A closed form formula for the normalizing constant in standard normal auto-regressive series? Let $Z_t = c_1Z_{t-1} + c_2Z_{t-2} + ... + c_nZ_{t-n} + c\epsilon_t$ 
where $Z_t, \epsilon_t \sim \mathtt{N}(0,1)$ are iid variables and $Z_s \sim \mathtt{N}(0,1)$ for all $s$.
Given the values of $c_i$ for $i = 1 ...n$ is there a closed form formula for $c$?
We can derive that $c = \sqrt{1 - c_1^2}$ for the case $n = 1$ and that $c = \sqrt{1 - c_1^2 - c_2^2 -\frac{2c_1^2c_2}{1-c_2}}$ for case $n=2$.
So it seems like there should be a nice closed form formula for $c$. However I was unable to work through the algebra for $n = 3$. 
But I think it might be that I don't know enough in this topic and perhaps this is a standard piece of work and someone has already worked it all out.
 A: It's not strange you didn't calculate the AR(3) case. It's rather complicated! And no, there is no closed form for the AR(n)-case. For the AR(3) we start with the Yule-Walker equationsAR-model wikipedia(where $\gamma_j=\gamma_{-j}$):


*

*$\gamma_1=c_1\gamma_0+c_2\gamma_{-1}+c_3\gamma_{-2}=c_1\gamma_0+c_2\gamma_{1}+c_3\gamma_{2}\\
 \gamma_2=c_1\gamma_1+c_2\gamma_0+c_3\gamma_{-1}=c_1\gamma_1+c_2\gamma_0+c_3\gamma_{1}\\
 \gamma_3=c_1\gamma_2+c_2\gamma_1+c_3\gamma_{0}$


*

*Then we multiply the AR(3) expression by $Z_t$ and take the expectation:
$ E[Z_t^2]=c_1E[Z_tZ_{t-1}]+c_2E[Z_tZ_{t-2}]+c_3E[Z_tZ_{t-3}]+c^2\Rightarrow \\
 \gamma_0=c_1\gamma_1+c_2\gamma_2+c_3\gamma_3+c^2$

*From Yule-Walker do we get:
$  \gamma_1=\frac{c_1+c_2c_3}{1-c_2-c_1c_3-c_3^2}\gamma_0\\
 \gamma_2=\left(\frac{c_1(c_1+c_2c_3)+c_3(c_1+c_2c_3)}{1-c_2-c_1c_3-c_3^2}+c_2\right)\gamma_0\\
 \gamma_3=\left(\frac{(c_1^ 2+c_3c_1+c_2)(c_1+c_2c_3)}{1-c_2-c_1c_3-c_3^2}+c_2+c_3\right)\gamma_0$

*Plugging these into the equation above (the second point) give what you want by setting $Var[Z_s]=\gamma_0=1$.

*In Hamiltons book on time series (p59) he writes that the solutions for $\gamma_j$ takes the form:
$\gamma_j=g_1\lambda_1^j+g_2\lambda_2^j+\cdots+g_p\lambda_p^j$
where the eigenvalues are the solutions of
$\lambda^p-c_1\lambda^{p-1}-c_2\lambda^{p-2}-\cdots-c_p=0$
(This is exactly the method outlined in the paper user Hunaphu has added).
So my guess is that these equations will be very complicated for $p=5$ or above since no formula exists for the solution of an equation of degree five or higher. It has to be solved by e.g. elliptic functions or theta functions. So to find a general solution for the AR(n)-case you must find a general solution for the n'th-degree algebraic equation which does not exist.
