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I want to know the mathematical reason why between-individual standard error is under-estimated, and conversely, why within-individual standard error is overestimated if we fail to take correlation into account.

My intuitive understanding is that when correlation is ignored, the model treats each covariate independently, neglecting the fact that the similarities between individuals may be because they belong to the same cluster which under-estimates the between-group variance. Within-group covariate can exhibit similar patterns - which may be attributed to correlation. If this is ignored, the model fails to account for this reduced variability = higher variance.

Apologies if this has been already asked before, but to my knowledge mathematical proof hasn't been exactly asked; the closest one I could find was this: Why does ignoring dependencies between observations *inflate* my standard errors?

Thanks for the help.

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  • $\begingroup$ Perhaps I will make this an answer later but the variance is not necessarily overestimated if you ignore the covariance between $i_{t = 1}$ and $i_{t = 2}$. While this covariance is generally positive, mathematically speaking nothing prevents it from being negative. $\endgroup$
    – num_39
    Commented Apr 23 at 11:31

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