Quantile Regression

Usage

rq(x, y, tau=-1, alpha=.1, dual=TRUE, int=TRUE, tol=1e-4, ci = TRUE, 
         method="score", interpolate=TRUE, tcrit=TRUE, hs=TRUE)
rq.formula(formula, data=list(), subset, na.action, tau=-1, 
         alpha = 0.10000000000000001, dual = TRUE, 
         tol = 0.0001, ci = TRUE, method="score", interpolate = TRUE, 
         tcrit = TRUE, hs=TRUE)

Arguments

x vector or matrix of explanatory variables. If a matrix, each column represents a variable and each row represents an observation (or case). This should not contain column of 1s unless the argument intercept is FALSE. The number of rows of x should equal the number of elements of y, and there should be fewer columns than rows. If x is missing, rq() computes the ordinary sample quantile(s) of y.
y response vector with as many observations as the number of rows of x.
tau desired quantile. If tau is missing or outside the range [0,1] then all the regression quantiles are computed and the corresponding primal and dual solutions are returned.
alpha level of significance for the confidence intervals; default is set at 10%.
dual return the dual solution if TRUE (default).
int flag for intercept; if TRUE (default) an intercept term is included in the regression.
tol tolerance parameter for rq computations.
ci flag for confidence interval; if TRUE (default) the confidence intervals are returned.
method if method="score" (default), ci is computed using regression rank score inversion; if method="sparsity", ci is computed using sparsity function.
interpolate if TRUE (default), the smoothed confidence intervals are returned.
tcrit if tcrit=T (default), a finite sample adjustment of the critical point is performed using Student's t quantile, else the standard Gaussian quantile is used.
hs logical flag to use Hall-Sheather's sparsity estimator (default); otherwise Bofinger's version is used.

Description

Perform a quantile regression on a design matrix, x, of explanatory variables and a vector, y, of responses.

Value

coef the estimated parameters of the tau-th conditional quantile function.
resid the estimated residuals of the tau-th conditional quantile function.
dual the dual solution (if dual=T).
h the index of observations in the basis.
ci confidence intervals (if ci=T).

METHOD

The algorithm used is a modification of the Barrodale and Roberts algorithm for l1-regression, l1fit in S, and is described in detail in Koenker and d"Orey(1987).

SEE ALSO

trq and qrq for further details and references.

Author(s)

Roger Koenker, roger@ysidro.econ.uiuc.edu, http://www.econ.uiuc.edu/~roger/research/rq/rq.html. Ported to R, and added rq.formula, by Kjetil Halvorsen.

References

[1] Koenker, R.W. and Bassett, G.W. (1978). Regression quantiles, Econometrica, 46, 33-50.

[2] Koenker, R.W. and d'Orey (1987). Computing Regression Quantiles. Applied Statistics, 36, 383-393.

[3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, Annals of Statistics, 20, 305-330.

[4] Koenker, R.W. and d'Orey (1994). Remark on Alg. AS 229: Computing Dual Regression Quantiles and Regression Rank Scores, Applied Statistics, 43, 410-414.

[5] Koenker, R.W. (1994). Confidence Intervals for Regression Quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349-359, Springer-Verlag, New York.

Examples

data(stackloss)
rq(stack.x, stack.loss, .5)  #the l1 estimate for the stackloss data
rq(stack.x, stack.loss, tau=.5, ci=T, method="score")  #same as above with 
	#regression rank score inversion confidence interval
rq(stack.x, stack.loss, .25)  #the 1st quartile, 
	#note that 8 of the 21 points lie exactly 
	#on this plane in 4-space
rq(stack.x, stack.loss, -1)   #this gives all of the rq solutions
rq(y=rnorm(10), method="sparsity")	#ordinary sample quantiles
data(Patacamaya)               # an example with formula
 z0.1 <- rq.formula(y ~ a+tipo, data=Patacamaya, na.action=na.omit, tau=0.1)
z0.1$coef
z0.1$ci


[Package Contents]