I want to measure the positioning of an active bond mutual fund vs. its benchmark via rolling linear regression of returns vs several factors. The intuition of using linear regression is that the returns of the index or of the fund ($r$) should be a linear function of the returns of several sub indices (i.e. rates $r_{\text{10yr}}$, credit spreads $r_{\text{cred spreads}}$, etc..).
The regression takes the form: $$r = \beta_1 \times r_{\text{10yr}} + \beta_2 \times r_{\text{cred spreads}} + \cdots$$
The R-squared of the regression of the index vs. the sub-indices is ~95%. Likewise the R-squared of the regression of the mutual fund vs. the sub indices is ~85%.
I want to know whether it is better to measure relative positioning using the difference in the Betas to each factor from the regressions, i.e $\beta_{\text{fund}}$ - $\beta_{\text{index}}$ or if it is better to run a single regression of excess returns of the fund over the benchmark vs. the same set of factors:
$$(r_{\text{fund}}-r_{\text{index}}) = \beta_1 \times r_{\text{10yr}} + \beta_2 \times r_{\text{cred spreads}} + \cdots $$
I believe running separate regressions allows for the error terms to be estimated with different variances but what other factors should I think about? A third possibility is to run a panel regression were sensitivities are estimated simultaneously but the marginal effect is captured through an interaction term for each variable/fund combo.
