According to the Wikipedia, the partial autocorrelation function of lag k is defined as:

$\alpha(k) = \operatorname{Cor}(z_{t+k} - P_{t,k}(z_{t+k}),\, z_t - P_{t,k}(z_t)),\text{ for }k\geq 2,$

where $P_{t,k}(x)$ denotes the projection of x onto the space spanned by $ x_{t+1}, \dots, x_{t+k-1}$.

To my best knowledge, partial correlation function is try to measure the linear association between $z_t$ and $z_{t+k}$ with the linear effect of $z_{t+1}, ...,z_{t+k-1}$ removed.

My problem lies on the term $z_t - P_{t,k}(z_t)$. Since $z_t$ occurs before $z_{t+1}, ...,z_{t+k-1}$, future variables should have no effect to a later variable. It would be unreasonable to remove the linear effect of $z_{t+1}, ...,z_{t+k-1}$ from $z_t$. Therefore, I expect $P_{t,k}(z_t)$ should be zero theoretically. Is it a correct understanding?


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