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ARMA models Definition 42 A time series {Y t } is an autoregressive-moving average process of order p,q, ARMA(p,q), model, if it is stationary and of the form Y t = φ 1 Y t−1 + φ 2 Y t−2 + · · · + φ p Y t−p + ε t + θ 1 ε t−1 + · · · + θ q ε t−q , (12) where φ 1 ,...,φ p ,θ 1 ,... ,θ q are constants with φ p ,θ q ≠ 0. Unless otherwise mentioned, we assume iid that ε t ∼ N(0,σ 2 ). A process with non-zero mean is obtained by replacing the Y s in (12) by Y t − µ, etc. □ We use the autoregressive and moving average operators to write (12) as φ(B)Y t = θ(B)ε t . The properties of the process are intimately tied to the polynomials φ(z),θ(z), where we take z ∈ C. Let D = {z ∈ C : |z| ≤ 1} denote the unit disk in the complex plane. □ We remove common factors from φ(B) and θ(B) to eliminate overparametrisation, also called parameter redundancy. Time Series Spring 2010 – slide 141 Causality, invertibility Definition 43 An ARMA(p,q) process φ(B)Y t = θ(B)ε t is causal if it can be written as a linear process ∞∑ Y t = ψ j ε t−j = ψ(B)ε t , where ∑ |ψ j | < ∞, and we set ψ 0 = 1. It is invertible if it can be written as where ∑ |π j | < ∞, and we set π 0 = 1, ε t = j=0 ∞∑ π j Y t−j = π(B)Y t , j=0 Theorem 44 (a) An ARMA(p,q) process φ(B)Y t = θ(B)ε t is causal iff φ(z) ≠ 0 within D. If so, then the coefficients of ψ(z) satisfy ψ(z) = θ(z)/φ(z) for z ∈ D. (b) The process is invertible iff θ(z) ≠ 0 for z ∈ D. If so, then the coefficients of π(z) satisfy π(z) = φ(z)/θ(z) for for z ∈ D. Thus the process is causal iff the roots of φ(z) lie outside D, and invertible iff the roots of θ(z) lie outside D. Time Series Spring 2010 – slide 142 Example Example 45 Investigate the properties of the process Y t = 0.4Y t−1 + 0.45Y t−2 + ε t + ε t−1 + 0.25ε t−2 . Time Series Spring 2010 – slide 143 133
Comments □ Why use ARMA processes – usually an empirical model, using φ 1 ,...φ p ,θ 1 ,...,θ q as summary statistics, but with no implication that the model has a ‘scientific’, explanatory, basis in terms of the underlying data generating mechanism – the spectrum of an ARMA process can take many forms without p or q being very large, so they provide a flexible and parsimonious way to approximate a wide range of second-order properties – they are useful for forecasting, or for other settings where the autocorrelation structure of the data is not of primary interest □ ARMA models are not usually useful when the focus is on understanding the underlying mechanism that generates the data □ AR and MA models separately may provide more interpretable models in such cases: – AR models have Markov structure, which may be interpretable – MA models stem from weighted moving averages, which may be interpretable Time Series Spring 2010 – slide 144 134
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