model.iv$phi

## Now the non-iv local linear estimator of E(y|z)

ll.mean <- fitted(npreg(y~z,regtype="ll"))

## For the plots, restrict focal attention to the bulk of the data
## (i.e. for the plotting area trim out 1/4 of one percent from each
## tail of y and z)

trim <- 0.0025

curve(phi,min(z),max(z),
      xlim=quantile(z,c(trim,1-trim)),
      ylim=quantile(y,c(trim,1-trim)),
      ylab="Y",
      xlab="Z",
      main="Nonparametric Instrumental Kernel Regression",
      lwd=2,lty=1)

points(z,y,type="p",cex=.25,col="grey")

lines(z,phi.iv,col="blue",lwd=2,lty=2)

lines(z,ll.mean,col="red",lwd=2,lty=4)

legend(quantile(z,trim),quantile(y,1-trim),
       c(expression(paste(varphi(z))),
         expression(paste("Nonparametric ",hat(varphi)(z))),
         "Nonparametric E(y|z)"),
       lty=c(1,2,4),
       col=c("black","blue","red"),
       lwd=c(2,2,2))
       

## End(Not run) 

Results