ARMA(p,q) Processes

Property 1: An ARMA(p, q) process

is stationary provided it is causal, i.e. the polynomial

for any z such that |z| ≤ 1.

Observation: Actually, we will only consider stationary ARMA(p, q) processes.

From Property 1, if z is a root of the polynomial 1 – φ₁z – φ₂z² – ··· – φ_pz^p, it follows that |z| > 1. As in the AR(p) case, this is equivalent to the fact that |w| < 1 for any w that satisfies the following equation

The causal property implies (and is equivalent to) the existence of constants ψ_j such that ψ₀ = 1 and

Thus all stationary ARMA processes can be expressed as an MA(∞) process. In fact, the ψ_j coefficients can be determined as in Property 2.

Property 2: Let

Then

which in turn results in

where θ₀ = 1, θ_j = 0 for j > q and ψ_j = 0 for j < 0.

Proof: See ARMA Proofs

Observation: We also restrict our attention to invertible ARMA(p, q) processes, i.e. those for which if 1 + θ₁z + θ₂z² + ··· + θ_pz^p = 0 then |z| > 1.

Under construction

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