@ARTICLE{17190791_1982,
author = {Kelejian, Harry H.},
keywords = {standard test, математическая модель, математический метод, эконометрическая модель},
title = {An extension of a standard test for heteroskedasticity to a systems
framework },
journal = {Journal of Econometrics},
year = {1982},
month = {},
volume = {20},
number = {2},
pages = {325-333},
url = {http://ecsocman.hse.ru/text/17190791/},
publisher = {},
language = {ru},
abstract = {The Glejser (1969) test for heteroskedasticity concerning the
disturbance terms of a regression model is widely referenced - see,
e.g., Goldfeld and Quandt (1972), Johnston (1972), Theil (1971) and
Maddala (1978) among others. As originally proposed, Glejser
suggested estimating the model's disturbance terms via least squares,
and then regressing their absolute values on certain known functions
of the regressors; the suggested test for heteroskedasticity then
relates to the significance of the ‘slope' coefficients.
Obvious modifications of this procedure, such as using squared
estimated disturbance terms, have been suggested - see Goldfeld and
Quandt (1972), and Kelejian and Oates (1974); still others have
suggested modifications of the procedure which involve iterations -
see Maddala (1978). Glejser noted that there may be shortcomings in
the procedure which arise due to the use of the estimated
disturbances in the second-stage regression. Goldfeld and Quandt
(1972), among others, also noted shortcomings in the procedure but
recognized that conditions may exist under which these shortcomings
are of no consequence in the relevant asymptotic distribution; they
went on to note, however, that at the time, a demonstration of such
conditions was not available. In recent papers Amemiya (1977) and
White (1980) gave results which justify the large sample version of
the Glejser test based on squared estimated disturbances. However,
their results assumed the absence of lagged dependent variables, and
were given in the single-equation context. The purpose of this paper
is to extend the Amemiya (1977) and White (1980) results to the case
of a simultaneous equation framework, which may or may not contain
lagged endogenous variables. We consider two cases. The first is the
one in which the researcher suspects that heteroskedasticity may
exist in only one of the system's equations. This extension is not
trivial due to, among other things, feedbacks involving the
endogenous regressors. Nevertheless, it turns out that if the
least-squares procedure in the first stage is replaced by virtually
any consistent procedure, such as two-stage least- squares -
henceforth 2SLS, no additional complexities arise. The importance of
this result is that under typical modelling specifications, a
computationally simple large sample test for heteroskedasticity,
which is associated with one equation of a system, can be carried out
in the context of that system. This test should be especially useful
in those cases in which the exact specification of the
‘suspected' heteroskedasticity is not known1 The second case we
consider is the one in which the researcher suspects that
heteroskedasticity may be associated with more than one equation of
the system. As expected, the resulting test is computationally more
‘demanding'. The model is specified in section 2, and the basic
results are given in section 3. Suggestions for further work are
given in section 4; technical details are relegated to the appendix. },
annote = {The Glejser (1969) test for heteroskedasticity concerning the
disturbance terms of a regression model is widely referenced - see,
e.g., Goldfeld and Quandt (1972), Johnston (1972), Theil (1971) and
Maddala (1978) among others. As originally proposed, Glejser
suggested estimating the model's disturbance terms via least squares,
and then regressing their absolute values on certain known functions
of the regressors; the suggested test for heteroskedasticity then
relates to the significance of the ‘slope' coefficients.
Obvious modifications of this procedure, such as using squared
estimated disturbance terms, have been suggested - see Goldfeld and
Quandt (1972), and Kelejian and Oates (1974); still others have
suggested modifications of the procedure which involve iterations -
see Maddala (1978). Glejser noted that there may be shortcomings in
the procedure which arise due to the use of the estimated
disturbances in the second-stage regression. Goldfeld and Quandt
(1972), among others, also noted shortcomings in the procedure but
recognized that conditions may exist under which these shortcomings
are of no consequence in the relevant asymptotic distribution; they
went on to note, however, that at the time, a demonstration of such
conditions was not available. In recent papers Amemiya (1977) and
White (1980) gave results which justify the large sample version of
the Glejser test based on squared estimated disturbances. However,
their results assumed the absence of lagged dependent variables, and
were given in the single-equation context. The purpose of this paper
is to extend the Amemiya (1977) and White (1980) results to the case
of a simultaneous equation framework, which may or may not contain
lagged endogenous variables. We consider two cases. The first is the
one in which the researcher suspects that heteroskedasticity may
exist in only one of the system's equations. This extension is not
trivial due to, among other things, feedbacks involving the
endogenous regressors. Nevertheless, it turns out that if the
least-squares procedure in the first stage is replaced by virtually
any consistent procedure, such as two-stage least- squares -
henceforth 2SLS, no additional complexities arise. The importance of
this result is that under typical modelling specifications, a
computationally simple large sample test for heteroskedasticity,
which is associated with one equation of a system, can be carried out
in the context of that system. This test should be especially useful
in those cases in which the exact specification of the
‘suspected' heteroskedasticity is not known1 The second case we
consider is the one in which the researcher suspects that
heteroskedasticity may be associated with more than one equation of
the system. As expected, the resulting test is computationally more
‘demanding'. The model is specified in section 2, and the basic
results are given in section 3. Suggestions for further work are
given in section 4; technical details are relegated to the appendix. }
}