The followings are copied from the PROC ARIMA doc. You might see different formulations from other software vendors. We are not aware of any issue with PROC ARIMA having sign switched. thanks
General Notation for ARIMA Models
Notation for Pure ARIMA Models
Mathematically the pure ARIMA model is written as
where
t
indexes time
is the response series or a difference of the response series
is the mean term
B
is the backshift operator; that is,
is the autoregressive operator, represented as a polynomial in the backshift operator:
is the moving-average operator, represented as a polynomial in the backshift operator:
is the independent disturbance, also called the random error
The series is computed by the IDENTIFY statement and is the series processed by the ESTIMATE statement. Thus, is either the response series Y or a difference of specified by the differencing operators in the IDENTIFY statement.
For simple (nonseasonal) differencing, . For seasonal differencing , where d is the degree of nonseasonal differencing, D is the degree of seasonal differencing, and s is the length of the seasonal cycle.
For example, the mathematical form of the ARIMA(1,1,1) model estimated in the preceding example is
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