Synthetic Control Methods and Difference-In-Differences

Guido Imbens

**Concept Graph & Resume using Claude 3.5 Sonnet | Chat GPT4o | Llama 3:**

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classDef methods fill:#d4f9d4, font-weight:bold, font-size:14px
classDef modeling fill:#d4d4f9, font-weight:bold, font-size:14px
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A[Synthetic Control Methods

and Difference-In-Differences] --> B[Panel

Data] A --> C[Estimation

Methods] A --> D[Modeling

Techniques] A --> E[Evaluation

Metrics] B --> B1[Multiple units

over

time. 1] B --> B2[Hypothetical treatment,

control

outcomes. 2] B --> B3[Difference between

treated, control

outcomes. 3] B --> B4[Treatment independent

of potential

outcomes. 4] B --> B5[Units adopt

treatment at

different times. 16] B --> B6[Units statistically

indistinguishable. 27] C --> C1[Weighted control

unit

combination. 5] C --> C2[Comparing changes

before, after

treatment. 6] C --> C3[Unit, time

effects control. 7] C --> C4[Probability of

treatment

assignment. 17] C --> C5[Combine outcome,

propensity score

models. 18] C --> C6[Prediction intervals

under

exchangeability. 28] D --> D1[Estimate missing

values using

approximation. 8] D --> D2[Penalty for low-rank

solutions. 9] D --> D3[Outcomes on pre-treatment

variables. 10] D --> D4[Pre-treatment on

control

outcomes. 11] D --> D5[Generalized two-way

fixed

effects. 12] D --> D6[Data into latent

factors,

loadings. 21] E --> E1[Select parameters

with held-out

data. 14] E --> E2[Measure of

prediction

accuracy. 15] E --> E3[Balance model

flexibility,

overfitting. 25] E --> E4[Variable correlation

at different

times. 26] E --> E5[Use prior distributions,

update

beliefs. 29] E --> E6[Functions balancing

covariates. 30] class A,B,B1,B2,B3,B4,B5,B6 panel class C,C1,C2,C3,C4,C5,C6 methods class D,D1,D2,D3,D4,D5,D6 modeling class E,E1,E2,E3,E4,E5,E6 evaluation

and Difference-In-Differences] --> B[Panel

Data] A --> C[Estimation

Methods] A --> D[Modeling

Techniques] A --> E[Evaluation

Metrics] B --> B1[Multiple units

over

time. 1] B --> B2[Hypothetical treatment,

control

outcomes. 2] B --> B3[Difference between

treated, control

outcomes. 3] B --> B4[Treatment independent

of potential

outcomes. 4] B --> B5[Units adopt

treatment at

different times. 16] B --> B6[Units statistically

indistinguishable. 27] C --> C1[Weighted control

unit

combination. 5] C --> C2[Comparing changes

before, after

treatment. 6] C --> C3[Unit, time

effects control. 7] C --> C4[Probability of

treatment

assignment. 17] C --> C5[Combine outcome,

propensity score

models. 18] C --> C6[Prediction intervals

under

exchangeability. 28] D --> D1[Estimate missing

values using

approximation. 8] D --> D2[Penalty for low-rank

solutions. 9] D --> D3[Outcomes on pre-treatment

variables. 10] D --> D4[Pre-treatment on

control

outcomes. 11] D --> D5[Generalized two-way

fixed

effects. 12] D --> D6[Data into latent

factors,

loadings. 21] E --> E1[Select parameters

with held-out

data. 14] E --> E2[Measure of

prediction

accuracy. 15] E --> E3[Balance model

flexibility,

overfitting. 25] E --> E4[Variable correlation

at different

times. 26] E --> E5[Use prior distributions,

update

beliefs. 29] E --> E6[Functions balancing

covariates. 30] class A,B,B1,B2,B3,B4,B5,B6 panel class C,C1,C2,C3,C4,C5,C6 methods class D,D1,D2,D3,D4,D5,D6 modeling class E,E1,E2,E3,E4,E5,E6 evaluation

**Resume: **

**1.-** Panel data: Data with multiple units observed over time, used to estimate causal effects of treatments.

**2.-** Potential outcomes: Hypothetical outcomes for each unit under treatment and control conditions.

**3.-** Average treatment effect: The average difference between treated and control outcomes across units.

**4.-** Unconfoundedness: Assumption that treatment assignment is independent of potential outcomes conditional on observed covariates.

**5.-** Synthetic control: Method using weighted combination of control units to estimate counterfactual for treated unit.

**6.-** Difference-in-differences: Method comparing outcome changes between treated and control groups before and after treatment.

**7.-** Two-way fixed effects: Regression model with unit and time fixed effects to control for time-invariant unit characteristics and time trends.

**8.-** Matrix completion: Technique to estimate missing values in a matrix using low-rank approximation.

**9.-** Nuclear norm regularization: Penalty added to objective function to encourage low-rank solutions in matrix completion.

**10.-** Horizontal regression: Regressing outcomes on pre-treatment variables using control units to predict counterfactuals.

**11.-** Vertical regression: Regressing pre-treatment outcomes on control unit outcomes to predict counterfactuals for treated units.

**12.-** Interactive fixed effects: More flexible generalization of two-way fixed effects using factor models.

**13.-** Low-rank matrix: Matrix that can be approximated well by a product of two smaller matrices.

**14.-** Cross-validation: Technique to select model parameters by evaluating performance on held-out data.

**15.-** Root mean squared error: Measure of prediction accuracy calculated as square root of average squared prediction errors.

**16.-** Staggered adoption: Treatment pattern where units adopt treatment at different times but remain treated once adopted.

**17.-** Propensity score: Probability of treatment assignment conditional on observed covariates, used for matching or weighting.

**18.-** Double robust methods: Estimation techniques combining outcome regression and propensity score models for improved robustness.

**19.-** Matching: Method pairing treated units with similar control units based on covariates or propensity scores.

**20.-** Inverse probability weighting: Technique using inverse of treatment probability to weight outcomes and balance covariate distributions.

**21.-** Factor models: Statistical models decomposing data into latent factors and factor loadings.

**22.-** Synthetic difference-in-differences: Method combining synthetic control and difference-in-differences approaches.

**23.-** Time weights: Weights assigned to different time periods in synthetic difference-in-differences to focus on relevant periods.

**24.-** Unit weights: Weights assigned to control units in synthetic control methods to create counterfactual for treated unit.

**25.-** Bias-variance tradeoff: Balance between model flexibility and overfitting in statistical learning.

**26.-** Autocorrelation: Correlation between a variable's values at different time points.

**27.-** Exchangeability: Assumption that units are statistically indistinguishable, often used in inference for panel data.

**28.-** Conformal inference: Method for constructing prediction intervals with guaranteed coverage under exchangeability assumptions.

**29.-** Bayesian methods: Statistical approaches using prior distributions and updating beliefs based on observed data.

**30.-** Balancing scores: Functions of covariates that, when conditioned on, balance covariate distributions between treated and control groups.

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