The R Book

Time Series Models

Time series models come in three kinds (Box and Jenkins 1976):

• Moving average (MA) models where

  

• autoregressive (AR) models where

  

• autoregressive moving average (ARMA) models where

  

A moving average of order q averages the random variation over the last q time periods. An autoregressive model of order p computes X_t as a function of the last p values of X, so, for a second-order process, we would use

Typically, we would use the partial autocorrelation plot (above) to determine the order. So, for the lynx data (p. 717) we would use order 2 or 4, depending on taste. Other things being equal, parsimony suggests the use of order 2. The fundamental difference is that a set of random components (_t–j) influences the current value of a MA process, whereas only the current random effect (_t) affects an AR process. Both kinds of effects are at work in an ARMA processes. Ecological ...