Question on use of FE with different levels of aggregation

[Elliot PJ]

I am trying to understand what the statistical issue would be with the following. Any help would be appreciated:

Suppose I have a panel of test scores that are recorded at the individual school-year level for all schools in the US. I merge this m:1 with a panel of state-year level observations containing the statewide educational expenditure. I regress test scores on state level expenditure and use school fixed effects.

I feel like there must be an issue with the fact that I am using school fixed effects on data that has been merged m:1, but I can't pinpoint what the exact issue would be called. My assumption is that this would not necessarily bias the coefficient on expenditures, but would lead to standard errors being smaller than they otherwise should be. Thanks!

[George Ford]

The state level expenditures is not telling you what's going on with individual schools and the budgets may vary substantially. I suspect the coefficient would be largely meaningless. You need to aggregate to state or get school/district budgets.