Linear Model - Time Series Models

Time Series Models

An example of a linear time series model is an autoregressive moving average model. Here the model for values {X_t} in a time series can be written in the form

where again the quantities ε_t are random variables representing innovations which are new random effects that appear at a certain time but also affect values of X at later times. In this instance the use of the term "linear model" refers to the structure of the above relationship in representing X_t as a linear function of past values of the same time series and of current and past values of the innovations. This particular aspect of the structure means that it is relative simple to derive relations for the mean and covariance properties of the time series. Note that here the "linear" part of the term "linear model" is not referring to the coefficients φ_i and θ_i, as it would be in the case of a regression model, which looks structurally similar.