2 Initial Concepts

Status: ported 2026-05-18. Reviewed by editor: pending.

Learning outcomes

By the end of this chapter the reader should be able to:

Explain what econometrics is and how an econometric model differs from a purely theoretical economic model.
Interpret the two components of an econometric model: the systematic part and the random error $u$.
Distinguish correlation from causation, and identify selection bias and omitted-variable bias in concrete examples.
Explain why randomisation eliminates selection bias on average, and contrast randomised experiments with quasi-experimental and observational designs.
Classify a dataset as cross-sectional, time-series, panel, or pooled cross-section, and recognise the research questions each structure can (and cannot) answer.
Use basic R commands (str(), summary(), head(), hist(), plot(), tapply(), cor()) to explore a microeconomic dataset and compute conditional means.

Motivating empirical question

Does going to university cause higher wages, or do the people who would have earned more anyway tend to go to university?

This single question drives the rest of the course. A correlation between schooling and wages is easy to compute; turning that correlation into a credible statement about the causal return to education is hard. The tools we develop — the linear model, the assumptions on the error term, randomisation, controls, and tests — are all in the service of answering questions of this form.

2.1 1.1 What is econometrics?

Econometrics is the application of statistical methods to economic data, with the goal of giving empirical content to economic relationships (Wooldridge 2020; Hill et al. 2017). Its main objective is the estimation of relationships between variables. Typical pairs of interest in this course include:

Education $\Rightarrow$ Salary
Class attendance $\Rightarrow$ Final grade
Tuition fees $\Rightarrow$ Enrolment
Income $\Rightarrow$ Happiness
Government spending $\Rightarrow$ Crime
Advertising $\Rightarrow$ Sales

In general we write $X \Rightarrow Y$, where $X$ is the independent variable (or regressor, explanatory variable) and $Y$ is the dependent variable (or response, outcome). Two questions then arise:

Is there an effect of $X$ on $Y$?
How large is that effect?

These “how much” questions are everywhere. A city council wonders how much violent crime will fall if it spends an additional million euros on policing. A local business must estimate the relationship between advertising spending and sales. A university must estimate how much enrolment will drop if it raises tuition by 300 euros. None of these questions can be answered by economic theory alone — they require data, a statistical model, and the discipline of econometrics.

2.2 1.2 The econometric model

An econometric model has two components:

A systematic component — the average or systematic behaviour of many individuals, firms, or other units.
A random error $u$ (sometimes called the disturbance) — the many factors that affect $Y$ but are not included in the systematic part.

Example: salary determination

The factors associated with an individual’s salary include age, gender, experience, and education. In general functional form:

\[ \text{Salary} = f(\text{Age},\,\text{Gender},\,\text{Experience},\,\text{Education}) + u. \]

A linear specification is

\[ \text{Salary} = \beta_0 + \beta_1\,\text{Age} + \beta_2\,\text{Gender} + \beta_3\,\text{Experience} + \beta_4\,\text{Education} + u. \]

The coefficients $\beta_0, \beta_1, \dots, \beta_4$ are unknown population parameters that we will estimate from data using an econometric technique (typically Ordinary Least Squares, introduced in Chapter 2).

The functional form is itself a hypothesis about how the variables are connected. A key challenge in any applied problem is choosing a functional form that is compatible both with economic theory and with what the data look like.

Common mistake: confusing $u$ with the residual $\hat u$

The error $u$ is a population object: it represents every determinant of $Y$ that is not in the model, and it is unobservable. The residual $\hat u_i = y_i - \hat y_i$ is its sample counterpart, obtained after estimating the model. Following Wooldridge’s notation, we keep $u$ and $\hat u$ strictly separate throughout the book.

2.3 1.3 Uses of econometrics

We use econometric models for three broad purposes:

Test economic theories and hypotheses. For example: does increasing the minimum wage reduce employment? Does class attendance improve grades?
Predict economic variables. For example: forecast next quarter’s GDP growth, or tomorrow’s electricity demand.
Estimate causal relationships. For example: does an additional year of education cause higher wages, or are educated workers simply different to begin with?

The third use — credible causal inference from observational data — is at the heart of the so-called Credibility Revolution in empirical economics. Hypothesis testing and prediction are sometimes called “Led Zeppelin” econometrics; causal inference is the modern emphasis (Hill et al. 2017).

2.4 1.4 Correlation does not imply causation

One of the most important lessons of the course — repeated, if necessary, on every page — is that correlation does not imply causation. The fact that two variables move together does not establish that one of them causes the other. Two reasons stand out: selection bias and omitted variables.

2.4.1 1.4.1 Selection bias

Example: ventilators and deaths

A hospital wants to know whether ventilators save lives. A naive comparison shows that patients on ventilators die more often than patients not on ventilators. Should we conclude that ventilators kill patients?

Obviously not. The problem is selection bias: the sickest patients are precisely the ones who are placed on a ventilator. These patients would have had a higher mortality rate regardless of whether they received a ventilator.

The observed correlation between ventilator use and death is positive; the true causal effect of ventilators on survival is negative. A naive comparison can therefore give us not just the wrong magnitude but the wrong sign of the effect.

The lesson is general: whenever treatment is not assigned randomly, simple comparisons of outcomes between treated and untreated units confound the causal effect of the treatment with the characteristics of the units that selected into it.

2.4.2 1.4.2 Omitted variable bias

Example: ice cream and drowning

Data show a strong positive correlation between ice-cream sales and drowning deaths. Should we ban ice cream to prevent drowning?

Of course not. The omitted variable is temperature. Hot weather causes both more ice-cream sales and more swimming (hence more drowning). Without controlling for temperature, the correlation between ice cream and drowning is a spurious correlation — a real statistical association with no causal content.

Whenever a variable that influences $Y$ is also correlated with $X$ but is left out of the model, the estimate of the effect of $X$ will be biased. We will study this formally as omitted variable bias in Chapter 3.

2.4.3 1.4.3 Why randomisation solves the problem

If we randomly assign individuals to a treatment group and a control group, then on average the two groups will have the same observable and unobservable characteristics before the treatment is administered. Any difference in outcomes after the treatment can therefore be attributed to the treatment itself.

In the ventilator example, if we could randomly assign ventilators (not based on severity of illness), then the treatment and control groups would have the same average severity. The difference in mortality between the two groups would be an unbiased estimate of the causal effect of ventilators.

A useful identity to keep in mind is

\[ \underbrace{\text{difference in outcomes}}_{\text{what we observe}} = \underbrace{\text{causal effect}}_{\text{what we want}} + \underbrace{\text{selection bias}}_{\text{nuisance}}. \]

Randomisation forces the second term on the right to zero on average. That is why Randomised Controlled Trials (RCTs) are considered the gold standard for establishing causality.

Example: private schools and test scores

A study finds that students at private schools score higher on standardised tests. Can we conclude that private schools cause better performance? No — students at private schools tend to come from wealthier families with more resources at home, more educated parents, and so on. The “treatment” (attending a private school) is not randomly assigned, so the comparison is contaminated by selection.

2.5 1.5 Types of experiments

2.5.1 1.5.1 Randomised experiments

A randomised experiment assigns participants to two or more groups by lot: one group receives the intervention (the treatment); another serves as the control and receives a placebo or standard treatment. Randomisation ensures the groups are comparable, eliminates selection bias on average, and permits causal inference.

Genuine randomised experiments are common in clinical medicine but rare in economics and business, for two reasons:

Ethical constraints — we cannot randomly assign individuals to poverty, unemployment, or disease.
Cost — well-designed field experiments are expensive and slow to run.

2.5.2 1.5.2 Quasi-experiments

In a quasi-experiment, assignment to treatment is not random but is based on some observable criterion — often a threshold, an eligibility rule, or a policy change that hits some units but not others.

Example: a regional lockdown

During the COVID-19 pandemic, lockdown (the treatment) was assigned by the Junta de Andalucía only to municipalities with infection rates above 1{,}000 per 100{,}000 inhabitants. Municipalities just above and just below the threshold are arguably similar, so comparing their outcomes can approximate a random experiment. We return to this kind of design later in the book.

2.5.3 1.5.3 Nonexperimental (observational) data

With nonexperimental data, all variables are collected simultaneously and the values are neither fixed nor repeatable by the researcher. Survey data are the classic example — the U.S. Current Population Survey (CPS) and the Spanish Encuesta de Población Activa (EPA) both belong to this category. Most of the datasets we use in this book are nonexperimental, which is precisely why the issue of selection bias matters so much.

2.6 1.6 Types of economic data

Economic data come in several “flavours”. Aggregation can be micro (households, firms, workers) or macro (regions, countries); variables can be quantitative (numbers) or qualitative (categories, e.g. employed / unemployed). For econometrics what matters most is the structure of the dataset: do we have one snapshot, a sequence over time, or repeated observations of the same units?

Definition: cross-sectional data

Data collected across sample units — individuals, firms, households, regions, countries — in a particular time period. Examples: income by Californian county in 2016; high-school graduation rates by U.S. state in 2015; the wage1 dataset we use in the lab below.

Definition: time-series data

Data collected over discrete intervals of time, with the same economic quantity recorded at a regular frequency. Examples: the annual price of wheat, daily General Electric stock prices, monthly Spanish unemployment rates from 1980 to 2024.

Definition: panel (longitudinal) data

Observations on individual micro-units that are followed over multiple time periods. The defining feature is that each unit is observed in several periods. If every unit is observed in every period, the panel is balanced; otherwise it is unbalanced. Typically the number of time periods is small relative to the number of units, but not always.

Definition: pooled cross-section

Independent cross-sections drawn in different periods and stacked together. Different units are sampled each time — there is no re-interview of the same individuals — but the periods can be compared. The Spanish EPA, which draws a fresh independent sample of households every quarter, yields a pooled cross-section when several waves are stacked.

The data structure matters because it determines the questions we can answer. With a cross-section we can study how outcomes vary across units at one moment in time. With a time-series we can study how a single quantity evolves over time. With a panel we can do both — and, crucially, we can control for time-invariant unobserved characteristics of each unit, an important tool in modern causal inference. With a pooled cross-section we can study how a relationship changes between periods (for example, before and after a policy reform).

2.7 1.7 Lab: Exploring economic data in R

The rest of the chapter is a guided exploration of the wage1 dataset shipped with the wooldridge R package. The dataset is a cross-section of 526 U.S. workers observed in May 1976, with information on hourly wages, education, experience, tenure, and a handful of demographic variables. (The mechanics of installing R and the wooldridge package are covered in Appendix A; the first formal regression with lm() is the opening lab of Chapter 2.)

We start by loading the package and looking at the data.

Code

library(wooldridge)
data("wage1")

2.7.1 Structure, size, and a first look

Code

str(wage1)

'data.frame':   526 obs. of  24 variables:
 $ wage    : num  3.1 3.24 3 6 5.3 ...
 $ educ    : int  11 12 11 8 12 16 18 12 12 17 ...
 $ exper   : int  2 22 2 44 7 9 15 5 26 22 ...
 $ tenure  : int  0 2 0 28 2 8 7 3 4 21 ...
 $ nonwhite: int  0 0 0 0 0 0 0 0 0 0 ...
 $ female  : int  1 1 0 0 0 0 0 1 1 0 ...
 $ married : int  0 1 0 1 1 1 0 0 0 1 ...
 $ numdep  : int  2 3 2 0 1 0 0 0 2 0 ...
 $ smsa    : int  1 1 0 1 0 1 1 1 1 1 ...
 $ northcen: int  0 0 0 0 0 0 0 0 0 0 ...
 $ south   : int  0 0 0 0 0 0 0 0 0 0 ...
 $ west    : int  1 1 1 1 1 1 1 1 1 1 ...
 $ construc: int  0 0 0 0 0 0 0 0 0 0 ...
 $ ndurman : int  0 0 0 0 0 0 0 0 0 0 ...
 $ trcommpu: int  0 0 0 0 0 0 0 0 0 0 ...
 $ trade   : int  0 0 1 0 0 0 1 0 1 0 ...
 $ services: int  0 1 0 0 0 0 0 0 0 0 ...
 $ profserv: int  0 0 0 0 0 1 0 0 0 0 ...
 $ profocc : int  0 0 0 0 0 1 1 1 1 1 ...
 $ clerocc : int  0 0 0 1 0 0 0 0 0 0 ...
 $ servocc : int  0 1 0 0 0 0 0 0 0 0 ...
 $ lwage   : num  1.13 1.18 1.1 1.79 1.67 ...
 $ expersq : int  4 484 4 1936 49 81 225 25 676 484 ...
 $ tenursq : int  0 4 0 784 4 64 49 9 16 441 ...
 - attr(*, "time.stamp")= chr "25 Jun 2011 23:03"

Code

head(wage1)

  wage educ exper tenure nonwhite female married numdep smsa northcen south
1 3.10   11     2      0        0      1       0      2    1        0     0
2 3.24   12    22      2        0      1       1      3    1        0     0
3 3.00   11     2      0        0      0       0      2    0        0     0
4 6.00    8    44     28        0      0       1      0    1        0     0
5 5.30   12     7      2        0      0       1      1    0        0     0
6 8.75   16     9      8        0      0       1      0    1        0     0
  west construc ndurman trcommpu trade services profserv profocc clerocc
1    1        0       0        0     0        0        0       0       0
2    1        0       0        0     0        1        0       0       0
3    1        0       0        0     1        0        0       0       0
4    1        0       0        0     0        0        0       0       1
5    1        0       0        0     0        0        0       0       0
6    1        0       0        0     0        0        1       1       0
  servocc    lwage expersq tenursq
1       0 1.131402       4       0
2       1 1.175573     484       4
3       0 1.098612       4       0
4       0 1.791759    1936     784
5       0 1.667707      49       4
6       0 2.169054      81      64

Code

nrow(wage1)

[1] 526

Code

ncol(wage1)

[1] 24

The output of str() tells us that wage1 is a data frame with 526 observations and 24 variables. Each row is one worker. The variables we care about most in this chapter are:

wage — hourly wage, in 1976 U.S. dollars,
educ — years of schooling,
exper — years of potential labour-market experience,
tenure — years with the current employer,
female — a dummy equal to 1 for women, 0 for men,
married — a dummy equal to 1 for married workers.

2.7.2 Summary statistics and distributions

Code

summary(wage1[, c("wage", "educ", "exper", "tenure")])

      wage             educ           exper           tenure      
 Min.   : 0.530   Min.   : 0.00   Min.   : 1.00   Min.   : 0.000  
 1st Qu.: 3.330   1st Qu.:12.00   1st Qu.: 5.00   1st Qu.: 0.000  
 Median : 4.650   Median :12.00   Median :13.50   Median : 2.000  
 Mean   : 5.896   Mean   :12.56   Mean   :17.02   Mean   : 5.105  
 3rd Qu.: 6.880   3rd Qu.:14.00   3rd Qu.:26.00   3rd Qu.: 7.000  
 Max.   :24.980   Max.   :18.00   Max.   :51.00   Max.   :44.000

Code

sd(wage1$wage)

[1] 3.693086

Code

sd(wage1$educ)

[1] 2.769022

summary() gives us min, quartiles, mean, and max for each variable. The mean wage is about $5.90 per hour and the median is about $4.65; the mean exceeds the median, which suggests a right-skewed distribution — a long upper tail of high earners. A histogram makes the shape obvious:

Code

hist(wage1$wage,
     main = "Distribution of hourly wages",
     xlab = "Wage (USD / hour)",
     col  = "lightblue",
     breaks = 30)

Distribution of hourly wages in wage1 (1976 USD).

Code

hist(wage1$educ,
     main = "Distribution of years of education",
     xlab = "Years of education",
     col  = "lightgreen",
     breaks = 15)

Distribution of years of education in wage1.

The education distribution piles up at 12 years (high-school completion) and 16 years (a four-year college degree), as one would expect for U.S. data from this era.

2.7.3 Frequency tables

For categorical or count-like variables, table() is more useful than summary():

Code

table(wage1$female)


  0   1 
274 252

Code

table(wage1$married)


  0   1 
206 320

Code

table(wage1$female) / nrow(wage1)


        0         1 
0.5209125 0.4790875

About 48% of the sample are women and 52% are men.

2.7.4 A first scatter plot

A natural starting question is: do workers with more education earn more?

Code

plot(wage1$educ, wage1$wage,
     xlab = "Years of education",
     ylab = "Hourly wage (USD)",
     main = "Wage vs. education",
     pch  = 16,
     col  = rgb(0, 0, 1, 0.3))

Wage versus years of education. Each dot is one of the 526 workers.

The cloud of points slopes upward: on average, more educated workers earn higher wages. We can quantify the linear association with the sample correlation:

Code

cor(wage1$wage, wage1$educ)

[1] 0.4059033

Code

cor(wage1$wage, wage1$exper)

[1] 0.1129034

Code

cor(wage1$educ, wage1$exper)

[1] -0.2995418

The correlation between wage and education is about $0.41$ — a clearly positive but far-from-perfect linear association. Two further patterns stand out:

The correlation between wage and experience is positive but smaller than the correlation with education.
The correlation between education and experience is negative. This is not because schooling destroys experience — it reflects how individuals allocate time across the life cycle: more years in school mean fewer years in the workforce, mechanically. We will see in Chapter 3 that ignoring this kind of correlation among regressors can severely bias a simple regression of wage on education.

2.7.5 Conditional means: a preview of regression

A cleaner way to look at the data is to compute the conditional mean of wages given another variable. tapply() does exactly that:

Code

tapply(wage1$wage, wage1$female,  mean)

       0        1 
7.099489 4.587659

Code

tapply(wage1$wage, wage1$married, mean)

       0        1 
4.843884 6.573469

Men earn about $7.10 per hour on average; women earn about $4.59. Married workers earn about $6.32 per hour; single workers about $5.16. These are raw, unconditional comparisons.

Common mistake: reading a conditional mean as a causal effect

The 2.51-dollar gender gap and the 1.16-dollar marriage premium are conditional means, not causal effects. The men and women in this sample differ in occupation, sector, experience, hours worked, and many other characteristics; married and single workers differ in age and tenure. Without holding those confounders fixed, neither gap can be interpreted as the ceteris paribus effect of gender or marriage on wages. Holding confounders fixed is what multiple regression (Chapter 3) is for.

A more striking selection-bias illustration uses the conditional mean of experience given education:

Code

tapply(wage1$exper, wage1$educ, mean)

       0        2        3        4        5        6        7        8 
32.00000 39.00000 51.00000 41.00000 34.00000 30.50000 31.25000 31.40909 
       9       10       11       12       13       14       15       16 
15.94118 14.83333 13.93103 18.14646 14.89744 16.13208 13.38095 12.35294 
      17       18 
13.33333 11.10526

Workers with little schooling have more labour-market experience, and workers with a college degree have less. So when we compare wages across education levels, we are simultaneously comparing workers who differ in experience — the comparison is confounded. This is the same kind of selection problem we saw in the ventilator and private-school examples, now in our actual dataset. We will fix it in Chapter 3 by including both educ and exper in a multiple regression.

Why wage1 is a cross-section

wage1 records each worker once, all in 1976. There is no time index, no person identifier observed in several years. It is a clean cross-section. With this dataset alone we cannot track how a given worker’s wage grows from one year to the next — that would require a panel, e.g. the U.S. Panel Study of Income Dynamics (PSID) or the Spanish Muestra Continua de Vidas Laborales (MCVL). Knowing what your data can and cannot tell you is half of applied econometrics.

Self-check

Eight short multiple-choice questions. Try each one before opening the answer.

Q1. What is an econometric model?

An econometric model differs from a purely economic (theoretical) model in that:

A. It assumes the relationship between variables is exact and deterministic.
B. It adds a stochastic error term and a functional form so that unknown parameters can be estimated from data.
C. It does not require a sample of data because the parameters are derived from theory.
D. It is only used in macroeconomic problems, not in microeconomic ones.

Answer: B. An econometric model couples a functional form with a random disturbance $u$, which makes the unknown parameters estimable from observed data.

Q2. The role of the error term

In the population regression model $y = \beta_0 + \beta_1 x + u$, the error term $u$:

A. Is the difference between $y_i$ and the OLS fitted value $\hat y_i$ in our sample.
B. Is always equal to zero in the population.
C. Captures every factor — other than $x$ — that influences $y$.
D. Is observable as soon as we have data on $y$ and $x$.

Answer: C. $u$ is a population, unobservable quantity collecting all omitted determinants of $y$. The residual $\hat u_i$ is its sample counterpart, computed only after estimation.

Q3. Cross-section or not?

A dataset with information on 526 workers interviewed in May 1976 (one observation per worker, all measured at the same time) is:

A. Cross-sectional data.
B. Time-series data.
C. Panel (longitudinal) data.
D. Pooled cross-section data.

Answer: A. Many units, one time period, no repeated observation of the same unit — the textbook definition of a cross-section.

Q4. Time series

Annual Spanish unemployment rates from 1980 to 2024 are an example of:

A. Cross-sectional data.
B. Time-series data.
C. Panel data.
D. Experimental data.

Answer: B. One quantity (the unemployment rate) recorded at a regular frequency over time.

Q5. Panel vs pooled cross-section

A dataset follows 1{,}000 firms during the years 2010–2024, recording sales, employment and exports for each firm in every year. This is:

A. Cross-sectional data, since each firm-year is one row.
B. Pooled cross-section, since different units are sampled in each period.
C. Time-series data, since it has a time dimension.
D. Panel (longitudinal) data: the same units observed repeatedly over time.

Answer: D. The same firms are tracked over multiple periods, which is exactly the panel structure.

Q6. What does correlation tell us?

A high positive correlation between two variables $X$ and $Y$ implies that:

A. $X$ causes $Y$.
B. $Y$ causes $X$.
C. There exists a deterministic linear relationship between them.
D. They tend to move together, but this association may be driven by a third variable, by reverse causality, or by selection.

Answer: D. Correlation is silent about causal direction or confounding. The ice-cream-and-drowning example in §1.4.2 is the canonical warning.

Q7. Selection bias

Selection bias arises when:

A. The units that select into the treatment group differ systematically from those that do not, even before treatment.
B. The researcher chooses regressors based on $t$-statistics.
C. The sample size is too small to detect a true effect.
D. The dependent variable is measured with error.

Answer: A. Selection bias is a property of the assignment of treatment, not of estimation or measurement. The ventilator example in §1.4.1 is the prototype.

Q8. Randomisation as a remedy

In which of the following research designs does randomisation by construction eliminate selection bias?

A. Comparing employed vs. unemployed individuals using cross-sectional survey data.
B. A randomised controlled trial (RCT) where the treatment is assigned by lottery.
C. Comparing students who chose to enrol in a private vs. a public school.
D. Comparing countries that adopted the euro vs. countries that did not.

Answer: B. Only random assignment guarantees that, on average, treated and untreated units have the same pre-treatment characteristics.

Exercises

Exercise 1.1 ★ — Classifying datasets. Classify each of the following datasets as cross-sectional, time-series, panel, or pooled cross-section.

Monthly Spanish CPI from 1980 to 2024.
wage1: 526 workers interviewed in May 1976, one row per worker.
The Spanish EPA: each quarter, a fresh independent sample of households is drawn (no household is revisited); several quarters are then stacked.
A dataset that follows the same 800 students from age 6 to age 18, recording their test scores every year.

Show answer

Time-series. (b) Cross-sectional. (c) Pooled cross-section — new units each period. (d) Panel: the same units are followed over multiple periods.

Exercise 1.2 ★ — Spotting selection bias. For each scenario, state whether selection bias is a likely problem, and identify which characteristic drives the selection.

A hospital compares mortality of patients who received a ventilator with patients who did not.
An economist compares the wages of workers who completed a job-training programme with the wages of workers who did not enrol.
Researchers compare countries that adopted the euro in 1999 with countries that did not.

Show answer

Yes — severity of illness drives both ventilator assignment and mortality. (b) Yes — self-selection into training is driven by motivation, prior earnings, and labour-market prospects, all of which also affect wages. (c) Yes — countries that joined the euro were chosen on the basis of macroeconomic fundamentals (the Maastricht criteria), so they are not a random sample of European countries.

Exercise 1.3 ★★ — Sign of an omitted-variable bias. Suppose you regress hourly wage on years of education in a sample of currently employed adults, without controlling for innate ability. Assume that (i) more able workers earn more, holding education fixed, and (ii) more able workers tend to acquire more education. Will the coefficient on educ be biased upward (too positive) or downward (too negative)? Explain in one sentence.

A full answer is given in the Instructor Edition.

Exercise 1.4 ★★ — Designing a randomised experiment. You want to know whether a free university preparation course (the treatment) causally raises the probability that high-school students from low-income families enrol in university. Sketch the design of a randomised experiment that would identify this causal effect. Be explicit about: the population, the assignment rule, the treatment and control groups, the outcome variable, and the time horizon. Briefly discuss one ethical objection your design will have to address.

A full answer is given in the Instructor Edition.

Exercise 1.5 ★★ — A naive omitted-variable model. A worker’s wage is generated by

\[ w = 5 + 2 \cdot \mathit{edu} + u, \qquad \mathbb{E}[u] = 0,\ \operatorname{Var}(u) = 1, \]

where $\mathit{edu}\in\{0,1,2,3\}$ are years of post-compulsory education. Suppose that ability (collected in $u$) is correlated with education: $\mathbb{E}[u\mid \mathit{edu}] = 0.5\cdot \mathit{edu}$.

Compute $\mathbb{E}[w\mid \mathit{edu}]$.
If we naively assume $\mathbb{E}[u\mid \mathit{edu}] = 0$ and read off the slope, what do we get? By how much do we miss the true marginal effect of an extra year of education?

A full answer is given in the Instructor Edition.

Exercise 1.6 ★★★ — Chocolate and Nobel prizes. It has been documented (Hill et al. 2017) that countries with higher per-capita chocolate consumption also win more Nobel prizes per capita. Is this evidence that chocolate consumption causes Nobel prizes? Identify (i) at least one plausible omitted variable, (ii) at least one alternative causal channel (reverse causation, selection of which countries get measured, etc.), and (iii) describe a hypothetical research design — experimental or quasi-experimental — that would allow you to learn the causal effect of chocolate on prize-winning. Be honest about why this design is unlikely to be feasible.

A full answer is given in the Instructor Edition.

Hill, R. Carter, William E. Griffiths, and Guay C. Lim. 2017. Principles of Econometrics. 5th ed. Wiley.

Wooldridge, Jeffrey M. 2020. Introductory Econometrics: A Modern Approach. 7th ed. Cengage Learning.