Glossary of economics research
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consistent:
An estimator for a parameter is consistent iff the estimator converges in
probability to the true value of the parameter; that is, the plim of the
estimator, as the sample size goes to infinity, is the parameter itself.
Another phrasing: an estimator is consistent if it has asymptotic
power of one.
"Consistency", without a modifier, is synonymous with weak
consistency.
From Davidson and Mackinnon, p. 79: If for any possible value of the
parameter q in a region of a parameter space the
power of a test goes to one as sample size n goes to infinity, that
test is said to be consistent against alternatives in that region of the
parameter space. That is, if as the sample size increases we can in the limit
reject every false hypothesis about the parameter, the test is consistent.
How does one prove that an estimator is consistent? Here are two ways.
(1) Prove directly that if the model is correct, the estimator has
power one in the limit to reject any alternative but the true
parameter.
(2) Sufficient conditions for proving that an estimator is consistent are (i)
that the estimator is asymptotically unbiased and (ii) that its variance
collapses to zero as the sample size goes to infinity. This method of proof
is usually easier than (1) and is commonly used.
The existence of a consistent estimator for a parameter is proof that
the parameter is identified. But a parameter could be identified
without there being a consistent estimator. For more on this see comment on consistency and identification.
Contexts: econometrics; statistics; estimation
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