Fixed effects in first-differenced model

[Xavier Pedros]

Hi!

I have a first-differenced model, where my units are municipalities. I am adding regional fixed effects (a unit higher than the municipality level). I understand that the latter captures regional trends (common for municipalities within each region), in both the outcome and the explanatory variable.

Is this understanding correct?

Many thanks!

[Fei Wang]

Yes, it's correct. Or to be more specific, the regional FEs in the first-differenced morel capture the regional linear trends of the original outcome.

[Xavier Pedros]

Yes, it's correct. Or to be more specific, the regional FEs in the first-differenced morel capture the regional linear trends of the original outcome.

Thanks Fei!

[Xavier Pedros]

Yes, it's correct. Or to be more specific, the regional FEs in the first-differenced morel capture the regional linear trends of the original outcome.

Apologies Fei, just one more question.

My full model is estimated with a panel data, where I have municipalities across multiple periods. The model writes:

ΔY_it = a + bΔX_it + Time FE_t + Regional FE_i + Error

where i is a municipality, and t is time. Δ is the change, in between time t.

Without regional FE, the parameter of my variable of interest (X) is identified on the basis of within-municipality variation, over time.

As you said, adding regional FE accounts for regional linear trends in the outcome. Alternatively, is it correct to say that, in this model, the parameter of my variable of interest is identified on the basis of municipalities within the same regions? E.g., within-municipality variation, across municipalities in the same region?

[Fei Wang]

Apologies Fei, just one more question.

My full model is estimated with a panel data, where I have municipalities across multiple periods. The model writes:

ΔY_it = a + bΔX_it + Time FE_t + Regional FE_i + Error

where i is a municipality, and t is time. Δ is the change, in between time t.

Without regional FE, the parameter of my variable of interest (X) is identified on the basis of within-municipality variation, over time.

As you said, adding regional FE accounts for regional linear trends in the outcome. Alternatively, is it correct to say that, in this model, the parameter of my variable of interest is identified on the basis of municipalities within the same regions? E.g., within-municipality variation, across municipalities in the same region?

I don't think so. Regional FEs are not the crucial part. The key is that the original model allows for municipality FEs and the first-differenced model eliminates them. In other words, you have actually controlled for municipality FEs which makes your identification mainly come from within-municipality variation. The inclusion of the regional FEs (or regional FEs * t in the original model) merely takes away some within-municipality variation, and leaves the rest for identification. To sum up, it's important to think of your original model for the source of identification.

[Xavier Pedros]

I don't think so. Regional FEs are not the crucial part. The key is that the original model allows for municipality FEs and the first-differenced model eliminates them. In other words, you have actually controlled for municipality FEs which makes your identification mainly come from within-municipality variation. The inclusion of the regional FEs (or regional FEs * t in the original model) merely takes away some within-municipality variation, and leaves the rest for identification. To sum up, it's important to think of your original model for the source of identification.

Understood. So, put it another terms: identification comes from within-municipality variation, controlling for common trends that are shared in municipalities of the same region.