Yesterday I tweeted about nested data, with multi-level models (MLM) versus OL + cluster-robust variance estimation (CRVE). This made me think about another confusion that arise, between what are called fixed versus random effects.
Let’s begin with a simple relationship between a covariate X and Y in nested data, e.g. students i nested in school j. We are interested in understanding the relationship between X and Y at the student level.
Approach 1: Assume the schools are fixed, but that students are a random sample within these schools. Assume the relationship between X and Y is the same in all schools. This often amounts to including a dummy variable for each school in the model. Here I use OLS to estimate β_1. 
Approach 2: Assume schools are fixed, but the relationship between X and Y can vary across schools. Now add in interactions. Here I use OLS to estimate separate relationships between X and Y for each school (η_1, …, η_J).
Approach 3: Assume the schools are a random sample from some population, but treat the relationship between X and Y as the same in all schools. Estimate β_1 using generalized least squares (GLS).
Approach 4: Assume schools are random, but now that the relationship between X and Y can vary across schools (also treated as random). Same as Approach 3 but change the last equation. Estimate both β_1 and the variation across schools.
Ok, here’s where the language gets confusing. Approach 3 is referred to as “Fixed” (as in ‘treat the relationship between X and Y as fixed’) in the MLM literature, but as a “Random Effects” model in economics!
In summary: Multi-level modelers use “fixed” or “random” for coefficients, while economists use “fixed” or “random” for models.
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