6.5 When does β₁ measure an effect?

We have a model and a slope. What has to be true for that slope to mean what we want it to mean?

The model so far is a decomposition. It says \(y\) can be split into a part that moves with \(x\) and a part that does not, which is true of any two variables whatever and therefore not yet useful.

To read \(\beta_1\) as the effect of \(x\) on \(y\), we need an assumption about how \(u\) and \(x\) are related. Two assumptions are usually stated; only the second restricts the world.

The error averages zero

\[E(u) = 0\]

This costs nothing. If the unobserved factors had a non-zero average, we could absorb that average into the intercept and start again.

Suppose \(u = e + c\) where \(c\) is some non-zero constant — a fixed effect of, say, generous marking. Then

\[ \begin{aligned} y &= \beta_0 + \beta_1 x + (e + c) \\[4pt] &= (\beta_0 + c) + \beta_1 x + e \\[4pt] &= \beta_0^{*} + \beta_1 x + e \end{aligned} \]

with \(E(e) = 0\). The intercept has changed and the slope has not. Since \(\beta_1\) is what we want, assuming \(E(u) = 0\) restricts nothing of importance.

The error is unrelated to x

We now ask a simple question. Among students with the same attendance, what is the average value of everything contained in \(u\)?

The assumption is that this average is the same at every value of \(x\), and equal to the overall average of zero:

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

In that case \(u\) is said to be mean independent of \(x\). Taken together with \(E(u) = 0\), this is the zero conditional mean assumption.

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

This is the key assumption for interpreting the slope as the effect of \(x\), and the only way to judge it is to state it in the language of the problem.

For the grades example it says

\[E(\textit{effort} \mid \textit{attendance} = \text{low}) = E(\textit{effort} \mid \textit{attendance} = \text{high}) = 0\]

— that students who attend more classes are, on average, no more and no less able, motivated, or well-taught than students who attend fewer.

Note carefully what is not required. The assumption does not say that students have identical effort — obviously they do not. It says only that effort does not systematically differ with attendance. Equal averages, not equal individuals.

Stated that concretely, the assumption is clearly doubtful here. Students who attend regularly are plausibly working harder in other ways too, and effort raises grades on its own.

We cannot check it. Effort is not in the dataset, which is precisely what “unobserved” means. There is no diagnostic, no test, and no amount of additional data that settles it — only an argument about how the data came to look the way it does.

When \(E(u \mid x) \neq 0\), \(\beta_1\) does not measure the effect of \(x\). It measures the effect of \(x\) plus the influence of whatever else differs between high and low values of \(x\).

This is omitted variable bias. Section 7.8 returns to it once we have the algebra to show exactly how large it is and in which direction it runs.

With the model written down and its assumptions stated, we can turn to the practical question: how do we estimate \(\beta_0\) and \(\beta_1\) from data?