Separating industry and market effects for an equity factor portfolio

I am running a regression to capture the risk factor exposures for a security and estimate its returns.

To explain the variation in the security's returns, the predictor variables include a "general equity" factor (an index such as S&P 500 representing general market movements common to all industries) as well as an industry-specific component (e.g a "US Financials" index consisting of bank/financials stocks within the S&P 500).

Naturally there will be a high degree of collinearity between the two factors, and overlap between the securities contained in each factor index. In the regression, it is difficult to disaggregate the competing effects of each factors since they move closer together.

Hence I need to adjust the industry factor so that it is a "pure" industry factor. What are some ways that I could do this while still preserving the economic interpretation of the factor? More generally, what is the procedure for adjusting factors so that they are "pure" factor portfolios and don't include incidental exposure to other factors in the analysis?

In your case, this would amount to starting with the market factor and then adjusting your industry factor by regressing it against the market factor and subtracting the beta-adjusted market returns from the industry returns ($r_{I,adj} = r_I - \beta_I r_m$). You can then regress the security on the adjusted industry returns and the market returns.