因果推断与商业应用

课程内容大纲 Contents

Causality: An introduction
Causal Diagrams
Classical Randomized Experiments
Instrumental Variables
Propensity Score Matching
Regression Discontinuity Design (RDD)
Difference in Differences
Synthetic Control

随机实验 Random Experient:

随机实验是最完美的。
要求：样本量足够大 + 随机分组

IV

Local Average Treatment Effect (ATE)
Always-taker, Never-taker, Complier

Propensity Score of Matching

Propensity Score Matching (PSM) is a statistical technique used to reduce selection bias in observational studies by matching individuals from treatment and control groups based on similar characteristics. It aims to simulate the conditions of a randomized experiment by balancing covariates between groups.

Probit, Logit model

A propensity score is the probability of an individual receiving treatment given a set of observed covariates. Mathematically, it is defined as:

e(X) = P(T = 1 | X) Copy Where:

T = 1 indicates the treatment group.

X represents the observed covariates.

e(X) is the propensity score.

Regression Discontinuity Design (断点回归)

DiD

Difference in Differences
DiDiD

Synthetic Control

任务要求：

每周给2-3篇文章，基于文章进行提问和完成结果复现。

Chapter 1 Causality: An introduction

Film: Run Lola Run. -- Potential Outcome

Causal Inference: Examples

Cause -> Effect Coffee Health Ice cream sales Drown & crime Chocolate Nobel Prize Police staffing Crime

Simpson's Paradox

Given that:

a/A > b/B, c/C > d/D

反而出现了：

(a+c)/(A+C) < (b+d)/(B+D)

In essence, this is due to omitted variable bias!

E(Y|D=1) - E(Y|D=0) > 0

but

E(Y|X,D=1) - E(Y|X,D=0) < 0

for each popolation characterized by

X

, the "effect" of

D

Y

takes one sign, while it takes the opposite sign for the whole population!

Example: Suppose for $X=1$ for female, $D=1$ for vegetarian, $Y$ is life span

Effect for male = -5;

$E(Y|X=0, D=0) = 65$ , $E(Y|X=0, D=1) = 60$

Effect for female = -5;

$E(Y|X = 1, D = 0) = 80$ , $E(Y|X=1, D=1) = 75$

P(X=1|D=0) = 0.2, \space P(X=1|D=1) = 0.7

The effect on overall population is 2.5 (positive).

Gender is not independent with treatment, thus is a confounder (confounding factor 混杂因子).

Linear regression model

Regression equation, i.e.,

y_i = \beta_0 + x_i\beta_1 + \epsilon_i, \space Y = X\beta + \epsilon

OLS estimator, i.e.,

\hat{\beta}_i = \frac{\Sigma_{i=1}^N(x_i - \bar{x})(y_i - \bar{y})}{\Sigma_{i=1}^N(x_i - \bar{x})^2}, \space \hat{\beta} = (X'X)^{-1}X'Y

ceteris paribus;
log value;
interaction term;
significance;
omitted variable bias: Confounding factor混杂因子;
heteroskedasticity: 异方差;

Assume $X$ is a random variable.

We can summarize the predictive powe rof $X$ on $Y$ , by:

E(y_i|X = x_i)

The law of iteracted expectation

E(y_i) = E(E(y_i|x_i))

CEF decomposition property

Y_i = E(Y_i|x_i) + \epsilon_i

Average Treatment Effect (ATE)

\tau_{ATE} = E_p(Y_i(1) - Y_i(0))

Conditional Independence Assumption(CIA, also called secltion on observables.)

Observed difference in earnings = Average treatment effect on the treated + selection bias

把会导致selction bias的变量，潜在的都control住后，就没有了；但是问题是，找到所有的这些变量不容易。