01-1: Univariate Regression: Introduction
Why do we want ceteris paribus causal impact?
Transcript
Let’s make ceteris paribus concrete with a decision you might actually face. The first three bullets set it up. You have been admitted to University A, described here as better and more expensive, and University B, described as worse and less expensive. You need to choose one, and the relevant benefit is the boost each school would give your future income. This is an individual causal question: how would your income differ if you attended A rather than B?
The table is observational evidence you found about former students. Each row is a university. “Average income” is the sample mean of graduates’ annual incomes, and “sample size” is the number of observed graduates. The displayed data contain five hundred graduates from each school. With the fixed simulation used for this slide, A’s mean income is 131.26 and B’s is 90.31, so the observed difference, A minus B, is 40.94 in the income units used here.
The question at the bottom asks whether you should treat that 40.94 as the expected boost that you personally would receive by attending A instead of B. Be careful about the comparison. The table compares two groups of different people who made different choices and passed different admissions processes. It does not observe the same person attending both schools. Your causal question instead compares two potential outcomes for you, one under A and one under B, with your own characteristics held fixed.
The raw difference could reflect what A does, but it could also reflect who attends A. A more selective university may enroll students who would have earned more regardless of where they studied. The table alone cannot separate those explanations. Keep 40.94 in mind, then flip to the next tab, where we state the comparison you actually want.
Quality of College
You
- have been admitted to University A (better, more expensive) and B (worse, less expensive)
- are trying to decide which school to attend
- are interested in knowing a boost in your future income to make a decision
You have found the following data
University | average income | sample size |
|---|---|---|
A | 131.26 | 500 |
B | 90.31 | 500 |
Question
Should you assume that the observed difference of 40.94 is the expected boost you would get if you are to attend University A instead of B?
Transcript
Here are two quantities, and the distinction between them is the heart of the lecture. Suppose your ability score is six out of ten. The first expression is expected income conditional on attending A, minus expected income conditional on attending B. Read expected income given A as the average income among people who attend A, and similarly for B. This is the raw group difference from the table, about 40.94.
The second expression adds ability equal to six to both conditioning sets. It compares expected income at A for people with ability six with expected income at B for people with ability six. School changes, but ability is deliberately held fixed at your level. That is the quantity relevant to your decision because you do not become a different person when we imagine sending you to the other school.
The first callout bullet explains why this restriction matters. Innate ability does not miraculously increase simply because you enroll at A. If A’s students tend to have greater ability before college, then the first expression changes school and ability together. The second callout bullet names the problem: the school effect is confounded with something else. A confounder affects income and is systematically related to school attendance, so its contribution becomes mixed into the raw school comparison.
The note at the bottom defines the notation. Expected Y given X means the average value of Y for units with a specified value of X. Conditioning on more variables creates a more like-for-like comparison. Here we condition on both school and ability so that we compare students of equal ability across schools. Of the two displayed quantities, you want number two, the ceteris paribus difference. The next tab shows why the two quantities differ in the observed data.
Let’s say your ability score is 6 out of 10 (the higher, the better),
\text{(1)}\;\; E[inc|A] -E[inc|B] \;\;\; \text{(the raw difference in the table)} \text{(2)}\;\; E[inc|A,ability=6] -E[inc|B,ability=6]
Which one would you like to know?
Transcript
Now look at what is actually in the data. Each dot represents one graduate. The horizontal axis is ability, with higher values farther to the right, and the vertical axis is annual income. Color identifies the university. The red cloud and red lines correspond to University A; the blue cloud and blue lines correspond to University B.
The sloped lines show expected income conditional on both university and ability, exactly as the note says. At any fixed ability, draw an imaginary vertical line upward. The red University A line is twenty income units above the blue University B line. Because the lines are parallel, that conditional gap is twenty at every displayed ability value. This is the ceteris paribus school effect built into the example.
The horizontal lines show something different: each university’s overall sample mean income, ignoring ability. The red horizontal line is around 131 and the blue horizontal line is around 90, so their separation is about 41. Those are the numbers from the previous table. Their gap is much larger than the vertical distance between the sloped lines.
The distribution of dots explains why. University A’s red observations are concentrated farther to the right, around ability seven, while University B’s blue observations are concentrated farther left, around ability five. Ability raises income within either school, as shown by the positive slope of both conditional lines. Consequently, A’s overall average income is high for two reasons: A has a positive school effect, and A enrolls students with higher average ability. The horizontal-line comparison mixes those reasons. The sloped-line comparison holds ability fixed and isolates the school effect. Flip to the next tab to put exact numbers on those two components.
Transcript
Let’s put numbers on the picture, because the size of the mistake matters. University A’s students have an average ability of about 7.0, and University B’s have an average of about 5.0. Within either school, each one-point increase in ability raises expected income by ten. A two-point average ability difference therefore contributes roughly twenty income units to the raw difference between the groups.
At any fixed ability, compare the formulas behind the sloped lines. For A, expected income is sixty plus ten times ability. For B, it is forty plus ten times ability. Subtract B from A while holding ability constant. The ten-times-ability terms cancel, leaving sixty minus forty, which is twenty. That is the ceteris paribus effect of attending A in this example.
Now read the three bullets. Number one is the gap between the horizontal group-mean lines, about 40.9 in this realized sample. That is what you see when you compare graduates without controlling for ability. Number two is the twenty-unit gap between the sloped conditional-mean lines. That is what attending A does for a student of any given ability. The remaining 20.9 is the raw gap minus twenty. It reflects the groups’ different ability composition plus sampling noise, not the effect of the school itself.
The exact decomposition is especially clear at the population means: roughly twenty comes from A’s school premium and roughly twenty comes from its two-point ability advantage. The random noise in observed incomes makes the sample values differ slightly from those round numbers. The fragment gives the practical consequence. If you credit A with the entire raw difference, you value it at about twice its true twenty-unit effect. The rest of the course develops research designs and estimators that try to recover the conditional, like-for-like comparison from data that initially offer only raw differences.
A’s students average an ability of 7, B’s average 5, and each point of ability is worth 10 of income. At the same ability, A pays 60 and B pays 40.
- (1) the gap between the two horizontal lines is 40.9 ← what you see
- (2) the gap between the two sloped lines is 20 ← what attending A does for you
- the remaining 20.9 is ability, not school
Going back to the college-income example
Transcript
Now bring the zero conditional mean condition back to the college question. Read the model as “income equals beta zero, plus beta one times College A, plus u.” College A is an indicator, or dummy variable: it equals one for someone attending University A and zero for someone attending University B. The variable’s name includes a space on the slide, but conceptually it is one binary explanatory variable.
Substitute each possible value to see the parameters. For a B student, College A equals zero, so income equals beta zero plus u. For an A student, College A equals one, so income equals beta zero plus beta one plus u. Holding u fixed and subtracting the B outcome from the A outcome leaves beta one. Thus beta one is the ceteris paribus difference between A and B. Under zero conditional mean, beta zero is the expected income for the B group and beta zero plus beta one is the expected income for the A group, so beta one also equals the difference in their conditional means.
But u contains every other determinant of income omitted from this one-variable model, including ability. The question in the lower half asks whether expected u, and particularly the ability component of u, is zero conditional on college attendance. For a binary x, that requires the average error to be zero in both the A group and the B group after the intercept normalization. In substantive terms, the people who attend A cannot systematically have higher or lower omitted income determinants than those who attend B.
That is doubtful here because college attendance is selected twice. University A may admit applicants partly on ability, and admitted students decide whether to enroll based on characteristics related to future earnings. If ability is related to both attendance and income, then the College A indicator is correlated with u. The next figure returns to the data and shows that failure directly.
Income = \beta_0+\beta_1 College\;\; A + u
where College\;\; A is 1 if attending college A, 0 if attending college B, and u is the error term that includes ability.
Zero conditional mean satisfied?
E[u(ability)|college A] = 0?
That is, are attending college A and ability systematically related with each other? Or, is college choice (and acceptance of course) correlated with ability?
Transcript
Use the figure to answer the zero conditional mean question with your eyes. Each point is a graduate, the horizontal axis is ability, and the vertical axis is annual income. Red identifies University A and blue identifies University B. The scatter around each sloped line reflects the random component of income beyond the systematic school and ability terms.
The two horizontal lines are the sample mean incomes for the universities. As the callout says, their vertical gap is about 40.9, the raw difference from the table. If you regress income only on the College A indicator, the estimated coefficient is exactly this difference in sample means. That is what the available one-variable model would report.
The sloped lines hold ability in view. At any chosen position on the ability axis, the red conditional-income line for A is twenty units above the blue line for B. Their common positive slope means that one additional ability point is associated with ten additional income units within either university. The twenty-unit vertical separation is the school effect for otherwise equally able people.
The two gaps differ because the groups occupy different parts of the horizontal axis. The red A cloud is centered farther right, around ability seven, while the blue B cloud is centered around five. Higher-ability students are more likely to be admitted to or choose A, and ability also predicts greater income. Because ability is omitted from the displayed regression model, it sits in u. Its conditional average is higher for College A equal to one than for College A equal to zero, so expected u given College A is not zero.
The zero conditional mean condition therefore fails. The raw beta-one estimate combines roughly twenty units of school effect with roughly twenty additional units due to ability differences. That is omitted-variable confounding. Flip to the final tab to see the same model when the two schools receive comparable students.
Transcript
This is the counterfactual world in which college choice and ability are unrelated. The axes and colors are the same as on the previous figure: ability is on the horizontal axis, annual income is on the vertical axis, red is University A, and blue is University B. Each school again has one thousand simulated students in this version.
First inspect the point clouds along the horizontal axis. They now cover essentially the same ability range and have nearly the same center. In the realized data, mean ability is about 4.98 at A and 5.02 at B. Unlike the earlier figure, the A students are not systematically shifted toward higher ability. The two sloped lines are still parallel and twenty income units apart, because the causal school effect built into the data remains twenty at every fixed ability.
Now inspect the horizontal group-mean lines. Their realized separation is about 17.68, which the callout rounds to 18. That is not exactly twenty because these are finite random samples: the schools have slightly different realized average ability and different realized income shocks. “Up to sampling noise” is therefore important. Across repeated samples from this design, the raw difference is centered on the true twenty-unit effect rather than systematically inflated by ability.
Read the callout’s conclusion in terms of the error. Ability is still omitted from the one-variable college model, and individual income still contains noise. We have not suddenly observed or eliminated those factors. What changed is their relationship with the College A indicator. When both groups are drawn from the same ability distribution and assignment is unrelated to the other components of u, expected u given College A equals zero after normalization. The difference in group means then isolates the causal effect.
This is the central benefit of random assignment: it makes treated and comparison groups comparable on average, including on characteristics the researcher did not measure. Observational studies seek other credible ways to achieve the same identifying logic. The exercise asks you to diagnose that problem in a setting of your own.
This is what it would look like if college choice and ability were not correlated:
