Nonparametric Instrumental Regression

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Description

crsiv 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, exogenous variables X, and instruments W. The function phi is the solution of an ill-posed inverse problem.

When method="Tikhonov", crsiv uses the approach of Darolles, Fan, Florens and Renault (2011) modified for regression splines (Darolles et al use local constant kernel weighting). When method="Landweber-Fridman", crsiv uses the approach of Horowitz (2011) using the regression spline methodology implemented in the crs package.

Usage

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crsiv(y,
      z,
      w,
      x = NULL,
      zeval = NULL,
      weval = NULL,
      xeval = NULL,
      alpha = NULL,
      alpha.min = 1e-10,
      alpha.max = 1e-01,
      alpha.tol = .Machine$double.eps^0.25,
      deriv = 0,
      iterate.max = 1000,
      iterate.diff.tol = 1.0e-08,
      constant = 0.5,
      penalize.iteration = TRUE,
      smooth.residuals = TRUE,
      start.from = c("Eyz","EEywz"),
      starting.values = NULL,
      stop.on.increase = TRUE,
      method = c("Landweber-Fridman","Tikhonov"),
      opts = list("MAX_BB_EVAL"=10000,
                  "EPSILON"=.Machine$double.eps,
                  "INITIAL_MESH_SIZE"="r1.0e-01",
                  "MIN_MESH_SIZE"=paste("r",sqrt(.Machine$double.eps),sep=""),
                  "MIN_POLL_SIZE"=paste("r",sqrt(.Machine$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

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

alpha

a numeric scalar that, if supplied, is used rather than numerically solving for alpha, when using method="Tikhonov"

alpha.min

minimum of search range for alpha, the Tikhonov regularization parameter, when using method="Tikhonov"

alpha.max

maximum of search range for alpha, the Tikhonov regularization parameter, when using method="Tikhonov"

alpha.tol

the search tolerance for optimize when solving for alpha, the Tikhonov regularization parameter, when using method="Tikhonov"

iterate.max

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

iterate.diff.tol

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

constant

the constant to use when using method="Landweber-Fridman"

method

the regularization method employed (default "Landweber-Fridman", see Horowitz (2011); see Darolles, Fan, Florens and Renault (2011) for details for "Tikhonov")

penalize.iteration

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

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 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)

opts

arguments passed to the NOMAD solver (see snomadr for further details)

deriv

an integer l (default deriv=0) specifying whether to compute the univariate lth partial derivative for each continuous predictor (and difference in levels for each categorical predictor) or not and if so what order. Note that if deriv is higher than the spline degree of the associated continuous predictor then the derivative will be zero and a warning issued to this effect (see important note below)

...

additional arguments supplied to crs

Details

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 (i.e. unfeasible) 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, though it does require iteration that may be equally or even more computationally demanding in terms of total computation time.

When using method="Landweber-Fridman", an optimal stopping rule based upon ||E(y|w)-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 non-trivial problems).

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.

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).

demo(crsiv), demo(crsiv_exog), and demo(crsiv_exog_persp) provide flexible interactive demonstrations similar to the example below that allow you to modify and experiment with parameters such as the sample size, method, and so forth in an interactive session.

Value

crsiv returns a crs object. The generic functions fitted and residuals extract (or generate) estimated values and residuals. Furthermore, the functions summary, predict, and plot (options mean=FALSE, deriv=i where i is an integer, ci=FALSE, plot.behavior=c("plot","plot-data","data")) support objects of this type.

See crs for details on the return object components.

In addition to the standard crs components, crsiv returns components phi and either alpha when method="Tikhonov" or phi, phi.mat, num.iterations, norm.stop, norm.value and convergence when method="Landweber-Fridman".

Note

Using the option deriv= computes (effectively) the analytical derivative of the estimated phi(Z,X) and not that using crsivderiv, which instead uses the method of Florens and Racine (2012). Though both are statistically consistent, practitioners may desire one over the other hence we provide both.

Note

This function should be considered to be in ‘beta test’ status until further notice.

Author(s)

Jeffrey S. Racine racinej@mcmaster.ca, Samuele Centorrino samuele.centorrino@univ-tlse1.fr

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 (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.

See Also

npreg, crs

Examples

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## 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

## Create an evaluation dataset sorting on z (for plotting)

evaldata <- data.frame(y,z,w)
evaldata <- evaldata[order(evaldata$z),]

## Compute the non-IV regression spline estimator of E(y|z)

model.noniv <- crs(y~z,opts=opts)
mean.noniv <- predict(model.noniv,newdata=evaldata)

## Compute the IV-regression spline estimator of phi(z)

model.iv <- crsiv(y=y,z=z,w=w)
phi.iv <- predict(model.iv,newdata=evaldata)

## 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 Spline Regression",
      sub=paste("Landweber-Fridman: iterations = ", model.iv$num.iterations,sep=""),
      lwd=1,lty=1)

points(z,y,type="p",cex=.25,col="grey")

lines(evaldata$z,evaldata$z^2 -0.325*evaldata$z,lwd=1,lty=1)

lines(evaldata$z,phi.iv,col="blue",lwd=2,lty=2)

lines(evaldata$z,mean.noniv,col="red",lwd=2,lty=4)

legend(quantile(z,trim),quantile(y,1-trim),
       c(expression(paste(varphi(z),", E(y|z)",sep="")),
         expression(paste("Nonparametric ",hat(varphi)(z))),
         "Nonparametric E(y|z)"),
       lty=c(1,2,4),
       col=c("black","blue","red"),
       lwd=c(1,2,2))

## End(Not run)