Nonparametric Instrumental Regression

npregiv computes 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 function \(\varphi\) 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,
        xeval = NULL,
        alpha = NULL,
        alpha.iter = NULL,
        alpha.max = 1e-01,
        alpha.min = 1e-10,
        alpha.tol = .Machine$double.eps^0.25,
        bw = NULL,
        constant = 0.5,
        iterate.diff.tol = 1.0e-08,
        iterate.max = 1000,
        iterate.Tikhonov = TRUE,
        iterate.Tikhonov.num = 1,
        method = c("Landweber-Fridman","Tikhonov"),
        nmulti = NULL,
        optim.abstol = .Machine$double.eps,
        optim.maxattempts = 10,
        optim.maxit = 500,
        optim.method = c("Nelder-Mead", "BFGS", "CG"),
        optim.reltol = sqrt(.Machine$double.eps),
        p = 1,
        penalize.iteration = TRUE,
        random.seed = 42,
        return.weights.phi = FALSE,
        return.weights.phi.deriv.1 = FALSE,
        return.weights.phi.deriv.2 = FALSE,
        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.

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.

alpha

a numeric scalar that, if supplied, is used rather than numerically solving for alpha, when using method="Tikhonov".

alpha.iter

a numeric scalar that, if supplied, is used for iterated Tikhonov rather than numerically solving for alpha, when using method="Tikhonov".

alpha.max

maximum of search range for \(\alpha\), the Tikhonov regularization parameter, when using method="Tikhonov".

alpha.min

minimum of search range for \(\alpha\), the Tikhonov regularization parameter, when using method="Tikhonov".

alpha.tol

the search tolerance for optimize when solving for \(\alpha\), the Tikhonov regularization parameter, when using method="Tikhonov".

bw

an object which, if provided, contains bandwidths and parameters (obtained from a previous invocation of npregiv) required to re-compute the estimator without having to re-run cross-validation and/or numerical optimization which is particularly costly in this setting (see details below for an illustration of its use)

constant

the constant to use when using method="Landweber-Fridman".

iterate.diff.tol

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

iterate.max

an integer indicating the maximum number of iterations permitted before termination occurs when using method="Landweber-Fridman".

iterate.Tikhonov

a logical value indicating whether to use iterated Tikhonov (one iteration) or not when using method="Tikhonov".

iterate.Tikhonov.num

an integer indicating the number of iterations to conduct when using method="Tikhonov".

method

the regularization method employed (defaults to "Landweber-Fridman", see Horowitz (2011); see Darolles, Fan, Florens and Renault (2011) for details for "Tikhonov").

nmulti

integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points.

optim.abstol

the absolute convergence tolerance used by optim. Only useful for non-negative functions, as a tolerance for reaching zero. Defaults to .Machine$double.eps.

optim.maxattempts

maximum number of attempts taken trying to achieve successful convergence in optim. Defaults to 100.

optim.maxit

maximum number of iterations used by optim. Defaults to 500.

optim.method

method used by optim for minimization of the objective function. See ?optim for references. Defaults to "Nelder-Mead".

the default method is an implementation of that of Nelder and Mead (1965), that uses only function values and is robust but relatively slow. It will work reasonably well for non-differentiable functions.

method "BFGS" is quasi-Newton method (also known as a variable metric algorithm), specifically that published simultaneously in 1970 by Broyden, Fletcher, Goldfarb and Shanno. This uses function values and gradients to build up a picture of the surface to be optimized.

method "CG" is a conjugate gradients method based on that by Fletcher and Reeves (1964) (but with the option of Polak-Ribiere or Beale-Sorenson updates). Conjugate gradient methods will generally be more fragile than the BFGS method, but as they do not store a matrix they may be successful in much larger optimization problems.

optim.reltol

relative convergence tolerance used by optim. The algorithm stops if it is unable to reduce the value by a factor of 'reltol * (abs(val) + reltol)' at a step. Defaults to sqrt(.Machine$double.eps), typically about 1e-8.

p

the order of the local polynomial regression (defaults to p=1, i.e. local linear).

penalize.iteration

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

random.seed

an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.

return.weights.phi

a logical value (defaults to FALSE) indicating whether to return the weight matrix which when postmultiplied by the response \(y\) delivers the instrumental regression

return.weights.phi.deriv.1

a logical value (defaults to FALSE) indicating whether to return the weight matrix which when postmultiplied by the response \(y\) delivers the first partial derivative of the instrumental regression with respect to \(z\)

return.weights.phi.deriv.2

a logical value (defaults to FALSE) indicating whether to return the weight matrix which when postmultiplied by the response \(y\) delivers the second partial derivative of the instrumental regression with respect to \(z\)

smooth.residuals

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

...

additional arguments supplied to npksum.

Details

Tikhonov regularization requires computation of weight matrices of dimension \(n\times n\) 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 \(||E(y|w)-E(\varphi_k(z,x)|w)||^2 \). 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.

When using method="Landweber-Fridman", 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.

The option bw= would be useful, say, when bootstrapping is necessary. Note that when passing bw, it must be obtained from a previous invocation of npregiv. For instance, if model.iv was obtained from an invocation of npregiv with method="Landweber-Fridman", then the following needs to be fed to the subsequent invocation of npregiv:

    model.iv <- npregiv(\dots)

    bw <- NULL
    bw$bw.E.y.w <- model.iv$bw.E.y.w
    bw$bw.E.y.z <- model.iv$bw.E.y.z
    bw$bw.resid.w <- model.iv$bw.resid.w
    bw$bw.resid.fitted.w.z <- model.iv$bw.resid.fitted.w.z
    bw$norm.index <- model.iv$norm.index

    foo <- npregiv(\dots,bw=bw)
  

If, on the other hand model.iv was obtained from an invocation of npregiv with method="Tikhonov", then the following needs to be fed to the subsequent invocation of npregiv:

    model.iv <- npregiv(\dots)

    bw <- NULL
    bw$alpha <- model.iv$alpha
    bw$alpha.iter <- model.iv$alpha.iter
    bw$bw.E.y.w <- model.iv$bw.E.y.w
    bw$bw.E.E.y.w.z <- model.iv$bw.E.E.y.w.z
    bw$bw.E.phi.w <- model.iv$bw.E.phi.w
    bw$bw.E.E.phi.w.z <- model.iv$bw.E.E.phi.w.z

    foo <- npregiv(\dots,bw=bw)

  

Or, if model.iv was obtained from an invocation of npregiv with either method="Landweber-Fridman" or method="Tikhonov", then the following would also work:

    model.iv <- npregiv(\dots)

    foo <- npregiv(\dots,bw=model.iv)

  

When exogenous predictors x (xeval) are passed, they are appended to both the endogenous predictors z and the instruments w as additional columns. If this is not desired, one can manually append the exogenous variables to z (or w) prior to passing z (or w), and then they will only appear among the z or w as desired.

Value

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

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

In addition, if any of return.weights.* are invoked (*=1,2), then phi.weights and phi.deriv.*.weights return weight matrices for computing the instrumental regression and its partial derivatives. Note that these weights, post multiplied by the response vector \(y\), will deliver the estimates returned in phi, phi.deriv.1, and phi.deriv.2 (the latter only being produced when p is 2 or greater). When invoked with evaluation data, similar matrices are returned but named phi.eval.weights and phi.deriv.eval.*.weights. These weights can be used for constrained estimation, among others.

When method="Landweber-Fridman" is invoked, bandwidth objects are returned in bw.E.y.w (scalar/vector), bw.E.y.z (scalar/vector), and bw.resid.w (matrix) and bw.resid.fitted.w.z, the latter matrices containing bandwidths for each iteration stored as rows. When method="Tikhonov" is invoked, bandwidth objects are returned in bw.E.y.w, bw.E.E.y.w.z, and bw.E.phi.w and bw.E.E.phi.w.z.

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, Samuele Centorrino samuele.centorrino@univ-tlse1.fr

Note

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

See also

npregivderiv,npreg

Examples

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

set.seed(42)
n <- 500

## 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("topright",
       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),
       bty="n")
       
} # }