...-
help for ^htest^, ^szroeter^, and ^white^ (Jeroen Weesie/ICS
Dec 10, 1996) ..-
Tests for heteroscedasticity after fit
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^htest^ [varlist] [^, r^hs ^u^nivar ]
^szroeter^ varname
^white^ [^, p^reserve ]
^htest^, ^szroeter^, and ^white^ are for use after ^fit^; see help
^fit^.
See the notes below for a discussion of the interpretation of test
results in case of weighted regression.
Description
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^htest^, ^szroeter^, and ^white^ provide tests for the assumption of
the linear regression model that the residuals e are homoscedastic,
i.e., have constant variance. The tests differ with respect to the
specification of residual variances under the alternative hypothesis.
^white^ considers the general (unrestricted) alternative hypothesis
in which no assumptions is made on the residual variances (White
1980).
^szroeter^ considers the alternative hypothesis that the residual
variances are monotonically increasing in some variable (Szroeter
1978).
^htest^ performs standard score tests for H: b=0 for two parametric
forms of heteroscedasticity:
^multiplicative heteroscedastcity^ (Cook & Weinberg 1983)
var(y) = s^^2 * exp( b1.z1 + b2.z2 + .. + bk.zk)
^random coefficient heteroscedasticity^ (REF)
var(y) = s^^2 + b1.z1^^2 + b2.z2^^2 + .. + bk.zk^^2
If no varlist is specified, ^htest^ uses the fitted values of the
regression.
^szoeter^ and ^htest^ test homoscedasticity against specific models
(monotonicity; multiplicative and random coefficient heteroscedasticty)
under the alternative hypothesis. Due to the general structure of the
alternative, ^white^ is not very powerfull against any particular form
of heteroscedasticity.
Options (^htest^)
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^rhs^ specifies that the independent variables of the ^fit^ted
regression should be used.
^univar^ specifies that univariate score tests for each of the
variables is displayed in addition to the "combined" test.
If ^htest^ suggest multiplicative forms of heteroscedasticity,
regression models can be estimated using ^regmh^ or ^regmhv^ . The
random coefficient model is currently not supported.
Options (^white^)
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^preserve^ specifies that the ^white^ may ^preserve^ the data in memory
and drop all variables and cases that are not needed in the
calculations. This is costly for large data sets. However, ^white^
has to generate k(k+1)/2 temporary variables, where k = #rhs-vars
in the ^fit^ted model.
Examples
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. ^fit income edu sex exp exp2 industry^ (fit a
regression model)
. ^white^
. ^szroeter edu^ (variance
increases in edu?)
. ^htest^ (homoscedasticity in
fitted values) . ^htest edu sex exp exp2 industry^
(homoscedasticity in rhs-vars) . ^htest, rhs^
(shorthand for the previous command) . ^htest edu exp age, univar^
(display univariate tests as well)
Details (^white^)
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White (1980) noted that the classical estimate
Vc = s^2 inv(X'X)
of the variance V of OLS regression estimates and the
Huber/White/robust estimate
Vr = X' inv(sum e_i^2 x_i x_i') X
of V are likely to be close under homoscedasticity. White (1980)
proposed a Hausmann-type test of the matrix difference Vc-Vr as a test
for homoscedasticy against general forms of heteroscedasticity.
(Actually, White showed that Vc and Vr may also be close under
particular forms of hetero-scedasticty as well. See White (1980) for
details.)
Details (^szroeter^)
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Szoeter's class of tests for homoscedasticity against the alternative
that the residual variance increases in some variable x are defined in
terms of
sum_i h(x(i)) e(i)^^2
H = -------------------- i=1..n
sum_i e(i)^^2
where h(x) is some weight function that increases in x. Note that H can
be interpreted as a weighted avarege of the h(x), weighted by the
squares residuals. Under homoscedasticty, H should be approximately
equal to the unweighted average of h(x). Large values of H suggest that
e(i)^^2 tends to be large where h(x) is large, i.e., the variance
indeed increases in x, while small values of H suggest that the
variance actually decreases in x.
King (1982) suggest to use h(x(i)) = rank(x(i) in x(1)..x(n)).
The Q-statistic displayed by ^szroeter^ is a transformation of H that
is asymptotically standard normal distributed under homoscedasticity.
^szroeter^ displays a Q-statistic (Judge 1985:452). Under
homoscedasticity, Q is approximately N(0,1) distributed. Q>2 suggests
that the residual variance increases in varname, while Q<-2 suggests
that it decreases in varname.
Homoscedastity and weighted regression
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References
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Cook, R.D., and S. Weisberg (1983) Diagnostics for
heteroscedasticity in regression. Biometrica 70: 1-10.
Judge, G.G. et al. (1985) The Theory and Practice of Econometrics.
2nd Ed. New York: Wiley.
Szroeter, J. (1978) A Class of Parametric Tests for
Heteroscedasticity in Linear Econometric Models. Econometrica
46, 1311-28.
White, H. (1980) A Heteroskedasticity-Consistent Covariance Matrix
Estimator and a Direct Test for Heteroskedasticity,
Econometrica, 48: 817-838.
Saved results
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^htest^ saves results for the final (multivariate) score tests in
scalar ^S_3^ #df for the asymptotic chi-square distribution under
Ho. scalar ^S_4^ score statistic
scalar ^S_5^ #df for the asymptotic chi-square distribution under
Ho. scalar ^S_6^ score statistic
^white^ saves results in
scalar ^S_1^ test statistic
scalar ^S_2^ #df for the asymptotic chi-square distribution under Ho.
^szroeter ^ saves results in
scalar ^S_1^ test statistic
Also See
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Manual: [R] fit
On-line: help for @fit@, @regmh@, @probith@, @_tscore@