March 8, 2019

- hypotheses/empirical predictions
- fundamental problem of causal inference

statements about what we should **observe** if the causal claim is true, so stated in terms of independent and dependent **variables**

- multiple
**hypotheses**based on the overall causal claim**and**the causal logic - Hypotheses state relationships between \(X\) and
**potential outcomes**of \(Y\)- When \(X\) is present(absent) for a case, \(Y\) is present(absent)
- When \(X\) is present(absent) for a case, \(Y\) is more(less) likely
- When \(X\) increases(decreases) for a case, \(Y\) increases(decreases)

Trump's causal claim (implicitly): "The wall caused El Paso to have fewer murders".

**independent variable**(\(X\)): presence of wall (\(1\)) or absence of wall (\(0\)) \(Wall_{El \ Paso}\)**dependent variable**(\(Y\)): number of murders \(Murders_{El \ Paso}\)

potential outcomes of \(Murders_{El \ Paso}\) lower when \(Wall_{El \ Paso} = 1\)

\[Murders_{El \ Paso}^{Wall = Yes} < Murders_{El \ Paso}^{Wall = No}\]

causal claim: **"Ownership of firearms by citizens reduces crime"**

**independent variable**(\(X\)): fraction of citizens owning firearms \(Gun \ Rate_i\)**dependent variable**(\(Y\)): crime victimization rate per 100 thousand \(Crime \ Rate_i\)

Across communities, \(i\) in \(1 \ldots n\), on average, potential outcomes of \(Crime \ Rate_i\) lower for higher values of \(Gun \ Rate_i\)

\[Crime \ Rate_i^{Gun \ Rate = High} < Crime \ Rate_{i}^{Gun \ Rate = Low}\]

**Hypotheses** are **empirical predictions**: they are about what we should **observe** if \(X\) causes \(Y\).

**Counterfactual causality** means \(X\) causes \(Y\) in some specific case, *only if* **potential outcomes** of \(Y\) are different across levels of \(X\).

But for each case, only one **potential outcome** becomes "factual" and observable; the other potential outcome(s) remain **counterfactual** and **unobservable**

We can never observe all of the potential outcomes of \(Y\) for a case under different exposures to \(X\). We can only observe the potential outcome of a case that becomes **factual** where \(X\) takes on one, specific value. All other potential outcomes of \(Y\) remain **counterfactual** and **unobservable**.

For a single case, we canneverempirically observe whether \(X\) causes \(Y\)

causal claim: **"Befriending an immigrant causes people to have positive attitudes of immigrants"**

**independent variable**(\(X\)): person has immigrant friend \(Friend_i\): either \(1\) (yes) or \(0\) (no)**dependent variable**(\(Y\)): self-reported "overall attitude" towards immigrants \(Attitude_i\) is \((-2,-1,0,1,2)\) for strongly negative, negative, neutral, positive, strongly positive

We collect data on 5 people.

Across persons, \(i\) in \(1 \ldots 5\), on average, potential outcomes of \(Attitude_i\) are higher for higher values of \(Friend_i\)

We collect this data, and we observe this:

\(Person_i\) | \(Friend_i\) | \(Attitude_i\) |
---|---|---|

1 | Yes |
Positive (1) |

2 | Yes |
Very Positive (2) |

3 | No |
Neutral (0) |

4 | No |
Neutral (0) |

5 | No |
Negative (-1) |

Can we conclude \(Friend\) causes more positive \(Attitude\)s toward immigrants?

No, we've forgotten **potential outcomes**

\(Person_i\) | \(Friend_i\) | \(Attitude_i^{Yes}\) | \(Attitude_i^{No}\) |
---|---|---|---|

1 | Yes |
Positive (1) |
? |

2 | Yes |
Very Positive (2) |
? |

3 | No |
? |
Neutral (0) |

4 | No |
? |
Neutral (0) |

5 | No |
? |
Negative (-1) |

We want to know the difference in **potential outcomes** due to \(Friend\)ship: but FPCI means it is **unobservable**

\(Person_i\) | \(Friend_i\) | \(Attitude_i^{Yes}\) | \(Attitude_i^{No}\) | \(Attitude_i^{Yes} - Attitude_i^{No}\) |
---|---|---|---|---|

1 | Yes |
Positive (1) |
? |
? |

2 | Yes |
Very Positive (2) |
? |
? |

3 | No |
? |
Neutral (0) |
? |

4 | No |
? |
Neutral (0) |
? |

5 | No |
? |
Negative (-1) |
? |

If we were **omniscient deities**, maybe we could know all **potential outcomes**: actually, **no effect**

\(Person_i\) | \(Friend_i\) | \(Attitude_i^{Yes}\) | \(Attitude_i^{No}\) | \(Attitude_i^{Yes} - Attitude_i^{No}\) |
---|---|---|---|---|

1 | Yes |
Positive (1) |
Positive (1) |
0 |

2 | Yes |
Very Positive (2) |
Very Positive (2) |
0 |

3 | No |
Neutral (0) |
Neutral (0) |
0 |

4 | No |
Neutral (0) |
Neutral (0) |
0 |

5 | No |
Negative (-1) |
Negative (-1) |
0 |

We **make assumptions** that let us treat \(Y\) for cases that we observe with on value of \(X\) as **counterfactuals** for cases with different values of \(X\).

How **reasonable**/**believable** are these assumptions?

We cannot know the causal effect of \(X\) on \(Y\) for a specific case.

Let us **estimate** (draw **inferences****average** causal effect of \(X\) on \(Y\) for a group of cases.

Three assumptions (you only need to know (**1**))

**Random Assignment**to "Treatment" and "Control"**Exclusion Restriction**(only one thing is changing: \(X\))**SUTVA**(If I receive the treatment, it doesn't affect you)

Why do experiments work?

- In example with immigrant friendship and attitudes about immigrants, people with immigrant friends
**already different**in their potential outcomes (people who hate immigrants don't befriend them) - Experiments,
**through random assignment**, ensure that potential outcomes of treated/untreated cases are the same (except for**random sampling error**)

**Last year's PR referendum**

**How many of you voted?****Would have voted if tagged in a post like this?**

"Voting records are public! Records indicate that [name], [name], [name], [name], [name], [name], [name], [name], [name], and [name] have not yet voted in this yearâ€™s proportional representation referendum! Voting ends Friday. Click here for voting locations or how to vote from home. [Link]"