Summary methods for Quantile Regression
Returns a summary list for a quantile regression fit. A null value will be returned if printing is invoked.
## S3 method for class 'rq' summary(object, se = NULL, covariance=FALSE, hs = TRUE, U = NULL, gamma = 0.7, ...) ## S3 method for class 'rqs' summary(object, ...)
object |
This is an object of class |
se |
specifies the method used to compute standard standard errors. There are currently seven available methods:
If |
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 |
gamma |
parameter controlling the effective sample size of the'bag
of little bootstrap samples that will be |
... |
Optional arguments to summary, e.g. bsmethod to use bootstrapping.
see |
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.
a list is returned with the following components, when object
is of class "rqs" then there is a list of such lists.
coefficients |
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. |
cov |
the estimated covariance matrix for the coefficients in the model,
provided that |
Hinv |
inverse of the estimated Hessian matrix returned if |
J |
Unscaled Outer product of gradient matrix returned if |
B |
Matrix of bootstrap realizations. |
U |
Matrix of bootstrap randomization draws. |
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.
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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