# Rolling intercept estimation for a data panel

I am working with a panel containing $$T$$ daily observations (one for each $$t = 1, 2, ..., T$$) for each element $$i$$ ($$i = 1, 2, ..., N$$) of the dependent variable $$y$$. The panel is unbalanced because some of the $$y_{it}$$ were not observed for all days. Specifically, $$y_{it}$$ would be the return of company $$i$$ on day $$t$$.

I am interested in calculating, for each element $$i$$ and for each month $$t'$$ ($$t' = 1, 2, ..., T'$$), the intercept coefficient from a regression of the $$y_{it}$$ on a common factor $$x_t$$ (think a market index return). $$x_t$$ is not specific to any $$i$$ element (because the market index is unique), but can of course be expressed in the form $$X_{it}$$ by simply replicating its values on each day for each element $$i$$ in the sample.

I have though of two possible procedures to achieve my objective:

1. Run the following time series regression for each $$i$$ ($$i = 1, 2, ..., N$$) and each month $$t'$$ ($$t' = 1, 2, ..., T'$$) in the sample: $$y_{it} = \alpha_i + b_ix_t + u_{it}$$. This should result in a series of $$T'$$ $$\alpha_i$$ values, one for each element $$i$$ and each month $$t'$$.
2. Alternatively, run a fixed-effects panel regression at each month $$t'$$: $$Y_{it} = \beta X_{it} + \alpha_i + u_{it}$$, where $$\alpha_i$$ would be the time-invariant (within each month $$t'$$) individual effect (for each $$i$$-th individual firm).

First, are the two procedures correctly defined? Second, how can I determine which of the two is more appropriate for my analysis?

A third alternative would be to consider one single equation with $$\alpha_{i,t'}$$: $$Y_{i,t}=\beta X_{i,t}+\alpha_{i,t'}+u_{i,t}$$
• I see what you mean and I agree it would be optimal to estimate everything at once, but how would you do it in practice? What I mean is: if you run a fixed effects regression using daily data over the entire period you would get a single $\alpha_i$ for each firm $i$ over the entire period. How can you actually estimate a different $\alpha_{i,t'}$ for each firm $i$ and at each month $t'$? Commented Jul 24, 2019 at 8:58