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Why don't we use restricted maximum likelihood to estimate parameters in non-mixed models?

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    $\begingroup$ You use it: REML is for estimating variance parameters, and in non-mixed models there is only one such, $\sigma^2$. When dividing by $n-1$ to get a variance estimate, you are using REML. See stats.stackexchange.com/questions/48671/… $\endgroup$ – kjetil b halvorsen May 4 '15 at 17:02
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For REML under fixed effects the variance estimator equivalent to $S^2=\frac{\sum_{i=1}^n (y_i - x_i \hat{\beta})^2}{n-p}$ where $\hat{\beta}$ is the OLS estimator, but as a bonus with OLS you also get to use Gauss-Markov Theorem for $\hat{\beta}$.

With REML you don't get any coefficient estimates.

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