# How to derive the first order autocorrelation coefficient of an AR(1) process?

I want to derive the first-order autocorrelation coefficient of an AR(1) process, where the AR(1) process can be expressed as:

$$y_t = \Theta y_{t-1} + u_t$$ with $|\Theta|<1$.

Please give me a hint on how to start with this problem?

• Le Max, please check that the edits people have made to improve your question leave it asking what you want. Jun 27, 2013 at 22:50
• @Glen_b I edited the question body because I think the clarity of the question was being dimmed by the (mis)use of the words "that" and "of" in the original statement of the question body. Hopefully the OP confirms that the edit(s) correspond to and make clear what he/she was asking. Jun 27, 2013 at 23:16
• @GraemeWalsh I approved your edit. However, the addition of "first order correlation coefficient" - which may well be the intent - is what I was worried about the addition of. The original didn't actually say what problem was to be solved. I think your edit has made a reasonable but not sure-to-be-correct assumption. I simply want the OP to check. Jun 27, 2013 at 23:19
• @Glen_b Thanks for the approval. That said, I suppose the only way to get to the bottom of the matter is to await for the OP's response. Jun 27, 2013 at 23:26

## 2 Answers

1. Find a mean and variance ($\gamma_0$) of $y_t$. How does the condition $|\Theta|<1$ help you to do that? Remember the assumptions of how residuals are distributed and similar.
2. Autocorrelation coefficient $\rho_1=\dfrac{\gamma_1}{\gamma_0}$, i.e. now you are missing only $\gamma_1$. Once you write down its expression the hints from the previous step are sufficient here too.

Multiply both sides by $y_{t-1}$ and take expectation. Exploit the fact that $u_t$ and $y_{t-1}$ are not correlated.

To calculate variance simply square both sides and then take expectation.