Nonparametric Instrumental Derivatives
Description
crsivderiv
uses the approach of Florens and Racine (2012) to
compute the partial derivative of a 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 derivative
function phi' is the solution of an illposed inverse
problem, and is computed using LandweberFridman regularization.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  crsivderiv(y,
z,
w,
x = NULL,
zeval = NULL,
weval = NULL,
xeval = NULL,
iterate.max = 1000,
iterate.diff.tol = 1.0e08,
constant = 0.5,
penalize.iteration = TRUE,
start.from = c("Eyz","EEywz"),
starting.values = NULL,
stop.on.increase = TRUE,
smooth.residuals = TRUE,
opts = list("MAX_BB_EVAL"=10000,
"EPSILON"=.Machine$double.eps,
"INITIAL_MESH_SIZE"="r1.0e01",
"MIN_MESH_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
"MIN_POLL_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
"DISPLAY_DEGREE"=0),
...)

Arguments
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 predictors. 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 predictors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof 
zeval 
a pvariate data frame of endogenous predictors on which the
regression will be estimated (evaluation data). By default, evaluation
takes place on the data provided by 
weval 
a qvariate 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 rvariate data frame of exogenous predictors on which the
regression will be estimated (evaluation data). By default,
evaluation takes place on the data provided by 
iterate.max 
an integer indicating the maximum number of iterations permitted before termination occurs when using LandweberFridman iteration 
iterate.diff.tol 
the search tolerance for the difference in the stopping rule from iteration to iteration when using LandweberFridman (disable by setting to zero) 
constant 
the constant to use when using LandweberFridman iteration 
penalize.iteration 
a logical value indicating whether to
penalize the norm by the number of iterations or not (default

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 
smooth.residuals 
a logical value (defaults to 
opts 
arguments passed to the NOMAD solver (see 
... 
additional arguments supplied to 
Details
For LandweberFridman iteration, an optimal stopping rule based upon
E(yw)E(phi(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=5
for crs
when
using cv="nomad"
or instead use cv="exhaustive"
if
possible (this may not be feasible for nontrivial problems).
When using LandweberFridman iteration, 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.
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).
Value
crsivderiv
returns components phi.prime
, phi
,
phi.prime.mat
, num.iterations
, norm.stop
,
norm.value
and convergence
.
Note
This function currently supports univariate z
only.
This function should be considered to be in ‘beta test’ status until
further notice.
Author(s)
Jeffrey S. Racine racinej@mcmaster.ca
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 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 (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, 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.
See Also
npreg
, crsiv
, crs
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72  ## Not run:
## This illustration was made possible by Samuele Centorrino
## <samuele.centorrino@univtlse1.fr>
set.seed(42)
n < 1000
## For trimming the plot (trim .5% from each tail)
trim < 0.005
## 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,u,v)
ivdata < ivdata[order(ivdata$z),]
rm(y,z,w,u,v)
attach(ivdata)
model.ivderiv < crsivderiv(y=y,z=z,w=w)
ylim <c(quantile(model.ivderiv$phi.prime,trim),
quantile(model.ivderiv$phi.prime,1trim))
plot(z,model.ivderiv$phi.prime,
xlim=quantile(z,c(trim,1trim)),
main="",
ylim=ylim,
xlab="Z",
ylab="Derivative",
type="l",
lwd=2)
rug(z)
## End(Not run)

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