bickel

This function implements the method of
 Bickel78;textualskedastic for testing for
 heteroskedasticity in a linear regression model, with or without the
 scale-invariance modification of
 Carroll81;textualskedastic.

Implements numerous methods for testing for, modelling, and
correcting for heteroskedasticity in the classical linear regression
model. The most novel contribution of the
package is found in the functions that implement the as-yet-unpublished
auxiliary linear variance models and auxiliary nonlinear variance
models that are designed to estimate error variances in a heteroskedastic
linear regression model. These models follow principles of statistical
learning described in Hastie (2009) <doi:10.1007/978-0-387-21606-5>.
The nonlinear version of the model is estimated using quasi-likelihood
methods as described in Seber and Wild (2003, ISBN: 0-471-47135-6).
Bootstrap methods for approximate confidence intervals for error variances
are implemented as described in Efron and Tibshirani
(1993, ISBN: 978-1-4899-4541-9), including also the expansion technique
described in Hesterberg (2014) <doi:10.1080/00031305.2015.1089789>. The
wild bootstrap employed here follows the description in Davidson and
Flachaire (2008) <doi:10.1016/j.jeconom.2008.08.003>. Tuning of
hyper-parameters makes use of a golden section search function that is
modelled after the MATLAB function of Zarnowiec (2022)
<https://www.mathworks.com/matlabcentral/fileexchange/25919-golden-section-method-algorithm>.
A methodological description of the algorithm can be found in Fox (2021,
ISBN: 978-1-003-00957-3).
There are 25 different functions that implement hypothesis tests for
heteroskedasticity. These include a test based on Anscombe (1961)
<https://projecteuclid.org/euclid.bsmsp/1200512155>, Ramsey's (1969)
BAMSET Test <doi:10.1111/j.2517-6161.1969.tb00796.x>, the tests of Bickel
(1978) <doi:10.1214/aos/1176344124>, Breusch and Pagan (1979)
<doi:10.2307/1911963> with and without the modification
proposed by Koenker (1981) <doi:10.1016/0304-4076(81)90062-2>, Carapeto and
Holt (2003) <doi:10.1080/0266476022000018475>, Cook and Weisberg (1983)
<doi:10.1093/biomet/70.1.1> (including their graphical methods), Diblasi
and Bowman (1997) <doi:10.1016/S0167-7152(96)00115-0>, Dufour, Khalaf,
Bernard, and Genest (2004) <doi:10.1016/j.jeconom.2003.10.024>, Evans and
King (1985) <doi:10.1016/0304-4076(85)90085-5> and Evans and King (1988)
<doi:10.1016/0304-4076(88)90006-1>, Glejser (1969)
<doi:10.1080/01621459.1969.10500976> as formulated by
Mittelhammer, Judge and Miller (2000, ISBN: 0-521-62394-4), Godfrey and
Orme (1999) <doi:10.1080/07474939908800438>, Goldfeld and Quandt
(1965) <doi:10.1080/01621459.1965.10480811>, Harrison and McCabe (1979)
<doi:10.1080/01621459.1979.10482544>, Harvey (1976) <doi:10.2307/1913974>,
Honda (1989) <doi:10.1111/j.2517-6161.1989.tb01749.x>, Horn (1981)
<doi:10.1080/03610928108828074>, Li and Yao (2019)
<doi:10.1016/j.ecosta.2018.01.001> with and without the modification of
Bai, Pan, and Yin (2016) <doi:10.1007/s11749-017-0575-x>, Rackauskas and
Zuokas (2007) <doi:10.1007/s10986-007-0018-6>, Simonoff and Tsai (1994)
<doi:10.2307/2986026> with and without the modification of Ferrari,
Cysneiros, and Cribari-Neto (2004) <doi:10.1016/S0378-3758(03)00210-6>,
Szroeter (1978) <doi:10.2307/1913831>, Verbyla (1993)
<doi:10.1111/j.2517-6161.1993.tb01918.x>, White (1980)
<doi:10.2307/1912934>, Wilcox and Keselman (2006)
<doi:10.1080/10629360500107923>, Yuce (2008)
<https://dergipark.org.tr/en/pub/iuekois/issue/8989/112070>, and Zhou,
Song, and Thompson (2015) <doi:10.1002/cjs.11252>. Besides these
heteroskedasticity tests, there are supporting functions that compute the
BLUS residuals of Theil (1965) <doi:10.1080/01621459.1965.10480851>, the
conditional two-sided p-values of Kulinskaya (2008) <arXiv:0810.2124v1>,
and probabilities for the nonparametric trend statistic of Lehmann (1975,
ISBN: 0-816-24996-1). For handling heteroskedasticity, in addition to the
new auxiliary variance model methods, there is a function
to implement various existing Heteroskedasticity-Consistent Covariance
Matrix Estimators from the literature, such as those of White (1980)
<doi:10.2307/1912934>, MacKinnon and White (1985)
<doi:10.1016/0304-4076(85)90158-7>, Cribari-Neto (2004)
<doi:10.1016/S0167-9473(02)00366-3>, Cribari-Neto et al. (2007)
<doi:10.1080/03610920601126589>, Cribari-Neto and da Silva (2011)
<doi:10.1007/s10182-010-0141-2>, Aftab and Chang (2016)
<doi:10.18187/pjsor.v12i2.983>, and Li et al. (2017)
<doi:10.1080/00949655.2016.1198906>.

Thomas Farrar

skedastic

Handling Heteroskedasticity in the Linear Regression Model

University of the Western Cape 

bickel function

<dl><dt>mainlm</dt>
<dd>Either an object of <code><a href="/link/class?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::class">class</a></code> <code>"lm"</code>
(e.g., generated by <code><a href="/link/lm?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::lm">lm</a></code>), or
a list of two objects: a response vector and a design matrix. The objects
are assumed to be in that order, unless they are given the names
<code>"X"</code> and <code>"y"</code> to distinguish them. The design matrix passed
in a list must begin with a column of ones if an intercept is to be
included in the linear model. The design matrix passed in a list should
not contain factors, as all columns are treated 'as is'. For tests that
use ordinary least squares residuals, one can also pass a vector of
residuals in the list, which should either be the third object or be
named <code>"e"</code>.</dd>
<dt>fitmethod</dt>
<dd>A character indicating the method to be used to fit the
regression model. This can be "OLS" for ordinary least squares (the
default) or "robust" in which case a robust fitting method is called
from <code><a href="/link/rlm?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::rlm">rlm</a></code>.</dd>
<dt>a</dt>
<dd>A character argument specifying the name of a function to be
applied to the fitted values, or an integer \(m\) in which case the
function applied is \(f(x) = x^m\). Defaults to <code>"identity"</code> for
<code><a href="/link/identity?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::identity">identity</a></code>.</dd>
<dt>b</dt>
<dd>A character argument specifying a function to be applied to the
residuals. Defaults to Huber's function squared, as recommended by
Carroll81;textualskedastic. Currently the only supported
functions are <code>"hubersq"</code> (for Huber's function squared) and
<code>"tanhsq"</code> (for \(b(x)=\mathrm{tanh}(x)^2\).)</dd>
<dt>scale_invariant</dt>
<dd>A logical indicating whether the scale-invariance
modification proposed by Carroll81;textualskedastic
should be implemented. Defaults to <code>TRUE</code>.</dd>
<dt>k</dt>
<dd>A double argument specifying a parameter for Huber's function
squared; used only if <code>b == "hubersq"</code>. This is not to be confused
with the argument <code>k2</code> that could be passed to
<code><a href="/link/rlm?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::rlm">rlm</a></code> if the regression is fitted using robust methods.
<code>k</code> defaults to 1.345.</dd>
<dt>statonly</dt>
<dd>A logical. If <code>TRUE</code>, only the test statistic value
is returned, instead of an object of <code><a href="/link/class?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::class">class</a></code>
<code>"htest"</code>. Defaults to <code>FALSE</code>.</dd>
<dt>...</dt>
<dd>Optional arguments to be passed to <code><a href="/link/rlm?package=skedastic&version=2.0.2" data-mini-rdoc="skedastic::rlm">rlm</a></code></dd></dl>

Arguments

Bickel's Test for Heteroskedasticity in a Linear Regression Model — bickel

<dl>

<dt>mainlm</dt>
<dd>Either an object of <code><a href='https://rdrr.io/r/base/class.html'>class</a></code> <code>"lm"</code>
(e.g., generated by <code><a href='https://rdrr.io/r/stats/lm.html'>lm</a></code>), or
a list of two objects: a response vector and a design matrix. The objects
are assumed to be in that order, unless they are given the names
<code>"X"</code> and <code>"y"</code> to distinguish them. The design matrix passed
in a list must begin with a column of ones if an intercept is to be
included in the linear model. The design matrix passed in a list should
not contain factors, as all columns are treated 'as is'. For tests that
use ordinary least squares residuals, one can also pass a vector of
residuals in the list, which should either be the third object or be
named <code>"e"</code>.</dd>


<dt>fitmethod</dt>
<dd>A character indicating the method to be used to fit the
regression model. This can be "OLS" for ordinary least squares (the
default) or "robust" in which case a robust fitting method is called
from <code><a href='https://rdrr.io/pkg/MASS/man/rlm.html'>rlm</a></code>.</dd>


<dt>a</dt>
<dd>A character argument specifying the name of a function to be
applied to the fitted values, or an integer \(m\) in which case the
function applied is \(f(x) = x^m\). Defaults to <code>"identity"</code> for
<code><a href='https://rdrr.io/r/base/identity.html'>identity</a></code>.</dd>


<dt>b</dt>
<dd>A character argument specifying a function to be applied to the
residuals. Defaults to Huber's function squared, as recommended by
Carroll81;textualskedastic. Currently the only supported
functions are <code>"hubersq"</code> (for Huber's function squared) and
<code>"tanhsq"</code> (for \(b(x)=\mathrm{tanh}(x)^2\).)</dd>


<dt>scale_invariant</dt>
<dd>A logical indicating whether the scale-invariance
modification proposed by Carroll81;textualskedastic
should be implemented. Defaults to <code>TRUE</code>.</dd>


<dt>k</dt>
<dd>A double argument specifying a parameter for Huber's function
squared; used only if <code>b == "hubersq"</code>. This is not to be confused
with the argument <code>k2</code> that could be passed to
<code><a href='https://rdrr.io/pkg/MASS/man/rlm.html'>rlm</a></code> if the regression is fitted using robust methods.
<code>k</code> defaults to 1.345.</dd>


<dt>statonly</dt>
<dd>A logical. If <code>TRUE</code>, only the test statistic value
is returned, instead of an object of <code><a href='https://rdrr.io/r/base/class.html'>class</a></code>
<code>"htest"</code>. Defaults to <code>FALSE</code>.</dd>


<dt>...</dt>
<dd>Optional arguments to be passed to <code><a href='https://rdrr.io/pkg/MASS/man/rlm.html'>rlm</a></code></dd>

</dl>

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bickel: Bickel's Test for Heteroskedasticity in a Linear Regression Model

Description

Usage

Value

Arguments

Details

References

See Also

Examples