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Community Bot
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This is because the linear regression estimator $\bar \beta$ makes use of the information that $X$ and $Y$ are centered whereas the estimator $\bar \rho$ for the correlation coefficient has to subtract the empirical mean of $X$ and $Y$. In this case, the variance of $\bar \beta$ is lower than the variance of $\bar \rho$ (in a similar fashion as to estimate the variance when the mean is knowin a similar fashion as to estimate the variance when the mean is know).

This is because the linear regression estimator $\bar \beta$ makes use of the information that $X$ and $Y$ are centered whereas the estimator $\bar \rho$ for the correlation coefficient has to subtract the empirical mean of $X$ and $Y$. In this case, the variance of $\bar \beta$ is lower than the variance of $\bar \rho$ (in a similar fashion as to estimate the variance when the mean is know).

This is because the linear regression estimator $\bar \beta$ makes use of the information that $X$ and $Y$ are centered whereas the estimator $\bar \rho$ for the correlation coefficient has to subtract the empirical mean of $X$ and $Y$. In this case, the variance of $\bar \beta$ is lower than the variance of $\bar \rho$ (in a similar fashion as to estimate the variance when the mean is know).

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ThePawn
ThePawn
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This is because the linear regression estimator $\bar \beta$ makes use of the information that $X$ and $Y$ are centered whereas the estimator $\bar \rho$ for the correlation coefficient has to subtract the empirical mean of $X$ and $Y$. In this case, the variance of $\bar \beta$ is lower than the variance of $\bar \rho$ (in a similar fashion as to estimate the variance when the mean is know).