6.2 Recursive relation

Similarly to how it was done for the pure additive model in Section 5.2, we can show what the recursive relation will look like for the pure multiplicative one (the logic here is the same, the main difference is in working with logarithms instead of the original values): \[\begin{equation} \begin{aligned} \log y_{t+h} = & \mathbf{w}_{m_1}^\prime \mathbf{F}_{m_1}^{\lceil\frac{h}{m_1}\rceil-1} \log \mathbf{v}_{t} + \mathbf{w}_{m_1}^\prime \sum_{j=1}^{\lceil\frac{h}{m_1}\rceil-1} \mathbf{F}_{m_1}^{j-1} \log \left(\mathbf{1}_k + \mathbf{g}_{m_1} \epsilon_{t+m_1\lceil\frac{h}{m_1}\rceil-j}\right) + \\ & \mathbf{w}_{m_2}^\prime \mathbf{F}_{m_2}^{\lceil\frac{h}{m_2}\rceil-1} \log \mathbf{v}_{t} + \mathbf{w}_{m_2}^\prime \sum_{j=1}^{\lceil\frac{h}{m_2}\rceil-1} \mathbf{F}_{m_2}^{j-1} \log \left(\mathbf{1}_k + \mathbf{g}_{m_2} \epsilon_{t+m_2\lceil\frac{h}{m_2}\rceil-j}\right) + \\ & \dots \\ & \mathbf{w}_{m_d}^\prime \mathbf{F}_{m_d}^{\lceil\frac{h}{m_d}\rceil-1} \log \mathbf{v}_{t} + \mathbf{w}_{m_d}^\prime \sum_{j=1}^{\lceil\frac{h}{m_d}\rceil-1} \mathbf{F}_{m_d}^{j-1} \log \left(\mathbf{1}_k + \mathbf{g}_{m_d} \epsilon_{t+m_d\lceil\frac{h}{m_d}\rceil-j}\right) + \\ & \log \left(1 + \epsilon_{t+h}\right) \end{aligned}. \tag{6.7} \end{equation}\] In order to see how this recursion works, we can take the example of ETS(M,N,N), for which \(m_1=1\) and all the other lags are equal to zero: \[\begin{equation} y_{t+h} = \exp\left(\mathbf{w}_{1}^\prime \mathbf{F}_{1}^{h-1} \log\mathbf{v}_{t} + \mathbf{w}_{1}^\prime \sum_{j=1}^{h-1} \mathbf{F}_{1}^{j-1} \log \left(\mathbf{1}_k + \mathbf{g}_{1} \epsilon_{t+h-j}\right) +\log \left(1 + \epsilon_{t+h}\right)\right) , \tag{6.8} \end{equation}\] or after inserting \(\mathbf{w}_{1}=1\), \(\mathbf{F}_{1}=1\), \(\mathbf{v}_{t}=l_t\), \(\mathbf{g}_{1}=\alpha\) and \(\mathbf{1}_k=1\): \[\begin{equation} y_{t+h} = l_t \prod_{j=1}^{h-1} \left(1 + \alpha \epsilon_{t+h-j}\right) \left(1 + \epsilon_{t+h}\right) . \tag{6.9} \end{equation}\]

This recursion is useful to understand how the states evolve, and in the case of ETS(M,N,N), it allows obtaining the conditional expectation and variance. Unfortunately, in general, for models with trend and/or seasonality, the recursion (6.7) cannot be used to calculate conditional moments, like the one for the pure additive ADAM ETS. This is discussed in the next Section 6.3.