npregivderiv {np} | R Documentation

## Nonparametric Instrumental Derivatives

### Description

`npregivderiv` uses the approach of Florens, Racine and Centorrino (forthcoming) 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 ill-posed inverse problem, and is computed using Landweber-Fridman regularization.

### Usage

    npregivderiv(y,
                 z,
                 w,
                 x = NULL,
                 zeval = NULL,
                 weval = NULL,
                 xeval = NULL,
                 random.seed = 42,
                 iterate.max = 1000,
                 iterate.break = TRUE,
                 constant = 0.5,
                 start.from = c("Eyz","EEywz"),
                 starting.values = NULL,
                 stop.on.increase = TRUE,
                 smooth.residuals = TRUE,
                 ...)

### Arguments

`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 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 `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 regressors on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by `x`.
`random.seed` | an integer used to seed R's random number generator. This ensures replicability of the numerical search. Defaults to 42.
`iterate.max` | an integer indicating the maximum number of iterations permitted before termination occurs for Landweber-Fridman iteration.
`iterate.break` | a logical value indicating whether to compute all objects up to `iterate.max` or to break when a potential optimum arises (useful for inspecting full stopping rule profile up to `iterate.max`)
`constant` | the constant to use for Landweber-Fridman iteration.
`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 phi'[-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-phi(z) on w or for the regression of phi(z) on w during Landweber-Fridman iteration.
`...` | additional arguments supplied to `npreg` and `npksum`.

### Details

Note that Landweber-Fridman iteration presumes that phi[-1]=0, and so for derivative estimation we commence iterating from a model having derivatives all equal to zero. Given this starting point it may require a fairly large number of iterations in order to converge. Other perhaps more reasonable starting values might present themselves. When `start.phi.zero` is set to `FALSE` iteration will commence instead using derivatives from the conditional mean model E(y|z). Should the default iteration terminate quickly or you are concerned about your results, it would be prudent to verify that this alternative starting value produces the same result. Also, check the norm.stop vector for any anomalies (such as the error criterion increasing immediately).

Landweber-Fridman iteration 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=1` for `npreg`.

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.

### Value

`npregivderiv` returns a list with components `phi.prime`, `phi`, `num.iterations`, `norm.stop` 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 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 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, 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.

### See Also

`npregiv,npreg`

### Examples

    ## Not run:
    ## This illustration was made possible by Samuele Centorrino
    ## <samuele.centorrino@univ-tlse1.fr>

    set.seed(42)
    n <- 1500

    ## 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 <- npregivderiv(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)

    ## End(Not run)

[Package np version 0.60-9 Index]