(1) Factors are first standardized. These are time series, so standardization for a factor involves differencing the time-t value from the time series mean and dividing by the standard deviation of the macro factor. (2) Then a time series for each stock's returns is regressed on the standardized factors (also called factor exposures) and the estimated factor sensitivities (also referred to as beta, factor return, or factor loading) are saved. (3) This time series regression is then repeated for the universe of stocks. The resulting population factor sensitivities are saved. (4) Individual factor sensitivities are standardized, that is, the standardized factor sensitivity for the ith asset on the jth factor is given by (bij – bj)/sbj where bj and sbj are the estimated sensitivity and standard error for the jth factor, using the universe of stocks.
A factor sensitivity with value zero means that the stock has average sensitivity to that particular factor. Standardized factor sensitivities can be positive or negative. Output from this model is called a score. For example, output is the sum of the standardized factors and sensitivities products for a firm. For the ith firm,
where denotes the score, βik are the factor returns or sensitivities, and the factors, fi, are ...