Autoregressive Integrated Moving Average - Forecasts Using ARIMA Models

Forecasts Using ARIMA Models

ARIMA models are used for observable non-stationary processes that have some clearly identifiable trends:

a constant trend (i.e. zero average) is modeled by
a linear trend (i.e. linear growth behavior) is modeled by
a quadratic trend (i.e. quadratic growth behavior) is modeled by

In these cases the ARIMA model can be viewed as a "cascade" of two models. The first is non-stationary:

$Y_t = \left( 1-L \right)^d X_t$

while the second is wide-sense stationary:

$\left( 1 - \sum_{i=1}^p \phi_i L^i \right) Y_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \, .$

Now standard forecasts techniques can be formulated for the process, and then (having the sufficient number of initial conditions) can be forecast via opportune integration steps.

Read more about this topic: Autoregressive Integrated Moving Average

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