Causality

• Correlation doesn’t tell us enough about causality.
• Solutions
  • Observe the same individual at different points in time.
  • Observe two nearly identical individuals and treat differently.

$$\widehat{ATE}\approx {\mathbb{E}}_N[y_i\mid t_i=1]-{\mathbb{E}}_N[y_i\mid t_i=0]-\text{Selection Bias}$$ $$\text{Selection Bias}=\mathbb{E}[Y(1)\mid T=0]-\mathbb{E}[Y(0)\mid T=0]$$

• Randomized Controlled Experiment (RCE)

$$Y_i\approx {\hat{\beta }}_0+{\hat{\beta }}_1{T}_i$$

• Regression model
  • ${\hat{\beta }}_0$: average outcome in the control group
  • ${\hat{\beta }}_0+{\hat{\beta }}_1$: average outcome in the control group
  • ${\hat{\beta }}_1$: the Average Treatment Effect $\widehat{\mathrm{ATE}}$
• Propensity score: represents the conditional probability of receiving a particular treatment given a set of observed covariates.

Procedures

• Compute a propensity score model using logistic regression, which predicts receiving the treatment based on observed covariates.
• Calculate propensity score of each observation, match the ones with similar scores, can be based on KNN or similar methods.
• Use the matched sample to make a fair comparison between treated and non-treated and examine the impact.