Causality Cheat Sheet

Describing Matching

• The goal of matching is reduce selection bias (due to confounding) before estimating causal effects

  • Usually, for controlled experiments, but sometimes natural experiments
  • As an example, we first could use matching to reduce selection bias, then afterwards use regression or synthetic control modeling to estimate causal effects
• Matching is a form of stratification

• Matching methods match treated units to similar control units

  • This enables a more accurate comparison between treated units and non-treated units
  • Specifically, matching involves sampling, so each treatment observation corresponds to a control observation with identical covariate values

Describing Regression Discontinuity Design

• The goal of RDD is to identify and estimate causal effects

  • Usually, for natural experiments

• RDD stitches the potential outcomes $Y^{0}$ that are actually observed (i.e. when $D=0$) with the potential outcomes $Y^{1}$ that are actually observed (i.e. when $D=1$)

  • Essentially, $Y^{0}$ is before the cutoff point $c_{0}$ and $Y^{1}$ is after the cutoff point
  • As a result, we can identify causal effects if there is significant discontinuity at the cutoff point $c_{0}$

• Typically, RDD is used in natural experiments

  • A natural experiment refers to an experiment without a controlled treatment assignment $D$ (not randomly assigned)
  • Rather, treatment is assigned by some random, external factor (e.g. $X$)
  • Whereas, a controlled experiment refers to an experiment where all factors are held constant except for one
  • As a result, RDD uses a running variable $X$ (don't mistake this for an ordinary covariate)
  • Thus, we usually use a continuous $X$ instead of binary treatment variable DD in RDD, since RDDs are usually used in natural experiments

• The following assumptions are made about RDD:

  • RDD requires that every variable (excluding the treatment and outcome variables) is continuous at the cutoff point $c_{0}$

    • This implies observed and unobserved predictors must be continuous at the cutoff point $c_{0}$
    • To be clear, the treatment and outcome variables can be discontinuous at the cutoff point $c_{0}$
  • RDD requires that all of the expected potential outcomes for $Y^{0}$ are continuous at the cutoff point
  • RDD requires that all of the expected potential outcomes for $Y^{1}$ are continuous at the cutoff point

Describing Difference-in-Difference

• The goal of a synthetic control is to identify and estimate causal effects

  • Usually, for natural experiments
• Diff-in-diff is a comparison between the differential effect of a treatment on a treatment group versus a control group in a natural experiment
• There are many of assumptions made about the diff-in-diff model

• All the assumptions of the OLS model apply equally to diff-in-diff:

  • No autocorrelation (e.g. a sine wave)
  • No heterscedasticity (i.e. constant variance)
  • No collinearity (i.e. this is rarely true)
  • Data $Y$ is linearly related (i.e. sometimes true)
  • Conditional means of errors should be $0$ (i.e. rarely true)
  • Conditional variance of errors should be constant
  • Errors conditional on regressors should be normally distributed
  • Observations should be iid

• Additionally, diff-in-diff requires a parallel trends assumption

  • Essentially, it's saying there must be constant differences in outcomes between:

    • The difference between the pre-intervention control and pre-intervention treatment groups
    • The difference between the post-intervention control and post-intervention treatment groups

• Roughly, the diff-in-diff model is similar to a synthetic control model

  • But, a diff-in-diff model treats the time points (pre-intervention and post-intervention) as predictors
  • Whereas, a synthetic control model treats the time points as rows

Describing Synthetic Control

• The goal of a synthetic control is to identify and estimate causal effects

  • Usually, for natural experiments

• A synthetic control involves weighting multiple sets of control groups and comparing it with a single treatment group

  • This comparison is used to estimate what would have happened to the treatment group if it had not received the treatment

• Unlike diff-in-diff, this method can account for the effects of confounders changing over time

  • It does this by weighting the control group to better match the treatment group before the intervention

• The following assumptions are made about synthetic controls:

  1. Existence of weights

     • Implying, there must be enough similarities with the control units to create a synthetic control

  2. Weakly stationary process

     • Implying, the mean and variance must be roughly fixed over time

  3. We must be aware beforehand that controls in synthetic control group don't receive treatment

     • Otherwise, this could open up backdoor path
     • Implying, we must have a good understanding of the natural experiment

Refrences

• Wiki about Regression Discontinuity Design
• Wiki about Difference-in-Difference
• Wiki about Synthetic Controls