quantreg (version 3.03)

rq: Quantile Regression

Description

Returns an object of class "rq" or "rq.process" that represents a quantile regression fit.

Usage

rq(formula, tau=.5, data, weights, na.action,
   method="br", contrasts, ...)

Arguments

formula

a formula object, with the response on the left of a ~ operator, and the terms, separated by + operators, on the right.

tau

the quantile to be estimated, this is generally a number between 0 and 1, but if specified outside this range, it is presumed that the solutions for all values of tau in (0,1) are desired. In the former case an object of class

data

a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument. If this is missing, then the variables in the formula should be on the search list. This may also be a single number to handl

weights

vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative

na.action

a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with na.fail) is to create an error if any missing values are found. A possible alternative is

method

the algorithmic method used to compute the fit. There are currently three options: The default method is the modified version of the Barrodale and Roberts algorithm for $l_1$-regression, used by l1fit in S, and is described i

contrasts

a list giving contrasts for some or all of the factors default = NULL appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any

...

additional arguments for the fitting routines (see rq.fit.br and rq.fit.fn and the functions they call).

Value

• See rq.object and rq.process.object for details.

Method

The function computes an estimate on the tau-th conditional quantile function of the response, given the covariates, as specified by the formula argument. Like lm(), the function presumes a linear specification for the quantile regression model, i.e. that the formula defines a model that is linear in parameters. For non-linear quantile regression see the function nlrq(). [To appear real soon now on a screen near you.] The function minimizes a weighted sum of absolute residuals that can be formulated as a linear programming problem. As noted above, there are three different algorithms that can be chosen depending on problem size and other characteristics. For moderate sized problems ($n \ll 5,000, p \ll 20$) it is recommended that the default "br" method be used. There are several choices of methods for computing confidence intervals and associated test statistics. Using "br" the default approach produces confidence intervals for each of the estimated model parameters based on inversion of a rank test. See the documentation for rq.fit.br for further details and options. For larger problems, the "fn" and "pfn" are preferred, and there are several methods of computing standard errors and associated test statistics described in the help files for rq.fit.fn, and summary.rq.

References

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

[2] Koenker, R.W. and d'Orey (1987, 1994). Computing regression quantiles. Applied Statistics, 36, 383--393, and 43, 410--414.

[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. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349--359, Springer-Verlag, New York.

There is also recent information available at the URL: http://www.econ.uiuc.edu.

See Also

summary.rq, rq.object, rq.process.object

Examples

data(stackloss)
rq(stack.loss ~ stack.x,.5)  #median (l1) regression  fit for the stackloss data. 
rq(stack.loss ~ stack.x,.25)  #the 1st quartile, 
        #note that 8 of the 21 points lie exactly on this plane in 4-space 
rq(stack.loss ~ stack.x, tau=-1)   #this returns the full rq process
rq(rnorm(50) ~ 1, ci=F)    #ordinary sample median --no rank inversion ci
rq(rnorm(50) ~ 1, weights=runif(50),ci=F)  #weighted sample median