PACF for MA(1) process I have MA(1) process:
$X_t=\epsilon_t+\theta\epsilon_{t-1}$
I want to prove the equation for PACF for $n\geq2$
$\alpha(n)=\phi_{nn} = \frac{\theta^n(-1)^{n+1}}{1+\theta^2+...+\theta^{2n}}$
I found the tip that if I show this for any $j \in \{1, ...,n-1\}$ 
$\phi_{n,n-j}=(-\theta)^{-j}(1+\theta^2+...+\theta^{2j})\phi_{nn}$
the result I want to obtain will be obvious. I did prove the equation above but I don't get that it implies the equation for $\alpha(n)$. 
Any suggestions?
 A: In another edition of Brockwell and Davis this problem is 3.20. The calculation of $\phi_{nn}$ relies on solving the system 
$$\Gamma_n(\phi_{11},...,\phi_{nn})'=(\gamma(1),...,\gamma(n))',$$
where  $\Gamma_n$ is the covariance matrix of the vector $(X_1,...,X_n)$, and $\gamma(i)=cov(X_{t},X_{t+i})$.
Now for MA(1) process the matrix $\Gamma_n$ is tridiagonal, and the $\gamma(j)=0$ for $j>1$. So you have the resursive system, from which you can calculate the $\phi_{nn}$.
A: I tried it via the Durbin-Levinson recursion on page 70. In general, 
$$ \phi_{nn} = \left[\gamma(n)-\sum_{j=1}^{n-1}\phi_{n-1,j}\gamma(n-j)\right]v_{n-1}^{-1},$$
where the $\gamma$ are the autocovariances, $v_{n}=v_{n-1}[1-\phi_{nn}^2]$ and $v_0=\gamma(0)$. Now, since $\gamma(n)=0$ for $n>1$ in the case of an $MA(1)$ process the expression simplifies to
$$ \phi_{nn} = -\phi_{n-1,n-1}\gamma(1)v_{n-1}^{-1}$$
for our purposes. Say,
$$ \phi_{22} = -\phi_{11}\gamma(1)v_{1}^{-1} = -\frac{\phi_{11}\gamma(1)}{\gamma(0)[1-\phi_{11}^2]}=-\frac{\phi_{11}\rho(1)}{1-\phi_{11}^2}=-\frac{\rho(1)^2}{1-\rho(1)^2},$$
where I used $\rho(1)=\phi_{11}$. Plugging in, using $\rho(1)=\theta/(1+\theta^2)$, gives me the right result.
In general, I get 
$$ \phi_{nn} = -\frac{\phi_{n-1,n-1}\rho(1)}{\prod_{j=1}^{n-1}(1-\phi_{jj}^2)},$$
but I do not see how to verify the general result from there. 
EDIT: I thought a general solution to this equation would maybe be known and posted a question here. In the answer, you can read about the completion of this alternative derivation.
