In the post that derives the least squares estimator, we make use of the following statement:

This post shows how one can prove this statement. Let’s start from the statement that we want to prove:

Note that is symmetric. Hence, in order to simplify the math we are going to label as A, i.e. .

Let’s compute the partial derivative of with respect to .

Instead of stating every single equation, one can state the same using the more compact matrix notation:

plugging in for A

Now let’s return to the derivation of the least squares estimator.

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Hi, thanks for the proof, I appreciate it. I just want to point out to a typo. When writing \hat{\beta}^\prime A \hat{\beta} as a number (i.e. sum of sums), two errors occur at the last row:

1) The index of very first beta in the row should be k, not 1.

2) The plus sign at the end of the row is redundant

Thanks Adam, you are right! I corrected the mistakes. Thanks a lot! Cheers, ad.