Applying ARMA to External Factors

Omitting the delay $d$ in expression (1.22) we write

$$\nu(t)=\sum_i^p (a_{1i} \nu(t-i)+a_{2i} \eta(t-i))+\epsilon(t),\ p < t < T_0$$ (26)

Including this expression into the ARMA model (1.6) one obtains the following expression

$$\nu(t)=\sum_i^p (a_{1i} \nu(t-i)+a_{2i} \eta(t-i))+$$

$$\sum_{i=1}^q b_{i} \epsilon_{t-i} +\epsilon_t ,\ p < t < T_0$$ (27)

Extending expression (1.28) to $M$ external factors

$$w^M(t)= \sum_{m=1}^{M}\sum_i^p a_{i}^m w^m(t-i) +$$

$$\sum_{i=1}^q b _{i} \epsilon_{t-i} +\epsilon_t ,\ p < t < T_0.$$ (28)

In expression (1.29) the predicted and the external factors are denoted by the same letter $w$ with different upper indices. This case can be represented as the one-dimensional time series of the special type

$$w(Mt)=\sum_{m=1}^{M } \sum_i^p \alpha_{Mi-(M-m)} w(M(t-i)-(M-m))+$$

$$\sum_{i=1}^q b_i \epsilon_{M(t-i)} +\epsilon_{Mt} ,\ p < t < T_0$$ (29)

Here $\alpha_{Mi-(M-m)}=a_{i}^m,\ w(M(t-i)-(M-m))=w^m(t),\ m=1,...,M$ and the index $mi=m*i$. This expression gives the possibility to apply the software developed for the one-dimensional ARMA model to the case of $M$ external factors by representing the data file in the way corresponding to the expression (1.30).