10.1 White Noise and Autocorrelation

In this chapter, the authors extend ARIMA models to include other information by allowing the errors from a regression to contain autocorrelation. The resulting model has two error terms, with only the ARIMA model errors assumed to be white noise.

WHITE NOISE

To incorporate autocorrelation into regression errors, the authors replace the traditional error term, $\epsilon_t$, with $\eta_t$, which is assumed to follow an ARIMA model.

$$y_t=\beta_0 + \beta_1 x_{1,t} + ... + \beta_k x_{k,t} + \eta_t$$

$\eta_t$ follows an ARIMA(1,1,1) model:

$$(1-\phi_1 B)(1-B)\eta_t(1+\theta_1B)\epsilon_t$$

A regression model with ARIMA errors is equivalent to a regression model in differences with ARMA errors.

• all of the variables in the model must first be stationary