Characteristic Equation of a Process in Time Series Analysis

The characteristic equation of a time series process helps determine its stationarity and stability by analyzing the roots of its characteristic polynomial. This is particularly important for AR(p), MA(q), and ARMA(p, q) processes.

An Auto-Regressive (AR) process of order p, denoted as AR(p), is defined as:

X_t​ = ϕ_1​X_t−1 ​+ ϕ_2​X_t−2 ​+...+ ϕ_p​X_t−p ​+ ϵ_t​

• X_t is the current value,
• ϕ₁, ϕ₂,...,ϕ_p​ are AR coefficients,
• ϵ_t​ is white noise.

The AR(p) process can be rewritten as below by replacing X_t​ with its backward shift operator or lag operator (L):

(1 − ϕ_1​L − ϕ_2​L² −...− ϕ_p​L^p) X_t​ = ϵ_t​

Characteristic Polynomial
The term in parentheses:

1 − ϕ₁​L − ϕ₂​L² − ... − ϕ_p​L^p

The characteristic equation is obtained by setting the characteristic polynomial to 0 :

1 − ϕ₁​L − ϕ₂​L² − ... − ϕ_p​L^p = 0

Example: AR(2) Process
For an AR(2) process:
X_t = ϕ₁X_{t −1} + ϕ₂X_t−2 + ϵ_t
The characteristic equation is:
1 − ϕ₁L − ϕ₂L² = 0

For an AR(p) process to be stationary, the roots of the characteristic equation must lie outside the unit circle (i.e., their absolute values must be greater than 1).

Example 1: AR(1) Process
X_t​ = ϕX_t−1​ + ϵ_t​
Characteristic equation:
1 − ϕL = 0
=> L = 1 / ϕ

• Stationary if ∣ L ∣ > 1 (root outside the unit circle).
• Non-stationary if ∣ L ∣ ≤ 1 (root inside or on the unit circle)

• Stationary if ∣ ϕ ∣ < 1 .
• Non-stationary if ∣ ϕ ∣ ≥. 1

Example 2: AR(2) Process
X_t = ϕ₁X_t−1 − ϕ₂X_t−2 + ϵ_t
Characteristic equation:
1 − ϕ₁L - ϕ₂L² = 0
To check stationarity, we solve for the roots of:
ϕ₂L² + ϕ₁L - 1 = 0
Using the quadratic formula:
L = −ϕ₁ ​± (ϕ₁²​ + 4ϕ_2​)^(1/2) / (2ϕ_2​) ​​​​
These roots determine whether the AR(2) process is stationary:

• If both roots satisfy ∣ L ∣ > 1, the process is stationary.
• If any root satisfies ∣ L ∣ ≤ 1, the process is non-stationary.

An Moving Average (MA) process of order q, denoted as MA(q), is defined as:

X_t​ = θ_1​ϵ_t−1 ​+ θ_2​ϵ_t−2 ​+...+ θ_q​ϵ_t−q ​+ ϵ_t​

• X_t is the current value
• θ₁, θ₂,...,θ_q​ are MA coefficients
• ϵ_t​ is white noise.

The MA(q) process can be rewritten as below by replacing ϵ_t​ with its backward shift operator or lag operator (L):

(1 − θ_1​L − θ_2​L² −...− θ_p​L^p) ϵ_t​ = ϵ_t​

The characteristic equation is:

1 + θ₁​L + θ₂​L² + ... + θ_q​L^q = 0

Example: MA(2) Process
For an MA(2) process:
X_t = θ₁ϵ_{t −1} + θ₂ϵ_t−2 + ϵ_t
The characteristic equation is:
1 + θ₁L + θ₂L² = 0