**1. Fundamental Problem of Causal Inference**

- causal hypotheses
- independent/dependent variables

**2. Correlation**

- scatterplots
- problems with correlation

March 16, 2021

- causal hypotheses
- independent/dependent variables

- scatterplots
- problems with correlation

In February 2019, Donald Trump held a rally in El Paso, TX. Argued that migrants were dangerous.

- “A wall is a very good thing, not a bad thing. Itâ€™s a moral thing”
- “[reducing numbers of immigrants in detention would be] cutting loose dangerous criminals into our country.”
- “The Democrat Party [is] becoming the party of socialism, late-term abortion,
**open borders and crime.**”

In August 2019, an armed man killed 22 people at a Walmart in El Paso, TX. In advance of his attack, he issued a manifesto that stated he was motivated in response to an alleged “Hispanic invasion of Texas.”

- It has been suggested that there is a causal link between these events.

A causal claim:

“Trumpâ€™s rally in El Paso increased the likelihood of hate crimes against immigrants.”

In Groupsâ€¦

- what is the implied counterfactual claim?
- If the causal claim were
**true**, what would be the potential outcomes of hate crimes in El Paso in 2019?

**Counterfactual claim:**

“If Trump had not held a rally in El Paso (in 2019), then there would have been fewer hate crimes against immigrants.”

**Potential Outcomes:**

\(\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{Rally}) >\) \(\color{red}{\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{No \ Rally})}\)

\(\mathrm{Black}\) is factual; \(\color{red}{\mathrm{Red}}\) is counterfactual

We make causal claims testable by generatingâ€¦

**causal hypotheses** (or empirical predictions):

statements about what we should **observe** if the causal claim is true.

Hypotheses state the **potential outcomes** implied by the causal claim. Put another way, what is implied relationship between **independent** (cause) and **dependent** (outcome) variables.

- If \(X\) were present(absent), then \(Y\) would be more(less) likely
- If \(X\) were to increase(decrease), then \(Y\) would increase(decrease)

Because causal hypotheses are **observable**, stated in terms of **variables**.

**Independent variable:**

The variable capturing the alleged cause in a causal claim.

- often called the “IV” or “X” or “right-hand variable”

**Dependent variable:**

The variable capturing the alleged outcome (what is affected) in a causal claim.

- often denoted as “DV” or “Y” or “left-hand variable”

**Potential Outcomes** are the values of **dependent variable** a case would take different values of the **independent variable**

“Trump rallies in a community increase the likelihood of hate crimes against immigrants.”

What could be an **independent variable** used to test this causal claim?

What could be a **dependent variable** used to test this causal claim?

\(\mathrm{City}_i\) | \(\mathrm{Rally}_i\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{Rally})\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{No \ Rally})\) |
---|---|---|---|

El Paso | Yes |
> 1 |
? |

How would we find the “\(?\)”?

For any case, we can **only** observe the potential outcome of \(Y\) for the value of \(X\) that the case is actually exposed to.

We can **never observe** the other, **counterfactual** potential outcomes of \(Y\) for different possible values of \(X\) that the case did not experience.

Causality is counterfactual, so for any specific case, we can never empirically observe whether \(X\) causes \(Y\).

If Trumpâ€™s rally cause an increase hate crimes in El Paso, we need to see this:

\(\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{Rally}) >\) \(\color{red}{\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{No \ Rally})}\)

While the value in \(\mathrm{Black}\) is factual; The value in \(\color{red}{\mathrm{Red}}\) is counterfactual and can **never** be known.

We **cannot observe**: \(\color{red}{\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{No \ Rally})}\)

But we **can** observe, e.g.: \(\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})\)

If we **assume**: \(\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})\) \(=\) \(\color{red}{\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{No \ Rally})}\)

Then, we can test our causal claim, to see if:

\(\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{Rally}) >\) \(\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})\)

\(\mathrm{City}_i\) | \(\mathrm{Rally}_i\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{Rally})\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{No \ Rally})\) |
---|---|---|---|

El Paso | Yes |
\(\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{Rally})\) | \(\color{red}{\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{No \ Rally})}\) |

\(\mathbf{\Uparrow}\) | |||

Austin | No |
\(\color{red}{\mathrm{Hate \ Crimes}_{Austin}(\mathrm{Rally})}\) | \(\boxed{\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})}\) |

\(\mathrm{City}_i\) | \(\mathrm{Rally}_i\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{Rally})\) | \(\mathrm{Hate \ Crimes}_i(\mathrm{No \ Rally})\) |
---|---|---|---|

El Paso | Yes |
\(\mathrm{Hate \ Crimes}_{El \ Paso}(\mathrm{Rally})\) | \(\boxed{\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})}\) |

\(\mathbf{\Uparrow}\) | |||

Austin | No |
\(\color{red}{\mathrm{Hate \ Crimes}_{Austin}(\mathrm{Rally})}\) | \(\mathrm{Hate \ Crimes}_{Austin}(\mathrm{No \ Rally})\) |

Every solution to the FPCI involves:

Comparing the

**observed**values of outcome \(Y\) in cases that**actually**have different values of cause \(X\)Making assumptions that let us treat

**factual**(observed) potential outcomes from some cases as**counterfactual**(unobserved) potential outcomes of other cases.

**Correlation** is the degree of association/relationship between the observed values of \(X\) (the independent variable) and \(Y\) (the dependent variable)

- There are formal mathematical definitions.
- We use the term loosely to describe observed relationship between \(X\) and \(Y\)

**All empirical evidence for causal claims** relies on **correlation** between the independent and dependent variables.

But, youâ€™ve all heard this:

- Many different ways of doing this (compare group means, regression) are all fundamentally about correlation.
- correlations have a
**direction**:- positive: implies that as \(X\) increases, \(Y\) increases
- negative: \(X\) increases, \(Y\) decreases

- correlations have
**strength**(has nothing to do**size of effect**):**strong**: \(X\) and \(Y\) almost**always**move together**weak**: \(X\) and \(Y\) do not move together very much

**mathematically**: correlation is the degree of **linear** association between \(X\) and \(Y\)

- Takes values between \(-1\) and \(1\)
- Values close to \(1\) or \(-1\) suggest
**high**degree of*linear*association - Values close to \(0\) suggest
**low**degree of*linear*association - Value of correlation does
**not**tell us**how much**\(Y\) changes with \(X\)

**negative correlation**: (correlation \(< 0\)) values of \(X\) and \(Y\) move in opposite direction:

- higher values of \(X\) appear with lower values of \(Y\)
- lower values of \(X\) appear with higher values of \(Y\)

**positive correlation**: (correlation \(> 0\)) values of \(X\) and \(Y\) move in same direction:

- higher values of \(X\) appear with higher values of \(Y\)
- lower values of \(X\) appear with lower values of \(Y\)