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10.4 Stability and forecastability conditions of ADAMX

It can be shown that any ADAMX{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 ADAMX 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 ADAMX{S} will always be unstable, but can be forecastable. However, this is unreasonable from the model building point of view: if for example, the ETS part of the model is stable but the model does not pass stability check just because of the X part, then the check itself is incorrect. Furthremore, there are no issues constructing ARIMAX models, but claim that there are issues with stability of ETSX. 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 separate transition matrix, persistence and measurement vectors and calculating the discount matrix for the ETS part in order to check already discussed stability and forecastability conditions.

When it comes to the dynamic ADAMX, 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 (5.9) 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): $$$\mathbf{v}_{t} = \prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0} + \sum_{j=0}^{t-1} \prod_{i=0}^{j} \mathbf{D}_{t-i} y_{t-j}, \tag{10.23}$$$ 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 ADAMX 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 ADAMX{D} it should be (based on and the same logic as in section 6.4): $$$\lim\limits_{t\rightarrow\infty}\left(\mathbf{w}'_{t}\prod_{j=1}^{t-1}\mathbf{D}_{t-j} \mathbf{v}_{0}\right) = \text{const} . \tag{10.24}$$$ However, for the reasons discussed earlier in this subsection, 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}, we recommend checking it separately for the dynamic part of the model.

References

• Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.