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6.2 Recursive relation

A useful thing that can be derived from the pure additive model (6.2) is the recursive value, which can be used in several important aspects. First, when we produce forecast for \(h\) steps ahead, it is important to understand what the actual value \(h\) steps ahead might be, given all the information we have on the observation \(t\): \[\begin{equation} \begin{aligned} {y}_{t+h} = &\mathbf{w}' \mathbf{v}_{t-h_\mathbf{l}} + \epsilon_{t+h} \\ \mathbf{v}_{t+h} = &\mathbf{F} \mathbf{v}_{t-h_\mathbf{l}} + \mathbf{g} \epsilon_{t+h} \end{aligned}, \tag{6.5} \end{equation}\] where \(\mathbf{v}_{t-h_\mathbf{l}}\) is the vector of previous states, given the lagged values \(\mathbf{l}\). In order to obtain the recursion, we need to split the measurement and persisitence vectors together with the transition matrix into parts for the same lags of components, leading to the following transition equation in (6.5): \[\begin{equation} \begin{aligned} {y}_{t+h} = &(\mathbf{w}_{m_1}' + \mathbf{w}_{m_2}' + \dots + \mathbf{w}_{m_d}') \mathbf{v}_{t-h_\mathbf{l}} + \epsilon_{t+h} \\ \mathbf{v}_{t+h} = &(\mathbf{F}_{m_1} + \mathbf{F}_{m_2} + \dots + \mathbf{F}_{m_d}) \mathbf{v}_{t-h_\mathbf{l}} + (\mathbf{g}_{m_1} + \mathbf{g}_{m_2} + \dots \mathbf{g}_{m_d}) \epsilon_{t+h} \end{aligned}, \tag{6.6} \end{equation}\] where \(m_1, m_2, \dots, m_d\) are the distinct seasonal frequencies. So, for example, in case of ETS(A,A,A) model on quarterly data (periodicity is equal to four), \(m_1=1\), \(m_d=4\), leading to \(\mathbf{F}_{m_1} = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) and \(\mathbf{F}_{m_2} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). This split of matrices and vectors into distinct sub matrices and subvectors is needed in order to get the correct recursion and obtain the correct conditional expectation and variance. By substituting the values in the transition equation of (6.6) by their previous values until we reach \(t\), we get: \[\begin{equation} \begin{aligned} \mathbf{v}_{t-h_\mathbf{l}} = & \mathbf{F}_{m_1}^{\lceil\frac{h}{m_1}\rceil-1} \mathbf{v}_{t} + \sum_{j=1}^{\lceil\frac{h}{m_1}\rceil-1} \mathbf{F}_{m_1}^{j-1} \mathbf{g}_{m_1} \epsilon_{t+m_1\lceil\frac{h}{m_1}\rceil-j} + \\ & \mathbf{F}_{m_2}^{\lceil\frac{h}{m_2}\rceil-1} \mathbf{v}_{t} + \sum_{j=1}^{\lceil\frac{h}{m_2}\rceil-1} \mathbf{F}_{m_2}^{j-1} \mathbf{g}_{m_2} \epsilon_{t+m_2\lceil\frac{h}{m_2}\rceil-j} + \\ & \dots \\ & \mathbf{F}_{m_d}^{\lceil\frac{h}{m_d}\rceil-1} \mathbf{v}_{t} + \sum_{j=1}^{\lceil\frac{h}{m_d}\rceil-1} \mathbf{F}_{m_d}^{j-1} \mathbf{g}_{m_d} \epsilon_{t+m_d\lceil\frac{h}{m_d}\rceil-j} \end{aligned}. \tag{6.7} \end{equation}\] Inserting (6.7) in the measurement equation of (6.6), we will get: \[\begin{equation} \begin{aligned} y_{t+h} = & \mathbf{w}_{m_1}' \mathbf{F}_{m_1}^{\lceil\frac{h}{m_1}\rceil-1} \mathbf{v}_{t} + \mathbf{w}_{m_1}' \sum_{j=1}^{\lceil\frac{h}{m_1}\rceil-1} \mathbf{F}_{m_1}^{j-1} \mathbf{g}_{m_1} \epsilon_{t+m_1\lceil\frac{h}{m_1}\rceil-j} + \\ & \mathbf{w}_{m_2}' \mathbf{F}_{m_2}^{\lceil\frac{h}{m_2}\rceil-1} \mathbf{v}_{t} + \mathbf{w}_{m_2}' \sum_{j=1}^{\lceil\frac{h}{m_2}\rceil-1} \mathbf{F}_{m_2}^{j-1} \mathbf{g}_{m_2} \epsilon_{t+m_2\lceil\frac{h}{m_2}\rceil-j} + \\ & \dots \\ & \mathbf{w}_{m_d}' \mathbf{F}_{m_d}^{\lceil\frac{h}{m_d}\rceil-1} \mathbf{v}_{t} + \mathbf{w}_{m_d}' \sum_{j=1}^{\lceil\frac{h}{m_d}\rceil-1} \mathbf{F}_{m_d}^{j-1} \mathbf{g}_{m_d} \epsilon_{t+m_d\lceil\frac{h}{m_d}\rceil-j} + \\ & \epsilon_{t+h} \end{aligned}. \tag{6.8} \end{equation}\] Substituting the specific values of \(m_1, m_2, \dots, m_d\) in (6.8) will simplify the equation and make it easier to understand. For example, for ETS(A,N,N) \(m_1=1\) and all the other frequencies are equal to zero, so the recursion (6.8) simplifies to: \[\begin{equation} y_{t+h} = \mathbf{w}_{1}' \mathbf{F}_{1}^{h-1} \mathbf{v}_{t} + \mathbf{w}_{1}' \sum_{j=1}^{h-1} \mathbf{F}_{1}^{j-1} \mathbf{g}_{1} \epsilon_{t+h-j} + \epsilon_{t+h} , \tag{6.9} \end{equation}\]

which is the recursion obtained by Hyndman et al. (2008).

References

Hyndman, Rob J., Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.