pDtrend

This function computes \(\Pr(D \le k)\) (\(\Pr(D \ge k)\)), i.e.
lower (upper) cumulative probabilities for
\(D=\sum_{i=1}^{n} (R_i - i)^2\), the nonparametric trend statistic
proposed by Lehmann75;textualskedastic, under the assumption
that the ranks \(R_i\) are computed on a series of \(n\) independent and
identically distributed random variables with no ties. The function may be
used to compute one-sided \(p\)-values for the nonparametric test for
heteroskedasticity of Horn81;textualskedastic. Computation
time is extremely slow for \(n &gt; 10\) if <code>usedata</code> is set to
<code>FALSE</code>; thus a normal approximation is implemented, including a
continuity correction.

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package is found in the functions that implement the as-yet-unpublished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 

pDtrend function

<dl><dt>k</dt>
<dd>An integer of <code>length</code> \(\ge 1\)
or a character <code>"all"</code> indicating that the cumulative
distribution function should be applied to the entire support of \(D\).
The latter is only acceptable when <code>exact</code> is <code>TRUE</code>, since
the distribution is otherwise continuous.</dd>
<dt>n</dt>
<dd>A positive integer representing the number of observations in the
series.</dd>
<dt>lower.tail</dt>
<dd>A logical. Should lower tailed cumulative probability be
calculated? Defaults to <code>TRUE</code>. Note that both lower- and upper-
tailed cumulative probabilities are computed inclusive of <code>k</code>.</dd>
<dt>exact</dt>
<dd>A logical. Should exact distribution of \(D\) be used by
calling <code>dDtrend</code>? If <code>FALSE</code>, a normal approximation
is used. If <code>tiefreq</code> is not <code>NA</code> (ties are present),
normal approximation is used regardless of the value of <code>exact</code>.
By default, <code>exact</code> is set to <code>TRUE</code> provided that
<code>n &lt;= 10</code>. Setting <code>exact</code> to <code>TRUE</code> for <code>n &gt; 11</code>
results in an error unless <code>override</code> is set to <code>TRUE</code>.</dd>
<dt>tiefreq</dt>
<dd>A double vector corresponding to the value of \(d_i\)
in Lehmann75;textualskedastic. These are the frequencies
of the various tied ranks. If ties are absent, <code>NA</code> (the default).</dd>
<dt>override</dt>
<dd>A logical. By default, the <code>dDtrend</code> function aborts if
\(n &gt; 11\) due to the prohibitively slow computation (which may cause
some systems to crash). Setting this argument to <code>TRUE</code> overrides
the abort. Ignored unless <code>exact</code> is <code>TRUE</code>.</dd></dl>

Arguments

Cumulative distribution function of nonparametric trend statistic \(D\) — pDtrend

<dl>

<dt>k</dt>
<dd>An integer of <code>length</code> \(\ge 1\)
or a character <code>"all"</code> indicating that the cumulative
distribution function should be applied to the entire support of \(D\).
The latter is only acceptable when <code>exact</code> is <code>TRUE</code>, since
the distribution is otherwise continuous.</dd>


<dt>n</dt>
<dd>A positive integer representing the number of observations in the
series.</dd>


<dt>lower.tail</dt>
<dd>A logical. Should lower tailed cumulative probability be
calculated? Defaults to <code>TRUE</code>. Note that both lower- and upper-
tailed cumulative probabilities are computed inclusive of <code>k</code>.</dd>


<dt>exact</dt>
<dd>A logical. Should exact distribution of \(D\) be used by
calling <code>dDtrend</code>? If <code>FALSE</code>, a normal approximation
is used. If <code>tiefreq</code> is not <code>NA</code> (ties are present),
normal approximation is used regardless of the value of <code>exact</code>.
By default, <code>exact</code> is set to <code>TRUE</code> provided that
<code>n &lt;= 10</code>. Setting <code>exact</code> to <code>TRUE</code> for <code>n &gt; 11</code>
results in an error unless <code>override</code> is set to <code>TRUE</code>.</dd>


<dt>tiefreq</dt>
<dd>A double vector corresponding to the value of \(d_i\)
in Lehmann75;textualskedastic. These are the frequencies
of the various tied ranks. If ties are absent, <code>NA</code> (the default).</dd>


<dt>override</dt>
<dd>A logical. By default, the <code>dDtrend</code> function aborts if
\(n &gt; 11\) due to the prohibitively slow computation (which may cause
some systems to crash). Setting this argument to <code>TRUE</code> overrides
the abort. Ignored unless <code>exact</code> is <code>TRUE</code>.</dd>

</dl>

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pDtrend: Cumulative distribution function of nonparametric trend statistic $D$

Description

Usage

Value

Arguments

References

See Also

Examples