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Let $\rho_{partial}(n) = Cor(Y_t, Y_{t-n}|Y_{t-1}=\mu,\cdots Y_{t-2}=\mu, Y_{t-n+1}=\mu)$ where $\mu$ is the mean of stationary process.

I know that $\rho_{partial}(1)= \rho(1)$ and that $\rho_{partial}(2)=\frac{\rho(2)-\rho^2(1)}{1-\rho^2(1)}$

I heard that all partial auto-correlations can be recurrently represented with auto-correlations and partial auto-correlations of lower order.

Could you give me a reference and/or explain how partials of higher order can be obtained?

UPD This answer suggests using Durbin-Lewinson rule, that contains a toepliz matrix. Is it true that the matrix entries are somehow special such that no matter how big the matrix is, the answers for $\rho_{partial(k)}$ depends only on lower order $\rho_{partials}$?

\begin{eqnarray} \left(\begin{array}{cccc} \rho(0) & \rho(1) & \cdots & \rho(k-1) \\ \rho(1) & \rho(0) & \cdots & \rho(k-2) \\ \vdots & \vdots & \vdots & \vdots \\ \rho(k-1) & \rho(k-2) & \cdots & \rho(0) \\ \end{array}\right) \left(\begin{array}{c} \phi_{k1} \\ \phi_{k2} \\ \vdots \\ \phi_{kk} \\ \end{array}\right) = \left(\begin{array}{c} \rho(1) \\ \rho(2) \\ \vdots \\ \rho(k) \\ \end{array}\right) \,, \end{eqnarray}

UPD2 Some hand-waving We can treat any stationary process as an AR process, and then the partial autocorrelations will be coefficients $\phi_n$. When we write $\rho(k)$ we get a sum of rhos with coefficients so the matrix arises.

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I think in this very good answer you should find all the clarifications needed, especially the one on how you calculate the higher-order PACF simultaneously up to lag k (see the regression). When you see it, you also get the conceptual answer to the question marked in bold (regardless the choose of a method for their calculation, regression or Durbin-Lewinson): indeed, I rephrase you question as “should the PACF(k) depend on the only lower order ACs and PACs?” The answer is yes, by the meaning of PACF, that shows the correlation between the process and its k-th leg after accounting for and removing the effects of the autocorrelations at lower-order lags. So that at each lag k the PACF(k) must depend only on the first k-1 lags, because it measures the serial correlation between the process at current time (at lag 0) and its lag k, after cleaning up the effects of the other intermediate k-1 lags being themself autocorrelated among each other. I hope it is clear enough.

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