Returns a summary list for a quantile regression fit. A null value will be returned if printing is invoked.
# S3 method for rq
summary(object, se = NULL, covariance=FALSE, hs = TRUE, U = NULL, gamma = 0.7, ...)
# S3 method for rqs
summary(object, ...)a list is returned with the following components, when object
is of class "rqs" then there is a list of such lists.
a p by 4 matrix consisting of the coefficients, their estimated standard errors, their t-statistics, and their associated p-values, in the case of most "se" methods. For methods "rank" and "extreme" potentially asymetric confidence intervals are return in lieu of standard errors and p-values.
the estimated covariance matrix for the coefficients in the model,
provided that cov=TRUE in the called sequence. This option
is not available when se = "rank".
inverse of the estimated Hessian matrix returned if cov=TRUE and
se %in% c("nid","ker") , note that for se = "boot" there
is no way to split the estimated covariance matrix into its sandwich
constituent parts.
Unscaled Outer product of gradient matrix returned if cov=TRUE and se
!= "iid". The Huber sandwich is cov = tau (1-tau) Hinv %*% J %*% Hinv.
as for the Hinv component, there is no J component when
se == "boot". (Note that to make the Huber sandwich you need to add the
tau (1-tau) mayonnaise yourself.)
Matrix of bootstrap realizations.
Matrix of bootstrap randomization draws.
This is an object of class "rq" or "rqs" produced by
a call to rq(), depending on whether one or more taus are
specified.
specifies the method used to compute standard standard errors. There are currently seven available methods:
"rank" which produces confidence intervals for the
estimated parameters by inverting a rank test as described in
Koenker (1994). This method involves solving a parametric linear
programming problem, and for large sample sizes can be extremely
slow, so by default it is only used when the sample size is less
than 1000, see below. The default option assumes that the errors are
iid, while the option iid = FALSE implements a proposal of Koenker
Machado (1999). See the documentation for rq.fit.br for additional arguments.
"iid" which presumes that the errors are iid and computes
an estimate of the asymptotic covariance matrix as in KB(1978).
"nid" which presumes local (in tau)
linearity (in x) of the
the conditional quantile functions and computes a Huber
sandwich estimate using a local estimate of the sparsity.
If the initial fitting was done with method "sfn" then use
of se = "nid" is recommended. However, if the cluster
option is also desired then se = "boot" can be used
and bootstrapping will also employ the "sfn" method.
"ker" which uses a kernel estimate of the sandwich
as proposed by Powell(1991).
"boot" which implements one of several possible bootstrapping
alternatives for estimating standard errors including a variate of the wild
bootstrap for clustered response. See boot.rq for
further details.
"BLB" which implements the bag of little bootstraps method
proposed in Kleiner, et al (2014). The sample size of the little bootstraps
is controlled by the parameter gamma, see below. At present only
bsmethod = "xy" is sanction, and even that is experimental. This
option is intended for applications with very large n where other flavors
of the bootstrap can be slow.
"conquer" which is invoked automatically if the fitted
object was created with method = "conquer", and returns the
multiplier bootstrap percentile confidence intervals described in
He et al (2020).
"extreme" which uses the subsampling method of Chernozhukov
Fernandez-Val, and Kaji (2018) designed for inference on extreme quantiles.
If se = NULL (the default) and covariance = FALSE, and
the sample size is less than 1001, then the "rank" method is used,
otherwise the "nid" method is used.
logical flag to indicate whether the full covariance matrix of the estimated parameters should be returned.
Use Hall Sheather bandwidth for sparsity estimation If false revert to Bofinger bandwidth.
Resampling indices or gradient evaluations used for bootstrap,
see boot.rq.
parameter controlling the effective sample size of the'bag
of little bootstrap samples that will be b = n^gamma where
n is the sample size of the original model.
Optional arguments to summary, e.g. bsmethod to use bootstrapping.
see boot.rq. When using the "rank" method for confidence
intervals, which is the default method for sample sizes less than 1000,
the type I error probability of the intervals can be controlled with the
alpha parameter passed via "...", thereby controlling the width of the
intervals plotted by plot.summary.rqs. Similarly, the arguments
alpha, mofn and kex can be passed when invoking the "extreme" option
for "se" to control the percentile interval reported, given by estimated
quantiles [alpha/2, 1 - alpha/2]; kex is a tuning parameter for the
extreme value confidence interval construction. The size of the bootstrap
subsamples for the "extreme" option can also be controlled by passing
the argument mofm via "...". Default values for kex, mofn and
alpha are 20, floor(n/5) and 0.1, respectively.
When the default summary method is used, it tries to estimate a sandwich
form of the asymptotic covariance matrix and this involves estimating
the conditional density at each of the sample observations, negative
estimates can occur if there is crossing of the neighboring quantile
surfaces used to compute the difference quotient estimate.
A warning message is issued when such negative estimates exist indicating
the number of occurrences -- if this number constitutes a large proportion
of the sample size, then it would be prudent to consider an alternative
inference method like the bootstrap.
If the number of these is large relative to the sample size it is sometimes
an indication that some additional nonlinearity in the covariates
would be helpful, for instance quadratic effects.
Note that the default se method is rank, unless the sample size exceeds
1001, in which case the rank method is used.
There are several options for alternative resampling methods. When
summary.rqs is invoked, that is when summary is called
for a rqs object consisting of several taus, the B
components of the returned object can be used to construct a joint covariance
matrix for the full object.
Chernozhukov, Victor, Ivan Fernandez-Val, and Tetsuya Kaji, (2018) Extremal Quantile Regression, in Handbook of Quantile Regression, Eds. Roger Koenker, Victor Chernozhukov, Xuming He, Limin Peng, CRC Press.
Koenker, R. (2004) Quantile Regression.
Bilias, Y. Chen, S. and Z. Ying, Simple resampling methods for censored quantile regression, J. of Econometrics, 99, 373-386.
Kleiner, A., Talwalkar, A., Sarkar, P. and Jordan, M.I. (2014) A Scalable bootstrap for massive data, JRSS(B), 76, 795-816.
Powell, J. (1991) Estimation of Monotonic Regression Models under Quantile Restrictions, in Nonparametric and Semiparametric Methods in Econometrics, W. Barnett, J. Powell, and G Tauchen (eds.), Cambridge U. Press.
rq
bandwidth.rq
data(stackloss)
y <- stack.loss
x <- stack.x
summary(rq(y ~ x, method="fn")) # Compute se's for fit using "nid" method.
summary(rq(y ~ x, ci=FALSE),se="ker")
# default "br" alg, and compute kernel method se's
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