Econometrics In a study relating college grade point average to time spent in va

Question

Econometrics In a study relating college grade point average to time spent in various activities, you distribute a survey to several students. The students are asked how many hours they spend each week in four activities: studying, sleeping, working, and leisure. Any activity is put into one of the four categories, so that for each student, the sum of hours in the four activities must be 168. In the model GPA = beta 0 + beta 1Study + beta 2sleep + beta 3work + beta4 leisure + u, does it make sense to hold sleep, work, and leisure fixed, while changing study? Explain why this model violates Assumption MLR.3. How could you reformulate the model so that its parameters have a useful interpretation and it satisfies Assumption MLR.3?

Explanation / Answer

i) Remember that there are 168 hours in a week. If you hold sleep, work and leisure fixed, while changing study, you implicitly allow the number of hours in a week to vary. This is doesn’t make sense.

ii) It’s important to realize that if there are 4 categories that students can lump their time into, and if the number of hours in a week is fixed (which it obviously is) then once we know how much time is spent on 3 of the categories, we already know how much is spent on the fourth. For example, if sleep=56, work=20, and leisure=62 then we already have enough information to know what study is. study must be 30. In other words, there is a perfect linear relationship (i.e. 168- (sleep+work+leisure) between the first three x variables and the fourth. In fact, there’s a perfect linear relationship between any three of the x variables and the fourth. If we include all four, then we violate the assumption of “no perfect collinearity,(MLR.3).

iii) We could omit one of the four variables. For instance, we could omit leisure from the model, so that we restate the model as GPA = 0 + 1study + 2sleep + 3work + u Now when we think about one of these variables changing by one hour, we’re implicitly letting leisure vary (also by one hour, but in the opposite direction) to preserve the 168 hours in a week. For instance, beta1 tells us what effect substituting one more hour of study for one less hour of leisure has on GPA. This is arguably a useful interpretation and the respecified model satisfies MLR.3.

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