crsivderiv: Nonparametric Instrumental Derivatives

In crs: Categorical Regression Splines

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, ...)

## Default S3 method:
crsivderiv(y,
           z,
           w,
           x = NULL,
           zeval = NULL,
           weval = NULL,
           xeval = NULL,
           constant = 0.5,
           display.nomad.progress = TRUE,
           display.warnings = TRUE,
           iterate.diff.tol = 1.0e-08,
           iterate.max = 1000,
           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_FRAME_SIZE"=paste("r",1,sep=""),
                       "DISPLAY_DEGREE"=0),
           penalize.iteration = TRUE,
           smooth.residuals = TRUE,
           start.from = c("Eyz","EEywz"),
           starting.values = NULL,
           stop.on.increase = TRUE,
           ...)

Arguments

Data, Model Inputs And Formula Interface

These arguments identify the response, endogenous variables, instruments, and exogenous covariates.

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
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 one-column data frame of continuous endogenous predictors. The current implementation of crsivderiv supports univariate continuous z only

Evaluation Inputs

These arguments identify evaluation data for the derivative fit.

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
zeval	a one-column data frame of continuous endogenous predictors on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by z

Landweber-Fridman Iteration Controls

These arguments control iteration, residual smoothing, starting values, and stopping behavior.

constant	the constant to use 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)
iterate.max	an integer indicating the maximum number of iterations permitted before termination occurs 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)
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
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)

Warnings And Progress

These arguments control warnings and displayed optimizer progress.

display.nomad.progress	a logical value indicating whether to display the progress of the NOMAD solver (default display.nomad.progress=TRUE)
display.warnings	a logical value indicating whether to display warnings (default display.warnings=TRUE)

Additional Arguments

Further NOMAD and CRS controls are passed through to lower-level routines.

...	additional arguments supplied to crs
opts	arguments passed to the NOMAD solver (see snomadr for further details)

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).

Note that for subsequent Landweber-Fridman iterations, a “warm start” strategy is employed. The optimal parameters (spline degree, number of segments, and bandwidths or inclusion indicators) from the previous iteration are used as starting values for the current iteration. The user-supplied nmulti is respected for all iterations. For iterations after the first successful one, these optimal parameters serve as the first of the multiple initial points (a warm start), while any remaining restarts are cold starts. If nmulti is not explicitly supplied by the user, it defaults to the crs default (5) for the first iteration and to 1 for all subsequent iterations. This strategy provides a balance between computational efficiency and robustness, allowing the NOMAD solver to refine the structural parameters as the residuals evolve incrementally while still guarding against local optima.

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).

The current implementation supports a single continuous endogenous regressor only. Instrument and exogenous regressor data may still be mixed continuous and categorical.

Value

crsivderiv returns a crsivderiv object (which inherits from the crs class). The generic functions print, summary, fitted, residuals, predict, and plot support objects of this type.

For the plot function, the options are plot.data=FALSE (a logical value indicating whether to plot the data as a scatter plot) and phi=FALSE (a logical value indicating whether to plot the reconstructed structural function rather than its derivative). Note that the plot method for crsivderiv objects currently only supports a univariate continuous endogenous predictor z.

See crs for details on the return object components.

In addition to the standard crs components, crsivderiv returns components phi.prime, phi, phi.prime.mat, phi.mat, num.iterations, norm.stop, norm.value and convergence.

Note

This function currently supports univariate z only. This function should be considered to be in ‘beta test’ status until further notice.

Author(s)

Jeffrey S. Racine [e-mail]

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)

## Setting cv.threshold = 0 forces NOMAD search instead of exhaustive search
## when no categorical predictors are present. This avoids unnecessary
## evaluation of all degree/segment combinations in the examples and, for
## crsiv() and crsivderiv(), ensures that the warm-start strategy is used.
model.ivderiv <- crsivderiv(y=y,z=z,w=w,cv.threshold=0)

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)

## End(Not run) 

crs documentation built on May 2, 2026, 1:06 a.m.