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

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?

• It seems that you are reading some textbook; if so, you could post the fragment or equations that you are struggling with. This way, you will probably have more chances to get an answer. – javlacalle Oct 20 '16 at 7:46
• 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
• I would recommend you to play around with these expressions, e.g., you can 1) simulate stationary and non-stationary processes and compare how they look like in a plot; 2) obtain the coefficients of the infinite MA representation of an ARMA process and see if they converge or not; 3) analyse the roots related to a given process, its modulus, angular frequency,... In order to do these exercises you will have to review some mathematical concepts, which will help you to become familiar with them, understand the information within them and eventually may give you the intuition you are looking for. – javlacalle Oct 25 '16 at 6:42