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I am interested in a One-Way ANOVA model:

                                   enter image description here

I wrote the model out as enter image description here and got $X$ as a matrix that looks like:

                                                      enter image description here

Which is rank 2 (6 x 4 Matrix, so not full rank)

Does the least squares estimator for β = (µ, α1, α2)^T exist? If it exists, is it unique? Why?

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    $\begingroup$ I see three parameters ($\alpha_1, \alpha_2,\mu$) in your model but four columns in $X.$ What does that extra column represent?? $\endgroup$ – whuber Feb 11 at 15:35
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Here is a hint for your self-study question:

Is the design matrix full rank? Also, please add the self-study tag. What do you know about full-rank vs. non-full rank matrices and the estimability of the parameters?

Also, have you tried fitting a model with a full rank design matrix and one without a full rank design matrix in your statistical software program? Some programs will generate a helpful warning that might help you answer this question.

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