Chapter 11 Estimation of ADAM
Now that we have discussed the properties of ETS, ARIMA, ETSX, and ARIMAX models, we need to understand how to estimate them. As mentioned earlier, when we apply a model to the data, we assume that it is suitable and we need to see how it fits the data and produces forecasts to assess this suitability. In this case, all the model parameters are substituted by their estimates (observed in the sample), and the error term becomes an estimate of the true one. In general, this means that the state space model (7.1) is substituted by: \[\begin{equation} \begin{aligned} {y}_{t} = &w(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) + r(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) e_t \\ \hat{\mathbf{v}}_{t} = &f(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) + \hat{g}(\hat{\mathbf{v}}_{t-\boldsymbol{l}}) e_t \end{aligned}, \tag{11.1} \end{equation}\] implying that the initial values of components and the smoothing parameters of the model are estimated. An example is the ETS(A,A,A) model applied to the data: \[\begin{equation} \begin{aligned} \hat{y}_{t} = & \hat{l}_{t-1} + \hat{b}_{t-1} + \hat{s}_{t-m} \\ e_t = & y_t - \hat{y}_{t} \\ \hat{l}_t = & \hat{l}_{t-1} + \hat{b}_{t-1} + \hat{\alpha} e_t \\ \hat{b}_t = & \hat{b}_{t-1} + \hat{\beta} e_t \\ \hat{s}_t = & \hat{s}_{t-m} + \hat{\gamma} e_t \end{aligned}, \tag{11.2} \end{equation}\] where the initial values \(\hat{l}_0, \hat{b}_0\), and \(\hat{s}_{-m+2}, ... \hat{s}_0\) are estimated and then influence all the future values of components via the recursion (11.2) and \(e_t = y_t -\hat{y}_t\) is the one step ahead in-sample forecast error, also known in statistics as the residual of the model. The set of equations (11.2) allows constructing the model by applying equations one by one (you can even do that in MS Excel by creating five columns for the respective five equations).
The estimation itself does not happen on its own, a complicated process of minimisation/maximisation of the pre-selected loss function by changing the values of parameters is involved. Typically, there is no analytical solution for estimates of ADAM parameters because of the model’s recursive nature. As a result, numerical optimisation is used to obtain the estimates of parameters. The results of the estimation will differ depending on:
- The assumed distribution;
- The used loss function;
- The initial values of parameters that are fed to the optimiser;
- The parameters of the optimiser (such as sensitivity, number of iterations etc.);
- The sample size;
- The number of parameters to estimate and;
- The restrictions imposed on parameters.
The aspects above are covered in this chapter.