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For the autoregressive AR(1) process $x_t = \delta + \phi x_{t-1} + \eta_t$, I am trying to prove that the variance is:

$\sigma_x^2 = \sigma_\eta^2/(1-\phi^2)$

And that the first-order covariance is:

$\gamma_{1,x} = \phi \sigma_x^2$.

I have tried many manipulations but I cannot succeed. I have the feeling that I didn't find the correct form yet in which I should write the process before I take expectations. Could anyone please help? Thanks in advance.

Carl
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rbm
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    Technically, it doesn't have to actually *be* homework, just equivalent to homework. Please don't take offense, we get a lot of people posting their homework here hoping someone will do it for them. So it's a sensitive topic. – gung - Reinstate Monica Jun 15 '14 at 12:06

2 Answers2

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$Var(X_t)=var(\delta+\phi x_{t-1}+\eta_t)=0+var(\phi x_{t-1})+var(\eta_t)\\ =\phi^2var(x_{t-1})+\sigma^2_{\eta}=\frac{var(\eta_{t})}{1-\phi^2}=\frac{\sigma^2_{\eta}}{1-\phi^2}$

frisbee
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    Please note that - as explained in the [tag wiki info](http://stats.stackexchange.com/tags/self-study/info) of the `self-study` tag, it's requested that for questions that are routine bookwork (the sort of question that might be set for an assignment or an exercise for example, whether it is assigned or not) that we don't provide full solutions, but lean more toward hints and guidance. – Glen_b Jun 15 '14 at 14:11
  • (PS I am not suggesting that your answer should be changed or removed; it's a suggestion for similar questions in future) – Glen_b Jun 15 '14 at 14:54
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\begin{align} \mathrm{Var}\left[X_t\right] &= \mathrm{E}\left[X_t\left(\delta+\phi X_{t-1}+\eta_t\right)\right]-\mathrm{E}^2\left[X_t\right]\\ &= \delta\mathrm{E}\left[X_t\right]+\phi\mathrm{E}\left[X_tX_{t-1}\right]+\mathrm{E}\left[X_t\eta_t\right]-\frac{\delta^2}{\left(1-\phi\right)^2}\\ &= \delta\mathrm{E}\left[X_t\right]+\phi\left(\mathrm{Cov}\left[X_t,X_{t-1}\right]+\mathrm{E}\left[X_t\right]\mathrm{E}\left[X_{t-1}\right]\right)+\mathrm{E}\left[X_t\eta_t\right]-\frac{\delta^2}{\left(1-\phi\right)^2}\\ &= -\frac{\phi\delta^2}{\left(1-\phi\right)^2}+\phi\left(\gamma_{1,x}+\frac{\delta^2}{\left(1-\phi\right)^2}\right)+\sigma_\eta^2\\ &= \phi^2\sigma_x^2+\sigma_\eta^2\\ &= \phi^2\mathrm{Var}\left[X_t\right]+\sigma_\eta^2\\ &\\ \mathrm{Var}\left[X_t\right] &= \frac{\sigma_\eta^2}{1-\phi^2}\\ \end{align}

assumednormal
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    Please note that - as explained in the [tag wiki info](http://stats.stackexchange.com/tags/self-study/info) of the `self-study` tag, it's requested that for questions that are routine bookwork (the sort of question that might be set for an assignment or an exercise for example, whether it is assigned or not) that we don't provide full solutions, but lean more toward hints and guidance. – Glen_b Jun 15 '14 at 14:12
  • (PS I am not suggesting that your answer should be changed or removed; it's a suggestion for similar questions in future) – Glen_b Jun 15 '14 at 14:55