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I'm trying to find a reference (including the full formula) for the following. If $X_n = a_1 X_{n-1} + \cdots a_p X_{n-p} + e(n)$ where $\{e(n)\}$ is a white noise, then

$$ X_n=g(e_0,e_1,\ldots,e_n)+\sum_{k=1}^{p}r_k^n\cdot Q_k,\ \mathrm{with}\ \begin{pmatrix} Q_1 \\ Q_2 \\ \vdots \\ Q_p \end{pmatrix} = V^{-1} \begin{pmatrix} X_0 \\ X_1 \\ \vdots \\ X_{p-1} \end{pmatrix} $$

where $V$ is the same matrix as in section 2 in this article, and $g$ is a linear function of $e_0,\ldots, e_n$ with some combination of the coefficients $a_1, \ldots, a_p$. For instance, if $p = 1$, we have $g(e_0,\ldots,e_n) = \sum_{k=0}^{n-1}a_1^k\cdot e_{n-k}$.

For an arbitrary $p$, the $k$-th term in the above sum (for the function $g$) is a polynomial of degree $k$, with $p$ variables $a_1, \ldots, a_p$. This formula also allows you to identify the auto-correlation structure, whether or not the time series is stationary or not.

Note

The matrix $V$ is a $p \times p$ Vandermonde matrix (its elements are the powers of the roots $r_1,\ldots,r_p$ of the characteristic polynomial $x^p = a_1 x^{p-1} + \cdots + a_{p-1} x + a_p$. The formula assumes that these roots are different (no multiple root) otherwise the formula must be adjusted.

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  • $\begingroup$ Does this have to do with AR vs. MA representations of a general ARMA model? There are known conditions when you can represent a general ARMA by a pure AR or a pure MA. Keywords: invertibility (for MA), causality (for AR). $\endgroup$ – Richard Hardy May 29 '19 at 17:24
  • $\begingroup$ Not really, the formula in question is valid in the context of AR($p$), though of course, it applies to any more sophisticated process equivalent to an AR($p$). A more general version of this formula is probably available for ARMA. $\endgroup$ – Vincent Granville May 29 '19 at 17:58
  • $\begingroup$ Does the formula have some name or interpretation for those of us who are not that math-headed? I wonder what its function is supposed to be and for what it can be useful. $\endgroup$ – Richard Hardy May 29 '19 at 18:19
  • $\begingroup$ I added a "note" at the bottom of my question, not sure if it helps. The formula can be used to compute the lag-$k$ auto-correlations $\mbox{Correl}[X_n, X_{n-k}]$ ($k=1, 2, \cdots$) although there are other ways to solve this problem. $\endgroup$ – Vincent Granville May 29 '19 at 18:23
  • $\begingroup$ The general idea is to come up with a unified framework that solves most statistical problems involving linear algebra: linear regression, PCA, pseudo-inverse matrices, Markov chains and so on, all using the same matrix $V$ in one way or another. See datasciencecentral.com/profiles/blogs/… $\endgroup$ – Vincent Granville May 29 '19 at 18:27
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I could not find a reference, but I worked a bit more on it, and here is the full formula (the details about the function $g$, for $n\geq $p):

$g(e_p, e_{p+1},\cdots,e_n) = \sum_{k=0}^{n-p}A_k e_{n-k}, \mbox{ with } A_k =\sum_{t=1}^p a_tA_{k-t}$.

It is now published in my recent article, here (see Appendix.) The function $g$ does not depend on $e_0, e_1, \cdots, e_{p-1}$, only on $e_p, e_{p+1},\cdots e_n$. The initial conditions for $A_k$ are

$A_0=1$ and $ A_{-1}=A_{-2}= \cdots =A_{-(p-1)} = 0$.

Illustration ($p=1$) including variance computation is provided in my article.

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