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What happens in linear regression $Y\sim X$ when the $Y$'s are not independently sampled and, particularly, may be autocorrelated?

I believe the estimator will still work. But what will happen to the t-stat and the R-squared that measures the goodness of fit?

Any other implications I might not be aware of?

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  • $\begingroup$ If the observations are not independent, then the least-squares estimates are still unbiased, however, the standard errors of your estimates can be seriously biased and very misleading, and lead to seriously biased incorrect conclusions from statistical tests. $\endgroup$
    – StatsStudent
    Mar 19, 2015 at 23:20
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    $\begingroup$ @StatsStudent you should post that as an answer. Answers don't need to be technically involved to be helpful and correct $\endgroup$
    – shadowtalker
    Mar 19, 2015 at 23:31

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As @ssdecontrol indicated, here's an answer without much explanation:

If the observations are not independent, then the least-squares estimates are still unbiased, however, the standard errors of your estimates can be seriously biased and very misleading, and lead to seriously biased incorrect conclusions from statistical tests.

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