6.3 From averages to observations

The PRF describes averages. Individual students are not averages. How do we write down the difference?

Return to the forty-nine students who attended exactly 25 classes. Their conditional mean was 6.10, and two of them scored 1.58 and 9.08.

The conditional expectation function accounts for the 6.10. It has nothing to say about why one student was four and a half points below it and another three points above.

So an individual’s grade is the conditional mean plus a departure from it. Give that departure a name.

For each individual \(i\),

\[y_i = E(Y \mid X = x_i) + u_i\]

where \(u_i\) is the error term or stochastic disturbance — the amount by which individual \(i\) departs from the average for their value of \(x\).

Substituting the linear conditional mean from Section 6.2 gives the simple linear regression model:

\[y_i = \beta_0 + \beta_1 x_i + u_i\]

This is not a new model. It is simply another way of writing the population regression function once we give the unexplained part a name.

The two students above become

\[1.58 = \beta_0 + \beta_1(25) + u_1 \qquad\qquad 9.08 = \beta_0 + \beta_1(25) + u_2\]

with a large negative \(u_1\) and a large positive \(u_2\). Both students attended the same twenty-five classes, so the first two terms are identical for each of them; everything that distinguishes them sits in \(u\).

We now have a model. Before asking how to estimate it, we should be clear about what that last term contains.