`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,
z,
w,
x = NULL,
zeval = NULL,
weval = NULL,
xeval = NULL,
iterate.max = 1000,
iterate.diff.tol = 1.0e-08,
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$double.eps,
"INITIAL_MESH_SIZE"="r1.0e-01",
"MIN_MESH_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
"MIN_POLL_SIZE"=paste("r",1,sep=""),
"DISPLAY_DEGREE"=0),
...)
```

y

a one (1) dimensional numeric or integer vector of dependent data, each
element \(i\) corresponding to each observation (row) \(i\) of
`z`

z

a \(p\)-variate data frame of endogenous predictors. 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 predictors. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof

zeval

a \(p\)-variate data frame of endogenous predictors on which the
regression will be estimated (evaluation data). By default, evaluation
takes place on the data provided by `z`

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 `w`

xeval

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`

iterate.max

an integer indicating the maximum number of iterations permitted before termination occurs when using Landweber-Fridman iteration

iterate.diff.tol

the search tolerance for the difference in the stopping rule from iteration to iteration when using Landweber-Fridman (disable by setting to zero)

constant

the constant to use when using Landweber-Fridman iteration

penalize.iteration

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

)

start.from

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)

starting.values

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)

stop.on.increase

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)

smooth.residuals

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

opts

arguments passed to the NOMAD solver (see `snomadr`

for
further details)

...

additional arguments supplied to `crs`

`crsivderiv`

returns components `phi.prime`

, `phi`

,
`phi.prime.mat`

, `num.iterations`

, `norm.stop`

,
`norm.value`

and `convergence`

.

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=5`

for `crs`

when
using `cv="nomad"`

or instead use `cv="exhaustive"`

if
possible (this may not be feasible for non-trivial problems).

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).

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.

# NOT RUN { ## This illustration was made possible by Samuele Centorrino ## <samuele.centorrino@univ-tlse1.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(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$z),] rm(y,z,w,u,v) attach(ivdata) model.ivderiv <- crsivderiv(y=y,z=z,w=w) ylim <-c(quantile(model.ivderiv$phi.prime,trim), quantile(model.ivderiv$phi.prime,1-trim)) plot(z,model.ivderiv$phi.prime, xlim=quantile(z,c(trim,1-trim)), main="", ylim=ylim, xlab="Z", ylab="Derivative", type="l", lwd=2) rug(z) # } # NOT RUN { # } # NOT RUN { <!-- % end dontrun --> # }