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One common way to construct portfolio is a high - low factor portfolio. First you sort the asset classes based on a particular factor. For example if the regression co-ef is positive implying positive risk premia, you sort them in ascending order of the factor, and opposite for negative co-ef. After that you percentile this sorted series and decide a ...

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You are right that it is a "very arbitrary procedure". More charitably it is a "hack" that gives a practical solution without addressing the fundamental issue. The very fact that when you do an OLS regression of x vs y you get a different result than when you regress y vs x tells you that OLS regression is probably not the right tool to construct a hedged ...

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What I would do: run the regression twice. The first time use only the time periods with $I_{t-1} = 0$ The resulting Alpha and Beta will be estimates of $\alpha_0,\beta_0$ Now run the regression a second time, using the other data points, those with $I_{t-1} = 1$. The resulting Alpha and Beta are estimates of $\alpha_0+\alpha_B,\beta_0+\beta_B$. By ...

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