ARIMA Equations

Equation for a p-th order autoregressive (AR) model — that is, AR(p) model:

Where {y_t} is the data on which the ARMA model is to be applied. That means, the series is already power-transformed and differenced, in that order. The parameters φ₁, φ₂ and so on are AR coefficients.

Equation for a q-th order moving average (MA) model — that is, MA(q) model:

Where {y_t} is as defined previously and θ₁, θ₂ and so on are MA coefficients.

Equation for an ARMA(p,q) model:

Where {y_t}, φ₁, φ₂..., θ₁, θ₂... are as defined previously.

Equation for a SARMA(p,q)(P,Q) model (seasonal):

Where {y_t}, {φ}, and {θ} are as defined previously, and {Φ} and {Θ} are the seasonal counterparts.