Deterministic Trends Forecast

Properties

For a deterministic trend model $Y_t={\mu }_t+X_t$, the forecast is: ${\hat{Y}}_t(l)={\mu }_{t+l}$

The forecast error is $e_t(l)=X_{t+l}$, which is unbiased ($E[e_t(l)]=0$). The error variance is $Var(e_t(l))={\gamma }_0$.

Specific Models:

• Linear trend (${\mu }_t={\beta }_0+{\beta }_{1}t$): ${\hat{Y}}_t(l)={\beta }_0+{\beta }_1(t+l)$
• Seasonal model (${\mu }_t={\mu }_{t+12}$): ${\hat{Y}}_t(l)={\hat{Y}}_t(l+12)$