\( \newcommand{\mathbbm}[1]{\boldsymbol{\mathbf{#1}}} \)

10.1 ADAMX: Model formulation

As discussed previously, there are two types of error terms in ADAM:

  1. Additive, discussed in Chapter 5 in the case of ETS and Chapter 9 for ARIMA;
  2. Multiplicative, covered in Chapter 6 for ETS and in Subsection 9.1.4 for ARIMA.

The inclusion of explanatory variables in ADAMX is determined by the type of the error, so that in case of (1) the measurement equation of the model is: \[\begin{equation} {y}_{t} = a_{0,t-1} + a_{1,t-1} x_{1,t} + a_{2,t-1} x_{2,t} + \dots + a_{n,t-1} x_{n,t} + \epsilon_t , \tag{10.1} \end{equation}\] where \(a_{0,t-1}\) is the point value based on all ETS components (for example, \(a_{0,t-1}=l_{t-1}\) in case of ETS(A,N,N)), \(x_{i,t}\) is the \(i\)-th explanatory variable, \(a_{i,t-1}\) is its parameter, and \(n\) is the number of explanatory variables. We will denote the estimated parameters of such models as \(\hat{a}_{i,t-1}\). In the simple case, the transition equation for such a model would imply that the parameters \(a_{i,t}\) do not change over time and are equal to some fixed value: \[\begin{equation} a_{i,t} = a_{i,t-1} = \dots = a_{i,0} \text{ for all } i = 1, \dots, n . \tag{10.2} \end{equation}\] Various complex mechanisms for the states update can be proposed instead of (10.2), but we do not discuss them at this point. Typically, the initial values of parameters would be estimated at the optimisation stage, either based on likelihood or some other loss function, so the index \(t\) can be dropped, substituting \(a_{i,t}=a_{i}\) for all \(i=1,\dots,n\).

When it comes to the multiplicative error model, it should be formulated differently. The most straight forward would be to formulate the model in logarithms in order to linearise it: \[\begin{equation} \log {y}_{t} = \log a_{0,t-1} + a_{1,t-1} x_{1,t} + a_{2,t-1} x_{2,t} + \dots + a_{n,t-1} x_{n,t} + \log(1+ \epsilon_t). \tag{10.3} \end{equation}\]

Remark. If a log-log model is required, all that needs to be done, is to substitute \(x_{i,t}\) with \(\log x_{i,t}\).

The model (10.3) aligns with both pure multiplicative ETS and ARIMA, discussed, respectively, in Chapter 6 and in Subsection 9.1.4.

The compact form of the ADAMX model implies that the explanatory variables \(x_{i,t}\) are included in the measurement vector \(\mathbf{w}_{t}\), making it change over time. The parameters are then moved to the state vector, and a diagonal matrix is added to the existing transition matrix to reflect the updating mechanism (10.2). Finally, the persistence vector for the parameters of explanatory variables should contain zeroes, because for now we assume that the parameters do not change over time. The pure additive state space model, in that case, can be represented as: \[\begin{equation} \begin{aligned} & {y}_{t} = \mathbf{w}'_t \mathbf{v}_{t-\boldsymbol{l}} + \epsilon_t \\ & \mathbf{v}_t = \mathbf{F} \mathbf{v}_{t-\boldsymbol{l}} + \mathbf{g} \epsilon_t \end{aligned} , \tag{10.4} \end{equation}\] while the pure multiplicative models is: \[\begin{equation} \begin{aligned} {y}_{t} = & \exp\left(\mathbf{w}'_t \log \mathbf{v}_{t-\boldsymbol{l}} + \log(1 + \epsilon_t)\right) \\ \log \mathbf{v}_t = & \mathbf{F} \log \mathbf{v}_{t-\boldsymbol{l}} + \log(\mathbf{1}_k + \mathbf{g} \epsilon_t) \end{aligned}. \tag{10.5} \end{equation}\] So, the only thing that changes in these models in comparison with the conventional ones in Chapters 5 and 6 is the time varying measurement vector \(\mathbf{w}'_t\) instead of the fixed one. For example, in the case of ETSX(A,Ad,A) we will have: \[\begin{equation} \begin{aligned} \mathbf{F} = \begin{pmatrix} 1 & \phi & 0 & 0 & \dots & 0 \\ 0 & \phi & 0 & 0 & \dots & 0 \\ 0 & 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & \dots & 1 \end{pmatrix}, & \mathbf{w}_t = \begin{pmatrix} 1 \\ \phi \\ 1 \\ x_{1,t} \\ \vdots \\x_{n,t} \end{pmatrix}, & \mathbf{g} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \\ & \mathbf{v}_{t} = \begin{pmatrix} l_t \\ b_t \\ s_t \\ a_{1,t} \\ \vdots \\ a_{n,t} \end{pmatrix}, & \boldsymbol{l} = \begin{pmatrix} 1 \\ 1 \\ m \\ 1 \\ \vdots \\ 1 \end{pmatrix} \end{aligned}, \tag{10.6} \end{equation}\] which is equivalent to the set of equations: \[\begin{equation} \begin{aligned} & y_{t} = l_{t-1} + \phi b_{t-1} + s_{t-m} + a_{1,t-1} x_{1,t} + \dots + a_{n,t-1} x_{n,t} + \epsilon_t \\ & l_t = l_{t-1} + \phi b_{t-1} + \alpha \epsilon_t \\ & b_t = \phi b_{t-1} + \beta \epsilon_t \\ & s_t = s_{t-m} + \gamma \epsilon_t \\ & a_{1,t} = a_{1,t-1} \\ & \vdots \\ & a_{n,t} = a_{n,t-1} \end{aligned}. \tag{10.7} \end{equation}\] Alternatively, the state, measurement, and persistence vectors and transition matrix can be split, each into two parts, separating the ETS and X parts in the state space equations: \[\begin{equation} \begin{aligned} & {y}_{t} = \mathbf{w}' \mathbf{v}_{t-\boldsymbol{l}} + \mathbf{x}'_{t} \mathbf{a}_{t-1} + \epsilon_t \\ & \mathbf{v}_{1,t} = \mathbf{F} \mathbf{v}_{t-\boldsymbol{l}} + \mathbf{g} \epsilon_t \\ & \mathbf{a}_{t} = \mathbf{a}_{t-1} \end{aligned} , \tag{10.8} \end{equation}\] where \(\mathbf{w}\), \(\mathbf{F}\), \(\mathbf{g}\) and \(\mathbf{v}_{t}\) contain the elements of the conventional components of ADAM, and \(\mathbf{a}_{t}\) is the vector of parameters for the explanatory variables.

When all the smoothing parameters of the ETS part of the model are equal to zero, the ETSX reverts to a deterministic model, becoming just a multiple linear regression. For example, in case of ETSX(A,N,N) with \(\alpha=0\) we get: \[\begin{equation} \begin{aligned} & y_{t} = l_{t-1} + a_{1,t-1} x_{1,t} + \dots + a_{n,t-1} x_{n,t} + \epsilon_t \\ & l_t = l_{t-1} \\ & a_{1,t} = a_{1,t-1} \\ & \vdots \\ & a_{n,t} = a_{n,t-1} \end{aligned}, \tag{10.9} \end{equation}\] where \(l_t=a_0\) is the intercept of the model. (10.9) can be rewritten then in the conventional way, dropping the transition part of the state space model: \[\begin{equation} y_{t} = a_0 + a_{1} x_{1,t} + \dots + a_{n} x_{n,t} + \epsilon_t . \tag{10.10} \end{equation}\] In the case of models with trend and/or seasonal components, the model becomes equivalent to the regression with deterministic trend and/or seasonality. This means that, in general, ADAMX implies that we are dealing with a regression with a time-varying intercept, where the principles of this variability are defined by the ADAM components (e.g. intercept can vary seasonally). Similar properties are obtained with the multiplicative error model. The main difference is that the specific impact of explanatory variables on the response variable will vary with the intercept changes. The model, in this case, combines the strengths of the multiplicative regression and the dynamic model, where the variability of the response variable changes with the change of the baseline model (ADAM ETS and/or ADAM ARIMA in this case).