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comment factor models and using cross section regression
It is an immensely popular / common approach (and commercially successful too as the good folks at Barra would tell you). Of course, I do find while on one hand it's easy to generalize, on the other hand, it's very difficult to have a one-glove-fits-all kind of approach. For most fund managers, assessing portfolio risk cross-sectionally makes far greater sense to estimate portfolio [incidental or otherwise] biases and exposures.
comment factor models and using cross section regression
Indeed. let's say you have a pool of factors which you believe may have crossover information. the goal is to strip the factors of this linked information and therefore using regressed residuals as clean factor scores.
comment factor models and using cross section regression
Speaking from equity quant factor building experience, it is a common practice to build multi-factor models by regressing one component against other(s) and using the residual scores. This is done to avoid bias as you mentioned - these biases could be from the factor itself (in different regimes, Quality / Momentum influencing each other - or earnings, value bringing opposite extremes to the portfolio ranking etc.). Robeco have done a few papers on regressed residual factors (mostly momentum ). one of the papers can be found here. papers.ssrn.com/sol3/papers.cfm?abstract_id=1911449
comment How to see the impact of one variable on a set of other variables?
Also, if the end goal is to look at building a Model that incorporates macro impacts, there are several papers about their impact onto the equity portfolios / market. A more recent paper is (papers.ssrn.com/sol3/papers.cfm?abstract_id=2445086)
comment Components of an index in a specific date
To add to Chris's information on historical changes, you could look at their quarterly index review announcements - these list the additions / deletions from their benchmark indices. This [link] (ftse.com/Indices/UK_Indices/Index_Reviews.jsp) here has the announcements for FTSE UK indices and they provide this information in excel files (HTML pre 2004, I think). The information on this website however does not go back to the 90s so you'll have to combine the PDF mentioned above with these files to work backwards into point in time constituents