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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: $$ \text{Cost}_{it} = \beta_0 + \beta_1 \text{number_of_visits}_{it} + \\\beta_2 \text{number_of_events}_{it} + \beta_3 \text{percentage_age65}_{it} + \\\beta_4 \text{area}_i + \beta_5 \text{taxable_income}_{it} + \\\beta_6 \text{percentage_private_healthcare}_{it} + \epsilon_{it} $$

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    What does your model look like? What are the dependent and explanatory variables? – Andy Jun 20 '14 at 13:47
  • @Andy I added some additional information. – Eric_B Jun 20 '14 at 14:17
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    Okay, but how do you measure economies of scale in this context? I guess the hypothesis is something like whether larger counties are more efficient in the provision of healthcare? – Andy Jun 20 '14 at 15:09
  • @Andy Yes, exactly. Simply using population might work a size measure also, apart from the two stated above. – Eric_B Jun 20 '14 at 15:18

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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.

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Andy
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  • Thanks! You gave me some important things to think through. Putting it in per capita terms is definitely a good idea. I think I wasn’t clear enough in my post though. My idea was that the number of medical care events and the number of physician visits, as production measures, would identify scale effects . Area was actually to partly control for geography. Would I then still need to relate variables of interest to area or has this changed? If I understand you correctly, your point is that variables should be related to the variable identifying scale effect. What is the reason for this though? – Eric_B Jun 20 '14 at 18:57
  • If the hypothesis is whether larger counties are more efficient in the provision of healthcare, you need the interaction with area because it's part of your hypothesis. As I said, you cannot have area alone in the fixed effects regression as this will be absorbed into the county fixed effects (since area doesn't change over time). – Andy Jun 20 '14 at 19:10
  • The reason for interacting number of visits and events with area is that this allows for $\beta_1$ and $\beta_2$ to depend on the size of the county. If larger counties have a lower $\beta_1$ than smaller ones, this would indicate that they are more efficient in providing health care because an additional patient in the large county then does not increase cost by as much as an additional patient in the small county. – Andy Jun 20 '14 at 19:12
  • Ok, I see what you mean. As I see it, area isn’t related to the central size I had in mind, which is the size of the production, i.e. the amount of medical care events and visits, and therefore not part of the hypothesis. It is only a control variable for costs incurred due to geographical differences. In this sense it can be left out of the FE regression. From your point of view though, I see the need for interaction with area. My idea is that a negative parameter of the number of events or visits in a regression would indicate economies of scale. – Eric_B Jun 20 '14 at 22:36
  • Ah ok, now I see what you want to test. It can't work this way though because the number of events or visits will always increase health care cost. A negative coefficient implies that an additional patient reduces health care spending but economically this cannot be true. I need to think more about this. Your estimation problem is far from trivial. – Andy Jun 20 '14 at 22:41
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    So the typical approach to estimating scale effects is to estimate a [Cobb-Douglas production function](http://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function). So you would regress the log of health expenditures (as proxy for health services produced) on the log of capital and the log of labor employed in producing health services. If their coefficients sum to more than one, then there are increasing returns to scale; a sum equal to one means constant returns; a sum less than one means decreasing returns to scale. – Andy Jun 20 '14 at 22:57
  • If you were happy with the answer and the feedback in general, please don't forget to vote/accept. Everyone who helps others on this site does so by using their spare time, hence this feature of the site is to reward such answers/efforts. Thanks! – Andy Jun 21 '14 at 09:21
  • Absolutely, I was just about to do that! I think the approach you suggested would be the ideal method. However, I don't have the data on labor and capital so I think I have to stick to the original approach. But I am pondering whether it is a worthwile model or not. I think it is possible that larger health care units with a higher number of medical care events will have lower per capita cost. Surprisingly, I have a negative and statistically significant coefficient in the regression. I am not sure how robust the results are though. – Eric_B Jun 21 '14 at 09:34
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    Try the regression with squares of the variables, i.e. $\text{number of visits} ^2$ and $\text{number of events} ^2$. Then the main coefficient should be positive and the coefficients of the squared variables negative - this would imply that costs fall for higher number of patients – Andy Jun 21 '14 at 09:48
  • Ok somewhat conflciting results. For events, both the main coefficient and the coefficient of the squared variable is negative. For visits the main coefficient is positive and the coefficient of the squared variable negative. All 4 of them are significant. – Eric_B Jun 21 '14 at 10:32
  • No, wait! Both of the visits variables are significant at the 5% level, in contrast to events where the main coefficient is only signficant at the 10% level while the coefficient of tis squared form lacks significance. Perhaps this could imply that costs fall for higher number of patients. Also, I don't know if it matters but the main coefficient of visits was not significant before its squared form was added. – Eric_B Jun 21 '14 at 10:49
  • Would you say that the functional form with squared x’s better captures the true relationship between size and cost per capita? Taken together, what can these results suggest? Perhaps it is too rash to say that they lend support to the idea of economies of scale. – Eric_B Jun 21 '14 at 11:14
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/15251/discussion-between-andy-and-eric-b). – Andy Jun 21 '14 at 11:56