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I have two variables $X$ and $Y$.

Consider that there is one sample with 1000 observations, we can get the standard error of coefficient by this equation: $se(\hat\beta) = \sqrt{\sigma^2(X^TX)^{-1}}$.

Then consider that we have 100 samples, each sample has 1000 observations. By running 100 regressions, we can get 100 $\hat\beta$. Then, we can compute the standard error of the mean: $SEM=\sum(\hat\beta-\bar{\beta})^2/100$.

So what is the relationship between the two standard errors?

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  • $\begingroup$ What is the relationship among the various design matrices $X$ for each of the 100 samples? $\endgroup$
    – whuber ♦
    Jun 21, 2021 at 15:39
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    $\begingroup$ Hi, $X$ is generated from normal distribution. And I replicated it 100 times. So each of the sample follows the same distribution. $\endgroup$
    – Xuanqi Liu
    Jun 23, 2021 at 10:42

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