Difference between two-way fixed effects and interaction of effects

[Sonia Chen]

I have a question about the differences between the formation of fixed effects in a regression model shown below:

Y_ikt = \beta X_ikt + group of fixed effects + error term

I would like to know the relationship between two equations with fixed effects below:

Y_ikt = \beta X_ikt + \alpha_it + \alpha_kt + \alpha_ik + error term (1)

Y_ikt = \beta X_ikt + \alpha_i + \alpha_t + \alpha_k + error term (2)

What I mean is, does interactive fixed-effect term [\alpha_it] have the same effect as [\alpha_i + \alpha_t]? If they are unequal, who would have more effect in the model?

Could anyone help with this question? Thank you!

[Maxence Morlet]

If you interact unit and time fixed effects for instance, you allow time shocks to be unit specific (sort of what you do in equation (1)). In equation two, you control for systematic differences and time shocks, simply.

However, be careful with interacting fixed effects as this creates a plethora of parameters to estimate, and all your regressors may be omitted because of collinearity (perfect prediction problem) depending on the number of levels and sources of variation you have in your data.

[Sonia Chen]

If you interact unit and time fixed effects for instance, you allow time shocks to be unit specific (sort of what you do in equation (1)). In equation two, you control for systematic differences and time shocks, simply.

However, be careful with interacting fixed effects as this creates a plethora of parameters to estimate, and all your regressors may be omitted because of collinearity (perfect prediction problem) depending on the number of levels and sources of variation you have in your data.

Appreciate it, Maxence! Regarding your saying: "you allow time shocks to be unit specific", do you mean the unit-time fixed effect term is more strict than the pure time fixed effect, it controls some variations between unit-time that both \alpha_i and \alpha_t cannot control?

In other words, I wonder which type of formula written below correctly describes the relationship between [\alpha_it] and [\alpha_i + \alpha_t], in terms of their control functions in the regression model?

Type (1): [\alpha_i + \alpha_t] ⊆ [\alpha_it]

Type (2): [\alpha_i + \alpha_t] ∩ [\alpha_it] ≠ Ø


Regarding the collinearity issue you mentioned, I have not met with this problem during my study related to event study and the gravity model. The gravity model generally uses three two-way interactive fixed-effect terms, as shown in equation (1) from the original post. I post this question because I am unclear why the literature does not include the single fixed effect term [like \alpha_t] other than those two-way fixed effects. I guess probably because the pure time shock can be controlled using the two-way unit-time interactive term, but I haven't found any literature proving or talking about this content specifically.

Thanks again!

[Maxence Morlet]

That’s interesting, I’m not too familiar with gravity models so if it’s standard there, then you should follow the prevailing literature.

What I mean is that if you include unit and time fixed effects individually, you are constraining any time shocks to having an identical effect on all of your units. So for instance you’d be assuming that COVID19-related time shocks have an identical effect on all of your countries (if you have countries as units and months as time). In this sense, yes it is “more strict”.

hope the example helps

[Sonia Chen]

That’s interesting, I’m not too familiar with gravity models so if it’s standard there, then you should follow the prevailing literature.

What I mean is that if you include unit and time fixed effects individually, you are constraining any time shocks to having an identical effect on all of your units. So for instance you’d be assuming that COVID19-related time shocks have an identical effect on all of your countries (if you have countries as units and months as time). In this sense, yes it is “more strict”.

hope the example helps

Thank you Maxence, that's very clear to me!