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Suppose the values of $-1≤ x(i) ≤ 1$ are at your disposal in the simple linear regression model $Y(i) = α + βx(i) + ε(i)$, where the $ε(i)$ are i.i.d. $N(0, σ^2)$.

How would you choose the $x(i)$ if the aim is to minimize the variance of the least squares estimator $β$ of the slope? What are the minimum value of $\text{var}(β)$ and the corresponding $β$?

Since the variance-covariance matrix of the predictors is $σ^2X'X^{-1}$. My approach is the take the values of $x(i)$ to minimize the term corresponding to $\text{var}(β)$. Does that mean that choosing the $x(i)$ with the largest magnitude is the right approach?

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  • $\begingroup$ This is a routine textbook-style question as is commonly set as an exercise (indeed, I recall answering a version of it for an exercise when I was a student). You should add the self-study tag and follow the guidelines at the self-study tag wiki on asking such questions. In particular you should ask about specific problems you have encountered in your initial efforts -- what did you try, and where did you get stuck? $\endgroup$ – Glen_b Apr 26 '16 at 8:25
  • $\begingroup$ You don't need the full variance-covariance matrix of parameters, just the variance for the coefficient of the linear term (look again at your first two paragraphs). What is that variance? (you can find it in many expositions of simple linear regression) $\endgroup$ – Glen_b Apr 26 '16 at 10:01

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