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).
npregiv(y,
z,
w,
x = NULL,
zeval = 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.iter = NULL,
alpha.min = 1e-10,
alpha.max = 1e-01,
alpha.tol = .Machine$double.eps^0.25,
iterate.Tikhonov = TRUE,
iterate.Tikhonov.num = 1,
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,
return.weights.phi = FALSE,
return.weights.phi.deriv.1 = FALSE,
return.weights.phi.deriv.2 = FALSE,
bw = NULL,
…)a one (1) dimensional numeric or integer vector of dependent data, each
element \(i\) corresponding to each observation (row) \(i\) of
z.
a \(p\)-variate data frame of endogenous regressors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
a \(q\)-variate data frame of instruments. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
an \(r\)-variate data frame of exogenous regressors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
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.
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.
the order of the local polynomial regression (defaults to
p=1, i.e. local linear).
integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points.
an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.
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 a 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.
maximum number of attempts taken trying to achieve successful
convergence in optim. Defaults to 100.
the absolute convergence tolerance used by optim. Only useful
for non-negative functions, as a tolerance for reaching
zero. Defaults to .Machine$double.eps.
relative convergence tolerance used by optim. The algorithm
stops if it is unable to reduce the value by a factor of 'reltol *
(abs(val) + reltol)' at a step. Defaults to
sqrt(.Machine$double.eps), typically about 1e-8.
maximum number of iterations used by optim. Defaults
to 500.
a numeric scalar that, if supplied, is used rather than numerically
solving for alpha, when using method="Tikhonov".
a numeric scalar that, if supplied, is used for iterated Tikhonov
rather than numerically solving for alpha, when using
method="Tikhonov".
minimum of search range for \(\alpha\), the Tikhonov
regularization parameter, when using method="Tikhonov".
maximum of search range for \(\alpha\), the Tikhonov
regularization parameter, when using method="Tikhonov".
the search tolerance for optimize when solving for
\(\alpha\), the Tikhonov regularization parameter,
when using method="Tikhonov".
a logical value indicating whether to use iterated Tikhonov (one
iteration) or not when using method="Tikhonov".
an integer indicating the number of iterations to conduct when using
method="Tikhonov".
an integer indicating the maximum number of iterations permitted
before termination occurs when using method="Landweber-Fridman".
the search tolerance for the difference in the stopping rule from
iteration to iteration when using method="Landweber-Fridman"
(disable by setting to zero).
the constant to use when using method="Landweber-Fridman".
the regularization method employed (defaults to
"Landweber-Fridman", see Horowitz (2011); see Darolles,
Fan, Florens and Renault (2011) for details for
"Tikhonov").
a logical value indicating whether to
penalize the norm by the number of iterations or not (default
TRUE)
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
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)
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)
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)
a logical value (defaults to FALSE) indicating whether to
return the weight matrix which when postmultiplied by the response
\(y\) delivers the instrumental regression
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 \(z\)
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 \(z\)
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)
additional arguments supplied to npksum.
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.
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 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=1 for npreg.
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.
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 (forthcoming), “Nonparametric instrumental derivatives,” Journal of Nonparametric Statistics.
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.
# NOT RUN {
## 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")
# }
# NOT RUN {
# }
# NOT RUN {
<!-- % enddontrun -->
# }
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