Equal regression coefficients in a mean zero stationary process Problem
I have a stationary stochastic process $\{Z_t\}$, where $\mathbb{E}(Z_t) = 0$, $\forall t=0,1,2,\cdots$. 
Say I would like to perform linear regression on $Z_{t+k}$ by $Z_{t+k-1}, \cdots Z_{t+1}$, for some integer(lag) $k>0$, i.e.
$$
\mathbb{E} (Z_{t+k} | Z_{t+k-1}, \cdots Z_{t+1}) = \xi_1 Z_{t+k-1} + \cdots + \xi_{k-1} Z_{t+1}
$$
And I would also like to perform linear regression on $Z_t$ by $Z_{t+1}, \cdots Z_{t+k-1}$, 
$$
\mathbb{E} (Z_t | Z_{t+1}, \cdots Z_{t+k-1}) = \psi_1 Z_{t+1} + \cdots + \psi_{k-1} Z_{t+k-1}
$$
then, I would like to show
$$
\xi_i = \psi_i
$$
for $i=1, \cdots, k-1$.

Try
I have noticed that $(\xi_1, \cdots, \xi_{k-1})$ minimizes the error
$$
\mathbb{E} \left[ (Z_{t+k} -  \xi_1 Z_{t+k-1} - \cdots - \xi_{k-1} Z_{t+1})^2 \right]
$$
and $(\psi_1, \cdots, \psi_{k-1})$ minimizes the error
$$
\mathbb{E} \left[ (Z_t -  \psi_1 Z_{t+1} -\cdots - \psi_{k-1} Z_{t+k-1})^2 \right]
$$
but I'm stuck at how I should proceed.
 A: Note that
$$
\begin{aligned}
\mathbb{E} \left[ (Z_{t+k} - \xi_1 Z_{t+k-1} - \cdots - \xi_{k-1} Z_{t+1})^2 \right]  &=\mathbb{E} \left[ Z_{t+k}^2 -  \xi_1 Z_{t+k}Z_{t+k-1} - \cdots + \xi_{k-1}^2 Z_{t+1}^2 \right] \\
&= \mathbb{E} \left[ Z_{t+k}^2 \right] - \xi_1 \mathbb{E}\left[  Z_{t+k}Z_{t+k-1}\right] - \cdots + \xi_{k-1}^2\mathbb{E}\left[ Z_{t+1}^2 \right] \\
&= \mathbb{E} \left[ Z_t^2 \right] -\xi_1 \mathbb{E}\left[  Z_tZ_{t+1}\right] - \cdots + \xi_{k-1}^2 \mathbb{E}\left[ Z_{t+k-1}^2 \right] \\
&= \mathbb{E} \left[ (Z_t -  \xi_1 Z_{t+1} -\cdots - \xi_{k-1} Z_{t+k-1})^2 \right]
\end{aligned}
$$
So let us suppose that $\exists! (\xi_1^\ast, \cdots, \xi_{k-1}^\ast)$ that minimizes the the above error. 
Then the parameter vector also minimizes 
$$
\mathbb{E} \left[ (Z_t -  \psi_1 Z_{t+1} -\cdots - \psi_{k-1} Z_{t+k-1})^2 \right]
$$
thus, if we denote $\exists! (\psi_1^\ast, \cdots, \psi_{k-1}^\ast)$ as the parameter vector that minimizes the above error, we have
$$
(\xi_1^\ast, \cdots, \xi_{k-1}^\ast) = (\psi_1^\ast, \cdots, \psi_{k-1}^\ast)
$$
