Predefined Variables for the ARIMA Model

For ARIMA models, you can include a predefined trend or seasonal dummy variables. In the following list of options, let t represent the observation count from the start of the period of fit for the model, and let Xt be the value of the time trend variable at observation t.

LINEAR

a linear trend, with x sub t end sub , equals t negative c. Click image for alternative formats.

QUADRATIC

a quadratic trend, with x sub t , equals . open t minus c close squared. Click image for alternative formats.

CUBIC

a cubic trend, with x sub t , equals . open t minus c close cubed. Click image for alternative formats.

INVERSE

an inverse trend, with x sub t , equals , 1 over t. Click image for alternative formats.

SEASONAL

seasonal dummies. For a seasonal cycle of length s, the seasonal dummy regressors include x sub i comma t end sub . colon 1 less than or equal to i less than or equal to open s minus 1 close comma 1 less than or equal to t less than or equal to n. Click image for alternative formats. for models that include an intercept term, and x sub i comma t end sub . colon 1 less than or equal to i less than or equal to open s close comma 1 less than or equal to t less than or equal to n. Click image for alternative formats. for models that do not include an intercept.

Each element of a seasonal dummy regressor is either zero or one.

x sub i comma t end sub . equals . left brace . table with 2 rows and 1 column , row1 column 1 , 1 w h e n i equals t mod of s , row2 column 1 , 0 o t h e r w i s e , column 2 , end table .. Click image for alternative formats.

If the model includes an intercept term, the number of seasonal dummy regressors is one less than s to ensure that the linear system is full rank.

Last updated: March 16, 2026