## 10.4 Stability and forecastability conditions of ADAMX

It can be shown that any **static** ADAMX is not *stable* (as it was defined for pure additive ETS models in Section 5.4), meaning that the weights of such models do not decline to zero over time. To see this, we can draw an analogy with a deterministic model, discussed in the context of pure additive ETS (from Section 5.4). For example, we have already discussed that when \(\alpha=0\), the ETS(A,N,N) model becomes equivalent to the global level, loses the stability condition, but still can be forecastable. It becomes a simple but still useful and efficient model:
\[\begin{equation}
y_{t} = a_0 + \epsilon_t .
\tag{10.27}
\end{equation}\]
Similarly, the X part of ADAMX will always be unstable, but can be useful. For example, with \(\alpha=0\), ETSX(A,N,N) reverts back to the linear regression:
\[\begin{equation}
y_{t} = a_0 + a_{1} x_{1,t} + \dots + a_n x_{n,t} + \epsilon_t ,
\tag{10.28}
\end{equation}\]
which is not stable, because its weights do not decline over time, and the very first observation (a set of initial states with model parameters) impacts the final forecast. This does not make the model inappropriate in any way, but according to the conventional approach to ETS, if the dynamic part of the model is stable, but the overall model does not pass the stability check just because of the X part, then the whole model will be considered unstable and potentially dangerous to use. This is absurd. Following the same logic, we would need to avoid regression models in forecasting because they are not stable. Furthermore, there are no issues constructing ARIMAX models, but Osman and King (2015) argue that there are some with ETSX, which does not make sense if we recall the connection between ETS and ARIMA (discussed in Section 8.4). This only means that the stability/forecastability conditions should be checked for the dynamic part of the model (ETS or ARIMA) separately, ignoring the X part. Technically, this implies creating a separate transition matrix, persistence and measurement vectors, and calculating the discount matrix for the ETS/ARIMA part to check already discussed stability and forecastability conditions (Section 5.4).

When it comes to the **dynamic** ADAMX, the situation changes because now the smoothing parameters for the model coefficients determine how weights decline over time. It can be shown based on (5.10) that the values of the state vector on the observation \(t\) can be calculated via the recursion (here we provide a formula for the non-seasonal case, keeping in mind that in the case of the seasonal one, the derivation and the main message will be similar):
\[\begin{equation}
\mathbf{v}_{t} = \prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0} + \sum_{j=0}^{t-1} \prod_{i=0}^{j} \mathbf{D}_{t-i} y_{t-j},
\tag{10.29}
\end{equation}\]
where \(\mathbf{D}_t=\mathbf{F} -\mathbf{z}_t \mathbf{g} \mathbf{w}_{t}'\) is the time varying discount matrix. The main issue in the case of dynamic ADAMX is that the stability condition varies over time together with the values of explanatory variables in \(\mathbf{z}_t\). So, it is not possible to derive the stability condition for the general case. In order to make sure that the model is stable, we need for all eigenvalues of each \(\mathbf{D}_{j}\) for all \(j=\{1,\dots,t\}\) to lie in the unit circle.

Alternatively, we can introduce a new condition. We say that the model is **stable on average** if the eigenvalues of the geometric mean \(\mathbf{\bar{D}}=\sqrt[t]{\prod_{j=1}^t\mathbf{D}_j}\) all lie in the unit circle. This way, some of the observations might have a higher impact on the final value, but they will be cancelled out by those with much lower weights in the product in (10.29). This condition can be checked during the model estimation, similar to how the conventional stability condition is checked.

As for the **forecastability** condition, for the ADAMX{D} it should be (based on the same logic as in Section 5.4):
\[\begin{equation}
\lim\limits_{t\rightarrow\infty}\left(\mathbf{w}'_{t}\prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0}\right) = \text{const} .
\tag{10.30}
\end{equation}\]
However, this condition will always be violated for the ADAMX models, just because the explanatory variables in \(\mathbf{w}_{t}\) have their own variability and typically do not converge to a stable value with the increase of the sample size. So, if a forecastability condition needs to be checked for either ADAMX{D} or ADAMX{S}, it should be checked separately for the dynamic part of the model, dropping the X part.