March 6, 2019

Causal Logics

Plan for Today:

(1) Causal Theories

  • Causal Logics
  • Independent/Dependent Variables

(2) Testing Causal Theories

  • hypotheses/empirical predictions
  • fundamental problem of causal inference

Causal Logics

Causal Theory

To evaluate causal claims, we turn them into causal theories

A causal theory identifies systematic/structural causes that operate across space and time

  • It is a kind of general knowledge
  • Patterns/regularities within complexity
  • and it is testable

Causal Logics

One part of a causal theory is:

causal logic:

Is a set of statements about how and why a cause \(C\) produces its (claimed) effect \(E\).

  • causal chain connects cause \(C\) to the effect \(E\): \(C \xrightarrow{} e_1 \xrightarrow{} e_2 \xrightarrow{} e_3 \xrightarrow{} E\)
  • Each step/mechanism is causal: \(e_1\) is a cause of \(e_2\)

Assumptions/assertions

  • Each step in the logic assumes a theory of how the world works/might work (why \(e_1 \to e_2\))

Making a causal logic

Causal logic for claim that \(C \xrightarrow{causes} E\):

  1. Must start with cause \(C\), end with the effect \(E\)
  2. Each step is related to the next through a causal relationship
  3. Not a chronological list of specific events. It is a general sequence through which one things causes another (structural causes)

Why causal logics?

  1. Tell us how and when cause creates effect
    • this can help refine the scope (generality) of the causal claim
  2. Help us test causal claim
    • we can test the main cause/effect, but also each step in the mechanism
  3. Suggest other causes
    • If \(C_1 \to e_1 \to e_2 \to E\), then \(C_2\) might \(\to e_2 \to E\)
  4. Improve prescriptive claims
    • If we can't change \(C\), can we change \(e_2\)?
    • Policies require specifics: causal logics more specific on how

Causal Logics and "Scope"

"Increasing wealth causes countries to be less likely to experience civil war."

Under what conditions?

A causal logic

  1. Wealth \(\xrightarrow{}\) Larger defense budget \(\xrightarrow{}\) Rivals deterred \(\xrightarrow{}\) Less civil war

Causal Logics and "Testing"

"Increasing wealth causes countries to be less likely to experience civil war."

What tests are there?

A different causal logic

  1. Wealth \(\xrightarrow{}\) Higher standard of living \(\xrightarrow{}\) Fewer grievances \(\xrightarrow{}\) Weak support for change \(\xrightarrow{}\) Less civil war

Causal Logics and "What to do"

You are a rebel (trying to start a civil war). You can't change the wealth of the country.

What would you do if 1 were true? 2? 3?

Three competing causal logics

  1. Wealth \(\xrightarrow{}\) Larger defense budget \(\xrightarrow{}\) Rivals deterred \(\xrightarrow{}\) Less civil war
  2. Wealth \(\xrightarrow{}\) Higher standard of living \(\xrightarrow{}\) Fewer grievances \(\xrightarrow{}\) Weak support for change \(\xrightarrow{}\) Less civil war
  3. Wealth \(\xrightarrow{}\) Lots of job opportunities \(\xrightarrow{}\) Fewer listless young men \(\xrightarrow{}\) Fewer rebel recruits \(\xrightarrow{}\) Less civil war

Causal Logics and "What to do"

You are the ruler (trying to prevent a civil war). You can't change the wealth of the country.

What would you do if 1 were true? 2? 3?

Three competing causal logics

  1. Wealth \(\xrightarrow{}\) Larger defense budget \(\xrightarrow{}\) Rivals deterred \(\xrightarrow{}\) Less civil war
  2. Wealth \(\xrightarrow{}\) Higher standard of living \(\xrightarrow{}\) Fewer grievances \(\xrightarrow{}\) Weak support for change \(\xrightarrow{}\) Less civil war
  3. Wealth \(\xrightarrow{}\) Lots of job opportunities \(\xrightarrow{}\) Fewer listless young men \(\xrightarrow{}\) Fewer rebel recruits \(\xrightarrow{}\) Less civil war

Causal Theories:

To turn a causal claim into a testable causal theory, we need:

  1. causal logic connecting cause to effect: ✓
  2. a statement of the direction of the effect: ✓
    • can't just say "\(C\) affects \(E\)" without saying whether it causes it to/to not happen; more or less of something.
  3. restate causal claim in terms of what is observable:
    • turn concepts into variables (exactly like with descriptive claims)
    • independent variables
    • dependent variables

Variables and Causal Claims:

Independent variable:

The variable capturing the purported cause in a causal claim.

  • often denoted as "IV" or "\(X\)" or "right-hand variable"

Dependent variable:

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

  • often denoted as "DV" or "\(Y\)" or "left-hand variable"

Why these terms?

Statistical/Mathematical expression: \(Y = f(X)\) or \(DV = f(IV)\)

Variables and Causal Claims:

Causal Claim:

"Guns don't kill people, people kill people"

Which implies: "Increasing the prevalence of firearms in a region causes no change in human mortality"

How would you turn this claim into something observable?

Independent Variable (\(X\)): ?

Dependent Variable (\(Y\)): ?

Variables and Causal Claims:

Causal Claim:

"Exposure to anti-immigrant rhetoric on social media causes an increase in violence against immigrants"

Independent Variable (\(X\)): ?

Dependent Variable (\(Y\)): ?

Causal Theories:

To turn a causal claim into a testable causal theory, we need:

  1. causal logic connecting cause to effect: ✓
  2. a statement of the direction of the effect: ✓
    • can't just say "\(C\) affects \(E\)" without saying whether it causes it to/to not happen; more or less of something.
  3. restate causal claim in terms of what is observable: ✓
    • turn concepts into variables (exactly like with descriptive claims)
    • independent variables
    • dependent variables

Testing Causal Theories

Testing!

We turn causal claims into causal theories which are testable. We test causal theories by making…

hypotheses/empirical predictions

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

  • multiple hypotheses based on the overall causal claim and the causal logic
  • Hypotheses state relationships between variables
    • When \(X\) is present(absent), \(Y\) is present(absent)
    • When \(X\) is present(absent), \(Y\) is more(less) likely
    • When \(X\) increases(decreases), \(Y\) increases(decreases)

Testing!

Preceding definition of causal hypotheses is incomplete.


We will revisit this.

Examples:

claim: "Ownership of firearms by citizens reduces crime"


independent variable: fraction of citizens owning firearms

dependent variable: crime victimization rate per 100 thousand


What is an empirical prediction/hypothesis?

Examples:

Examples:

claim: "Increased immigration causes more crime"


independent variable: immigrant share of population

dependent variable: crime victimization rate per 100 thousand


What is an empirical prediction/hypothesis?

Examples:

Examples: What about this?

What are we missing?

Causality is Counterfactual


If \(X\) had been different for the same case and the same time, \(Y\) would have been different


hypotheses from causal theories must be about potential outcomes

  • recall: potential outcomes are values the dependent variable \((Y)\) would take for a case under different values of the independent variable \((X)\)

New hypotheses:

Restate hypotheses in terms of counterfactuals


If there were more guns (in a specific place,time), there would have been less crime


If there were fewer immigrants (in a specific place, time), there would have been less crime


If there were fewer Nick Cage films release (in a country, year), there would have been fewer pool drownings

A big problem

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

The counterfactual approach to causality means: if \(X\) causes \(Y\) in some specific case, the potential outcomes of \(Y\) are different across levels of \(X\).

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

Fundamental Problem of Causal Inference