# Is Least Squares estimator for linear model the unique minimum variance unbiased estimator for a linear model?

I am following Linear Modfels in Statistics, Rencher & Schaalje, 2nd Edicition for the proof of Gauss-Markov Theorem (Theorem 7.3d, Page 146). I understand how least squares estimator $$\mathbf{\hat{\beta}_{LS}} = (\mathbf{X}^{\mathsf{T}}\mathbf{X})^{-1}\mathbf{X}^{\mathsf{T}}\mathbf{y}$$ achieves minimum variance among all linear unbiased estimators for $$\beta_1, \ldots, \beta_p$$. I am not able to figure out whether $$\mathbf{\hat{\beta}_{LS}}$$ is the only linear unbiased estimator which achieves this minimum variance for each $$\beta_1, \ldots, \beta_p$$.

• Have you looked at the Wikipedia page, in particular remarks on proof section which seems to show uniqueness Jan 24, 2021 at 11:17

Well, there are other estimators, but they must by equal to $$\hat{\boldsymbol\beta}_{LS}$$ :)
If you look for $$\arg\min_\beta(\mathbf{y}-\mathbf{X}\boldsymbol{\beta})^T(\mathbf{y}-\mathbf{X}\boldsymbol{\beta})$$ then you find $$\hat{\boldsymbol{\beta}}_{LS}$$.
If you assume normality and look for a maximum likelihood estimator you get $$\hat{\boldsymbol{\beta}}_{ML}=\hat{\boldsymbol{\beta}}_{LS}$$. BTW, the Gauss-Markov theorem does not assume normality and states that $$\hat{\boldsymbol{\beta}}_{LS}$$ is the best linear unbiased estimator, but if you assume normality then $$\hat{\boldsymbol{\beta}}_{ML}$$ is the best - linear or nonlinear - unbiased estimator (e.g. Seber & Lee, Linear Regression Analysis, John Wiley & Sons, 2003, §3.5).
If you assume that $$\mathbf{X}$$ is stochastic, i.e. that the population model is $$y=\mathbf{x}\boldsymbol\beta+\varepsilon$$ where $$\mathbf{x}$$ is a random row vector, you can premultiply by $$\mathbf{x}'$$, take expectations assuming $$E[\mathbf{x}'u]=\mathbf{0}$$ and get $$\boldsymbol\beta=E[\mathbf{x}'\mathbf{x}]^{-1}E[\mathbf{x}'y]$$. Replacing the population moments $$E[\mathbf{x}'\mathbf{x}]^{-1}$$ and $$E[\mathbf{x}'y]$$ with the corresponding sample averages you get a method of moments estimator $$\hat{\boldsymbol{\beta}}_{MM}=\hat{\boldsymbol{\beta}}_{LS}$$ (e.g. Wooldridge, Econometric Analysis of Cross Section and Panel Data, MIT Press, 2010, §4.2.1).