# first order auto regressive

Hello how to do the following:

Suppose that $X =(X_1,X_2,...,X_n)$ follows the following:

$X_t - \mu = \eta (X_{t-1} - \mu) + \epsilon_t,$ $t= 1,2,...$

where $\mu \in R$ and $\eta \in (-1,1)$ unknown and $\epsilon_t$'s are iid from $N(0,1)$. Let $\theta = (\mu,\eta)$. Suppose $X_0 = 0$

What is the joint density function of $(X_1,X_2,X_3,..,X_n)$?

What is the maximum likelihood estimator for $(\mu,\eta)$?

Thanks a lot!

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And... what did you try? – Did Mar 4 at 8:24
i tried to solve that recursion but got lost.. i tried to see a pattern for X_1 X_2 ... X_n but didnt work out.. – Salih Ucan Mar 4 at 8:25
i tried to solve that recursion but got lost... Really? $X_t-\mu=\epsilon_t+\eta$ times something $=\epsilon_t+\eta\epsilon_{t-1}+\eta^2$ times something $=...$ Can you continue? – Did Mar 4 at 8:29
No because it asks for joint distribution would you help if you could figure it out? Thank you! – Salih Ucan Mar 4 at 8:37
I would, gladly, IF ONLY YOU TRIED something yourself instead of refusing to follow the lead I gave you simply because this lead does not produce the result at once. – Did Mar 4 at 8:41