General formula for AR($p$) auto-regressive time series

by Vincent Granville   Last Updated May 29, 2019 17:19 PM - source

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

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

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

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