crsivderiv uses the approach of Florens and Racine (2012) 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.
crsivderiv(y, ...)# S3 method for formula
crsivderiv(y, data = NULL, subset, na.action, ...)
# S3 method for default
crsivderiv(y,
z,
w,
x = NULL,
zeval = NULL,
weval = NULL,
xeval = NULL,
constant = 0.5,
display.nomad.progress = TRUE,
display.warnings = TRUE,
iterate.diff.tol = 1.0e-08,
iterate.max = 1000,
opts = list("MAX_BB_EVAL"=10000,
"EPSILON"=.Machine$double.eps,
"INITIAL_MESH_SIZE"="r1.0e-01",
"MIN_MESH_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
"MIN_FRAME_SIZE"=paste("r",1,sep=""),
"DISPLAY_DEGREE"=0),
penalize.iteration = TRUE,
smooth.residuals = TRUE,
start.from = c("Eyz","EEywz"),
starting.values = NULL,
stop.on.increase = TRUE,
...)
crsivderiv returns a crsivderiv object (which inherits
from the crs class). The generic functions
plot support objects of this type.
For the plot function, the options are
plot.data=FALSE or data_overlay=FALSE (logical values
indicating whether to plot the data as a scatter plot),
phi=FALSE (a logical value indicating whether to plot the
reconstructed structural function rather than its derivative), and
output=c("plot","data","plot-data") or behavior
(whether to draw, return the plot data, or both). Bootstrap,
asymptotic interval, surface-rendering, rug, legend, and regression
gradient plot controls are not supported for this curve route and
fail fast when supplied. See plot.crs for the shared CRS
plot-output and data-overlay conventions. Note that the plot
method for crsivderiv objects currently only supports a univariate
continuous endogenous predictor \(z\).
See crs for details on the return object components.
In addition to the standard crs components,
crsivderiv returns components phi.prime, phi,
phi.prime.mat, phi.mat, num.iterations,
norm.stop, norm.value and convergence.
num.iterations is the selected positive-index state, and
phi.prime and phi are taken from that same column of their
respective matrices. The matrices and norm.stop contain all
evaluated states, so their common length can exceed
num.iterations when a later evaluated state triggers stopping or
the stopping-rule selector chooses an earlier state.
fitted() and predict() without newdata return the
selected structural state phi; predict(..., deriv=1) without
newdata returns the matching selected phi.prime. On a
training-grid fit, residuals() returns the original response minus
that same selected structural state. summary() returns a structured
summary.crsivderiv object. New objects also contain a namespaced
iv component recording the public IV call, role labels, row map,
original response, evaluation grid, and selected-state metadata; inherited
CRS fields are retained unchanged.
These arguments identify the response, endogenous variable, instruments, and
exogenous covariates. The formula interface uses
y ~ z | w or y ~ z | w | x. Each partition may contain
multiple additive terms and scalar transformations. Dot notation,
interaction operators, offsets, and matrix-valued terms are not supported.
an optional data frame, list, or environment containing the variables in
the IV formula. When data is supplied, every formula variable must
be present in it; ambient variables do not fill missing columns
an optional specification of observations to retain when using the formula interface
a function specifying the action for missing observations when using the
formula interface. With na.exclude, fitted() and
residuals() restore omitted rows as NA
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 predictors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof
either a one-dimensional numeric or integer response vector, with element
\(i\) corresponding to row \(i\) of z, or an IV formula of the
form y ~ z | w or y ~ z | w | x
a one-column data frame of continuous endogenous predictors. The
current implementation of crsivderiv supports univariate
continuous \(z\) only
These arguments identify evaluation data for the derivative fit.
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
an \(r\)-variate data frame of exogenous predictors on which the
regression will be estimated (evaluation data). By default,
evaluation takes place on the data provided by x
a one-column data frame of continuous endogenous predictors on
which the regression will be estimated (evaluation data). By
default, evaluation takes place on the data provided by z
These arguments control iteration, residual smoothing, starting values, and stopping behavior.
the constant to use when using Landweber-Fridman iteration
the search tolerance for the difference in the stopping rule from iteration to iteration when using Landweber-Fridman (disable by setting to zero)
an integer indicating the maximum number of complete positive-index Landweber-Fridman states that may be evaluated. The initialization state \(N=0\) is not counted
a logical value indicating whether to
penalize the norm by the number of iterations or not (default
TRUE)
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 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)
optional derivative values for the initialization
state \(\varphi'_0\). They are used directly before the first
reported update; no hidden \(-1\) to \(0\) update is performed. When
NULL, the initialization is obtained from \(E(y|z)\) according to
start.from (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)
These arguments control warnings and displayed optimizer progress.
a logical value indicating whether to
display the progress of the NOMAD solver (default display.nomad.progress=TRUE)
a logical value indicating whether to
display warnings (default display.warnings=TRUE)
Further NOMAD and CRS controls are passed through to lower-level routines.
additional arguments supplied to crs. In formula calls,
weights and observation-indexed starting.values are aligned
through the same model frame as the response and IV roles. Formula-time
evaluation arguments (newdata, zeval, weval, and
xeval) are deliberately not supported; use the native vector
interface for evaluation-grid fitting
arguments passed to the NOMAD solver (see snomadr for
further details)
Jeffrey S. Racine racinej@mcmaster.ca
The formula method is a training-grid interface. It constructs one model
frame for the response, endogenous variable, instruments, optional
exogenous variables, weights, and observation-indexed starting values, then
calls the established native estimator once. Thus subset, missing
values, and row-indexed controls cannot acquire different row maps. Formula
fits support fitted(), residuals(), predict() without
newdata, predict(..., deriv=1) for the selected derivative,
summary(), and plot(). Arbitrary formula newdata is
deferred because evaluating a retained spline projection is not generally
the same operation as resolving the IV problem on a new grid. Native
objects retain their established post-fit CRS projection route.
The iteration index has a single meaning throughout the returned object.
The initialization consists of \(\varphi'_0\) and its centered
integral \(\varphi_0\). For \(N=1,2,\ldots\), one adjoint
update produces \(\varphi'_N\), integration and centering produce
\(\varphi_N\), and the residual at that same curve produces the
stopping value for state \(N\). Thus column \(N\) of
phi.prime.mat and phi.mat, and element \(N\) of
norm.stop, always describe one coherent state. With
penalize.iteration=TRUE, that element is \(N q_N\), where
\(q_N\) is the normalized squared residual criterion at
\(\varphi_N\).
For Landweber-Fridman iteration, an optimal stopping rule based upon
\(||E(y|w)-E(\varphi_k(z,x)|w)||^2 \)
is used to terminate iteration. 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=2 for crs when
using cv="nomad" or instead use cv="exhaustive" if
possible (this may not be feasible for non-trivial problems).
Note that for subsequent Landweber-Fridman iterations, a “warm
start” strategy is employed. The optimal parameters (spline degree,
number of segments, and bandwidths or inclusion indicators) 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 parameters 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
the crs default (2) for the first iteration and to 1 for
all subsequent iterations. This strategy provides a balance between
computational efficiency and robustness, allowing the NOMAD solver to
refine the structural parameters as the residuals evolve
incrementally while still guarding against local optima.
When using Landweber-Fridman iteration, 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.
When your problem is a simple one (e.g. univariate \(Z\), \(W\),
and \(X\)) you might want to avoid cv="nomad" and instead use
cv="exhaustive" since exhaustive search may be feasible (for
degree.max and segments.max not overly large). This will
guarantee an exact solution for each iteration (i.e. there will be no
errors arising due to numerical search).
The current implementation supports a single continuous endogenous regressor only. Instrument and exogenous regressor data may still be mixed continuous and categorical.
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.
npreg, crsiv, crs,
plot.crs
if (FALSE) {
## This illustration was made possible by Samuele Centorrino
##
set.seed(42)
n <- 500
## 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$z),]
rm(y,z,w,u,v)
model.ivderiv <- crsivderiv(y ~ z | w, data=ivdata)
plot(ivdata$z,model.ivderiv$phi.prime,
xlim=quantile(ivdata$z,c(trim,1-trim)),
main="",
xlab="Z",
ylab="Derivative",
type="l",
lwd=2)
rug(ivdata$z)
}
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