I've written about fixed effects before in the context of mixed models. But how are FE useful in the context of causal inference? What can we learn from a panel data using FE that we can't get from a standard regression with cross sectional data? Let's view this through a sort of parable, based largely on a very good set of notes produced by J. Blumenstock, used in a management statistics course (link).
Suppose we have a restaurant chain and have gathered some cross sectional data on the pricing and consumption of large pizzas for some portion of the day for some period 1 across three cities, as pictured below:
What we have is unobserved heterogeneity related to these specific individual effects. How can we account for this? Suppose we instead collected the same data for two periods, essentially creating a panel of data for pizza consumption:
See also: Difference-in-Difference models. These are a special case of fixed effects also used in causal inference.
Fixed Effects Models(Very Important Stuff)