March 19, 2018

Original Average \(\approx 60\)%

Similar across tutorial sections

\[ScaledPoints = (MidtermPoints - 27.4) \cdot 0.715 + 32.6\] Original Average: \(59.9\)% New Average: \(70.9\)%

It is possible

If \(X \rightarrow Y\):

- Values of \(X\) and \(Y\) should move together
- Values of \(X\) and \(Y\) should be correlated
- This correlation should not be
**by chance**

It is possible that \(X,Y\) are correlated, without causation

i | \(Y_i^0\) | \(Y_i^1\) | \(Y_i^1 - Y_i^0\) | \(X_i\) | \(W_i\) |
---|---|---|---|---|---|

1 | 5 | 9 | 4 | 1 | 1 |

2 | 4 | 8 | 4 | 1 | 1 |

3 | 3 | 7 | 4 | 0 | 0 |

4 | 2 | 6 | 4 | 0 | 0 |

True causal effect of \(X\): \(4\).

Apparent causal effect of \(X\):

\[\frac{9+8}{2} - \frac{3+2}{2} = \frac{12}{2} \neq 4\]

i | \(Y_i^0\) | \(Y_i^1\) | \(Y_i^1 - Y_i^0\) | \(X_i\) | \(W_i\) |
---|---|---|---|---|---|

1 | 5 | 9 | 4 | 1 | 1 |

2 | 4 | 8 | 4 | 1 | 1 |

3 | 3 | 7 | 4 | 0 | 0 |

4 | 2 | 6 | 4 | 0 | 0 |

- \(X = 0\) when \(W = 0\). \(X = 1\) when \(W = 1\)
- perfect correlation

- \(Y \approx 4.5\) when \(W = 0\). \(Y \approx 6.5\) when \(W = 1\)
- partial correlation

but \(W\) makes effect of \(X\) look too big

\(W\) is a **confounding variable**

We observe \(X,Y\) correlated…

- … but additional factors \(W_1 \ldots W_n\) affect \(X\) and \(Y\)
- So correlation is biased.
- Could be entirely spurious (no true causal effect of \(X\) on \(Y\))
- Or different from true effect of \(X\). Observed effect too big/too small
- These other factors \(W_1 \ldots W_n\) are called
**confounding variables**

**Comparative Method:**

If causal claim that \(X \rightarrow Y\), then:

We generate **empirical prediction**:

If we observe

two casesto be the same in all relevant respects except for value of \(X\), then we should observe that the two cases differ in the value of \(Y\)

- We don't need cases to be
**identical**on**all**attributes - They need to be the
**identical**on attributes related to both**cause**(\(X\)) and**outcome**(\(Y\))- In short: identical on factors that affect \(Y\) and are related to or affect \(X\)

- Examines whether \(Y\) changes when \(X\) changes
- Within pairs of cases where
**confounding**variables held constant

**Comparative method**

- Can address
**confounding** *But*works with only a few cases, patterns by chance?

**Correlation**

- Correlation uses many cases, can judge statistical significance
*But*does not address**confounding**/bias

Two approaches to make correlation work like the comparative method

**adjustment**-based- Identify possible
**confounding**variables (e.g. \(W\)) - Measure these variables
- explicitly
**adjust**our correlation between \(X\) and \(Y\) - "
**conditioning**" on confounding variables

- Identify possible
**design**-based- Carefully choose
**cases**for comparison **Structure of comparison**accounts for many or**all**confounding variables- Kind of comparison permits
**unconditional**comparison, given assumptions.

- Carefully choose

An experiment is **design**-based solution:

- We
**compare**a "treated" group to an "untreated" group - We test using
**unadjusted**correlation of treatment and outcome in these groups - We have
**good reason**to believe that correlation implies causation:- Not because we
**explicitly**identified and accounted for**confounding**variables - Because
**design**(random assignment of treatment) eliminates confounding - Wouldn't work if we compared randomly assigned treatment group non-random control

- Not because we

This extrapolates **comparative method** to correlation.

The goal is to:

- observe
**confounding variables** - examine the correlation of \(X\) and \(Y\)…
- holding confounding variables
**constant***ceteris parabis*, "all else being equal"

**conditioning**: the process of holding other variables constant while looking at relationship between \(X\) and \(Y\).

- Remaining correlation between \(X\) and \(Y\) after conditioning on other variables
**cannot**result from confounding by those variables

How to think about **adjustment**/**conditioning**:

It is **like**:

**many**applications of comparative method simultaneously- correlation of \(X\) and \(Y\) within groups where other variables \(W_1 \ldots W_k\) are held constant
- Change in \(X\) and \(Y\) cannot be due to change in variables we condition on
**because**these other variables are held constant

It is not this process **EXACTLY**, but similar

Does UN peace-keeping prevent re-occurrence of civil war?

**Exposure** to UN **peace-keeping** at the end of a conflict causes countries to experience **longer periods** of **peace**

Correlation is insufficient. Many variables may cause UN intervention AND ability to have durable peace

- Intensity of conflict (Deaths), Length of conflict, ethnic diversity, population, mountainous terrain, size of armed forces, democracy

Look at correlation between Peacekeeping and Peace **within** groups of cases that have **same values** of:

- Violence, War length, Diversity, Terrain, Army Size, Democracy

**DOES NOT** eliminate the causal link between confounding variables and **peacekeeping** by design. We **adjust** our correlation to account for confounders.

Example (1)

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | 20 | 10 | 1 | 0 | \(\mathbf{20}\) | 10 |

treated | 20 | 10 | 1 | 0 | \(\mathbf{20}\) | 10 |

treated | 15 | 5 | 1 | 1 | \(\mathbf{15}\) | 10 |

untreated | 20 | 10 | 0 | 0 | \(\mathbf{10}\) | 10 |

untreated | 15 | 5 | 0 | 1 | \(\mathbf{5}\) | 10 |

untreated | 15 | 5 | 0 | 1 | \(\mathbf{5}\) | 10 |

What is the **true** effect of peace-keeping?

\[\frac{10 + 10 + 10 + 10 + 10 + 10}{6} = 10\]

\(10\) years more peace.

What is the effect we see with **unadjusted** or **unconditional** correlation?

(Average in peacekeeping - Average in no-peacekeeping)

Example (1)

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | 20 | 10 | 1 | 0 | \(\mathbf{20}\) | 10 |

treated | 20 | 10 | 1 | 0 | \(\mathbf{20}\) | 10 |

treated | 15 | 5 | 1 | 1 | \(\mathbf{15}\) | 10 |

untreated | 20 | 10 | 0 | 0 | \(\mathbf{10}\) | 10 |

untreated | 15 | 5 | 0 | 1 | \(\mathbf{5}\) | 10 |

untreated | 15 | 5 | 0 | 1 | \(\mathbf{5}\) | 10 |

\[\frac{20 + 20 + 15}{3} - \frac{10 + 5 + 5}{3} = \frac{35}{3} \neq 10\]

We have upward **bias**: it looks like peace-keeping causes more peace than it does

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | 20 | 10 | \(\mathbf{1}\) | \(\mathbf{0}\) | 20 | 10 |

treated | 20 | 10 | \(\mathbf{1}\) | \(\mathbf{0}\) | 20 | 10 |

treated | 15 | 5 | 1 | 1 | 15 | 10 |

untreated | 20 | 10 | \(\mathbf{0}\) | \(\mathbf{0}\) | 10 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | 1 | \(\mathbf{0}\) | 20 | 10 |

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | 1 | \(\mathbf{0}\) | 20 | 10 |

treated | 15 | 5 | 1 | 1 | 15 | 10 |

untreated | \(\mathbf{20}\) | \(\mathbf{10}\) | 0 | \(\mathbf{0}\) | 10 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

- Intense wars \(\xrightarrow{reduces}\) Peacekeeping
- Intense wars \(\xrightarrow{reduces}\) Durable Peace

Result?

- Upward bias in effect of peacekeeping on durable peace

Conditioning! Calculate effect of peacekeeping where Intense War is \(1\)

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | 20 | 10 | 1 | 0 | 20 | 10 |

treated | 20 | 10 | 1 | 0 | 20 | 10 |

untreated | 20 | 10 | 0 | 0 | 10 | 10 |

treated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{1}\) | 1 | \(\mathbf{15}\) | 10 |

untreated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{0}\) | 1 | \(\mathbf{5}\) | 10 |

untreated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{0}\) | 1 | \(\mathbf{5}\) | 10 |

Effect of peacekeeping where war intensity is high (1):

\[\frac{15}{1} - \frac{5 + 5}{2} = \frac{10}{1} = 10\]

… and where war intensity is low (0)?

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{1}\) | 0 | \(\mathbf{20}\) | 10 |

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{1}\) | 0 | \(\mathbf{20}\) | 10 |

untreated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{0}\) | 0 | \(\mathbf{10}\) | 10 |

treated | 15 | 5 | 1 | 1 | 15 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

Effect of peacekeeping where war intensity is low (0):

\[\frac{20 + 20}{2} - \frac{10}{1} = \frac{10}{1} = 10\]

Within the groups of cases defined by values of war intensity:

- Estimated effect of peacekeeping (compare cases with to without) retrieves the
**true**effect **Conditioning**solved our problem of bias

Why does this work?

- Within groups of cases with same values on confounder (war intensity …
- Potential outcomes (years of peace) with and without UN peacekeepers are
**same**for "treated" and "untreated" - Same as: no other factor affects
**both**UN peacekeeping (\(X\)) and peace (\(Y\))

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{1}\) | 0 | 20 | 10 |

treated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{1}\) | 0 | 20 | 10 |

treated | 15 | 5 | 1 | 1 | 15 | 10 |

untreated | \(\mathbf{20}\) | \(\mathbf{10}\) | \(\mathbf{0}\) | 0 | 10 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

untreated | 15 | 5 | 0 | 1 | 5 | 10 |

\(Peace_i^{UN}\) | \(Peace_i^{noUN}\) | \(UN_i\) | \(WarIntense_i\) | \(Peace_i^{Obs}\) | \(Effect_i\) | |
---|---|---|---|---|---|---|

treated | 20 | 10 | 1 | 0 | 20 | 10 |

treated | 20 | 10 | 1 | 0 | 20 | 10 |

treated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{1}\) | 1 | 15 | 10 |

untreated | 20 | 10 | 0 | 0 | 10 | 10 |

untreated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{0}\) | 1 | 5 | 10 |

untreated | \(\mathbf{15}\) | \(\mathbf{5}\) | \(\mathbf{0}\) | 1 | 5 | 10 |

(Gilligan & Sergenti 2008)

Look at correlation of peacekeeping and peace after conflict:

- Matching countries on: intensity of conflict (Deaths), length of conflict, ethnic diversity, population, mountainous terrain, size of armed forces, democracy

(Lower proportional hazards => war less likely)

(Gilligan & Sergenti 2008)

**Conditioning** on other factors (holding them constant):

Peacekeeping makes war 85% less likely to re-emerge

The presence of guns causes an increase in violent crame

The **firearm ownership rate** causes an increase in the **violent crime rate**

Correlation between gun ownership and homicides is nearly \(0\).

- Does this mean no causal relationship?
- Lots of possible sources of bias.

- Look at US counties
- Use "regression" to adjust correlation between guns and violent crime
- Adjust for possible confounders:
- Region, urban/rural, population, poverty, ethnic diversity, unemployment, gender, poverty
- "Holding these factors constant", "ceteris parabis"