1.1 Causality and Potential Outcomes Framework

GOVT 300 GMU

Observation

information about a particular individual (at a particular time)

An arbitrary observation is denoted by

i

Observations are the

rows is most datasets

Unit of Observation

what our data defines as the individuals or entities for each observations (e.g., people, countries, districts)

Sample Size

the number of observations in our data

Sample Size is denoted by

n

Variable

contains the values of a particular characteristic of the observations

Variables correspond to the

columns in most datasets

Causality deals with the connection between two variables

Treatment variable (aka independent variable): X

Outcome variable (aka dependent variable, response variable): Y

Causality is directional

the treatment produces and influences the outcome: X → Y

That means X→Y is not the same as

Y → X, nor X ←→Y

X → Y (no intermediary)

X is direct/immediate/proximate cause of Y

X → a → b → Y

X is an indirect/remote cause of Y

(We can still say X → Y, even when the relationship is not direct)

Binary treatments

the treatment is present or not

Binary treatments (examples)

• Vaccinated/Not vaccinated

• College Graduate/Non College Graduate

• Incumbent/Non incumbent

Categorical (multinomial) treatment:

mutually exclusive categories

Categorical (multinomial) treatment examples:

• Pfizer/Moderna/Other/Not Vaccinated

• HS or less/Some college/ College / More than College

Numerical treatment

numeric variable with intrinsic meaning

Numerical treatment (examples):

• Number of doses

• Years of formal education

• Amount spend in a campaign

Define Xi

is the value of variable X for observation i

For a binary treatment

An individual can be in one of two conditions

• Treatment: is the condition under which i receives the treatment, Xi = 1

• Control: is the condition under which i doesn’t receive treatment, Xi = 0

Factual

what happened or what we observed

Counterfactual

What would have happened if some key condition were different, but all else remained the same

Under counterfactual causality: a treatment causes the outcome whenever the factual and the counterfactual differ from each other. Why?

Because all else was assumed the same; so if the factual and counterfactuals are different, it must be due to the treatment

All else the same (ceteris paribus) is really important! Why?

If something else were different, then maybe the treatment was not really the cause.

Formal version of counterfactual causality.

• Define two potential outcomes

Potential Outcome under Treatment: what would happen to i if i received the treatment.

• Denoted Yi (Xi = 1), or Yi (1) .

• Potential Outcome under Control: what would happen to i if i did not receive the treatment

• Denoted Yi (Xi = 0), or Yi (0).

All else prior and up to receiving or not receiving the treatment is the same (ceteris paribus)

Potential Outcome under Treatment

Yi (1)

Potential Outcome under Control

Yi (0)

Treatment Effect:

Measures the impact of the treatment:

ΔYi = Yi (1) - Yi (0)

ΔYi = 0

The treatment does not cause the outcome:

Yi (1) = Yi (0)

ΔYi > 0

The treatment causes the outcome (an increase):

Yi (1) > Yi (0)

ΔYi < 0

The treatment causes the outcome (a decrease):

Yi (1) < Yi (0)

Sample Average Treatment Effect (SATE)

Equivalent Formulas

where the line over means sample average

SATE measures

the average effect of the treatment on the outcome

If SATE = 0, we say that, on average, the treatment (X)

does NOT cause the outcome (Y)

If SATE > 0 we say that, on average, the treatment (X)

causes an increase of the outcome (Y)

If SATE < 0 we say that, on average, the treatment (X)

causes a decrease on the outcome (Y)

Population Average Treatment Effect (PATE)

Same as SATE, but for an entire population instead of a sample

Letting N

denote the size of the population

The interpretation of PATE is similar to SATE, but

focusing on the entire population instead of a sample

The Fundamental Problem of Causal Inference

• A unit of observation either gets the treatment or doesn’t!

• We only observe Yi (1) or Y (0 ) for any i !

• So, we cannot calculate ΔYi for any i !

• So, we cannot calculate SATE for any sample!

• And, we cannot calculate PATE for any population!