npregiv {np} | 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 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, weval = 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.min = 1e-10, alpha.max = 1e-01, alpha.tol = .Machine$double.eps^0.25, 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, ...)
y |
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. |
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 |
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 |
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 |
p |
the order of the local polynomial regression (defaults to
|
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. |
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 non-differentiable 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.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.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 (defaults to |
start.from |
a character string indicating whether to start from
E(Y|z) (default, |
starting.values |
a value indicating whether to commence
Landweber-Fridman assuming
phi[-1]=starting.values (proper
Landweber-Fridman) or instead begin from E(y|z) (defaults to
|
stop.on.increase |
a logical value (defaults to |
... |
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. 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(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=0
for npreg
.
When using method="Landweber-Fridman"
, iteration will terminate
when either the change in the value of
||(E(y|w)-E(phi(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(phi(z,x)|w))/E(y|w)||^2 stops falling in value and
starts rising.
npregiv
returns a list with components phi
and either
alpha
when method="Tikhonov"
or num.iterations
,
norm.stop
and convergence
when
method="Landweber-Fridman"
.
This function should be considered to be in ‘beta test’ status until further notice.
Jeffrey S. Racine racinej@mcmaster.ca, Samuele Centorrino samuele.centorrino@univ-tlse1.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 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.
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 <- 1500 ## 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(quantile(z,trim),quantile(y,1-trim), 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)) ## End(Not run)