Forecasting Process Details

Time Trend Curves

When you specify a time trend curve as a predictor in a forecasting model, the system computes a predictor series that is a deterministic function of time. This variable is then included in the model as a regressor, and the trend curve is fit to the dependent series by linear regression, in addition to other predictor series.

Some kinds of nonlinear trend curves are fit by transforming the dependent series. For example, the exponential trend curve is actually a linear time trend fit to the logarithm of the series. For these trend curve specifications, the series transformation option is set automatically, and you cannot independently control both the time trend curve and transformation option.

The computed time trend variable is included in the output data set in a variable named in accordance with the trend curve type. Let t represent the observation count from the start of the period of fit for the model, and let ${X_{t}}$ represent the value of the time trend variable at observation t within the period of fit. The names and definitions of these variables are as follows. (Note: These deterministic variables are reserved variable names.)

Linear trend: variable name _LINEAR_, with ${X_{t}=t-c}$
Quadratic trend: variable name _QUAD_, with ${X_{t}=(t-c)^{2}}$ . Note that a quadratic trend implies a linear trend as a special case and results in two regressors: _QUAD_ and _LINEAR_.
Cubic trend: variable name _CUBE_, with ${X_{t}=(t-c)^{3}}$ . Note that a cubic trend implies a quadratic trend as a special case and results in three regressors: _CUBE_, _QUAD_, and _LINEAR_.
Logistic trend: variable name _LOGIT_, with ${X_{t} = t}$ . The model is a linear time trend applied to the logistic transform of the dependent series. Thus, specifying a logistic trend is equivalent to specifying the logistic series transformation and a linear time trend. A logistic trend predictor can be used only in conjunction with the logistic transformation, which is set automatically when you specify logistic trend.
Logarithmic trend: variable name _LOG_, with ${X_{t}=ln(t)}$
Exponential trend: variable name _EXP_, with ${X_{t}=t}$ . The model is a linear time trend applied to the logarithms of the dependent series. Thus, specifying an exponential trend is equivalent to specifying the log series transformation and a linear time trend. An exponential trend predictor can be used only in conjunction with the log transformation, which is set automatically when you specify exponential trend.
Hyperbolic trend: variable name _HYP_, with ${X_{t}=1/t}$
Power curve trend: variable name _POW_, with ${X_{t}=ln(t)}$ . The model is a logarithmic time trend applied to the logarithms of the dependent series. Thus, specifying a power curve is equivalent to specifying the log series transformation and a logarithmic time trend. A power curve predictor can be used only in conjunction with the log transformation, which is set automatically when you specify a power curve trend.
EXP(A+B/TIME) trend: variable name _ERT_, with ${X_{t}=1/t}$ . The model is a hyperbolic time trend applied to the logarithms of the dependent series. Thus, specifying this trend curve is equivalent to specifying the log series transformation and a hyperbolic time trend. This trend curve can be used only in conjunction with the log transformation, which is set automatically when you specify this trend.