6.4 The error term

What is \(u\)?

Students meet the word “error” and assume something has gone wrong. Nothing has.

For the grades example, \(u\) contains everything other than attendance that affects a student’s GPA:

• how hard they worked outside class,
• how well prepared they were when the course began,
• the quality of teaching they happened to receive,
• whether they were ill during the examinations,
• whether they held a job that semester,
• how much they happened to like the subject,
• and luck on the day.

None of these is a mistake, and none of them is measurement noise. They are real influences on real outcomes that our model does not include — because they are not in the dataset, and in some cases could not be.

The error term is not a mistake and not measurement noise. It is the rest of the world.

That is why \(u\) is sometimes called the unobserved variable, and why econometricians spend so much of their time worrying about it. It is the part of the story we did not measure.

Whether those omitted influences are related to attendance is the central question of causal inference.

Systematic and stochastic

The model divides \(y\) into two parts, and it is worth naming them.

\[y_i = \underbrace{\beta_0 + \beta_1 x_i}_{\text{systematic}} + \underbrace{u_i}_{\text{stochastic}}\]

The systematic or deterministic part, \(\beta_0 + \beta_1 x\), is what attendance accounts for. It is identical for every student with the same attendance.

The stochastic part, \(u\), is everything else. It differs from person to person and is not observed.

What the two parameters mean

In the linear PRF \(E(Y \mid X = x) = \beta_0 + \beta_1 x\):

\(\beta_0\) is the conditional mean when \(x = 0\) — the average \(y\) among individuals with no \(x\) at all. It is called the intercept, and it is rarely central to the analysis; sometimes \(x = 0\) does not even occur in the data.

\(\beta_1\) is the amount by which the conditional mean changes when \(x\) increases by one unit, \(\partial y / \partial x\). It is called the slope, and it is almost always the parameter of interest.

The remaining names are worth having in one place. \(y\) is the dependent variable — semester GPA. \(x\) is the independent variable, also called the explanatory or control variable — attendance. \(u\) is the error or disturbance term.

For the grades example,

\[\textit{GPA} = \beta_0 + \beta_1\,\textit{attendance} + u\]

and \(\beta_1\) is the change in GPA associated with attending one more class.

Notice the phrase associated with. Whether \(\beta_1\) is the change attendance would cause is a separate question, and it is the subject of the next section.