Does a BLUE estimator need to be consistent?

[ancl9]

The characteristics of a BLUE estimator are:
- Efficient
- Linear
- Unbiased

Does an estimator need to be consistent in order to be BLUE?

Another question: study notes page 14 Stock and Watson Chapter 4:
- on the top section of the page it states that OLS estimators are only efficient under certain additional special conditions. This seems contradictory to Gauss-Markov Theorem who states that least-squares estimators are BLUE.

Any insights would be appreciated. Thank you,

 

[ami44]

Here it makes sense to differentiate between an estimator and a sequence of estimators. Often this distinction is very murky.
For example Theta_n (X) = 1/n * (X1 + X2 + ... + Xn)
is an estimator for the expected value for a fixed n. For a n = 1, 2, 3, ... We get an infinite sequence of estimators.
BLUE is a property of a single estimator. For a given n there is no better linear, unbiased estimator as function of X1, X2, ... Xn.
Consistency on the other hand is a property of a sequence of estimators. Theta_n(X) converges to the true expected value with increasing n. The estimators in the sequence do not necessarily have to be BLUE for the sequence to be consistent. There is a connection to the bias property though
see https://en.m.wikipedia.org/wiki/Consistent_estimator