Hi, I am trying to calculate the portfolio weights of "Pure Factor Portfolios" - which are described by Jose Menchero here: https://www.coursehero.com/file/29680439/Chapter-25pptx/. These are portfolios which have unit exposure to the target factor (e.g. USA, France, Utilities, P/B, Market Cap), and 0 exposure to all other factors.

However, I am having issues when trying to calculate "omega" or the stock weights in the pure factor portfolio, because the industry and country exposures seem to be equally weighted (and not market cap weighted). I wonder if there is a problem with my R, the restriction matrix, as I am confident in the matrix algebra formula used.

the formula i am using is

omega= R(R'X'VXR)^(-1)R'X'V

where:

R= the restriction matrix

X= exposure matrix, all stocks have a "1" for the market, [0, 1] for both all industries and countries and

V= a diagonal matrix, proportional to the SQRT(mcap) of each stock

My question is: If i have a market, country and industry factors (i.e. no style factors yet) - how should the restriction matrix look? Should it consist only of [-1, 0 and +1], or should these be scaled by the mcap weightings of the countries and industries?