Basic Data Analysis for Time Series with R

14 THE BACKSHIFT OPERATOR, THE IMPULSE RESPONSE FUNCTION, AND GENERAL ARMA MODELS

14.1 THE GENERAL ARMA MODEL

14.1.1 The Mathematical Formulation

Filtering is the key for fitting many useful models, but now generalization from AR(1) to all kinds of AR(m) and/or MA(l) models is required. The general notion of an ARMA(m,l) model is developed.

• The general AR(m) model: ϵ_j = a₁ϵ_{j − 1} + a₂ϵ_{j − 2} + ⋅⋅⋅ + a_mϵ_{j − m} + j.
• The general MA(l) model: ϵ_j = −b_lw_{j − l}… − b₂w_{j − 2} − b₁w_{j − 2} + w_j.

The ARMA(m,l) model: ϵ_j = ∑^m_{s = 1}α_sϵ_{j − s} + ∑^l_{r = 1} − b_rw_{j − r} + w_j where the white noise components have been combined. A rationale will be developed for arguing that, in practice, all such models can be treated as AR(∞), and approximated by AR(m), for some sufficiently large m.

14.1.2 The arima.sim() Function in R Revisited

The most general ARMA(m,l) models can be simulated in R using the arima.sim() function. For example, consider the ARMA(2,2) model given by ϵ_n = 0.6ϵ_{n − 1} − 0.25ϵ_{n − 2} + w_n − 1.1w_{n − 1} + 0.28w_{n − 2}. The command to simulate a series of 200 such errors in R, with a standard deviation of 2.0, is arima.sim(n = 200, list(ar =c(0.6, -0.25),ma = c(1.1, -0.28)), sd = 2.0). Notice that the sign convention in R, for the MA(l) part of the function, is the opposite of this book.

FIGURE 14.1 A representative collection of some ARMA(m,l) model errors.

14.1.3 Examples of ARMA(m,l) Models ...