Partial autocorrelation function - A doubt on a property The PACF of a stationary process $\{x_t\}$, denoted by $\phi_{hh}$ is equal to:


*

*$\rho(1)$, for $h=1$

*$Corr(x_{t+h}-\hat x_{t+h}, x_{t}-\hat x_{t})$ for $h>1$,
where $\hat x_{t+h}$ is the OLS regression of $x_{t+h}$ on $\{x_{t+h-1},...,x_{t+1}\}$, and $\hat x_{t}$ is the OLS regression of $x_{t}$ on $\{x_{t+1},...,x_{t+h-1}\}$.(no intercept)


This can be seen as the 'direct' or partial effect of $x_t$ on $x_{t+h}$.
What I don't understand is why should both OLS regressions give the same coefficients(although for different variables).
Any help would be appreciated
 A: PACF is usually defined in terms of linear projections. 
Suppose we have random variable $Y$, and a set of random variables $X_1,...,X_n$. Then the linear projection of $Y$ on $X_1,...,X_n$ is a random variable $P_{X_1,...,X_n}Y=\alpha_1X_1+...+\alpha_nX_n$, where coefficients $\alpha_1,...,\alpha_n$ minimize the expression:
$$E(Y-\alpha_1X_1-...-\alpha_nX_n)^2.$$
It is not hard to see that the coefficients satisfy the following system of linear equations:
\begin{align}
\begin{bmatrix}
EX_1^2 & EX_1X_2 & ...&EX_1X_n\\
... & ... & ... & ...\\
EX_nX_1 & EX_nX_2 & ...& EX_n^2
\end{bmatrix}
\begin{bmatrix}
\alpha_1\\
...\\
\alpha_n
\end{bmatrix} = 
\begin{bmatrix}
EYX_1\\
...\\
EYX_n
\end{bmatrix}
\end{align}
Now the partial correlation coefficient is defined for stationary time series $X_t$ as 
$$\alpha(k)=corr(X_{k+1}-P_{X_2,...,X_k}X_{k+1}, X_1-P_{X_2,...,X_k}X_{1}).$$
Now try writing the corresponding systems of linear equations for projections $P_{X_2,...X_k}X_{k+1}$ and $P_{X_2,...,X_k}X_1$. Notice that the matrix is the same, because we project on the same set of variables. Also the right hand side vectors will contain the same values but these values will be in a reversed order for different system of equations. Since the matrix is symmetric the solutions will have the same values, but again in a reversed order for different system. 
