Jacknife cue with many weak moments and nearly-singular design
Mehmet Caner
(North Carolina State University)
11/03/08
This paper analyzes many weak moment asymptotics under the possibility of similar moments.
The possibility of highly related moments arise when there are many of them.
Knight and Fu (2000), designates the issue of similar regressors as the ``nearly singular "
design in the least squares case.
In the nearly singular design, the sample variance converges to a singular limit term.
However Knight and Fu (2000) assume that, on the nullspace of the
limit term, the difference between the sample variance and the singular matrix converges
in probability to a positive definite matrix when multiplied by an appropriate rate.
We show that the nearly singular design affects the limit of the many weak moment asymptotics that is introduced
by Newey and Windmeijer (2007). Even though, the estimator is still consistent, there is a different variance
term in the asymptotically normal limit compared with Newey and Windmeijer (2007) case. This difference stems
from the limit of the second order partial derivative of the objective function. In Newey and Windmeijer (2007), this limit
converges to a ``signal" term. The bias term is zero since they are able to replace the sample variance with its
limit counterpart. We cannot do this because of the nearly singular design, hence our limit involves the
additional ``bias" term. In simulations we find that the interquartile range of the estimates is smaller for all
possibilities that are analyzed in nearly singular design case compared with Newey and Windmeijer (2007).
These findings and the limit can explain the findings of Altonji and Segal (1996). They show that in simulations, the
estimation of the weight matrix can affect the estimators' finite sample properties, and simply replacing the weight by its
sample estimate may not be a very good approximation at all.
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