# Choice of fixed effects versus random effects model

I have a question about the use of fixed versus random effects models. Let us consider the following example for motivation:

Suppose we are interested in the relationship between the performance of students on tests and their study time. We randomly select 99 students at a University and record the study time and score for each student in 9 tests.

Now, there are unobserved characteristics that we should control, for example general ability of the student and previous courses taken. These characteristics correlate with the study time so, in order to avoid omitted variable bias, one should use a fixed effects model. However, on the other hand the students where chosen randomly from a large population which suggests we should use a random effects model ...

How can one determine in this case what type of model is appropriate ? Statistically one could use the data and use the Hausman test I suppose, but I wonder whether there is a decision rule that is perhaps motivated a priori by the design of fixed / random effects models and the context of the study?

Thanks!

• "Students" will implicitly be treated as a random effect since the variance of the residuals will be a function of student differences. The harder question is whether you want to treat "tests" as a random effect. That is, do you want to generalize your results to tests other than the 9 used in the study? – David Lane Jan 23 '17 at 19:30
• If you think general ability and previous courses taken impact the study time, I think your experimental design should control for them. Only select students who have taken certain (or no) courses. If you have already collected the data, you might try a multiple regression with general ability, number or type of courses taken, and study time as predictors. Let me ask, will each student take the same 9 tests? If not, I do not know what conclusions you could reasonably draw from the data you collect. – Joel W. Jan 24 '17 at 1:57
• @JoelW. yes, all students take the same 9 tests so that the resulting data is in the form of panel data. – harlekin Jan 24 '17 at 8:39
• @DavidLane Thanks a lot for your comment! Assuming that general ability and prior knowledge of the student correlate with the explanatory variable (stuy time), how do I account for the omitted variable bias in the random effects model? (For simplicity the 9 tests are meant to be exhaustive) – harlekin Jan 24 '17 at 8:45
• @harlekin Can't you just include them in the regression model? – David Lane Jan 24 '17 at 17:58