Difficulty: Intermediate | Prerequisites: OLS coefficient interpretation, omitted variable bias formula (Ch. 3 Q1–8)
These exercises move from algebra to real data. Each one asks you to think about what belongs in a regression, what the numbers mean once it is estimated, and how adding or removing a variable changes the story. The birth weight exercise (BWGHT) is about controlling for confounders. The attendance exercise (ATTEND) practises interpreting intercepts and slopes. The wage exercise (WAGE2) lets you verify the omitted variable bias formula with actual regression output.
These three exercises teach the same lesson from different angles: what happens when you add or omit a relevant control variable. In the birth weight model, you check whether controlling for family income changes the estimated effect of smoking. In the attendance model, you practise interpreting regression output and making predictions. In the wage model, you verify the omitted variable bias formula numerically by running three regressions and checking that the algebra works.
bwght (birth weight)
Infant birth weight, typically measured in ounces. The dependent variable in the BWGHT dataset regression. Low birth weight is associated with health risks for the infant.
cigs (cigarettes per day)
Average number of cigarettes the mother smoked per day during pregnancy. Expected to have a negative effect on birth weight.
faminc (family income)
Annual family income, measured in thousands of dollars. Acts as a control variable that captures access to prenatal care, nutrition, and other resources.
Think of it as a proxy for the overall quality of the prenatal environment beyond smoking.
atndrte (attendance rate)
Percentage of classes attended. The dependent variable in the ATTEND dataset. Measured as a percentage (0 to 100).
priGPA (prior GPA)
Cumulative GPA before the current semester. A measure of past academic performance.
ACT (ACT score)
The student's ACT composite score. A standardised admissions test score.
IQ (intelligence quotient)
A standardised measure of cognitive ability. Used as a control variable in wage regressions.
log(wage)
The natural logarithm of hourly wage. Using the log transformation means coefficients are interpreted as approximate percentage changes in wages.
In simple terms, if the coefficient on educ is 0.06, each additional year of education is associated with roughly a 6% increase in wages.
Control variable
A variable included in a regression to isolate the effect of the variable of interest. By "controlling for" family income, you hold income constant so that the coefficient on cigs reflects the smoking effect alone.
Proxy variable
A variable used in place of an unobservable variable it is closely related to. IQ can serve as a proxy for innate ability in a wage equation.
The model is:
bwght = \beta_0 + \beta_1 cigs + \beta_2 faminc + ubeta-2 should be positive: higher family income means better prenatal care, better nutrition, and generally healthier pregnancies, all of which contribute to higher birth weight
They are likely negatively correlated
Higher-income families tend to smoke less (better health information, lower stress, different social norms)
Lower-income families are more likely to smoke and to smoke more heavily
This negative correlation is exactly what makes faminc a useful control: without it, the coefficient on cigs partly reflects the disadvantages of lower income, not just the direct effect of smoking
The key question is whether adding faminc substantially changes the coefficient on cigs:
If it does, that suggests the simple regression was picking up income effects through cigs (omitted variable bias)
If it does not change much, the bias from omitting faminc is small, possibly because the correlation between cigs and faminc, while real, is not strong enough to distort the cigs coefficient meaningfully
In practice (running the actual data), the coefficient on cigs barely changes, and R-squared increases only slightly. This tells you faminc does not confound the cigs-bwght relationship much in this particular dataset
The model is:
atndrte = \beta_0 + \beta_1 priGPA + \beta_2 ACT + uBefore estimating, the exercise asks for the minimum, maximum, and average of atndrte, priGPA, and ACT. This is good practice: knowing the range of your data helps you spot implausible predictions.
The intercept is the predicted attendance rate when priGPA = 0 and ACT = 0
Both values are outside the realistic range of the data (nobody has a 0 GPA and a 0 ACT score)
So the intercept has no meaningful real-world interpretation here
This is common in regression: the intercept is a mathematical anchor, not always a useful number
The coefficient on priGPA is expected to be positive: students with higher past GPAs tend to attend more
The coefficient on ACT could surprise you. In some samples, it is small or negative, which seems counterintuitive. One explanation: students with high ACT scores may feel they can skip class and still do well
The sign of a coefficient is an empirical question. Your prior expectation and the estimate may differ, and that is worth discussing
For priGPA = 3.65 and ACT = 20:
Plug the values into the estimated equation
If the predicted atndrte exceeds 100, that is a limitation of the linear model (it does not respect the 0-to-100 bound)
Comparing Student A (priGPA = 3.1, ACT = 21) with Student B (priGPA = 2.1, ACT = 26):
Predicted difference = beta-hat-1 × (3.1 - 2.1) + beta-hat-2 × (21 - 26)
= beta-hat-1 × 1.0 + beta-hat-2 × (-5)
The two effects may partly offset each other, depending on the signs and magnitudes of the coefficients
This exercise is a hands-on proof that the omitted variable bias formula works with real numbers.
IQ = \hat{\alpha}_0 + \tilde{\delta}_1 \cdot educSave the slope delta-tilde-1. This tells you how correlated IQ and education are in the data.
\log(wage) = \hat{\gamma}_0 + \tilde{\beta}_1 \cdot educSave the slope beta-tilde-1. This is the "naive" return to education that ignores ability.
\log(wage) = \hat{\alpha}_0 + \hat{\beta}_1 \cdot educ + \hat{\beta}_2 \cdot IQSave both slopes. beta-hat-1 is the return to education controlling for IQ. beta-hat-2 is the return to IQ controlling for education.
The claim is:
\tilde{\beta}_1 = \hat{\beta}_1 + \hat{\beta}_2 \cdot \tilde{\delta}_1Plug in your numbers from Steps 1 through 3
The left side (beta-tilde from the simple regression) should equal the right side (beta-hat-1 from the multiple regression plus beta-hat-2 times delta-tilde-1)
This confirms numerically that the simple regression coefficient equals the multiple regression coefficient plus the bias
IQ and education are positively correlated (delta-tilde-1 > 0): smarter people tend to get more education
IQ has a positive effect on wages (beta-hat-2 > 0)
So the bias is positive: the simple regression overstates the return to education by attributing some of the IQ effect to schooling
The multiple regression coefficient on educ (beta-hat-1) is smaller than the simple regression coefficient (beta-tilde-1), which is exactly what the formula predicts
The birth weight regression is a simplified version of models used in public health research to quantify the effect of maternal smoking on infant outcomes. Family income is one of many confounders researchers must control for. The wage-IQ exercise illustrates a debate that runs through all of labour economics: how much of the observed "return to education" is really a return to underlying ability? This question drives entire research agendas around instrumental variables and natural experiments.
Students sometimes think that if adding a control variable barely changes the coefficient of interest, the control was pointless. The exercise was still worth doing: you now know the original estimate was not badly biased, which is itself valuable information.
A predicted value outside the feasible range (attendance rate above 100%) does not mean the regression is wrong. Linear models do not enforce bounds. This is a known limitation, and in introductory econometrics you note it and move on.
Students often assume that a positive correlation between IQ and education means education causes higher IQ. The regression of IQ on educ is a correlation, not a causal claim. It could equally reflect selection (smarter people choose more schooling).
When verifying the OVB formula, students sometimes report that the two sides do not match exactly due to rounding. Use enough decimal places in each regression to confirm equality before rounding.
⚠️ Be prepared to discuss the expected sign of a coefficient before seeing any data. This is a common warm-up question.
⚠️ Exams may ask you to compare two regression outputs and comment on whether adding a variable changed a key coefficient. State the coefficients, note whether the change is large or small, and explain what that implies about bias.
⚠️ The OVB verification exercise is a favourite on exams: you may be given three sets of regression output and asked to show that beta-tilde = beta-hat + beta-hat-2 × delta-tilde.
⚠️ Know when an intercept has a useful interpretation and when it does not. If the values x = 0 are outside the data range, say so.
True or False: The expected sign of beta-2 (the coefficient on faminc) in the birth weight regression is negative. (False. Higher income is associated with higher birth weight, so the expected sign is positive.)
Fill in the blank: In the attendance regression, the intercept represents the predicted attendance rate when both priGPA and ACT equal ____. (zero)
True or False: If adding faminc to the birth weight regression barely changes the coefficient on cigs, this means faminc is irrelevant. (False. It means the omitted variable bias from excluding faminc is small for the cigs coefficient. faminc may still matter for predicting bwght.)
Fill in the blank: In the OVB verification exercise, the simple regression coefficient on educ equals the multiple regression coefficient on educ plus ____ times ____. (beta-hat-2, delta-tilde-1)
True or False: A predicted attendance rate of 105% means the regression failed. (False. Linear models can produce predictions outside feasible bounds. This is a known limitation, not a failure.)
Q: In the birth weight regression, why is faminc expected to have a positive coefficient?
A: Higher family income gives access to better prenatal care, nutrition, and healthier living conditions. All of these contribute to higher birth weight, so the coefficient on faminc should be positive.
Q: Suppose the coefficient on cigs barely changes when you add faminc to the model. What does this tell you about omitted variable bias?
A: The bias from omitting faminc is small for the cigs coefficient. Even though cigs and faminc may be correlated, the magnitude of the correlation or the effect of faminc on bwght (or both) is not large enough to distort the smoking estimate meaningfully.
Q: In the attendance model, the intercept is 75.7 and the coefficient on priGPA is 17.3. If a student has priGPA = 3.0 and ACT = 22, what is the structure of the prediction?
A: Predicted atndrte = 75.7 + 17.3(3.0) + beta-hat-2(22). You would need the coefficient on ACT to complete the calculation. The key skill is setting up the substitution correctly.
Q: You run log(wage) on educ alone and get beta-tilde-1 = 0.060. You run log(wage) on educ and IQ and get beta-hat-1 = 0.046 and beta-hat-2 = 0.0058. You run IQ on educ and get delta-tilde-1 = 2.41. Verify the OVB formula.
A: Right side = 0.046 + 0.0058 × 2.41 = 0.046 + 0.01398 ≈ 0.060. This matches beta-tilde-1, confirming the formula. The simple regression overstates the return to education by about 0.014 because it does not control for IQ.
The birth weight exercise connects to health economics and programme evaluation, both of which rely heavily on controlling for confounders. The attendance model is a precursor to models with limited dependent variables (Chapter 17), where the outcome is bounded and linear regression's limitations become more serious. The wage-IQ exercise ties directly to the omitted variable bias theory in the earlier part of Chapter 3, and foreshadows instrumental variables (Chapter 15), which provide a systematic solution when the omitted variable cannot be observed or proxied.
birth weight, BWGHT, smoking, cigarettes, faminc, family income, prenatal care, control variable, confounder, atndrte, attendance rate, priGPA, ACT score, intercept interpretation, out-of-range prediction, WAGE2, IQ, log wage, return to education, ability bias, proxy variable, omitted variable bias verification, Wooldridge Chapter 3 computer exercises, introductory econometrics