External Factors

Often we have to predict one main factor which depends on some external factors. This means that we are not interested in future values of external factors while predicting the main one. For example, we predict the stock rate as the main factor depending on the GNP as the external factor. We consider the cases when future values of all the factors are not known. Regarding the cases when future values of external factors are known the model should be modified accordingly.

Different symbols are used, regarding the ARMA model with external factors. The predicted value is denoted by $\nu(t)$ and the external factors as $\eta(t)=(\eta_1(t),\eta_2(t),...)$. An influence of some external factors may be delayed.

For illustration, consider two-dimensional case, omitting the Moving Average (MA) part. This means the two-dimensional Auto Regression (AR) model with an external factor $\eta(t)$

$$\nu(t)=\sum_i^p (a_{1i} \nu(t-i)+a_{2i} \eta(t-d-i))+\epsilon(t),\ p+d < t < T_0.$$ (21)

Here $d$ is the delay parameter. First we minimize the squared deviation $\sum_t \epsilon(t)^2$ at fixed delay $d$ and then chose the optimal delay $d$. The minimum of sum (1.22), as a function of parameters $a$, is defined by the system of linear equations. which are These are obtained from the condition that all the partial derivatives are zero

$$\sum_{t=p+d+1}^{T_0-1} ( \nu(t)-\sum_i^p (a_{1i} \nu(t-i)+a_{2i} \eta(t-d-i)) \nu(t-i)=0,$$ (22)

$$\sum_{t=p+d+1}^{T_0-1} ( \nu(t)-\sum_i^p (a_{1i} \nu(t-i)+a_{2i} \eta(t-d-i)) \eta(t-d-i)=0$$ (23)

$$\ i=1,...,p.$$

We have to solve $2p$ linear equations with $2p$ variables $a_{1i},a_{2i},\ i=1,...,p$ and obtain the least squares estimates $a_{1i}(d),a_{2i}(d),\ i=1,...,p$ at given $d$.

Sum (1.25), as a function of delay parameter $d$, is not necessarily uni-modal one. Thus to define the exact minimum

$$\nu(t)=\sum_i^p (a_{1i} (d)\nu(t-i)+a_{2i}(d) \eta(t-d-i))+\epsilon(t),\ p+d < t < T_0$$ (24)

one should consider all the $K(d)$ values of the integer $d$. Here $K(d)$ is a number of "interesting" values of $d$. Therefore expression (1.25) is an example of multi-modal function in the AR models. We mentioned the delay time just to show a way how to include this important factor into a time series model. Later on the delay factor will be omitted, to simplify the expressions.