# Dependent variable is a function of independent variables; can I sensibly include them in a regression?

We've created a survey asking students, among other things, their GPA (=weighted average of grades) and their marks in some specific courses (which count towards GPA).

We wanted to see which regressors influence the GPA using a simple OLS model. Is it sensible to use a formula like this?

GPA ~ grade_maths + grade_statistics + grade_privatelaw + ... + {other regressors, like study habits or origin}


Of course, the grade regressors turn out to be highly significant (some more than others, and not directly related to the weight they have in GPA), while few of the other ones are...

Is this a case of endogeneity, i.e. does a regression like this violate strict exogeneity?

With this regression we want to get an quick overview of which variables will likely be useful in the regressions that follow, which for example try to find out whether being good in quantitative courses tends to help get good grades in law and other ideas like this...

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I wouldn't say it's "not sensible" to do what you have done. However, because you already know the factors which determine GPA - namely the grades in various courses, it may make more sense to model those factors rather than GPA. Perhaps a "better question" might be, for example: What factors are most strongly associated with high math scores?. –  probabilityislogic May 28 '11 at 14:12
A factor Analyis seems to be more sensible than a regression. –  Manoel Galdino May 28 '11 at 18:14
@probabilitylogic You're right, I have the impression that using OLS in this case just leads to trivial assertions. Probably it would be a better idea to either use other techniques like @Manoel suggested, or just proceed to more interesting questions like the one you proposed (which is casually similar to one of those we had in mind). –  LCC May 28 '11 at 19:41
you better be careful about the phrase "@probabilitylogic You're right" - I could go and get a big head from it :). @manoel's suggestion of factor analysis does seem like a good way to go, pick up the "dimensions of school ability" or something like it. –  probabilityislogic May 29 '11 at 12:59

I see no problem with fitting the regression. We do regressions because we believe that the predictors may be related to the response, you just have more knowledge to begin with.

But what questions are you actually trying to answer? The fact that certain coefficients are significant is not surprising, so those were not really interesting questions to begin with. What may be interesting is if they differ from a value other than 0 (the weight in the GPA). That could tell you if they have an indirect effect in addition to the known effect, e.g. math score could be related to science score which is not in your model, but contributes to the GPA.

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But what questions are you actually trying to answer? The first question was whether such a regression would violate an OLS assumption, like someone with more statistical experience suggested. Your answer confirms my hunch, namely that it does not. Secondly, we wanted to get an quick overview of which variables will likely be useful in the regressions that followed, which for example try to find out whether being good in quantitative topics tends to help get good grades in law and other ideas like this. –  LCC May 28 '11 at 19:32

Another point to consider: what enables a student to do well in one course is related to what enables him/her to do well in another. There are overarching factors (cognitive, personality, circumstances) that play some role in determining each of the individual course grades. So to use regression--to see how X1 relates to GPA while controlling for X2, X3, X4, etc.--, will "cannibalize" each X-Y relationship, partialling a portion of the relationship right out of itself. The coefficients you obtain will be, in Tukey and Mosteller's words, "arbitrary nonsense." Here's how Elazar Pedhazur puts it (Multiple Regression in Behavioral Research, 3rd Ed., 170-2):

“Partial correlation is not an all-purpose method of control […] Controlling variables without regard to the theoretical considerations about the pattern of relations among them may yield misleading or meaningless results […] It makes no sense to control for one measure of mental ability, say, while correlating another measure of mental ability with academic achievement when the aim is to study the relation between mental ability and academic achievement. […] This is tantamount to partialling a relation out of itself and may lead to the fallacious conclusion that mental ability and academic achievement are not correlated.”

I'd recommend studying bivariate correlations among the variables over a regression predicting GPA. Predicting each individual course grade (@probabilityislogic's idea) also seems very worth doing. @Manoel's factor analysis idea makes me pause because you may not have all the variables necessary to map out the key underlying factors.

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Thanks, that’s an interesting point and I will keep that in mind. However, in this specific case, where the response variable (GPA) is just a linear combination of the regressors (marks), my logic intuition says that the coefficients should assume values similar to the weights used for calculating GPA, so they might not be useful but they shouldn’t be arbitrary… Am I right? –  LCC May 30 '11 at 0:01
Well, you're using some courses' grades, not all of them. And I wouldn't want you to underestimate the power of partial correlation to produce very different results than you would see in zero-order correlation. You might even get a negative coefficient for one or two course grades. –  rolando2 May 30 '11 at 4:17

Strong exogeneity is a term related to dynamic models, i.e. when there is time-series data involved. Since you are doing one-time survey, this term does not apply. What might be the problem with the regression though is omitted variable bias. Since GPA is a weighted average purely arithmetical formula applies:

$$GPA = w_1 G_1+...+ w_n G_n$$

where $w_i$ are the weights and $G_i$ are grades. This equation is not stochastic. However we can say that each grade is determined by student's ability plus stochastic term:

$$G_i=f_i(\mathbf{A})+\varepsilon_i$$

where $\mathbf{A}$ is vector of variables which determine students ability, and $f$ is the functional form of the relationship.

When viewed in this light it makes no sense including grades in the regression. What might be of interest how different functional relationships $f_i$ aggregate into functional relationship $f$:

$$GPA=f(\mathbf{A})+\varepsilon$$

But this does not answer the questions in your given example. So as other's suggested it is better to use factor analysis.

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