Nonparametric Instrumental Derivatives

npregivderiv uses the approach of Florens, Racine and Centorrino (2018) 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

npregivderiv(y,
             z,
             w,
             x = NULL,
             zeval = NULL,
             weval = NULL,
             xeval = NULL,
             constant = 0.5,
             iterate.break = TRUE,
             iterate.max = 1000,
             nmulti = NULL,
             random.seed = 42,
             smooth.residuals = TRUE,
             start.from = c("Eyz","EEywz"),
             starting.values = NULL,
             stop.on.increase = TRUE,
             ...)

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 regressors. 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 regressors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
zeval: a \(p\)-variate data frame of endogenous regressors 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 regressors on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by x.
constant: the constant to use for Landweber-Fridman iteration.
iterate.break: a logical value indicating whether to compute all objects up to iterate.max or to break when a potential optimum arises (useful for inspecting full stopping rule profile up to iterate.max)
iterate.max: an integer indicating the maximum number of iterations permitted before termination occurs for Landweber-Fridman iteration.
nmulti: integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points.
random.seed: an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.
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 Landweber-Fridman 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).
...: additional arguments supplied to npreg and npksum.

Details

Note that Landweber-Fridman iteration presumes that \(\varphi_{-1}=0\), and so for derivative estimation we commence iterating from a model having derivatives all equal to zero. Given this starting point it may require a fairly large number of iterations in order to converge. Other perhaps more reasonable starting values might present themselves. When start.phi.zero is set to FALSE iteration will commence instead using derivatives from the conditional mean model \(E(y|z)\). Should the default iteration terminate quickly or you are concerned about your results, it would be prudent to verify that this alternative starting value produces the same result. Also, check the norm.stop vector for any anomalies (such as the error criterion increasing immediately).

Landweber-Fridman iteration uses an optimal stopping rule based upon \(||E(y|w)-E(\varphi_k(z,x)|w)||^2 \). 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 the first iteration.

Note that for subsequent Landweber-Fridman iterations, a “warm start” strategy is employed. The optimal bandwidths 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 bandwidths 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 5 for the first iteration and to 1 for all subsequent iterations. This strategy provides a balance between computational efficiency and robustness, allowing the numerical optimizer to refine the structural bandwidths as the residuals evolve incrementally while still guarding against local optima.

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.

Value

npregivderiv returns a npregivderiv object. The generic functions print, summary, and plot support objects of this type.

npregivderiv returns a list with components phi.prime, phi, num.iterations, norm.stop and convergence.

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 and S. Centorrino (2018), “Nonparametric instrumental derivatives,” Journal of Nonparametric Statistics, 30 (2), 368-391.

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.

Author

Jeffrey S. Racine racinej@mcmaster.ca

Note

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

Examples

if (FALSE) { # \dontrun{
## This illustration was made possible by Samuele Centorrino
## <samuele.centorrino@univ-tlse1.fr>

set.seed(42)
n <- 1500

## 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 <- npregivderiv(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)     
} # }