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11.4 Stability and forecastability conditions of ETSX

It can be shown that any ETSX{S} is not stable (as it was defined for pure additive ETS models), meaning that the weights of such model do not decline exponentially to zero. This becomes apparent, when we compare the explanatory part of any ETSX with a deterministic model, discussed in the context of pure additive models. For example, we have already discussed that when $$\alpha=0$$ in ETS(A,N,N), then the model becomes equivalent to the global level, looses the stability condition, but still can be forecastable. Similarly, the X part of ETSX{S} will always be not stable, but forecastable. This means that the stability / forecastability conditions should be checked for the ETS part, ignoring the X.

When it comes to the dynamic ETSX, then the situation changes, because now the smoothing parameters for the coefficients of the model determine, how weights decline over time. It can be shown based on (6.8) that the values of the state vector on the observation $$t$$ can be calculated via the recursion (here we provide formula for the non-seasonal case, keeping in mind that in case of the seasonal one, the derivation and the main message will be similar): $$$\mathbf{v}_{t} = \prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0} + \sum_{j=0}^{t-1} \prod_{i=0}^{j} \mathbf{D}_{t-i} y_{t-j}, \tag{11.23}$$$

where $$\mathbf{D}_t=\mathbf{F} - \mathrm{diag}\left(\mathbf{w}_{t}\right)^{-1} \mathbf{g} \mathbf{w}_{t}'$$ is the time varying discount matrix. The main issue in the case of dynamic ETSX is that the stability condition varies over time together with the values of explanatory variables. So, it is not possible to derive it 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 $$\mathbf{\bar{D}}=\frac{1}{t}\sum_{j=1}^t\mathbf{D}_t$$ all lie in the unit circle. This way, some of observations might have a higher impact on the final value, but they will be canceled out by those that have much lower weights. 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 ETSX{D} it should be (based on and the same logic as in section 6.4): $$$\lim\limits_{t\rightarrow\infty}\left(\mathbf{w}'_{t}\prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0}\right) = \text{const} . \tag{11.24}$$$