# Fixed effects in linear model

I am fitting a linear model on some panel data

$$y_{it} = \sigma_{cr} + x_{it}^\top\beta + \mu_i + \delta_t+ \epsilon_{it}$$

where $$\mu_i$$ is individual fixed effect (in the sense used in econometrics) $$\delta_t$$ are time dummies and $$\sigma_{cr}$$ is a dummy for area of work and area of residence of individual $$i$$ at time $$t$$.

So $$c = c(i,t)$$ where $$c(i,t)$$ is the area of individual i in time t. So $$r = r(i,t)$$ where $$r(i,t)$$ is the area of individual i in time t.

The places of work and the places of residence are defined as political municipalities. Assume that the median number in each $$(c,r)$$-group at a given time $$t_0$$ is is around 5000 measuring group size as $$\#\{i\lvert c(i,t_0)=c,r(i,t_0)=r \}$$.

The dummies $$\sigma_{cr}$$ are only identified by individuals who change either area of residence or area of work. Still I am a bit concerned about the empirical identification. Are there any checks I should/could do to investigate this?

• So $\sigma_{cr}$ could have index $_{it}$ for completeness? – Richard Hardy Nov 5 '19 at 11:56
• yes that is correct – Stop Closing Questions Fast Nov 5 '19 at 12:57
• Just so that your setting is clear: is it possible (or not) that several people have the same area of work and home at a given time? – Alex. C-L - Reinstate Monica Nov 6 '19 at 9:16
• yes that is correct – Stop Closing Questions Fast Nov 6 '19 at 9:19