crs (version 0.15-31)

crsivderiv: Nonparametric Instrumental Derivatives

Description

crsivderiv uses the approach of Florens and Racine (2012) to compute the partial derivative of a nonparametric estimation of an instrumental regression function \(\varphi\) defined by conditional moment restrictions stemming from a structural econometric model: \(E [Y - \varphi (Z,X) | W ] = 0\), and involving endogenous variables \(Y\) and \(Z\) and exogenous variables \(X\) and instruments \(W\). The derivative function \(\varphi'\) is the solution of an ill-posed inverse problem, and is computed using Landweber-Fridman regularization.

Usage

crsivderiv(y,
           z,
           w,
           x = NULL,
           zeval = NULL,
           weval = NULL,
           xeval = NULL,
           iterate.max = 1000,
           iterate.diff.tol = 1.0e-08,
           constant = 0.5,
           penalize.iteration = TRUE,
           start.from = c("Eyz","EEywz"),
           starting.values = NULL,
           stop.on.increase = TRUE,
           smooth.residuals = TRUE,
           opts = list("MAX_BB_EVAL"=10000,
                       "EPSILON"=.Machine$double.eps,
                       "INITIAL_MESH_SIZE"="r1.0e-01",
                       "MIN_MESH_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
                       "MIN_POLL_SIZE"=paste("r",1,sep=""),
                       "DISPLAY_DEGREE"=0),
           ...)

Arguments

y

a one (1) dimensional numeric or integer vector of dependent data, each element \(i\) corresponding to each observation (row) \(i\) of z

z

a \(p\)-variate data frame of endogenous predictors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof

w

a \(q\)-variate data frame of instruments. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof

x

an \(r\)-variate data frame of exogenous predictors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof

zeval

a \(p\)-variate data frame of endogenous predictors on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by z

weval

a \(q\)-variate data frame of instruments on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by w

xeval

an \(r\)-variate data frame of exogenous predictors on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by x

iterate.max

an integer indicating the maximum number of iterations permitted before termination occurs when using Landweber-Fridman iteration

iterate.diff.tol

the search tolerance for the difference in the stopping rule from iteration to iteration when using Landweber-Fridman (disable by setting to zero)

constant

the constant to use when using Landweber-Fridman iteration

penalize.iteration

a logical value indicating whether to penalize the norm by the number of iterations or not (default TRUE)

start.from

a character string indicating whether to start from \(E(Y|z)\) (default, "Eyz") or from \(E(E(Y|z)|z)\) (this can be overridden by providing starting.values below)

starting.values

a value indicating whether to commence Landweber-Fridman assuming \(\varphi'_{-1}=starting.values\) (proper Landweber-Fridman) or instead begin from \(E(y|z)\) (defaults to NULL, see details below)

stop.on.increase

a logical value (defaults to TRUE) indicating whether to halt iteration if the stopping criterion (see below) increases over the course of one iteration (i.e. it may be above the iteration tolerance but increased)

smooth.residuals

a logical value (defaults to TRUE) indicating whether to optimize bandwidths for the regression of \(y-\varphi(z)\) on \(w\) or for the regression of \(\varphi(z)\) on \(w\) during iteration

opts

arguments passed to the NOMAD solver (see snomadr for further details)

...

additional arguments supplied to crs

Value

crsivderiv returns components phi.prime, phi, phi.prime.mat, num.iterations, norm.stop, norm.value and convergence.

Details

For Landweber-Fridman iteration, an optimal stopping rule based upon \(||E(y|w)-E(\varphi_k(z,x)|w)||^2 \) is used to terminate iteration. However, if local rather than global optima are encountered the resulting estimates can be overly noisy. To best guard against this eventuality set nmulti to a larger number than the default nmulti=5 for crs when using cv="nomad" or instead use cv="exhaustive" if possible (this may not be feasible for non-trivial problems).

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

When your problem is a simple one (e.g. univariate \(Z\), \(W\), and \(X\)) you might want to avoid cv="nomad" and instead use cv="exhaustive" since exhaustive search may be feasible (for degree.max and segments.max not overly large). This will guarantee an exact solution for each iteration (i.e. there will be no errors arising due to numerical search).

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.

See Also

npreg, crsiv, crs

Examples

# NOT RUN {
## This illustration was made possible by Samuele Centorrino
## <[e-mail]>

set.seed(42)
n <- 1000

## For trimming the plot (trim .5% from each tail)

trim <- 0.005

## 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,u,v)
ivdata <- ivdata[order(ivdata$z),]
rm(y,z,w,u,v)
attach(ivdata)

model.ivderiv <- crsivderiv(y=y,z=z,w=w)

ylim <-c(quantile(model.ivderiv$phi.prime,trim),
         quantile(model.ivderiv$phi.prime,1-trim))

plot(z,model.ivderiv$phi.prime,
     xlim=quantile(z,c(trim,1-trim)),
     main="",
     ylim=ylim,
     xlab="Z",
     ylab="Derivative",
     type="l",
     lwd=2)
rug(z)
# }
# NOT RUN {
 
# }
# NOT RUN {
<!-- % end dontrun -->
# }