# why to add the interaction of two fixed effects in regression

A general question raised during my study.

For example, we are estimating the effect of X on Y at individual level. Then it is intuitive for me to add state fixed effect and year fixed effect in the regression.

$$Y_{it} = X_{it} + \mu_{s} + \mu_{t} + \epsilon_{it}$$

While in a lot of papers, people also add the interaction of the two fixet effects. The specification then looks like

$$Y_{it} = X_{it} + \mu_{s} + \mu_{t} + \mu_{st} + \epsilon_{it}$$

My question is that why people do so? WHat variation is exactly captured by the interaction of the fixed effect?

Thank you a lot in advance!

Some people live in warmer places than others so that sandals can be worn for more months out of the year. That will get picked up by the $$\mu_s$$ terms: big and positive for Hawaiian stores and large and negative for Alaskan ones. This will pick up differences between stores across states that are time-invariant.
People everywhere tend to buy more sandals during warmer months. That will be picked up by $$\mu_t$$ terms: small and positive in spring, big and positive in summer, small and negative in fall, and big and negative in winter. This is just a fairly flexible way to model national time trends. You are allowing that some summers are colder than other summers since the effects depend on $$t$$ rather than dummies for the calendar month.
But you might expect seasonality to be pretty different in North Dakota than in Florida, so a national trend that is the same everywhere is unrealistic. The state-time interactions allow you to make it more flexible. The $$\mu_{st}$$ terms capture the unobserved shocks shared by stores from state $$s$$ at time $$t$$. So Florida might see a significant boost from spring break sales, while North Dakota may get a tiny bump that may not be distinguishable from zero.