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I understand that the OLS slope estimator of a simple linear regression model is unbiased and consistent when all the usual assumptions are satisfied (eg. random sample, E(u | X) = 0 etc.)

However, instead of using the entire sample in the OLS slope estimator, if I only use half the observations or a quarter of the observations, does this make the OLS estimator biased or inconsistent?

I would think that it will not be affected because the usual assumptions still hold..

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  • $\begingroup$ Regarding unbiasedness, does the concept refer to a sample size? Regarding consistency, what the half of a sample size of infinity? $\endgroup$
    – Christoph Hanck
    Feb 27, 2018 at 19:27
  • $\begingroup$ @ChristophHanck From my understanding, an unbiased estimator is unbiased regardless of sample size, thus the OLS slope estimator will still be unbiased even if half the observations of the sample are used. Sorry, I don't really get the part about consistency. $\endgroup$
    – kyusj
    Feb 28, 2018 at 1:28
  • $\begingroup$ Consistency asks what happens when the sample size goes to infinity. Your question asks what would happen if you only took every other or every fourth of these infinitely many observations. $\endgroup$
    – Christoph Hanck
    Feb 28, 2018 at 5:20
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    $\begingroup$ @Christoph You seem to be confounding bias (a property of any estimator based on even just a single observation) with consistency--or perhaps asymptotic bias--which is a limiting property of a class of estimators. It's a little hard to say what you're trying to assert because there's no such thing in practice (and rarely even in theory) as an infinite sample. $\endgroup$
    – whuber ♦
    Feb 28, 2018 at 23:38
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    $\begingroup$ stats.stackexchange.com/questions/38063 is a related question: it concerns how one can impose a result on a completely random set of data by a relatively simple, innocuous selection of the data. It shows why the answer to your question depends on how you choose the observations you will use. $\endgroup$
    – whuber ♦
    Feb 28, 2018 at 23:52

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Using a random subsample in general will not bias your OLS estimates but will render them more inefficient. Nonrandom subsample selection, however, can cause bias in your estimates.

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