# Fixed effects or random effects model

I am currently writing my Master's thesis in which I aim at two things: 1) I try to find out if there are efficiency differences between public, private and non-profit hospitals 2) If efficiency increased or deceresead after the introduction of a new payment system.

My dataset contains roughly 1500 hospitals for each of the 13 years. The panel is unbalanced. I estimated the efficiency scores using Data Envelopment analysis. Now I would like to do a regression with the efficiency score as the dependent and some external factors (including ownership form but also patient age, case severity, region etc.) as the independent variables.

I am a bit confused about which regression to use, as I am unsure about how to interpret the results of FE regression. Some of the hospitals changed ownership during the period that the data covered, but I am not interested in change of efficiency after the hospitals became public or private (or whatever) but I am, at first, generally interested in finding out about basic efficiency differences. That is why I am leaning towards random effects regression rather than fixed effects (I will use the cluster robust option in STATA). In order to answer the second question, I created dummy variables for the time periods representing the old and the new payment system. Again, I am not sure which type of regression would be the adequate one. I performed the robust Hausman test which spoke for fixed effects regression, however, I am not sure if fixed effects will lead to the answer that I am looking for.

I would very much appreciate some insights on this (sorry if this is a bit confusing, I can elaborate on my intentions, if necessary)! Thank you very much!

Choosing between RE and FE depends on your assumtions about the error term. FE tries to remove constant unobserved homogeneity, where as RE assume not unobserved factors and instead corrects for serial correlation.

Use RE only if you think that $cov(x_{itj},a_{i})=0$. Typically FE is a much more convincing, and the leading case for using RE is if a important variable is time constant - but then correlated random effects can be employed. If you willing to assume a very strict set of assumption, then you could use the hausman test to help you decide.

For an introduction to correlated random effects see this - *.pdf, from the master himself (Wooldridge).

• Thank you very much for both of your answers (@hsl)!! I might not have been clear what my exact problem is. My first research question aims at a cross-sectional analysis. Ownership type usually is a time-invariant variable (I have only a few observations that changed ownership) and that is why it is included into the FE model in the first place. My supervisor told me I should let the Hausmann Test decide and that's what got me confused because ownership is for most observations time invariant. It might be that I am overthinking this (I am at the point where I can't see the wood for the trees). – Maddie Jul 11 '15 at 14:54
• Technically you can include the variable, even if changes very few times. So I suggest you try that, and see what happens. Perhaps try correlated random effects (CRE), which is basically FE but where you can also include time constant factors. To estimate a CRE model, include time averages for each variable $x_{it}$, and estimate the model by random effects - simple. Since your supervisor suggested the hausman test - use it! But remember that the test is only valid under the full set of assumptions including no serial correlation and heteroskedasticity. – Repmat Jul 11 '15 at 15:32
• Thank you very much! I will definitiely look find more about CRE! – Maddie Jul 11 '15 at 15:49

Whether you use a fixed or random effects model depends mainly on whether a random effects model is necessary (i.e., does varying the slope significantly improve the fit of the model). So usually, you will fit a random-intercept-only model and compare it with a fixed-intercept-only model. Then compare the two models to see if the random-intercept model is better (compare AIC, BIC, and -2 log likelihood). This is how I usually do it. You'll also need theoretical reasons to justify a random-effects model. Hope this helps.

I think using fixed or random effects model won't really influence how you interpret your results. The main difference is that the standard errors (and the degrees of freedom) will be different for the two models.