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Chapter 12 Estimation of ADAM models

Now that we have discussed the properties and issues of some of ETS 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 see how it fits the data and produces forecasts. In this case all the parameters in the model are substituted by the estimated ones (observed in sample) and the error term becomes an estimate of the true one. In general this means that the state space model (6.1) is substituted by: \[\begin{equation} \begin{aligned} {y}_{t} = &w(\hat{\mathbf{v}}_{t-\mathbf{l}}) + r(\hat{\mathbf{v}}_{t-\mathbf{l}}) e_t \\ \hat{\mathbf{v}}_{t} = &f(\hat{\mathbf{v}}_{t-\mathbf{l}}) + \hat{g}(\hat{\mathbf{v}}_{t-\mathbf{l}}) e_t \end{aligned}, \tag{12.1} \end{equation}\] 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: \[\begin{equation} \begin{aligned} y_{t} = & \hat{l}_{t-1} + \hat{b}_{t-1} + \hat{s}_{t-m} + e_t \\ \hat{l}_t = & \hat{l}_{t-1} + \hat{b}_{t-1} + \hat{\alpha} e_t \\ \hat{b}_t = & \hat{b}_{t-1} + \hat{\beta} e_t \\ \hat{s}_t = & \hat{s}_{t-m} + \hat{\gamma} e_t \end{aligned}, \tag{12.2} \end{equation}\] 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 (12.2).

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. The results of this will differ depending on:

  1. what distribution you assume,
  2. what loss you use,
  3. what the initial values of parameters that you feed to the optimiser are,
  4. what the parameters of the optimiser are,
  5. what the sample size is,
  6. how many parameters you need to estimate and
  7. what restriction you impose on parameters.

All of these aspects will be covered in this chapter.

Note that the discussions in this chapter are widely applicable to other dynamic models, such as ARIMA, which will be discussed later in this textbook.