# summary.rq

##### Summary methods for Quantile Regression

Returns a summary list for a quantile regression fit. A null value will be returned if printing is invoked.

- Keywords
- regression

##### Usage

```
# S3 method for rq
summary(object, se = NULL, covariance=FALSE, hs = TRUE, U = NULL, gamma = 0.7, ...)
# S3 method for rqs
summary(object, ...)
```

##### Arguments

- object
This is an object of class

`"rq"`

or`"rqs"`

produced by a call to`rq()`

, depending on whether one or more taus are specified.- se
specifies the method used to compute standard standard errors. There are currently six 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.`"ker"`

which uses a kernel estimate of the sandwich as proposed by Powell(1990).`"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.

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.- covariance
logical flag to indicate whether the full covariance matrix of the estimated parameters should be returned.

- hs
Use Hall Sheather bandwidth for sparsity estimation If false revert to Bofinger bandwidth.

- U
Resampling indices or gradient evaluations used for bootstrap, see

`boot.rq`

.- gamma
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`

.

##### Details

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.

##### Value

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.

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.

##### References

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.

##### See Also

##### Examples

```
# NOT RUN {
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
# }
```

*Documentation reproduced from package quantreg, version 5.54, License: GPL (>= 2)*