# equities hedging betas for a cross-sectional risk model

This question is on equities risk models. I would like to know how to define betas when using a cross-sectional regression approach, rather than the time series approach. My goal is beta hedging of a portfolio of stocks.

Suppose that the risk factors of interest are GICS sectors factors and a few Fama-French like factors based on company characteristics (size, value,...).

Let's start with sector factors (discrete data):

• For a time series model, I could define a time series of returns for 10 sectors then compute a rolling beta for each portfolio stock independently. So I'd have 10 betas, to hedge the portfolio I'd have 10 linear constraints.
• For a cross-sectional model, the betas are known (covariates in the regression) whereas the returns are unknown. Since I care about beta hedging, I only need to specify the betas. I take sector membership dummy variables (1 if stock belong's to that sector, 0 otherwise). Beta hedging means that for each of the 10 sectors, the sum of portfolio weights is 0.

For Fama-French like factors (continuous company characteristic data):

• For a time series model, I create factors returns. For example, for a style factor, I compute returns on a portfolio formed by Long/Short ranking on each company's market cap. Then I compute betas from a rolling regression for each stock independently.

How would you define the cross-sectional betas in the Fama-French case? If I follow a Fama-McBeth approach, the betas are computed from the time series regression in the first step, so that's equivalent to the time series approach.

More generally, given any company characteristic, for example short interest, how do I define the cross-sectional beta (BARRA style)?

• In Barra style, it's stock x return = beta_1 * short_interest factor_return + beta2 * liquidity factor return + beta_3 * industry_factor_return + dotdotodt. So, you need the construct the loadings ( the betas ) and then, once you have those, the factor returns can be estimated. But, constructing the loadings is involved. See the book by Rudd and Clasing. I think it's called modern portfolio theory. Commented Sep 9, 2023 at 16:00
• This would be useful but it doesn't seem to be free. Atleast I couldn't obtain it.link.springer.com/referenceworkentry/10.1007/… Commented Sep 9, 2023 at 16:03
• first line of initial comment should have read stock X's return so some random stock called $X$. my apologies for confusion. it currently could be read as multiplication so X was a bad choice. Commented Sep 10, 2023 at 4:01
• Yes, I'm not clear how beta_1 is constructed when the factor is continuous. Commented Sep 10, 2023 at 8:36
• Hi: you should get your hands on rudd and clasing. In that book, a lot of details are given in terms of how the each loading is constructed ( each beta if you will ). How much that level of detail helps in terms of the model's ability to capture risk, who knows ? It could be more for "marketing". Commented Sep 10, 2023 at 13:57