These notes walk through three applied regression exercises: modelling CEO salary as a function of firm performance (CEOSAL2 dataset), demonstrating the partialling-out interpretation of OLS coefficients (wage1 dataset), and examining how school spending and poverty relate to maths scores (MEAP93 dataset). The common thread is learning to interpret coefficients, understand what happens when you add or remove variables, and reason about correlation among regressors in real data.
Difficulty: Intermediate | Prerequisites: Chapter 2 simple regression, Chapter 3 theory (OLS properties, omitted variable bias, partialling out). Familiarity with log transformations and elasticity interpretation.
The computer exercises in Chapter 3 bridge theory and practice. After learning about omitted variable bias, multicollinearity, and the partialling-out interpretation in the abstract, these exercises ask you to see those ideas play out in real datasets. You will estimate constant-elasticity models, work with logarithmic transformations, interpret intercepts when they correspond to unrealistic scenarios, and watch omitted variable bias appear (and disappear) as you add controls.
If you are not comfortable with interpreting log-level, log-log, and level-log specifications, revisit Chapter 2's discussion of functional form. You will also need to know what it means for a variable to be correlated with another regressor and how that affects coefficient estimates.
Constant elasticity model (log-log specification)
A regression where both the dependent variable and the independent variable are in natural-log form: log(y) = β₀ + β₁ log(x) + u. The coefficient β₁ is the elasticity: the percentage change in y for a 1% change in x.
Think of it as a model built for proportional thinking, where a 10% rise in x always has the same proportional effect on y, regardless of the level.
Level-log specification
A regression of the form y = β₀ + β₁ log(x) + u. Here β₁/100 gives the change in y for a 1% increase in x.
In simple terms, x enters on a percentage scale, but y stays in its original units.
Profits (in levels, not logs)
In the CEO salary exercises, profits can be negative (firms can lose money), so you cannot take their logarithm. This is a practical constraint that drives modelling decisions.
CEO tenure (ceoten)
The number of years a CEO has held the position. In the CEOSAL2 dataset, this enters in levels (not logs), so the coefficient gives the percentage return to one additional year of tenure when the dependent variable is log(salary).
Partialling out (Frisch-Waugh-Lovell theorem)
A two-step procedure: (1) regress x₁ on all other regressors and save the residuals r̂₁; (2) regress y on r̂₁. The coefficient on r̂₁ equals the coefficient on x₁ from the full multiple regression.
Think of it as stripping x₁ of everything the other variables can explain, then seeing what remains and how it connects to y.
lnchprg (lunch programme participation rate)
The percentage of students eligible for the free or reduced-price lunch programme. It serves as a proxy for school-level poverty in the MEAP93 dataset.
In simple terms, higher lnchprg means a poorer student body, which you would expect to be associated with lower test scores.
R-squared (R²)
The fraction of the total variation in y that is explained by the regression model. Ranges from 0 to 1. A low R² does not mean the model is useless; it means other unobserved factors also matter.
Think of it as: how much of the scatter in the data can the included variables account for?
Tags: CEO salary, log-log model, constant elasticity, profits variable, ceoten, market value, multicollinearity in practice
The model is: log(salary) = β₀ + β₁ log(sales) + β₂ log(mktval) + u
Both independent variables are in logs, so β₁ is the elasticity of salary with respect to sales and β₂ is the elasticity with respect to market value. A 1% increase in sales is associated with a β₁% increase in salary, holding market value fixed.
When you estimate this, write the results in equation form: report coefficients, standard errors (in parentheses beneath), sample size, and R².
Profits can be negative (loss-making firms exist in the sample), so log(profits) is undefined for those observations. You must include profits in levels, not logs.
The model becomes: log(salary) = β₀ + β₁ log(sales) + β₂ log(mktval) + β₃ profits + u
The coefficient on profits gives the proportional change in salary for a one-unit (one-thousand-dollar or one-million-dollar, depending on units) change in profits.
Do these firm performance variables explain most of the variation in CEO pay? Look at R². The typical finding is that R² is modest, meaning much of the variation in CEO salaries comes from factors not captured by firm sales, market value, and profits alone.
Adding ceoten (CEO tenure in years) to the model gives the percentage return to an additional year of tenure, holding firm performance fixed. Since the dependent variable is log(salary), the coefficient on ceoten, multiplied by 100, gives the approximate percentage change in salary per extra year.
Compute the sample correlation coefficient between log(mktval) and profits. These variables capture related aspects of firm performance, so some positive correlation is expected. If the correlation is high, it means the regressors are sharing information, which inflates standard errors (multicollinearity). Crucially, this does not bias the OLS coefficients; it only makes them less precisely estimated.
Tags: partialling out, Frisch-Waugh-Lovell, residuals, OLS interpretation, wage equation, education, experience, tenure
This exercise asks you to verify the partialling-out theorem by doing it step by step with real data.
Regress educ on exper and tenure. Save the residuals, r̂₁. These residuals represent the part of education that cannot be explained by experience and tenure.
Regress log(wage) on r̂₁. The coefficient on r̂₁ is the partialled-out estimate of the return to education.
Compare that coefficient with the coefficient on educ from the full multiple regression: log(wage) = β₀ + β₁ educ + β₂ exper + β₃ tenure + u.
The coefficient on r̂₁ in step 2 will be numerically identical to the coefficient on educ in step 3. This is the Frisch-Waugh-Lovell result in action.
The intuition: in the full regression, the coefficient on educ already reflects only the variation in education that is independent of experience and tenure. The partialling-out procedure makes that explicit by stripping away the common variation first.
This exercise reinforces a deep point about multiple regression. The coefficient β₁ on educ does not measure the raw association between education and wages. It measures the association after removing the confounding influence of the other regressors. Understanding this is essential for interpreting any multiple regression output.
Tags: education production function, log spending, lunch programme, omitted variable bias in practice, intercept interpretation, MEAP93
The model is: math10 = β₀ + β₁ log(expend) + β₂ lnchprg + u
math10 is the percentage of 10th-graders passing the maths exam. log(expend) is the log of per-pupil spending. lnchprg is the percentage eligible for the free lunch programme (a poverty proxy).
Expected signs:
β₁ should be positive: more spending should be associated with better outcomes, holding poverty fixed.
β₂ should be negative: higher poverty (more students eligible for free lunch) is associated with lower scores.
Setting both explanatory variables to zero means log(expend) = 0, which implies expend = 1 (one dollar per pupil), and lnchprg = 0 (no students eligible for free lunch). Spending of one dollar per pupil is absurd, so the intercept has no meaningful real-world interpretation. This is common in models with logged variables.
Run math10 on log(expend) alone. Compare the slope coefficient with β₁ from part (i).
The typical finding: the coefficient on log(expend) is larger in the simple regression. This is because the simple regression omits lnchprg, and the omitted variable bias pushes the spending coefficient upward (spending and poverty are negatively correlated, and poverty has a negative effect on scores; a negative times a negative gives a positive bias).
The correlation is typically negative: districts with more disadvantaged students tend to spend less per pupil (or at least not proportionally more). This negative correlation is what drives the upward bias in the simple regression.
The negative correlation from part (iv) explains the finding in part (iii). When you omit lnchprg:
lnchprg has a negative effect on math10 (β₂ < 0).
lnchprg is negatively correlated with log(expend).
The omitted variable bias on β₁ is: bias = β₂ × (negative correlation) = (negative) × (negative) = positive.
So the simple regression overstates the spending effect. Adding lnchprg as a control removes this confounding and gives a cleaner (and smaller) estimate of the spending effect.
Students often try to include profits in logarithmic form without checking for negative values. If any observation has profits ≤ 0, log(profits) is undefined. Always inspect the data before transforming.
Students sometimes interpret the intercept in a log model as meaningful. When log(expend) = 0 means spending is $1 per pupil, the intercept is an extrapolation far outside the data and has no practical interpretation.
A common mistake on the partialling-out exercise is to think the residuals from step 1 should be regressed on the original y, not log(y). Follow the original model specification exactly.
Students frequently confuse "the spending coefficient gets smaller when I add controls" with "spending does not matter." The coefficient getting smaller typically means the simple regression was biased upward by omitted variables, not that spending is irrelevant.
⚠️ Know when you can and cannot take logs. If a variable can be zero or negative, it stays in levels. This comes up constantly in applied work and on exams.
⚠️ Be able to interpret coefficients in log-log, log-level, and level-log models. Exams regularly ask: "what is the estimated effect of a 10% increase in X on Y?" and the answer depends entirely on the functional form.
⚠️ The partialling-out result is a favourite exam proof or demonstration. You may be asked to describe the procedure in words, or to show that the two-step coefficient equals the multiple regression coefficient.
⚠️ Omitted variable bias with a concrete sign calculation (as in the MEAP93 exercise) is a standard exam question. Practise stating the direction of bias: sign of the omitted variable's effect times the sign of the correlation between the omitted and included variable.
True or false: You can always take the log of any variable in a regression.
Fill in the blank: In a log-log model, the coefficient on log(x) is interpreted as the ______ of y with respect to x.
True or false: The partialling-out procedure gives a different coefficient from the full multiple regression.
True or false: If an omitted variable has a negative effect on y and is negatively correlated with an included regressor, the bias on that regressor's coefficient is positive.
Fill in the blank: When the intercept in a regression corresponds to setting log(expend) = 0, the implied spending level is ______ per pupil.
Answers: 1. False (the variable must be strictly positive). 2. Elasticity. 3. False (they are identical). 4. True. 5. $1 (since exp(0) = 1).
Q: Why can profits not be included in logarithmic form in the CEO salary regression?
A: Because some firms report negative profits (losses). The natural logarithm is only defined for strictly positive values, so log(profits) would be undefined for any firm with profits ≤ 0.
Q: In the model log(salary) = β₀ + β₁ log(sales) + β₂ log(mktval) + β₃ ceoten + u, what is the approximate percentage increase in salary for one additional year of CEO tenure?
A: 100 × β₃ percent. Since salary is in logs and ceoten is in levels, the coefficient gives the proportional change in salary per unit change in tenure.
Q: Describe the partialling-out procedure for estimating the return to education in a wage regression with experience and tenure as controls.
A: First, regress education on experience and tenure and save the residuals (the part of education uncorrelated with the other two variables). Then regress log(wage) on those residuals. The coefficient on the residuals equals the coefficient on education from the full multiple regression of log(wage) on education, experience, and tenure.
Q: In the MEAP93 regression, the coefficient on log(expend) is larger when lnchprg is omitted. Explain why, using the omitted variable bias formula.
A: lnchprg has a negative partial effect on math10, and lnchprg is negatively correlated with log(expend). The bias on the spending coefficient is (negative effect) × (negative correlation) = positive. So omitting lnchprg inflates the spending coefficient upward.
Q: If the sample correlation between log(mktval) and profits is 0.78, does this mean the OLS estimates in the CEO salary regression are biased?
A: No. Correlation between included regressors (multicollinearity) inflates standard errors, making estimates less precise, but it does not bias OLS coefficients. Both variables are in the model, so there is no omitted variable problem.
The CEO salary exercise connects to the broader literature on executive compensation in labour economics and corporate governance. The education production function (MEAP93) connects to public economics and policy evaluation: how much does money matter in education, once you control for student demographics?
The partialling-out result reappears in Chapter 4 when you construct t-statistics, and in later chapters when you encounter instrumental variables (where a first-stage regression has the same flavour of isolating exogenous variation). The functional form choices here (log-log, level-log) are foundational for every applied chapter that follows.
CEO salary regression, CEOSAL2, constant elasticity model, log-log specification, log-level regression, level-log regression, profits in regression, ceoten, CEO tenure, partialling out, Frisch-Waugh-Lovell theorem, residuals from first-stage regression, wage equation, education return, MEAP93, school spending, math10, lnchprg, free lunch programme, education production function, omitted variable bias direction, intercept interpretation log model, multicollinearity in applied regression, Wooldridge Chapter 3 computer exercises