5.3 Modeling the Errors

Up to this point, we have decomposed our time series into a trend component, a seasonal component, and a random component,

\[Y_t = \underbrace{f(x_t)}_\text{trend} + \underbrace{s(x_t)}_\text{seasonality} + \underbrace{\epsilon_t}_\text{noise}\]

With observed data, we can work with the residuals after removing the trend and seasonality,

\[e_t = y_t - \underbrace{\hat{f}(x_t)}_\text{trend} - \underbrace{\hat{s}(x_t)}_\text{seasonality} \] which is approximately the true error process,

\[\epsilon_t = Y_t - \underbrace{f(x_t)}_\text{trend} - \underbrace{s(x_t)}_\text{seasonality}\]

For the following time series models (AR, MA, and ARMA), we will use \(Y_t\) to represent the random variable for the residuals, \(e_t\).

In the next five sections, we’ll discuss the theory of these three models. To see a real data example and the R code to fit the models, go to the Real Data Example section.