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.