7.8 Two students

The slope says grades rise with attendance. Does attending raise grades?

Regression tells us how average outcomes differ across observations. It does not, by itself, tell us why they differ.

Student A attends thirty classes. Student B attends twenty.

Student A also studies harder outside class, enjoys the subject more, and came into the course better prepared.

If Student A earns a higher GPA, how much of the difference is due to attendance?

Regression has no difficulty describing the difference between these two students. The fitted line tells us that students attending more classes tend to earn higher grades on average, and it says so with a standard error and a confidence interval.

The harder question is whether attendance itself produced the difference.

What must be true

To read the slope as the effect of attendance, something more is required. The remaining influences on grades must not systematically differ across students with different attendance.

That is the assumption we met in Section 6.5:

\[E(u \mid X) = 0\]

For attendance and grades it is a demanding one. Students who attend regularly are also likely to be more motivated, better organised and more interested in the subject. Those differences belong in \(u\). If they are related to attendance, then

\[E(u \mid X) \neq 0\]

and the regression slope reflects both attendance and the omitted differences between students. This is omitted variable bias.

Section 7.4 showed where it enters mathematically: in the part of the estimator that depends on the errors. When the errors are related to \(x\), that term no longer averages away, and its expectation is the bias.

None of this appears in the output

The coefficient may be estimated very precisely. The standard error may be small. The \(p\)-value may be close to zero. The residuals sum to zero, the algebra checks out, and every diagnostic in this unit passes.

Every calculation in the output may be correct.

What is wrong is not the computation but the interpretation.

Regression answers a statistical question:

How do average outcomes differ as \(X\) changes?

Causal inference asks a different one:

What would happen if \(X\) were changed?

The first can be answered from the data alone. The second cannot.

Where to go from here

Multiple regression is the first attempt at the causal question, and it works by comparing observations that are similar in other observed respects. If motivation can be measured, students can be compared at the same level of it.

When the important differences cannot be observed, a different strategy is needed. The second half of this book develops those strategies.