# Variance of autocorrelation

At this link I have seen the following formula whereas $$r_k = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ $$Var(r_k) = \frac {n-k}{n*(n+2)}$$ where $$r_k$$ is the autocorrelation at relevant lag, $$n$$ is the number of points in the data set, and $$a_t$$ is the error.

I have searched the internet for the proof for variance equation, but I haven't found it. Could anyone help me prove the formula I mentioned above?

• In the provided link, it's not the variance, it's the definition of autocorrelation of model errors. – gunes Jun 20 '19 at 0:03
• Yes sorry for mistake. I have edited my question. Could you help me about it? – mertcan Jun 20 '19 at 5:58
• To be honest, I really would like to believe your knowledge is totally beyond my question but having not received any response from you just make me consider that you are so reluctant( except @Richard Hardy ) to spare your time, actually which is an ambivalence towards the aim of this forum. Therefore, please be more responsive and keep in mind that those questions are kind of which everybody may wonder and be keen on comprehending. I again ask : Could you help me about my question? – mertcan Jun 20 '19 at 18:51
• Hi @mertcan, I actually did look at the result in the original paper of Box; however he also seems to give no explanation. I just don't have much time nowadays and I couldn't think of a quick way to prove it. Actually, I found the result quite interesting, and follow the question. – gunes Jun 21 '19 at 12:50
• I have not even a clue to start. Could you give me at least the draft of proof? – mertcan Jun 21 '19 at 20:02