qregcdf: Nonparametric quantiles based on conditional CDF functions

Description

Estimates conditional quantile functions based on nonparametric conditional CDF functions

Usage

qregcdf(form,taumat=c(.10,.25,.50,.75,.90),hx=0,hy=0,nx=20,ny=100,
  targetx=0,targety=0,graph.target=FALSE,graph.yhat=FALSE, data=NULL)

Arguments

form

Model formula

taumat

Vector of quantiles. Default: taumat=c(.10, .25, .50, .75, .90)

Bandwidth for x in $K((X_i-x)/hx)$. Default: hx = 1.06*min(sd(x), (quantile(x,.75)-quantile(x,.25))/1.349)*(n^{-.2})

Bandwidth for y in $\Phi((y-Y_i)/hy)$ Default: hy = 1.06*min(sd(y), (quantile(y,.75)-quantile(y,.25))/1.349)*(n^(-.2))

Number of target points for x if using an evenly spaced grid. Default is nx = 20. If nx>0, then targetx <- seq(min(x),max(x),length=nx)

Number of target points for y if using an evenly spaced grid. Default is ny = 200. If ny>0, then $targety <- seq(min(y),max(y),length=ny)$

targetx

Vector of user-provided target values for x. An alternative to using nx to specify a uniformly spaced grid. Default: not specified.

targety

Vector of user-provided target values for y. An alternative to using nxy to specify a uniformly spaced grid. Default: not specified.

graph.target

If graph.target=T, graph of results is produced based on target values. Default: graph.target=F.

graph.yhat

If graph.yhat=T, graph of results is produced based on interpolations to actual values of x and y. Default: graph.yhat=F.

data

A data frame containing the data. Default: use data in the current working directory

Value

yhatMatrix of quantile predictions. Dimension is n x length(taumat)
yhat.targetMatrix of quantile predictions at target values of x. Dimension is length(targetx) x length(taumat).
targetxVector of target values for x.
targetyVector of target values for y.
taumatVector of target quantile values.
hxBandwidth for x.
hyBandwidth for y.

Details

Following Li and Racine (2007), equation 6.3, a smoothed version of the conditional CDF of y given a value of x can be written:

F (y | x) = \frac{\frac{1}{n} \sum_{i} Φ (\frac{y - y_{i}}{h_{y}}) h_{x}^{- 1} K (\frac{X_{i} - x}{h_{x}})}{h_{x}^{- 1} \sum_{i} K (\frac{X_{i} - x}{h_{x}})}

The estimation procedure begins by evaluating this expression at each point in the grid determined by the values of target.x and target.y The result is an nx x ny matrix of values for F(y|x). Let $f(x_j)$ represent the ny-vector of values of $F(y|x_j)$, and let $f_k$ indicate the entry of $F(y|x_j)$ associated with $y_k$, k = 1,...,ny. Finally, let $\tau$ represent an entry of taumat. Then the value of yhat.target associated with quantile $\tau$ and $x_j$ is the largest value of $f_k$ such that $f_k < \tau < f_k+1$. The resulting nx x length(taumat) matrix is available after estimation as yhat.target. The package akima is used to interpolate each column of yhat.target to span the full vector of original values of x. The result is the n x length(taumat) matrix yhat. Note: The default bandwidth may prove too small if there are regions where x is sparse. It may be necessary to experiment with larger bandwidths for hx and hy. The function qreglwr is more flexible, allowing nearest neighbor approaches as well as fixed bandwidths.

References

Li, Oi and Jeffrey Scott Racine. Nonparametric Econometrics: Theory and Practice. Princeton, NJ: Princeton University Press, 2007. Chapter 6.

Examples

Run this code

data(dupage99)
dupage99$ratio <- dupage99$av/dupage99$price
o <- order(dupage99$price)
dupage99 <- dupage99[o,]
attach(dupage99)
price <- price/1000

fit <- qregcdf(ratio~price)
ymin = min(fit$yhat)
ymax = max(fit$yhat)
plot(price, fit$yhat[,1],type="l",xlab="Sales Price (1000s)",ylab="Assessment Ratio",
  ylim=c(ymin,ymax),main="Nonparametric Conditional CDF Quantile Regression")
for (j in seq(2,5)) {
  lines(price,fit$yhat[,j])
}
fit$hx
fit$hy

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