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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?

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It can be a tough problem to make the factors independent while retaining the "economic interpretation" of the factor. People often used to start by using the Gram-Schmidt Orthonormalization technique.

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.

If you have more than two factors, then you can keep going with this approach and subtract other factors using this technique. However, this is where the problem with "economic interpretation" comes in. The more factors you have, the harder it is to interpret the final adjusted factor. This is because you would now be looking at betas against factors adjusted for market and other industry returns. The returns of the final adjusted factor also depend on the order in which the factors are adjusted. This means that you would be regressing against quite abstract factors rather than easily observable market indices.

Gram-Schmidt Orthonormalization can also be quite numerically unstable if there are a lot of highly-correlated factors. You can use other methods to adjust the factors such as PCA. Unfortunately, these methods typically make it even harder to interpret the final results.

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