Overview
The multi-index factor model is given as
{% r = \alpha + \sum_i \beta_i \times f_i + e %}
This equation is linear in the factors and hence can be estimated using
linear regression,
as in the single factor model.
When the underlying factors are correlated, you need to take care in interpreting the results of the regression. When this is the case, the analyst needs to consider the effects of multicollinearity.