Opening
Host
Welcome back. Today we are on regression, college choices, and the price of a famous name. Imagine choosing between an expensive private college and a less expensive public university. Graduates of the private college earn more. That sounds like useful evidence. But how much of their success came with them through the college gate?
Economist
Consider Nancy, a student described in the college study discussion. She was admitted to Harvard and the University of Texas. Texas offered generous financial help, and she chose Texas. She later became an economics professor. Her experience makes the choice interesting. It cannot tell us what Harvard would have changed in her life.
Host
We need more than an impressive personal story. Is an elite private college worth the extra tuition? I want to know whether the more expensive education creates higher earnings. A list of successful graduates would not answer that question.
Economist
By the end of this episode, you should be able to explain how regression makes controlled comparisons. You should also predict the direction of omitted variables bias. Finally, you should judge whether a control variable strengthens or damages a causal argument. Those skills help us ask better questions about education and many other policies.
Host
Our mind map follows The Question, The Investigation, and The Resolution. We will keep returning to the same college choice. We will also visit a computer exercise where the truth is known. That should help separate what the method calculates from what we have to assume.
Economist
Exactly. A computer can fit a regression very accurately while answering the wrong causal question. We need to understand the people being compared, the variables held fixed, and the remaining uncertainty. You can follow the episode without looking at the map. Its branches simply organize our route.
The Question
Host
The Question begins with the tuition decision itself. Paying more might buy smaller classes, helpful contacts, or a different learning environment. But it also means giving up money that could support other plans. What exactly are we trying to measure before we decide whether the cost is worthwhile?
Economist
Start with the extra tuition at the time described in the original discussion. Private college cost roughly 20,000 dollars more per year than an in-state public university. This is a historical comparison, not a current price quote. We want to compare the same person's later earnings under the different college choices.
Host
So the question is personal, even though the evidence must come from many people. How much would Nancy earn after Texas, compared with what she would earn after Harvard? We can observe the first path. The second path never happens for her.
Economist
That missing path is a counterfactual: an outcome under a choice that was not made. Each student has potential outcomes under public and private attendance. We see only the outcome associated with the actual choice. The earnings effect is the difference between those potential outcomes. It is not automatically the difference between different students.
Host
My prediction is that a large raw earnings gap will settle the tuition question. If private graduates earn much more, perhaps the extra tuition pays for itself. Even if we cannot observe both paths for Nancy, should a large enough survey give us the answer?
Economist
A larger survey cannot repair a systematically unfair comparison. It can measure an unfair comparison more precisely. Before calculating a financial payoff, we need a credible earnings effect. We also need to remember that earnings are only part of college value. Friendships, learning, and personal experience may matter without appearing in a wage measure.
Host
The naive evidence is the raw earnings gap. Private graduates have higher earnings, on average. But those students already passed a demanding admissions process. Their motivation or family connections might have raised their earnings at a public university too. We could be paying the college credit for advantages the students already had.
Economist
Yes. That is selection bias. People who select different treatments may also have different earnings prospects before treatment. Admissions decisions add another layer of selection. Colleges evaluate applicants, and applicants decide where to enroll. There is no admissions lottery in this investigation that balances all those differences for us.
Host
But the same admission letter seems useful. Nancy could have gone to Harvard. Someone who accepted Harvard's offer also could have gone there. Is comparing them more informative than comparing every Harvard graduate with every public university graduate?
Economist
Potentially. Applications show where students hoped to go, and admissions decisions summarize how colleges judged them. These signals can help us compare students with similar preparation and ambition. Still, admission to the same college does not make people identical. Their final choices may reflect differences that also affect later earnings.
Host
Then we need vocabulary for a careful comparison. I hear researchers say they control for something. They clearly cannot change a graduate's family background after the fact. What does a control actually do in the analysis?
Economist
A control variable is a measured characteristic we hold fixed in the statistical comparison. We might compare students with similar admission test scores or family income. Ceteris paribus means other things equal. In practice, the calculation equalizes the characteristics included in the model. It does not automatically equalize everything important in life.
Host
And short and long regressions? Those names sound as if one model takes more time to run. Do they describe the length of the computer program, or the list of characteristics we hold fixed?
Economist
They describe relative lists of controls. A short regression leaves out something included in a longer regression. The long version contains the short version's variables and adds others. The resulting change in the treatment coefficient helps us study omitted variables bias. A coefficient is the estimated number attached to a variable in the fitted model.
Host
I understand the possibility of missing characteristics. But why use the formal phrase conditional independence assumption? What extra idea is hidden behind those words? I would like a version I could explain without using the abbreviation.
Economist
It says that, among students with the same controls, the college choice is unrelated to their potential earnings. Conditional means within the groups defined by those controls. Independence means the remaining choice carries no systematic information about those potential outcomes. This is the claim that choice is effectively random inside comparable groups.
Host
Let me recap The Question. We want the earnings change caused by attending private college, rather than a gap between different kinds of students. Controls make particular measured traits equal. They help only if the remaining college choice no longer sorts students by their potential earnings.
Economist
That recap captures the challenge. The assumption concerns outcomes we cannot fully observe, so we cannot verify it directly. Good research explains why the controls are convincing and looks for weaknesses. Our investigation now asks how regression builds the comparison and how an omitted trait can distort it.
The Investigation
Host
The Investigation starts with selection without randomization. In Chapter one, a fair assignment procedure gave us comparable groups on average. Here, students make choices and admissions offices screen them. Can regression recover some of that fairness, or does it merely describe the selection problem in more detail?
Economist
Think of regression as an automated matching procedure. It compares students with similar measured characteristics who enrolled in different kinds of colleges. Within each comparison, the controls are held fixed. It then summarizes the differences. The hope is to remove the important selection differences without needing to measure every feature of each person's life.
Host
I predict that comparing students with similar applications will reduce the apparent private college advantage. Applications should reveal something about preparation and ambition. But if an entire matched group chooses private colleges, can that group still tell us what private attendance changed?
Economist
It cannot provide a within-group private versus public comparison. We need overlap: comparable students taking each path. A group containing only private students supplies no public outcome for that comparison. A group containing only public students has the opposite problem. Regression cannot create the missing contrast simply because the data set is large.
Host
That helps explain why the population matters. We are learning about students whose choices overlap in the observed data. We should hesitate before applying the answer to students who never had either option. The statistical comparison has a particular audience as well as a particular treatment.
Economist
Right. The original discussion groups students using applications and admissions. Some groups offer useful contrasts; others do not. Regression with group controls removes differences across those groups. With a separate indicator for each group, the private attendance estimate combines within-group contrasts. Groups with more students and more mixed attendance receive greater weight.
Host
The tool is regression with controls. Talk me through its ingredients without reading an equation. Which number represents college choice, and where do family background and test scores enter? I also want to understand what the prediction error means.
Economist
The outcome is later earnings. The treatment indicator records whether the student attended private college. Controls describe background characteristics and application groups. The model predicts earnings using a baseline, a private attendance term, and terms for the controls. The residual is actual earnings minus predicted earnings: whatever the fitted prediction misses.
Host
How does the computer choose those numbers? If some predictions are too high and others too low, could their errors cancel? A model might look accurate overall while being wrong for almost everyone.
Economist
Ordinary least squares prevents that cancellation by squaring each residual before adding them. It selects coefficients that make this total as small as possible. Large misses count more heavily than small misses. This defines how we fit the model. It does not establish that the private attendance coefficient measures a causal effect.
Host
Is it useful that fitted residuals are unrelated to the variables in the regression? I sometimes hear that the treatment must be safe because its relationship with the error is zero. Does the fitting procedure itself prove that selection has disappeared?
Economist
No. With an intercept, least squares residuals average zero and are uncorrelated with the included regressors by construction. Those are mathematical properties of the fitted model. Unmeasured ability can still contaminate the treatment coefficient. Removing sample correlation with a fitted residual does not prove independence from potential outcomes.
Host
So the conditional independence assumption does the causal work. We need overlap too. Is there also a modeling condition? Holding income fixed sounds straightforward, but income and earnings might have a curved relationship. A straight line could compare students poorly even when income is measured correctly.
Economist
That is why model specification matters. A conditional expectation is the average outcome for people sharing specified characteristics. The collection of those averages need not follow a straight line. Regression reproduces it when the chosen model represents it correctly. Otherwise, the fitted line gives a least squares approximation, which can miss important features.
Host
Then automated matching is an intuition with conditions. Using separate group indicators makes the within-group comparison especially clear. With continuous controls, we must check that the fitted relationship is suitable. And even an excellent prediction rule cannot replace a credible argument about the remaining selection.
Economist
Exactly. Another useful idea is regression anatomy. First predict private attendance using the other controls. Subtract that prediction from actual attendance. The leftover attendance variation is the part not explained by those controls. The controlled earnings comparison uses that leftover variation. If almost nothing remains, identifying the contrast becomes difficult.
Host
So first we remove attendance differences predicted by the controls. The remaining attendance variation supplies the comparison. How does that connect to calculating the slope?
Economist
Covariance measures how two quantities move together. Here we compare earnings with the leftover attendance variation. The slope divides that shared movement by the amount of leftover attendance variation. Regression anatomy applies this idea after accounting for the controls. It clarifies what information identifies the coefficient, rather than creating new information.
Host
The simulation lab gives us a known truth. We cannot look inside real students and measure every relevant talent. But we can create artificial students with ability recorded. What does the companion exercise build into the data, and what can it teach us about the limits of regression?
Economist
It creates 1,000 simulated students. Higher ability makes private attendance more likely and independently raises earnings. The programmer builds in a private attendance effect of 5,000 dollars. These are teaching data, not the real college survey. Knowing their rules lets us check whether a regression separates the school effect from ability.
Host
The short regression ignores ability. Private students have both the school term and better average ability. I expect that regression to assign some of ability's contribution to private college. What happens when the exercise adds ability as a control?
Economist
The short estimate is well above the effect built into the exercise. The long estimate moves close to 5,000 dollars. It need not match perfectly because random earnings noise remains in this finite sample. Including the correct confounder works here because the data were designed so that the remaining private choice is unrelated to earnings noise.
Host
I should not conclude that adding ability always solves a real college study. In the exercise, we know the complete rules. In life, a measured test score may capture only part of ability. A successful simulation demonstrates the mechanism. It does not certify the controls available in a real survey.
Economist
Exactly. The notebook also changes the rules to explore additional confounders and reversed selection. These exercises distinguish sampling variation from omitted variables bias. A well-designed estimator can fluctuate around the truth. A biased comparison systematically includes another relationship. More simulated observations would reduce random fluctuation without fixing a missing confounder.
Host
The omitted variables formula gives that distortion a precise structure. Can we work through a calculation using the original chapter's small teaching example? I would rather understand the mechanism slowly than memorize a formula with unexplained letters.
Economist
Use the example's informative students, grouped by applications and admissions. The short private attendance coefficient is 20,000 dollars. Adding the applicant-group control makes the long coefficient 10,000 dollars. These are dollar differences from a fictional classroom table, not findings from the real college survey. We are asking how much the coefficient changes.
Host
I subtract the controlled coefficient from the uncontrolled coefficient. That gives 10,000 dollars of omitted variables bias relative to the long model. The raw comparison attributed that extra amount to private college. Now explain how the missing group characteristic produces exactly the same difference another way.
Economist
We need two relationships. First, how strongly does membership in the omitted applicant group differ between private and public students? An auxiliary regression measures that relationship by predicting the omitted variable using treatment. Auxiliary simply means it helps interpret our main regression. Here, the relevant group contains two-thirds of private students but half of public students.
Host
The difference between two-thirds and one-half is one-sixth. So private students are more likely to belong to the higher-earning applicant group. The second relationship must tell us how much that group membership matters for earnings after accounting for private attendance.
Economist
Correct. In the long regression, that group coefficient is 60,000 dollars. Multiply one-sixth by 60,000 and the result is 10,000 dollars. That matches the difference between short and long. The missing trait travels with treatment and predicts the outcome. The short coefficient absorbs the product of those relationships.
Host
I can see why both factors are necessary. If the omitted trait predicts earnings but is equally distributed across the college choices, it does not distort this contrast. If it differs across choices but has no earnings relationship conditional on the other controls, there is no contribution through that factor either.
Economist
Yes. With existing controls, both relationships must account for those same controls. Predict the omitted trait using treatment and the shared controls. Then use its coefficient from the long outcome regression. The identity is exact for these least squares models in the same sample. Whether the long coefficient is causal remains a separate question. Calling this difference bias assumes the longer model is the benchmark we want. The algebra remains valid even when neither model identifies a causal effect. Also, separate regressions using different students need not obey the same comparison. Missing observations can change the sample when a new control is added. We must keep that possibility separate from the effect of the control itself.
Host
What about direction? With ability, higher ability makes private attendance more likely and raises earnings. Both relationships are positive, so omission pushes the private coefficient upward. If the signs differ, the bias should be negative. Does a pair of negative relationships produce upward bias too?
Economist
It does. The original discussion considers family size as an unobserved concern. Suppose private students come from smaller families after the existing controls. Also suppose larger families predict lower later earnings. Those are possible relationships, not verified findings here. Both factors would be negative, and their product would push the short coefficient upward.
Host
That sounds useful even when family size is missing from the data. We cannot calculate its actual bias, but we can state a plausible direction and explain our reasoning. We must not treat that reasoning as a direct test. Unobserved variables remain unobserved.
Economist
Precisely. Positive with positive gives upward bias, and negative with negative also gives upward bias. Opposite signs give downward bias. The simulation verifies the product identity using its recorded ability variable. The same reasoning lets us explore missing characteristics in real data without claiming we have measured them.
Host
The Dale and Krueger case study now takes us back to actual college choices. Their College and Beyond data include more than 14,000 former students. How do they use applications to learn about qualities that a test score alone might miss?
Economist
Applying to a demanding college signals ambition and confidence about admission. An offer provides another assessment of preparation. The analysis studies students who entered college in nineteen seventy-six. They were working full time in nineteen ninety-five. The outcome is pretax annual earnings. The sample excludes historically Black universities. Those historical graduates at the included institutions are the evidence base. The findings are not a direct survey of today's applicants.
Host
Before we discuss the result, I want to separate the designs. Does the matching approach literally require identical college names? And is that the same thing as the self-revelation model? Those descriptions sound close enough that I might accidentally treat them as interchangeable.
Economist
The motivating example matches identical application and admission sets. For the real matching analysis, the book groups schools using Barron's selectivity categories. That allows more students to be compared. It then keeps groups containing private and public attendance. This matched sample contains 5,583 students. Its controls describe application and admission selectivity patterns.
Host
So that sample supports within-group contrasts among students with similar options. The controls do not merely record how selective the attended college was. They describe the student's application and admission history. What is the raw earnings gap within this matched sample, before those group controls enter the regression?
Economist
The raw private premium is about 14 percent. After adding the applicant selectivity-group controls, the estimate is about 0.7 percent, close to zero. The matched-sample result is near zero already. These controls compare students with similar application and admission options, rather than only adjusting for the selectivity of the college attended.
Host
So the raw premium nearly disappears when we compare similar applicants. Are those exact percentage changes, or approximations from a different earnings measure? I do not want to confuse a percentage increase in earnings with a change in the percentage of students who attend private college.
Economist
They are approximate proportional earnings differences from a regression using log earnings. A logarithm expresses proportional changes conveniently. A small difference in log earnings is approximately a percentage difference in earnings. These are not percentage points of attendance. The interpretation is approximate, and precision does not remove the need for a valid selection argument.
Host
Now explain self-revelation separately. Exact or category-based matching can lose many observations because suitable comparisons are unavailable. How does the alternative model use more students without pretending that everybody has an identical admissions history?
Economist
It controls for the number of applications and the average selectivity of colleges applied to. Selectivity is measured using those colleges' average admission test scores. The idea is that applicants reveal information about themselves through their choices. This model uses 14,238 students. It is a different control scheme and a broader sample.
Host
Then its raw comparison cannot simply be inserted into the matched sample's sequence. What happens in the broader sample when those self-revelation controls are included? And what does the evidence say about prestigious classmates rather than the private label itself?
Economist
The broader sample's controlled private estimate is small and statistically insignificant, roughly three percent. The book also replaces private attendance with the attended college's average test score. Its strong uncontrolled earnings relationship disappears with self-revelation controls. That weakens the claim of a large earnings gain from more selective peers in this setting.
Host
Small and statistically insignificant does not mean exactly zero. We have a point estimate and uncertainty around it. The students also come from a particular collection of colleges. We should not announce that college quality never matters for any student, any outcome, or any historical period.
Economist
Correct. Standard errors measure how much an estimate varies across repeated samples under the statistical setup. They do not measure omitted variables bias. The observed results challenge a substantial average private earnings advantage under these controls. They leave uncertainty and possible differences across students. Earnings results alone cannot decide every family's best college choice. A confidence interval describes a range of effects compatible with the estimate and its sampling uncertainty under the statistical assumptions. A range containing zero can also contain positive or negative effects that matter in practice. Ask how wide that range is. Statistical significance and economic importance answer different questions. Neither one settles whether selection bias remains.
Host
Sensitivity analysis asks what happens when the controls change. Suppose we already include the application information. If we then add the student's own test score, parental income, and demographic characteristics, do we get a very different private attendance estimate?
Economist
The estimated private effect changes little. That stability is reassuring because those background characteristics are plausible confounders. The omitted variables formula explains why. Conditional on application information, private and public students have much smaller differences in these measured traits. The relationship between the omitted trait and treatment has become small.
Host
So stability does not mean test scores and family income are irrelevant to earnings. They could still predict earnings strongly. Their omission changes the treatment coefficient little because they no longer differ much across the treatment choices after the core controls. We have reduced the first factor in the bias formula.
Economist
Exactly. That is evidence in favor of the core controls, rather than a proof that every missing variable is harmless. Unmeasured traits could behave differently from measured traits. Controls could also share similar measurement weaknesses. A stable estimate is most informative when we have a substantive reason to trust the comparisons being tested.
Host
Does adding controls always make an estimate more precise? If they explain earnings, I would expect less prediction noise. But earlier you said the controlled comparison also depends on leftover variation in college attendance. Could better prediction come with less useful treatment variation?
Economist
Yes. Controls may reduce unexplained earnings variation, which improves precision. They may also remove much of the treatment variation, which reduces precision. Those forces can pull in different directions. Robust standard errors allow prediction noise to vary across people. They address that statistical uncertainty; they do not correct an invalid causal design.
Host
The danger of bad controls makes the choice of variables even more important. Imagine private attendance changes someone's occupation, and occupation changes earnings. If I hold occupation fixed, I might remove part of the effect I wanted to measure. Is that the main problem?
Economist
It is one problem. Occupation can be a pathway through which education changes earnings. Holding it fixed blocks that pathway. Occupation may also depend on other personal traits affecting pay. Selecting people with the same eventual occupation can therefore create new differences between the college groups. The result need not estimate the total college effect.
Host
My misconception would be that adding every available variable must improve the causal estimate. But the timing and role of each variable matter. A characteristic measured before enrollment is a candidate control. A job chosen after enrollment could be an outcome of the education itself.
Economist
Exactly. Pre-treatment timing helps us avoid controlling away a consequence of treatment. It is not a guarantee that every earlier variable is useful or safe. We need an argument about its role in the selection process. For policy, distinguish the total earnings effect of college from a different question about earnings within occupations.
Host
Regression across the wider course is now easier to recognize. Experiments compare assigned groups, and regression can summarize the difference. Later methods use instruments, admission thresholds, or changes over time. We will need their separate assumptions even if a regression program produces the final coefficient.
Economist
Regression is a calculation tool across those designs. It does not turn all of them into the same research strategy. Returns to additional schooling also differ from returns to a private college label. Changing years of education and changing college type are different treatments. Each requires its own comparison and interpretation. A single regression coefficient may also summarize different effects across students. In the group-control example, its weights depend on how much private and public attendance mix within groups. It need not give every student or every group equal weight. That is another reason to describe the comparison population before treating the estimate as a universal effect.
Host
What about the historical name? Regression to the mean sounds like a force pulling people back toward average outcomes. Should I think of that as another causal effect, or is the word regression connecting different statistical ideas?
Economist
Galton used the term in 1886 while studying the heights of parents and children. Exceptionally tall parents tended to have children whose heights were less exceptional. This is a pattern in correlated measurements, not proof of a causal force toward average height. His statistical contribution also coexisted with deeply harmful eugenic ideas.
Host
And Yule's contribution was closer to what we are doing with controls? Rather than treating an observed relationship as an automatic explanation, he tried to compare places after accounting for other measured characteristics. That sounds like the beginning of the modern controlled policy comparison.
Economist
In 1899, Yule studied English poverty and relief policy while controlling for population growth and age structure. His question was causal, even though including controls did not automatically settle it. To recap The Investigation, regression summarizes controlled comparisons. Omitted variables explain coefficient changes. Useful controls, overlap, suitable modeling, and credible selection assumptions make the comparison persuasive.
The Resolution
Host
The Resolution begins with a devil's advocate. Elite graduates earn more, so elite college pays. That statement still sounds persuasive in everyday conversation. If I present the controlled result, someone might reply that clever statistical adjustments are hiding an obvious benefit. How do we answer without overselling our own evidence?
Economist
Ask which students are being compared. The raw gap mixes the institution's effect with the students' earlier advantages. The application-based comparisons substantially reduce that gap in this evidence. Those adjustments have a reason: they aim to compare similar applicants. We should defend the comparison and its assumptions, rather than defend regression simply because it is technical.
Host
Here is my prediction for the other extreme. If I add enough controls, I will eventually prove there is no private college effect. The coefficient might become tiny and statistically insignificant. Would that count as success, provided I keep all the available observations?
Economist
No. A tiny estimate could reflect a good comparison, a bad control, or a poor specification. Statistical insignificance also leaves uncertainty; it does not establish an exact zero. The goal is a credible estimate of a defined effect. We must justify controls before celebrating where the coefficient ends up.
Host
And a family's decision still depends on more than the average coefficient. Financial aid can change the actual tuition difference. Students can value subjects, location, friendships, or other opportunities. The study challenges a particular earnings claim. It does not remove personal judgment from choosing a university.
Economist
That is the policy implication. Avoid treating an uncontrolled private graduate earnings gap as a guaranteed return on tuition spending. Also avoid declaring that every expensive college is a bad investment. The evidence concerns particular graduates, institutions, and earnings outcomes. A responsible recommendation states those limits and separates earnings from other benefits.
Host
Our takeaways start with a fair comparison. Regression holds measured controls fixed and summarizes the remaining treatment contrast. It cannot promise that unmeasured differences disappear. That is the central distinction between fitting a model and identifying a causal effect.
Economist
The bias formula adds a practical diagnostic. Ask how an omitted trait differs with treatment after the shared controls. Then ask how that trait predicts the outcome in the long model. Their product explains the short-to-long coefficient difference. Its signs suggest upward or downward distortion. Its interpretation still depends on the credibility of the longer comparison.
Host
Stability across reasonable models gives us another check. If important measured background traits no longer change the estimate after application controls, confidence grows. But simply collecting many similar controls can create a false sense of security. We need meaningful challenges to the design, including missing traits that might behave differently.
Economist
We also distinguish uncertainty from bias. A standard error quantifies sampling variation, not whether the controls remove selection. More observations can narrow uncertainty around a biased answer. A more complicated model can change precision in either direction. Report the estimate's uncertainty while separately explaining why its causal interpretation may be credible.
Host
Then the thesis is that regression delivers causal effects only when the right pre-treatment controls make treatment as good as random. We also need actual overlap and a suitable model for the controlled comparison. Those qualifications give the thesis content. They tell us what to inspect before trusting the treatment coefficient.
Economist
To recap The Resolution, private college's raw earnings premium does not establish its payoff. The relevant application controls largely remove it in the original evidence. Better controls are justified by the causal question, not by their quantity. A credible regression argument explains selection, modeling, uncertainty, and the limits of the population being compared.
Wrap-up
Host
Let us return to our learning goals. We can explain a controlled regression as a comparison of students with similar measured characteristics. We can predict omitted variables bias from the signs of its two factors. We can also distinguish useful background controls from treatment consequences that could damage the comparison.
Economist
Our central thesis remains: regression delivers causal effects only when the right pre-treatment controls make treatment as good as random. The calculation alone cannot establish that condition. In the college example, application information creates much more convincing comparisons. The resulting evidence challenges a large private earnings advantage, while leaving room for uncertainty and differences across people.
Host
Here is our closing think question. A researcher already controls for application patterns and prior test scores. They propose adding the graduate's eventual occupation. They say it will make students more comparable. If the goal is the total earnings effect of private attendance, should we welcome that control?
Economist
Pause the episode and think about when occupation is determined. Ask whether the college choice could change it. Also ask whether holding it fixed might select different kinds of private and public graduates. You do not need a numerical calculation to diagnose this problem.
Host
My answer is that occupation can be caused by the education. Holding it fixed could remove an earnings channel and introduce a new selection problem. The proposal changes the comparison. More controls do not guarantee a better estimate of the total effect.
Economist
That is the answer. We would need a different, carefully justified design to study a direct effect holding occupation fixed. For practice with the selection mechanism, use the companion Colab notebook. Change the simulated confounders and compare the short and long regressions. The lecture slides revisit the same reasoning and the application-based college example.
Host
Next, Chapter three, Instrumental Variables, asks what we can do when the treatment itself remains selected. It begins with people who ignore assigned treatment. We will separate what people were offered from what they actually received. That brings a different route to causal knowledge and a new set of assumptions.
Economist
Until then, ask what was held fixed, why it was held fixed, and what still differs. Those questions turn a regression result into a research argument you can examine. Thanks for listening, and goodbye.