npregiv  R Documentation 
npregiv
computes nonparametric estimation of an instrumental
regression function phi defined by conditional moment
restrictions stemming from a structural econometric model: E [Y  phi (Z,X)  W ] = 0, and involving
endogenous variables Y and Z and exogenous variables
X and instruments W. The function phi is the
solution of an illposed 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="LandweberFridman"
,
npregiv
uses the approach of Horowitz (2011) again using local
polynomial kernel regression (Horowitz uses Bspline weighting).
npregiv(y, z, w, x = NULL, zeval = NULL, xeval = NULL, p = 1, nmulti = 1, random.seed = 42, optim.maxattempts = 10, optim.method = c("NelderMead", "BFGS", "CG"), optim.reltol = sqrt(.Machine$double.eps), optim.abstol = .Machine$double.eps, optim.maxit = 500, alpha = NULL, alpha.iter = NULL, alpha.min = 1e10, alpha.max = 1e01, alpha.tol = .Machine$double.eps^0.25, iterate.Tikhonov = TRUE, iterate.Tikhonov.num = 1, iterate.max = 1000, iterate.diff.tol = 1.0e08, constant = 0.5, method = c("LandweberFridman","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, ...)
y 
a one (1) dimensional numeric or integer vector of dependent data, each
element i corresponding to each observation (row) i of

z 
a pvariate data frame of endogenous regressors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof. 
w 
a qvariate data frame of instruments. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof. 
x 
an rvariate data frame of exogenous regressors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof. 
zeval 
a pvariate data frame of endogenous regressors on which the
regression will be estimated (evaluation data). By default, evaluation
takes place on the data provided by 
xeval 
an rvariate data frame of exogenous regressors on which the
regression will be estimated (evaluation data). By default,
evaluation takes place on the data provided by 
p 
the order of the local polynomial regression (defaults to

nmulti 
integer number of times to restart the process of finding extrema of the crossvalidation 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. 
optim.method 
method used by 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 nondifferentiable functions. method method 
optim.maxattempts 
maximum number of attempts taken trying to achieve successful
convergence in 
optim.abstol 
the absolute convergence tolerance used by 
optim.reltol 
relative convergence tolerance used by 
optim.maxit 
maximum number of iterations used by 
alpha 
a numeric scalar that, if supplied, is used rather than numerically
solving for 
alpha.iter 
a numeric scalar that, if supplied, is used for iterated Tikhonov
rather than numerically solving for 
alpha.min 
minimum of search range for alpha, the Tikhonov
regularization parameter, when using 
alpha.max 
maximum of search range for alpha, the Tikhonov
regularization parameter, when using 
alpha.tol 
the search tolerance for 
iterate.Tikhonov 
a logical value indicating whether to use iterated Tikhonov (one
iteration) or not when using 
iterate.Tikhonov.num 
an integer indicating the number of iterations to conduct when using

iterate.max 
an integer indicating the maximum number of iterations permitted
before termination occurs when using 
iterate.diff.tol 
the search tolerance for the difference in the stopping rule from
iteration to iteration when using 
constant 
the constant to use when using 
method 
the regularization method employed (defaults to

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

smooth.residuals 
a logical value indicating whether to
optimize bandwidths for the regression of
(yphi(z)) on w (defaults to

start.from 
a character string indicating whether to start from
E(Yz) (default, 
starting.values 
a value indicating whether to commence
LandweberFridman assuming
phi[1]=starting.values (proper
LandweberFridman) or instead begin from E(yz) (defaults to

stop.on.increase 
a logical value (defaults to 
return.weights.phi 
a logical value (defaults to 
return.weights.phi.deriv.1 
a logical value (defaults to 
return.weights.phi.deriv.2 
a logical value (defaults to 
bw 
an object which, if provided, contains bandwidths and parameters
(obtained from a previous invocation of 
... 
additional arguments supplied to 
Tikhonov regularization requires computation of weight matrices of dimension n x n which can be computationally costly in terms of memory requirements and may be unsuitable for large datasets. LandweberFridman 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="LandweberFridman"
uses an optimal stopping rule based
upon E(yw)E(phi(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="LandweberFridman"
, iteration will terminate
when either the change in the value of
(E(yw)E(phi(z,x)w))/E(yw)^2 from iteration to iteration is
less than iterate.diff.tol
or we hit iterate.max
or
(E(yw)E(phi(z,x)w))/E(yw)^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="LandweberFridman"
, 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="LandweberFridman"
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.
npregiv
returns a list with components phi
,
phi.mat
and either alpha
when method="Tikhonov"
or norm.index
, norm.stop
and convergence
when
method="LandweberFridman"
, 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="LandweberFridman"
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
.
This function should be considered to be in ‘beta test’ status until further notice.
Jeffrey S. Racine racinej@mcmaster.ca, Samuele Centorrino samuele.centorrino@univtlse1.fr
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 56335751
Darolles, S. and Y. Fan and J.P. Florens and E. Renault (2011), “Nonparametric instrumental regression,” Econometrica, 79, 15411565.
Feve, F. and J.P. Florens (2010), “The practice of nonparametric estimation by solving inverse problems: the example of transformation models,” Econometrics Journal, 13, S1S27.
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, 233334, in Russian.
Horowitz, J.L. (2011), “Applied nonparametric instrumental variables estimation,” Econometrica, 79, 347394.
Landweber, L. (1951), “An iterative formula for Fredholm integral equations of the first kind,” American Journal of Mathematics, 73, 61524.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.
Li, Q. and J.S. Racine (2004), “Crossvalidated Local Linear Nonparametric Regression,” Statistica Sinica, 14, 485512.
npregivderiv,npreg
## Not run: ## This illustration was made possible by Samuele Centorrino ## <samuele.centorrino@univtlse1.fr> set.seed(42) n < 500 ## The DGP is as follows: ## 1) y = phi(z) + u ## 2) E(uz) != 0 (endogeneity present) ## 3) Suppose there exists an instrument w such that z = f(w) + v and ## E(uw) = 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 noniv local linear estimator of E(yz) 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,1trim)), ylim=quantile(y,c(trim,1trim)), 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(yz)"), lty=c(1,2,4), col=c("black","blue","red"), lwd=c(2,2,2), bty="n") ## End(Not run)
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