Analysing regression results This question is a bit embarrassing, but I do not have much experience of statistical analysis. The problem I have is that the analysis I do of the results in my paper is so meager. My results from a fixed effects regression basically consist in the effect of one or two central explanatory variables. I can comment on their sign, significance and magnitude. I can perhaps discuss whether there is a problem of omitted variable bias in the model. Is there something more I can do? As of now, the analysis won’t even be a page long. My knowledge is limited and I would appreciate all kind of advice or input here.
The research question is very narrow: are there economies of scale in health care at the state/county level? I have a cost measure as dependent variable and the number of medical care events and physician visits as the explanatory variables capturing the size of the health care. Then I try to control for geography, demographics and socioeconomic status.
A limited specification could look like this:
$$
\text{Cost}_{it} = \beta_0 + \beta_1 \text{number_of_visits}_{it} + \\\beta_2 \text{number_of_events}_{it} + \beta_3 \text{percentage_age65}_{it} + \\\beta_4 \text{area}_i + \beta_5 \text{taxable_income}_{it} + \\\beta_6 \text{percentage_private_healthcare}_{it} + \epsilon_{it}
$$
 A: Interpretation in fixed effects models is the same as with standard OLS, i.e. your coefficients give you the change in cost for a given one unit increase in the independent variable. So $\beta_1$ tells you by how much costs rise on average for one additional visit; $\beta_3$ tells you by how much health care costs rise on average for a 1 percentage point increase in the population over 65, and so on.
If you want to estimate scale effects, it would be a good idea to express all your variables in per capita terms. Given that larger counties have more people, they also have higher costs. Therefore a more meaningful measure is cost per capita, number of visits per capita, taxable income per capita, etc. For this you would just divide all your variables by each county's population in a given year. You don't need to transform the variables that are in percentage terms (population above age 65, private health care).
Secondly, if you want to identify scale effects you need to relate your variables of interest with the $\text{area}$ variable. This you can do by multiplying them with $\text{area}$. As a simplified example, if you have a regression equation like
$$\text{Cost}_{it} = \gamma_0 + \gamma_1 \text{percentage_age65}_{it} + \gamma_2 (\text{percentage_age65}_{it} \cdot \text{area}_{i}) + \epsilon_{it}$$
The effect of the number of visits now depends on a counties size. So when you assess the partial effect (which is like taking the partial derivative of the regression equation with respect to the number of visits), this will be
$$\gamma_1 + \gamma_2 \text{area}$$
so your effect of the elderly population on health care costs depends on size. You then need to plug in different values for $\text{area}$ to assess the overall effect. If larger counties are more efficient in providing health care for the elderly, you might expect a negative coefficient for $\gamma_2$ because then the additional costs of one more old person reduces if the county is larger. Or something like that may be a possible economic interpretation.
If you only have $\text{area}$ in your regression, there are two problems: the first is that $\text{area}$ alone will not identify scale effects. All you will learn from this is that larger counties have higher health care expenditure but this is probably just because they have more population. The second problem is that $\text{area}$ is fixed, i.e. a county's area does not change over time. You cannot include such variables in a fixed effects regression because the within estimator will kill them together with the unobserved time-invariant variables. How to keep time invariant variables in a fixed effects regression is a bit of a more advanced topic.
First of all you should be clear about which variables to use. As I said it's a good idea to express them in per capita terms because this will give you meaningful estimates for each variable. Then try to think about which variables affect health care costs, e.g. why would states with higher taxable income have more or less health care spending? Then be aware of the limitations of fixed effects which is mainly that you cannot keep variables in the regression that do not change over time. Then try to think about which variables should have a changing effect given a county's size. Those are the variables that you eventually might want to interact with $\text{area}$.
But most of all, don't get discouraged. Estimating scale effects is not the most trivial task and it requires a lot of thinking and learning. I hope this helps you in this initial research effort of yours. Please feel free to ask if you have any additional questions.
