Difference-in-Differences Methods

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1 Difference-in-Differences Methods Teppei Yamamoto Keio University Introduction to Causal Inference Spring 2016

2 1 Introduction: A Motivating Example 2 Identification 3 Estimation and Inference 4 Diagnostics and Extensions

3 Example: Minimum Wage and Employment Do higher minimum wages decrease employment? Card and Krueger (1994) consider impact of New Jersey s 1992 minimum wage increase from $4.25 to $5.05 per hour Compare employment in 410 fast-food restaurants in New Jersey and eastern Pennsylvania before and after the rise Survey data on wages and employment from two waves: Wave 1: March 1992, one month before the minimum wage increase Wave 2: December 1992, eight month after increase Teppei Yamamoto Difference in Differences Causal Inference 1 / 30

4 Location of Restaurants Teppei Yamamoto Difference in Differences Causal Inference 2 / 30

5 Wages Before Rise in Minimum Wage Teppei Yamamoto Difference in Differences Causal Inference 3 / 30

6 Wages After Rise in Minimum Wage Teppei Yamamoto Difference in Differences Causal Inference 4 / 30

7 1 Introduction: A Motivating Example 2 Identification 3 Estimation and Inference 4 Diagnostics and Extensions

8 Setup: Groups, Periods and Treatments Data structure: Two waves of randomly sampled cross-sectional observations Either panel or repeated cross sections Cross-sectional units: i {1,..., N} Time periods: t {0 (pre-treatment), 1 (post-treatment)} { 1 (treatment group) Group indicator: G i = 0 (control group) Treatment indicator: Z it {0, 1} Units in the treatment group receive treatment in t = 1: Time Period Group t = 0 t = 1 G i = 1 (treatment group) Z i0 = 0 (untreated) Z i1 = 1 (treated) G i = 0 (control group) Z i0 = 0 (untreated) Z i0 = 0 (untreated) Teppei Yamamoto Difference in Differences Causal Inference 5 / 30

9 Setup: Potential Outcomes Potential outcomes Y it (z): Y it (0): potential outcome for unit i in period t when not treated Y it (1): potential outcome for unit i in period t when treated Teppei Yamamoto Difference in Differences Causal Inference 6 / 30

10 Setup: Potential Outcomes Potential outcomes Y it (z): Y it (0): potential outcome for unit i in period t when not treated Y it (1): potential outcome for unit i in period t when treated Causal effect for unit i at time t is τ it = Y it (1) Y it (0) Teppei Yamamoto Difference in Differences Causal Inference 6 / 30

11 Setup: Potential Outcomes Potential outcomes Y it (z): Y it (0): potential outcome for unit i in period t when not treated Y it (1): potential outcome for unit i in period t when treated Causal effect for unit i at time t is τ it = Y it (1) Y it (0) Observed outcomes Y it are realized as Y it = Y it (0)(1 Z it ) + Y it (1)Z it Because Z i1 = G i in the post-treatment period, we can also write Y i1 = Y i1 (0)(1 G i ) + Y i1 (1)G i Teppei Yamamoto Difference in Differences Causal Inference 6 / 30

12 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Problem: Missing potential outcome: Teppei Yamamoto Difference in Differences Causal Inference 7 / 30

13 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Problem: Missing potential outcome: E[Y i1 (0) G i = 1], i.e. what is the average post-period outcome for the treated group in the absence of the treatment? Teppei Yamamoto Difference in Differences Causal Inference 7 / 30

14 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Before vs. After Use E[Y i1 G i = 1] E[Y i0 G i = 1] for τ ATT Assumes Teppei Yamamoto Difference in Differences Causal Inference 8 / 30

15 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Before vs. After Use E[Y i1 G i = 1] E[Y i0 G i = 1] for τ ATT Assumes E[Y i1 (0) G i = 1] = E[Y i0 (0) G i = 1] (No change in average potential outcome over time) Teppei Yamamoto Difference in Differences Causal Inference 8 / 30

16 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Treated vs. Control in Post-Period Use E[Y i1 G i = 1] E[Y i1 G i = 0] for τ ATT Assumes Teppei Yamamoto Difference in Differences Causal Inference 9 / 30

17 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Treated vs. Control in Post-Period Use E[Y i1 G i = 1] E[Y i1 G i = 0] for τ ATT Assumes E[Y i1 (0) G i = 1] = E[Y i1 (0) G i = 0] (Mean ignorability of treatment assignment) Teppei Yamamoto Difference in Differences Causal Inference 9 / 30

18 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Difference-in-Differences (DD) { } { } Use: E[Y i1 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 1] E[Y i0 G i = 0] Teppei Yamamoto Difference in Differences Causal Inference 10 / 30

19 Identification Strategies Estimand: ATT in the post-treatment period τ ATT = E[Y i1 (1) Y i1 (0) G i = 1] = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] Pre-Period (t = 0) Post-Period (t = 1) Treatment Group (G i = 1) E[Y i0 (0) G i = 1] E[Y i1 (1) G i = 1] Control Group (G i = 0) E[Y i0 (0) G i = 0] E[Y i1 (0) G i = 0] Control Strategy: Difference-in-Differences (DD) { } { } Use: E[Y i1 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 1] E[Y i0 G i = 0] Assumes: E[Y i1 (0) Y i0 (0) G i = 1] = E[Y i1 (0) Y i0 (0) G i = 0] (Parallel trends) Teppei Yamamoto Difference in Differences Causal Inference 10 / 30

20 Graphical Representation: Difference-in-Differences E[Y i1 G i = 1] E[Y i0 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 0] t = 0 t = 1 Teppei Yamamoto Difference in Differences Causal Inference 11 / 30

21 Graphical Representation: Difference-in-Differences E[Y i1 G i = 1] E[Y i0 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 0] t = 0 t = 1 Teppei Yamamoto Difference in Differences Causal Inference 11 / 30

22 Graphical Representation: Difference-in-Differences E[Y i1 G i = 1] E[Y i1 (0) G i = 1] E[Y i0 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 0] t = 0 t = 1 Teppei Yamamoto Difference in Differences Causal Inference 11 / 30

23 Graphical Representation: Difference-in-Differences E[Y i1 G i = 1] E[Y i1 (0) G i = 1] E[Y i0 G i = 1] E[Y i1 G i = 0] E[Y i0 G i = 0] τ ATT t = 0 t = 1 Teppei Yamamoto Difference in Differences Causal Inference 11 / 30

24 Identification with Difference-in-Differences Under the parallel trends assumption: E[Y i1 (0) Y i0 (0) G i = 1] = E[Y i1 (0) Y i0 (0) G i = 0] The ATT can be nonparametrically identified as: { } τ ATT = E[Y i1 G i = 1] E[Y i1 G i = 0] { } E[Y i0 G i = 1] E[Y i0 G i = 0] Teppei Yamamoto Difference in Differences Causal Inference 12 / 30

25 Identification with Difference-in-Differences Under the parallel trends assumption: E[Y i1 (0) Y i0 (0) G i = 1] = E[Y i1 (0) Y i0 (0) G i = 0] The ATT can be nonparametrically identified as: { } τ ATT = E[Y i1 G i = 1] E[Y i1 G i = 0] { } E[Y i0 G i = 1] E[Y i0 G i = 0] Proof: {E[Y i1 G i = 1] E[Y i1 G i = 0]} {E[Y i0 G i = 1] E[Y i0 G i = 0]} = {E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 0]} {E[Y i0 (0) G i = 1] E[Y i0 (0) G i = 0]} = E[Y i1 (1) G i = 1] E[Y i1 (0) G i = 1] +E[Y i1(0) G i = 1] }{{} = τ ATT E[Y i1 (0) G i = 0] E[Y i0 (0) G i = 1] + E[Y i0 (0) G i = 0] = τ ATT + {E[Y i1 (0) Y i0 (0) G i = 1] E[Y i1 (0) Y i0 (0) G i = 0]} }{{} = 0 under parallel trends = τ ATT Teppei Yamamoto Difference in Differences Causal Inference 12 / 30

26 Notes on the Parallel Trends Assumption What type of confounding does DD make us robust to? Teppei Yamamoto Difference in Differences Causal Inference 13 / 30

27 Notes on the Parallel Trends Assumption What type of confounding does DD make us robust to? Parallel trends is satisfied if unobserved confounding is time-invariant and additive Parallel trends is violated if there is unobserved time-varying confounding Teppei Yamamoto Difference in Differences Causal Inference 13 / 30

28 Notes on the Parallel Trends Assumption What type of confounding does DD make us robust to? Parallel trends is satisfied if unobserved confounding is time-invariant and additive Parallel trends is violated if there is unobserved time-varying confounding Parallel trends may be more plausible with pre-treatment covariates: E[Y i1 (0) Y i0 (0) G i = 1, X i = x] = E[Y i1 (0) Y i0 (0) G i = 0, X i = x] This assumes parallel trends within strata Under the conditional parallel trends assumption, the ATT is identified as τ ATT = [ x {E[Y i1 G i = 1, X i = x] E[Y i1 G i = 0, X i = x]} ] {E[Y i0 G i = 1, X i = x] E[Y i0 G i = 0, X i = x]} Pr(X i = x G i = 1) Teppei Yamamoto Difference in Differences Causal Inference 13 / 30

29 Notes on the Parallel Trends Assumption What type of confounding does DD make us robust to? Parallel trends is satisfied if unobserved confounding is time-invariant and additive Parallel trends is violated if there is unobserved time-varying confounding Parallel trends may be more plausible with pre-treatment covariates: E[Y i1 (0) Y i0 (0) G i = 1, X i = x] = E[Y i1 (0) Y i0 (0) G i = 0, X i = x] This assumes parallel trends within strata Under the conditional parallel trends assumption, the ATT is identified as τ ATT = [ x {E[Y i1 G i = 1, X i = x] E[Y i1 G i = 0, X i = x]} ] {E[Y i0 G i = 1, X i = x] E[Y i0 G i = 0, X i = x]} Pr(X i = x G i = 1) Note the parallel trends assumption is not invariant to nonlinear transformation of the outcome scale For example, parallel trends in Y it (z) implies non-parallel trends in log Y it (z) and vice versa Teppei Yamamoto Difference in Differences Causal Inference 13 / 30

30 1 Introduction: A Motivating Example 2 Identification 3 Estimation and Inference 4 Diagnostics and Extensions

31 Plug-in Estimation for Panel Data Estimand: { } τ ATT = E[Y i1 G i = 1] E[Y i1 G i = 0] { } E[Y i0 G i = 1] E[Y i0 G i = 0] Teppei Yamamoto Difference in Differences Causal Inference 14 / 30

32 Plug-in Estimation for Panel Data Estimand: { } τ ATT = E[Y i1 G i = 1] E[Y i1 G i = 0] { } E[Y i0 G i = 1] E[Y i0 G i = 0] A plug-in estimator ( difference in difference-in-means ): { 1 N 1 i=1 N G i Y i1 1 N 0 = { 1 N 1 i=1 N i=1 } { 1 (1 G i )Y i1 N G i {Y i1 Y i0 } 1 N 0 N 1 N i=1 N G i Y i0 1 N 0 i=1 N i=1 (1 G i ){Y i1 Y i0 } (1 G i )Y i0 } where N 1 and N 0 are the number of treated and control units respectively }, Teppei Yamamoto Difference in Differences Causal Inference 14 / 30

33 Plug-in Estimation for Panel Data Estimand: { } τ ATT = E[Y i1 G i = 1] E[Y i1 G i = 0] { } E[Y i0 G i = 1] E[Y i0 G i = 0] A plug-in estimator ( difference in difference-in-means ): { 1 N 1 N G i Y i1 1 N 0 i=1 = { 1 N 1 N i=1 } { 1 (1 G i )Y i1 N G i {Y i1 Y i0 } 1 N 0 i=1 N 1 N i=1 N G i Y i0 1 N 0 i=1 N i=1 (1 G i ){Y i1 Y i0 } (1 G i )Y i0 } where N 1 and N 0 are the number of treated and control units respectively Standard errors can be estimated by extending the diff-in-means variance formula using the same logic, assuming no clustering }, Teppei Yamamoto Difference in Differences Causal Inference 14 / 30

34 THE AMERICAN ECONO Example: Card and Krueger TABLE 3-AVERAGE EMPLOYMENT PER STO IN NEW JERSEY MINIM Variable PA (i) Stores by state NJ (ii) Difference, NJ-PA (iii) Wag $4. (iv 1. FTE employment before, all available observations 2. FTE employment after, all available observations 3. Change in mean FTE employment 4. Change in mean FTE employment, balanced sample of storesc Teppei Yamamoto Difference in Differences Causal Inference 15 / 30

35 Plugin-Estimation for Repeated Cross Sections Repeated cross-sectional data require slight change in notation: Period indicator is now a variable: T i {0, 1} Estimand: τ ATT = E[Y i (1) Y i (0) G i = 1, T i = 1] Identified as: τ ATT = E[Y i G i = 1, T i = 1] E[Y i G i = 0, T i = 1] {E[Y i G i = 1, T i = 0] E[Y i G i = 0, T i = 0]} N now refers to the size of the pooled sample Teppei Yamamoto Difference in Differences Causal Inference 16 / 30

36 Plugin-Estimation for Repeated Cross Sections Repeated cross-sectional data require slight change in notation: Period indicator is now a variable: T i {0, 1} Estimand: τ ATT = E[Y i (1) Y i (0) G i = 1, T i = 1] Identified as: τ ATT = E[Y i G i = 1, T i = 1] E[Y i G i = 0, T i = 1] {E[Y i G i = 1, T i = 0] E[Y i G i = 0, T i = 0]} N now refers to the size of the pooled sample The plug-in estimator is then written as: ˆτ ATT = { N i=1 G } it i Y N i i=1 N i=1 G (1 G i)t i Y i N it i i=1 (1 G i)t i { N i=1 G } i(1 T i )Y N i i=1 N i=1 G (1 G i)(1 T i )Y i i(1 T i ) N i=1 (1 G i)(1 T i ) Covariates X i can be incorporated via subclassification Teppei Yamamoto Difference in Differences Causal Inference 16 / 30

37 Regression Estimator for Repeated Cross Sections Because G i and T i are both binary, the same estimator can be calculated via regression: Ŷ i = ˆµ + ˆγG i + ˆδT i + ˆτG i T i where ˆµ, ˆγ, ˆδ and ˆτ are OLS regression estimates Teppei Yamamoto Difference in Differences Causal Inference 17 / 30

38 Regression Estimator for Repeated Cross Sections Because G i and T i are both binary, the same estimator can be calculated via regression: Ŷ i = ˆµ + ˆγG i + ˆδT i + ˆτG i T i where ˆµ, ˆγ, ˆδ and ˆτ are OLS regression estimates Easy to show that ˆτ = ˆτ ATT : After (T i = 1) Before (T i = 0) After - Before Treated G i = 1 ˆµ + ˆγ + ˆδ + ˆτ ˆµ + ˆγ ˆδ + ˆτ Control G i = 0 ˆµ + ˆδ ˆµ ˆδ Treated - Control ˆγ + ˆτ ˆγ ˆτ Teppei Yamamoto Difference in Differences Causal Inference 17 / 30

39 Regression Estimator for Repeated Cross Sections Because G i and T i are both binary, the same estimator can be calculated via regression: Ŷ i = ˆµ + ˆγG i + ˆδT i + ˆτG i T i where ˆµ, ˆγ, ˆδ and ˆτ are OLS regression estimates Easy to show that ˆτ = ˆτ ATT : After (T i = 1) Before (T i = 0) After - Before Treated G i = 1 ˆµ + ˆγ + ˆδ + ˆτ ˆµ + ˆγ ˆδ + ˆτ Control G i = 0 ˆµ + ˆδ ˆµ ˆδ Treated - Control ˆγ + ˆτ ˆγ ˆτ Covariates (X i ) can be added to the right-hand side, with the risk of possible misspecification bias Don t include X i that can be affected by the treatment! (post-ttt bias) Cluster standard errors at the level the treatment is assigned (Card & Krueger get this wrong) Teppei Yamamoto Difference in Differences Causal Inference 17 / 30

40 Regression Estimator for Panel Data For panel data, consider an additive linear model for potential outcomes: Y it (z) = α i + γt + τz + ε it where α i is a time-invariant unobserved effect for unit i that may be correlated with treatment. Teppei Yamamoto Difference in Differences Causal Inference 18 / 30

41 Regression Estimator for Panel Data For panel data, consider an additive linear model for potential outcomes: Y it (z) = α i + γt + τz + ε it where α i is a time-invariant unobserved effect for unit i that may be correlated with treatment. We can show: τ = τ ATE = τ ATT Parallel trends imply: E[Y i1 (0) Y i0 (0) G i = 1] = E[Y i1 (0) Y i0 (0) G i = 0] E[ε i1 ε i0 G i = d] = 0 for d {0, 1} Teppei Yamamoto Difference in Differences Causal Inference 18 / 30

42 Regression Estimator for Panel Data For panel data, consider an additive linear model for potential outcomes: Y it (z) = α i + γt + τz + ε it where α i is a time-invariant unobserved effect for unit i that may be correlated with treatment. We can show: τ = τ ATE = τ ATT Parallel trends imply: E[Y i1 (0) Y i0 (0) G i = 1] = E[Y i1 (0) Y i0 (0) G i = 0] E[ε i1 ε i0 G i = d] = 0 for d {0, 1} Therefore, the first-differenced regression of Y i1 Y i0 on G i can unbiasedly estimate τ ATT = τ ATE Notice that panel data allow for unit-level unobserved confounding unlike the repeated cross-section case, but it must be additive and time-invariant Can include time-varying covariates (X it ) with possible risk of post-ttt bias Teppei Yamamoto Difference in Differences Causal Inference 18 / 30

43 Example: Minimum Wage and Employment ck_data <- plm.data(ck_data, indexes = c("id", "postperiod")) firstdiff.mod <- (plm(emptot ~ postperiod * nj, data = ck_data, model = "fd")) summary(firstdiff.mod) Oneway (individual) effect First-Difference Model Call: plm(formula = emptot ~ postperiod * nj, data = ck_data, model = "fd") Unbalanced Panel: n=410, T=1-2, N=794 Coefficients : Estimate Std. Error t-value Pr(> t ) (intercept) * postperiod1:nj * Teppei Yamamoto Difference in Differences Causal Inference 19 / 30

44 1 Introduction: A Motivating Example 2 Identification 3 Estimation and Inference 4 Diagnostics and Extensions

45 Parallel Trends Violations: Selection The parallel trends assumption can be violated for various reasons 1 Selection and targeting: Treatment assignment may depend on time-varying factors Examples: Self-selection: participants in worker training programs experience a decrease in earnings before they enter the program Targeting: policies may be targeted at units that are currently performing best (or worst). Teppei Yamamoto Difference in Differences Causal Inference 20 / 30

46 More Parallel Trends Violations 2 Compositional Differences Across Time In repeated cross-sections, the composition of the sample may change between periods, i.e. due to migration. This may confound any DD estimate since effect may be attributable to change in population. 3 Long-term effects versus reliability Parallel trends assumption most likely to hold over shorter time-period In the long run, many things may happen that could confound effect of treatment. Teppei Yamamoto Difference in Differences Causal Inference 21 / 30

47 Functional form dependence 4 Functional form dependence: Magnitude or even sign of the DD effect may be sensitive to the functional form, when average outcomes for controls and treated are very different at baseline Training program for the young Employment for the young increases from 20% to 30% Employment for the old increases from 5% to 10% Positive DD effect: (30-20) - (10-5) = 5% increase Teppei Yamamoto Difference in Differences Causal Inference 22 / 30

48 Functional form dependence 4 Functional form dependence: Magnitude or even sign of the DD effect may be sensitive to the functional form, when average outcomes for controls and treated are very different at baseline Training program for the young Employment for the young increases from 20% to 30% Employment for the old increases from 5% to 10% Positive DD effect: (30-20) - (10-5) = 5% increase But if you consider log changes in employment, the DD is [log(30) log(20)] [log(10) log(5)] = log(1.5) log(2) < 0 DD estimates may be more reliable if treated and control are more similar at baseline Teppei Yamamoto Difference in Differences Causal Inference 22 / 30

49 Diagnostics for Parallel Trends 1 Pre-treatment trends in the outcome: Check if the trends are parallel in the pre-treatment periods Requires data on multiple pre-treatment periods (the more the better) Note this is only diagnostics, not a direct test of the assumption! 2 Placebo test using previous periods: Suppose DD with time periods t 1, t 2, t 3, where treatment occurs in t 3 Exclude data from t 3, assign t 2 as placebo" treatment period, and re-estimate DD 3 Placebo test using alternative groups: Re-code some control groups as treated Re-estimate DD with the placebo treated units & without actual treated units 4 Placebo outcomes: Find outcomes that, theoretically, should be unaffected by the treatment, but might Re-estimate DD on these outcomes Teppei Yamamoto Difference in Differences Causal Inference 23 / 30

50 Checking for Pre-Treatment Trends FTE Employment New Jersey Pennsylvania Time Teppei Yamamoto Difference in Differences Causal Inference 24 / 30

51 Checking for Pre-Treatment Trends FTE Employment New Jersey Pennsylvania Time Teppei Yamamoto Difference in Differences Causal Inference 24 / 30

52 Checking for Pre-Treatment Trends FTE Employment New Jersey Pennsylvania Time Teppei Yamamoto Difference in Differences Causal Inference 24 / 30

53 Checking for Pre-Treatment Trends FTE Employment New Jersey Pennsylvania Time Teppei Yamamoto Difference in Differences Causal Inference 25 / 30

54 Chiefs and Voting (de Kadt and Larreguy 2015) ANC Vote Share KwaZulu Bantustans excl. KZ Rest of South Africa Ethnic shift in ANC elite Election Year Teppei Yamamoto Difference in Differences Causal Inference 26 / 30

55 Extension: Triple Differences Triple differences (or difference in differences in differences ): Use a placebo DD to make parallel trends more plausible Teppei Yamamoto Difference in Differences Causal Inference 27 / 30

56 Extension: Triple Differences Triple differences (or difference in differences in differences ): Use a placebo DD to make parallel trends more plausible Example: In state A, all 9th-grade girls are given a free bicycle. 1st DD: G i {girl, boy} and T i {9, 10} in state A 2nd DD: G i {girl, boy} and T i {9, 10} in state B (placebo) The DDD estimator: ˆτ DDD = Ê[Y i girl, 10, A] Ê[Y i girl, 9, A] } {Ê[Yi boy, 10, A] Ê[Y i boy, 9, A] (Ê[Yi girl, 10, B] Ê[Y i girl, 9, B] } ) {Ê[Yi boy, 10, B] Ê[Y i boy, 9, B] Can use regression with a triple interaction May eliminate time-varying confounding that are common in states A and B (e.g. girls change more from grade 9 to 10 than boys) Teppei Yamamoto Difference in Differences Causal Inference 27 / 30

57 Muralidharan and Prakash (2013) Teppei Yamamoto Difference in Differences Causal Inference 28 / 30

58 Muralidharan and Prakash (2013) Teppei Yamamoto Difference in Differences Causal Inference 29 / 30

59 Summary and Remarks DD: An extremely popular strategy when there is longitudinal data (panel or repeated cross-sections) and the treatment is one-shot Parallel trends = a form of ignorability assumption, i.e., unobserved confounding must be additive and time-invariant An important generalization of DD is fixed effects regression with both time and unit fixed effects: y it = x it b + α i + η t + ε it where x it includes the treatment variable Z it Other extensions include: Nonlinear and semi-/nonparametric DD Difference in discontinuities: Combine RD with DD Matching before DD: Make estimates more robust to functional form misspecification Teppei Yamamoto Difference in Differences Causal Inference 30 / 30

150C Causal Inference

150C Causal Inference Difference in Differences Jonathan Mummolo Stanford Selection on Unobservables Problem Often there are reasons to believe that treated and untreated units differ in unobservable characteristics