Publication date: 08/13/2020

Statistical Details for ARIMA Models


For a response series {yi}, the general form for the ARIMA model is as follows:


t is the time index

B is the backshift operator defined as Byt = yt - 1

wt = (1 - B)d yt is the response series after differencing

μ is the intercept or mean term

φ(B) and θ(B) are the autoregressive operator and the moving average operator, respectively, and are written as follows:


at are the sequence of random shocks

The at are assumed to be independent and normally distributed with mean zero and constant variance.

The model can be rewritten as follows:

The constant estimate δ is given by the relation:

Seasonal ARIMA Model

In the case of Seasonal ARIMA modeling, the differencing, autoregressive, and moving average operators are the product of seasonal and nonseasonal polynomials:

where s is the number of observations per period. The first index on the coefficients is the factor number (1 indicates nonseasonal, 2 indicates seasonal) and the second is the lag of the term.

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