What is Characteristic Polynomial in Time-series analysis? [duplicate]

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I'm currently taking a course in time-series at the undergraduate level. The course content goes right to the characteristic polynomial function (and solving it like you solve for roots) to determine if certain processes (e.g AR, MA, etc) is stationary or invertible.

After doing some googling it seems that many text-books go straight to the characteristic polynomial without explaining much. The professor did explain that to understand why, it requires a much higher level understanding than the undergraduate level.

What is it? Can we understand it intuitively without the complicated math?

marked as duplicate by kjetil b halvorsen, whuber♦Jun 25 at 15:37

• Basically, the characteristic equation is the representation of an AR, MA or MA representation of an ARMA model as a polynomial in the lag operator $L$, defined such that $L^i=x_{t-i}$. For example, the characteristic equatoin of an AR model $x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + \varepsilon_t$ is: $1-\phi_1L -\phi_2L^2=0$. Solving for $L$ the roots are obtained, which provide information about the invertibility of a MA process, or about causality, stationarity of the process. – javlacalle Oct 20 '16 at 7:53
• Thanks @javlacalle for the explanation. I'm not struggling on one specific equation or concept but rather just curious of the math behind the characteristic polynomial. It seems like some core concepts (which may be more higher level math than undergraduate) were left out. For example, if the Lag operator $L$ was used, how is it possible that we can treat it as a variable when solving for the roots? And WHY do we solve for the roots, or why does solving the roots lead to these conclusion about the AR process? – Kevin Pei Oct 23 '16 at 0:03