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Let's say I have some regression model, say $\mathbb E[y]=\beta_0+\beta_1x_1+\beta_2x_2$. However, I am fitting this regression because I want to estimate $\beta_1$ while "controlling for" the effect of $x_2$, as is common.

A typical approach might be to fit the regression with a method like least squares or maximum likelihood estimation in order to estimate the entire parameter vector $\beta = (\beta_0,\beta_1,\beta_2)$. However, if I do not care about making an accurate estimate of $\beta_2$ or even $\beta_0$, is there a way to trade off some accuracy in the estimation of those coefficients in order to improve accuracy in estimating the $\beta_1$ of interest?

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    $\begingroup$ Not if you use a linear estimation technique - Gauss-Markov shows this. So, if such a method exists, it has to be nonlinear. $\endgroup$
    – jbowman
    Feb 28, 2023 at 19:14
  • $\begingroup$ @jbowman What if that linear estimator is biased for the coefficients of non-interest? Then Gauss-Markov does not apply, as we are not considering an unbiased estimator. $\endgroup$
    – Dave
    Feb 28, 2023 at 19:25
  • $\begingroup$ Good catch, my apologies. $\endgroup$
    – jbowman
    Feb 28, 2023 at 19:29
  • $\begingroup$ Perhaps you may simply find more (causal) paths linking your interested variable $x_1$ and $y$ via some other mediated variable(s) thus you can get more reliable and quality correlation coefficient data to fix your $\beta_1$ more accurately hopefully. $\endgroup$
    – cinch
    Feb 28, 2023 at 22:18

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