ECON 2041

Introductory Econometrics

Great expectations

Plan for the semester

1 What is econometrics?

2 Basic mathematical tools

3 Stats fundamentals

4 Stats fundamentals II

5 Simple regression

6 OLS properties & fit

7 Flavors of OLS & OVB Quiz

8 Causality

9 Regression inference

10 Inference, continued PS due 16 Oct

11 Diagnostics

12 Measurement error & IRL

13 Revision

Plan for today

  • Your grade forecasts, revealed
  • Income and life expectancy in the US
  • Discrete or categorical?
  • A correlation gallery
  • Where a covariance comes from

What you expect of yourself

What I asked you

Your own semester


  • Expected final mark (0-100)
  • Plans:
    • attend live?
    • study hours?
    • try tutorials before class?
  • The mark if you did "everything"

Three hypothetical students


  • Attends live lecture:
    rarely / some weeks / most
  • Study hours: <2 / 2-5 / 5+
  • Try tutorials first: no / yes


For each:

  • a mark
  • % chance they pass,
  • % chance they get a D

Nobody expects to fail

Full effort: you expect it to add about 7 points!

Last year's marks

Your forecast, given your plans

Attendance plans: flat

Study-hour plans: flat

Tutorials before class: where the action is

Confidence in the plan barely matters

Effort, student by student

What you expect of others

You DO think that attendance matters

You also think that study hours matter

Same story for tutorial work before class

Independence by design

Share of hypothetical students at each study-hours level, within each attendance level



Assigned study hours per week
Assigned attendance<22 - 55+N
Rarely44%30%26%50
Some weeks20%26%54%35
Most weeks42%26%32%38
All vignettes37%28%36%123

Predicted marks, given "rarely"

Predicted marks, given "some weeks"

Marks are not independent of attendance

Pass chance, given attendance & hours

Distinction \Rightarrow pass

For each hypothetical student, two chances:

  • P(\text{mark} \geq 50), the chance of a pass
  • P(\text{mark} \geq 75), the chance of a distinction


Every mark of 75 or more is also a mark of 50 or more

\{\text{mark} \geq 75\} \subset \{\text{mark} \geq 50\}, so P(\text{mark} \geq 75) \leq P(\text{mark} \geq 50)


10 of 120 answers put the probability of D > probability of P

Income and life expectancy

Life expectancy at birth

Life expectancy is based on an estimate of the average age that members of a particular population group will be when they die

Source:
Our World in Data

Life expectancy by current age

Source: Our World in Data

Guess the gap

US men, aged 40, 2001 - 2014

How many more years does a man in the richest 1% expect to live than a man in the poorest 1%?


The gap is 14.6 years

Men, age 40

  • Poorest 1%: 72.7 years
  • Richest 1%: 87.3 years

Women: a gap of 10.1 years

Y: life expectancy at 40
X: income percentile
Each dot is E[Y \mid X = x]

E[Y \mid X = \text{p20}] \approx 77

E[Y \mid X = \text{p80}] \approx 84

Source: Chetty et al. (2016)

We also condition on sex here

Y: life expectancy at 40
X: income percentile
Z: sex

Each line is
E[Y \mid X = x, Z = z]

Source: Chetty et al. (2016)

Unconditional vs. conditional expectation

E[Y] hides a lot of heterogeneity that we can see with E[Y | X]!

Looking beyond averages

Low-income men in the US have \approx LE as Zimbabwean men!

Is this a causal relationship?

Does this suggest that giving more money to those at the bottom of the income distribution would \uparrow their life expectancy?

Source: Chetty et al. (2016)

Descriptives can be incredibly powerful!

Place matters!

It's better to be poor in
New York than in Detroit

Source: Chetty et al. (2016)

Discrete or categorical?

A practical rule

Discrete: takes a finite set of values


Continuous: takes a continuum


But every column in every dataset has finitely many values

  • PISA escs, the socio-economic index: 62 distinct values in 12,136 rows
  • PISA math score: 3,858 distinct values


Ask what the variable measures in the world, not how many values the column has

Three kinds of variables in the world



In the world Example Arithmetic makes sense?
A count number of siblings: 0, 1, 2, ... yes: an average of 1.4 siblings
A measurement math score, socio-economic index, height yes: any value in a range, rounded when recorded
A label degree enrolled in, suburb, gender no: there is no average suburb

Watch out for numeric-looking codes

Education recorded as

  • 1 = completed primary
  • 2 = completed high school
  • 3 = completed a degree...


df['education'].mean() returns 2.3


2.3 is a fact about the codes, not about anyone's education

Where a covariance comes from

300 students, math against family background

Data: PISA 2022, Australia (OECD), 300 students drawn at random

Step 1: find the two means

Data: PISA 2022, Australia (OECD), 300 students drawn at random

Step 2: each student has two deviations

Step 3: the sign of the product is the quadrant

Step 4: add them up

Nonsense units, so we standardize

\text{cov}(X, Y) = \frac{1}{n}\sum_i (x_i - \bar{x})(y_i - \bar{y}) = 29 \;\text{index points} \times \text{score points}


r = \frac{\text{cov}(X, Y)}{s_X \, s_Y} = \frac{29}{0.78 \times 91.7} = 0.41

Some take-aways

  • A conditional expectation is a function: one average for every value of X
  • Averaging hides the ends: 14.6 years, inside one country
  • Correlation sees straight lines only, and a sample's r is itself random
  • The covariance is a vote by quadrant; dividing by the SDs removes the units