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

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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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