15.2 Modeling the trend

• The simplest models for trend are regressions of the variable \(y\) with respect to time..

• A linear trend model is:

\[t_{t}=\alpha_{0}+\alpha_{1} t, \,\, \,\, \forall t=1,2, \ldots, T.\]

The parameter \(\alpha_{1}\) represents the slope and describes the expected growth between two periods. The forecast of the variable \(y\) for the period \(T+k\) is:

\[\hat{y}_{T}(k)=\hat{\alpha}_{0}+\hat{\alpha}_{1}(T+k)\]

• A quadratic trend model is:

\[t_{t}=\alpha_{0}+\alpha_{1} t+\alpha_{2} t^{2}, \,\,\,\, \forall t=1,2, \ldots, T.\]

• An exponential trend model is:

\[ t_t = \exp \left ( \alpha_0 +\alpha_1 t \right ) .\]

By taking logarithms is:

\[\log t_{t} = \alpha_{0} + \alpha_{1} t + \varepsilon_{t}\]