Fixed effect model with household level and state level data

Question

I have the annual cross-sectional household data with the following variables that I am interested in:

a) Body Mass Index (BMI) for each head of household b) State of residence for household c) Education level for each head of household d) Age in years for each head of household

I merge the annual cross-sectional data by State with the state unemployment rate (UR). I repeat this process for years 1990-2010 and pooled all the resulting merged annual data. My objective is to examine the impact of state unemployment rate on Body Mass Index of household. Following is the model: \begin{equation} BMI_{ist}=\beta_{ur}UR_{st}+ \beta_{age}age_{ist}+\beta_{edu}education_{ist}+\text{state fixed effects} + \text{year fixed effects}+e_{ist} \end{equation} where, $i$ is household, $s$ is state, and $t$ is year. I would like to know whether we can call this model as fixed effect model and use fixed effect estimator (because we have state fixed effects). I am asking this question because I am using dependent variable which is measured at the household level and the main explanatory variable which is measured at the state level. I also would like to know if UR is assumed to be exogenous, is it okay to say that the coefficient on UR, $\beta_{ur}$, measures the causal effect of state unemployment rate on household level BMI.

Answer 1

This is a fixed effects model. you should probably cluster your standard errors at the state level. I think it is reasonable to assume the unemployment rate is exogenous. Roughly speaking, any single state resident cannot significantly influence the unemployment rate while the unemployment rate can have significant influence on any single resident's behavior. Education, however could be endogenous since both BMI and education could be linked to an unobserved motivation factor.

If education is endogenous, unless $\hat \beta_{edu}$ and $\hat \beta_{ur}$ are completely uncorrelated, $\hat \beta_{ur}$ will be a biased estimate of the causal effect. from here you could either

1. Find a REALLY good reason for why education is exogenous (I don't know if this is possible)
2. include other covariates to control for unobserved confounders, male/female indicators, mother's education, father's education, income, etc.
3. Find a good instrument for education. Though it's outdated, Angrist and Krueger (1991) use season of birth to instrument education. Labor economists have both criticized and revised on this instrument but it's a start.
4. Construct some sort of structural equation, such as a simultaneous system, to account for the endogeneity of both BMI and education.

Overall, unless you are trying to publish something, I would just go with (2) from above.