Characteristic Equation for AR(p) Processes

Basic Properties

Property 1: An AR(p) process is stationary provided all the roots of the following polynomial equation (called the characteristic equation) have an absolute value greater than 1.

This is equivalent to saying that if z satisfies the characteristic equation then |z| > 1.

In fact, setting w = 1/z, this is equivalent to saying that |w| < 1 for any w that satisfies the following equation

By the Fundamental Theorem of Algebra, any p^th degree polynomial has p roots; i.e. there are p values of z that satisfy the above equation. Unfortunately, not all of these roots need to be real; some can involve “imaginary” numbers such as , which is usually abbreviated by the letter i. For example, the equation z² + 1 has the roots i and –i as can be seen by substituting either of these values for z in the equation z² + 1.

We now give three properties of imaginary numbers, which will help us avoid discussing imaginary numbers in any further detail:

• all values which involve imaginary numbers can be expressed in the form a + bi where a and b are real numbers
• if a + bi is a root of a p^th degree polynomial, then so is a – bi
• if z = a + bi then the absolute value of z is defined by |z| =

Since a and b are real numbers, not involving , we only need to deal with real numbers.

Property 2: An AR(1) process is stationary provided |φ₁| < 1

Property 3: An AR(2) process is stationary provided

|φ₂| < 1 and |φ₁| + φ₂ < 1

Examples

Example 1: Determine whether the following AR(2) process is stationary.

The roots of w² – 2w + .5 = 0 are

This process is not stationary since 1 + √1.5 ≥ 1. You get the same result via Property 3 since |φ₁| + φ₂ = 2 – .5 = 1.5 ≥ 1.

Example 2: Determine whether the following AR(2) process is stationary.

Since

the roots of the reverse characteristic equation are not real. In fact

Thus

and so we see that this AR(2) process is stationary. We get the same result via Property 3 since

Observation

It turns out that by Property 4 of Basic AR Concepts, for any k, ρ_k can be expressed as a linear combination

where w₁, …, w_p are the unique roots of the reverse characteristic equation

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack supplies the following array function where R1 is a p × 1 range containing the phi coefficients of the polynomial where φ_p is in the first position and φ₁ is in the last position.

ARRoots(R1): returns a p × 3 range where each row contains one root, and where the first column consists of the real part of the roots, the second column consists of the imaginary part of the roots and the third column contains the absolute value of the roots

This function calls the ROOTS function described in Roots of a Polynomial. Note that just like in the ROOTS functions, the ARRoots function can take the following optional arguments:

ARRoots(R1, prec, iter, r, s)

prec = the precision of the result, i.e. how close to zero is acceptable. This value defaults to 0.00000001.

iter = the maximum number of iteration performed when performing Bairstow’s Method. The default is 50.

r, s = the initial seed values when using Bairstow’s Method. These default to zero.

References

Alonso, A. M., Garcia-Martos, C. (2012) Time series analysis: autoregressive, MA and ARMA processes
https://www.academia.edu/35659911/Time_Series_Analysis_Autoregressive_MA_and_ARMA_processes

Greene, W. H. (2002) Econometric analysis. 5th Ed. Prentice-Hall
https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=1243286

Gujarati, D. & Porter, D. (2009) Basic econometrics. 5th Ed. McGraw Hill
http://www.uop.edu.pk/ocontents/gujarati_book.pdf

Hamilton, J. D. (1994) Time series analysis. Princeton University Press
https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis

Wooldridge, J. M. (2009) Introductory econometrics, a modern approach. 5th Ed. South-Western, Cegage Learning
https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf