Víšek, J. Á. : Instrumental weighted variables under heteroscedasticity. Part I. Consistency.
|Author(s):|| prof. RNDr. Jan Ámos Víšek CSc., |
|Type:||Articles in journals with impact factor|
|ISSN / ISBN:||0023-5954|
|Published in:||Kybernetika 53, 1 - 25.|
|Keywords:||Weighting of order statistics of the squared residuals, consistency of the instrumental weighted variables, heteroscedasticity of disturbances, numerical study.|
|Grants:||GAČR 13-01930S Robust methods for nonstandard situations, their diagnostics and implementations|
|Abstract:||The proof of consistency of instrumental weighted variables, the robust version of the classical
instrumental variables is given. It is proved that all solutions of the corresponding normal
equations are contained, with high probability, in a ball, the radius of which can be selected -
asymptotically - arbitrarily small. Then also
n-consistency is proved. An extended numerical
study (the Part II of the paper) oers a picture of behavior of the estimator for nite samples
under various types and levels of contamination as well as various extent of heteroscedasticity.
The estimator in question is compared with two other estimators of the type of \robust instrumental
variables" and the results indicate that our estimator gives comparatively good results
and for some situations it is better.
The discussion on a way of selecting the weights is also oered. The conclusions show the
resemblance of our estimator with the M-estimator with Hampel's -function. The dierence
is that our estimator does not need the studentization of residuals (which is not a simple task)
to be scale- and regression-equivariant while the M-estimator does. So the paper demonstrates
that we can directly compute - moreover by a quick algorithm (reliable and reasonably quick
even for tens of thousands of observations) - the scale- and the regression-equivariant estimate
of regression coecients.