Definition of Residuals

One of the advantages of residual minimization is that one may see directly how the objective depends on unknown parameters. Using equalities (1.1) we define residuals by recurrent expressions:

$$\epsilon_1 = w_1$$

$$\epsilon_2 = w_2-a_1 w_1 - b_1 \epsilon_1)$$

$$..........................................$$

$$\epsilon_t = w_t-a_1 w_{t-1} - ... -a_p w_{t-p} - b_1 \epsilon_{t-1} - ... - b_q \epsilon_{t-q}.$$

Next the sum

$$f(x)=\log f_m(x),\ \ f_m(x)= \sum_{t=1}^T \epsilon_t^2$$ (4)

is minimized.

The logarithm is used to decrease the objective variation by improving the scales.