150C Causal Inference

Transcription

1 150C Causal Inference Panel Methods Jonathan Mummolo Stanford Jonathan Mummolo (Stanford) 150C Causal Inference 1 / 59

2 Unobserved Group Effects 7 6 SD(Economic Output) for all data = 346 Range(Economic Output) for all data = 1,034 Country 5 Democracy (1 7 Scale) Country 2 Country 3 Country 4 2 Country Economic Output ($ billions) Jonathan Mummolo (Stanford) 150C Causal Inference 2 / 59

3 Outline Panel Setup 1 Panel Setup 2 Unobserved Effects Model and Pooled OLS 3 Fixed Effects Regression 4 Modeling Time 5 Modeling Dynamic Effects Jonathan Mummolo (Stanford) 150C Causal Inference 3 / 59

4 Panel Setup Panel Setup Let y and x (x 1, x 2,..., x K ) be observable random variables and c be an unobservable random variable We are interested in the partial effects of variable x j in the population regression function E[y x 1, x 2,..., x K, c] We observe a sample of i = 1, 2,..., N cross-sectional units for t = 1, 2,...T time periods (a balanced panel) For each unit i, we denote the observable variables for all time periods as {(y it, x it ) : t = 1, 2,..., T } x it (x it1, x it2,..., x itk ) is a 1 K vector Typically assume that cross-sectional units are i.i.d. draws from the population: {y i, x i, c i } N i=1 i.i.d. (cross-sectional independence) y i (y i1, y i2,..., y it ) and x i (x i1, x i2,..., x it ) Consider asymptotic properties with T fixed and N Jonathan Mummolo (Stanford) 150C Causal Inference 4 / 59

5 Panel Setup Panel Setup Single unit: y i = y i1. y it. y it T 1 X i = x i,1,1 x i,1,2 x i,1,j x i,1,k.... x i,t,1 x i,t,2 x i,t,j x i,t,k.... x i,t,1 x i,t,2 x i,t,j x i,t,k T K Panel with all units: y = y 1. y i. y N NT 1 X = X 1. X i. X N NT K Jonathan Mummolo (Stanford) 150C Causal Inference 5 / 59

6 Outline Unobserved Effects Model and Pooled OLS 1 Panel Setup 2 Unobserved Effects Model and Pooled OLS 3 Fixed Effects Regression 4 Modeling Time 5 Modeling Dynamic Effects Jonathan Mummolo (Stanford) 150C Causal Inference 6 / 59

7 Unobserved Effects Model and Pooled OLS Unobserved Effects Model: Farm Output For a randomly drawn cross-sectional unit i, the model is given by y it = x it β + c i + ε it, t = 1, 2,..., T y it : output of farm i in year t x it : 1 K vector of variable inputs for farm i in year t, such as labor, fertilizer, etc. plus an intercept β: K 1 vector of marginal effects of variable inputs c i : farm effect, i.e. the sum of all time-invariant inputs known to farmer i (but unobserved for the researcher), such as soil quality, managerial ability, etc. often called: unobserved effect, unobserved heterogeneity, etc. ε it : time-varying unobserved inputs, such as rainfall, unknown to the farmer at the time the decision on the variable inputs x it is made often called: idiosyncratic error What happens when we regress y it on x it? Jonathan Mummolo (Stanford) 150C Causal Inference 7 / 59

8 Unobserved Effects Model and Pooled OLS Pooled OLS When we ignore the panel structure and regress y it on x it we get y it = x it β + v it, t = 1, 2,..., T with composite error v it c i + ε it Main assumption to obtain consistent estimates for β is: Jonathan Mummolo (Stanford) 150C Causal Inference 8 / 59

9 Unobserved Effects Model and Pooled OLS Pooled OLS When we ignore the panel structure and regress y it on x it we get y it = x it β + v it, t = 1, 2,..., T with composite error v it c i + ε it Main assumption to obtain consistent estimates for β is: E[v it x i1, x i2,..., x it ] = E[v it x it ] = 0 for t = 1, 2,..., T x it are strictly exogenous: the composite error v it in each time period is uncorrelated with the past, current, and future regressors But: labour input x it likely depends on soil quality c i and so we have omitted variable bias and ˆβ is not consistent No correlation between x it and v it implies no correlation between unobserved effect c i and x it for all t Violations are common: whenever we omit a time-constant variable that is correlated with the regressors Jonathan Mummolo (Stanford) 150C Causal Inference 8 / 59

10 Unobserved Effects Model and Pooled OLS Unobserved Effects Model: Program Evaluation Program evaluation model: y it = prog it β + c i + ε it, y it : log wage of individual i in year t t = 1, 2,..., T prog it : indicator coded 1 if individual i participants in training program at t and 0 otherwise β: effect of program c i : sum of all time-invariant unobserved characteristics that affect wages, such as ability, etc. What happens when we regress y it on prog it? Jonathan Mummolo (Stanford) 150C Causal Inference 9 / 59

11 Unobserved Effects Model and Pooled OLS Unobserved Effects Model: Program Evaluation Program evaluation model: y it = prog it β + c i + ε it, y it : log wage of individual i in year t t = 1, 2,..., T prog it : indicator coded 1 if individual i participants in training program at t and 0 otherwise β: effect of program c i : sum of all time-invariant unobserved characteristics that affect wages, such as ability, etc. What happens when we regress y it on prog it? ˆβ not consistent since prog it is likely correlated with c i (e.g. ability) Always ask: Is there a time-constant unobserved variable (c i ) that is correlated with the regressors? If yes, pooled OLS is problematic Additional problem: v it c i + ε it are serially correlated for same i since c i is present in each t and thus pooled OLS standard errors are invalid Jonathan Mummolo (Stanford) 150C Causal Inference 9 / 59

12 Outline Fixed Effects Regression 1 Panel Setup 2 Unobserved Effects Model and Pooled OLS 3 Fixed Effects Regression 4 Modeling Time 5 Modeling Dynamic Effects Jonathan Mummolo (Stanford) 150C Causal Inference 10 / 59

13 Fixed Effects Regression Fixed Effect Regression Our unobserved effects model is: y it = x it β + c i + ε it, t = 1, 2,..., T If we have data on multiple time periods, we can think of c i as fixed effects or nuisance parameters to be estimated Jonathan Mummolo (Stanford) 150C Causal Inference 11 / 59

14 Fixed Effects Regression Fixed Effects Regression Estimating a regression with the time-demeaned variables ÿ it y it ȳ i and ẍ it x it x i is numerically equivalent to a regression of y it on x it and unit specific dummy variables. Fixed effects estimator is often called the within estimator because it only uses the time variation within each cross-sectional unit. Even better, the regression with the time-demeaned variables is consistent for β even when Cov[x it, c i ] 0, because time-demeaning eliminates the unobserved effects: y it = x it β + c i + ε it ȳ i = x i β + c i + ε i (y it ȳ i ) = (x it x i )β + (c i c i ) + (ε it ε i ) ÿ it = ẍ it β + ε it Jonathan Mummolo (Stanford) 150C Causal Inference 12 / 59

15 Fixed Effects Regression Fixed Effects Regression: Main Results Identification assumptions: 1 E[ε it x i1, x i2,..., x it, c i ] = 0, t = 1, 2,..., T regressors are strictly exogenous conditional on the unobserved effect allows x it to be arbitrarily related to c i 2 regressors vary over time for at least some i and are not collinear Fixed effects estimator: 1 Demean and regress ÿ it on ẍ it (need to correct degrees of freedom) 2 Regress y it on x it and unit dummies (dummy variable regression) 3 Regress y it on x it with canned fixed effects routine R: plm(y~x, model = within, data = data) Properties (under assumptions 1-2): ˆβ FE is consistent: plim ˆβ FE,N = β N ˆβ FE also unbiased conditional on X Jonathan Mummolo (Stanford) 150C Causal Inference 13 / 59

16 Fixed Effects Regression Fixed Effects Regression: Main Issues Inference: Standard errors have to be clustered by panel unit (e.g. farm) to allow correlation in the ε it s for the same i. R: coeftest(mod, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) Yields valid inference as long as number of clusters is reasonably large Typically we care about β, but unit fixed effects c i could be of interest plm routine demeans the data before running the regression and therefore does not estimate ĉ i Jonathan Mummolo (Stanford) 150C Causal Inference 14 / 59

17 Fixed Effects Regression Example: Direct Democracy and Naturalizations Do minorities fare worse under direct democracy than under representative democracy? Hainmueller and Hangartner (2016, AJPS) examine data on naturalization requests of immigrants in Switzerland, where municipalities vote on naturalization applications in: referendums (direct democracy) elected municipality councils (representative democracy) Annual panel data from 1,400 municipalities for the period y it : naturalization rate = # naturalizations it / eligible foreign population it 1 x it : 1 if municipality used representative democracy, 0 if municipality used direct democracy in year t Jonathan Mummolo (Stanford) 150C Causal Inference 15 / 59

18 Fixed Effects Regression Naturalization Panel Data Long Format > d <- read.dta("swiss_panel_long.dta") > print(d[30:40,],digits=2) muniid muni_name year nat_rate repdem 30 2 Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Bonstetten Bonstetten Jonathan Mummolo (Stanford) 150C Causal Inference 16 / 59

19 Pooled OLS Fixed Effects Regression > summary(lm(nat_rate~repdem,data=d)) Call: lm(formula = nat_rate ~ repdem, data = d) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** repdem <2e-16 *** Jonathan Mummolo (Stanford) 150C Causal Inference 17 / 59

20 Fixed Effects Regression Time-Demeaning for Fixed Effects: y it ÿ it > library(plyr) > d <- ddply(d,.(muniid), transform, + nat_rate_demean = nat_rate - mean(nat_rate), + nat_rate_mean = mean(nat_rate), + repdem_demean = repdem - mean(repdem)) > > print(d[20:38, + c("muniid","muni_name","year","nat_rate","nat_rate_mean","nat_rate_demean","repdem","repdem_demean") + ],digits=2) muniid muni_name year nat_rate nat_rate_mean nat_rate_demean repdem repdem_demean 20 2 Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Affoltern A.A Jonathan Mummolo (Stanford) 150C Causal Inference 18 / 59

21 Fixed Effects Regression Fixed Effects Regression with Demeaned Data > summary(lm(nat_rate_demean~repdem_demean,data=d)) Call: lm(formula = nat_rate_demean ~ repdem_demean, data = d) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 6.266e e repdem_demean 3.023e e <2e-16 *** Jonathan Mummolo (Stanford) 150C Causal Inference 19 / 59

22 Fixed Effects Regression Fixed Effects Regression with Canned Routine > library(plm) > library(lmtest) > d <- plm.data(d, indexes = c("muniid", "year")) > mod_fe <- plm(nat_rate~repdem,data=d,model="within") > coeftest(mod_fe, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) repdem < 2.2e-16 *** --- Jonathan Mummolo (Stanford) 150C Causal Inference 20 / 59

23 Fixed Effects Regression Fixed Effects Regression with Dummies > mod_du <- plm(nat_rate~repdem+as.factor(muniid),data=d,model="pooling") > coeftest(mod_du, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e e e+01 < 2.2e-16 *** repdem e e e+01 < 2.2e-16 *** as.factor(muniid) e e e+07 < 2.2e-16 *** as.factor(muniid) e e e+07 < 2.2e-16 *** as.factor(muniid) e e e+07 < 2.2e-16 *** as.factor(muniid) e e e+07 < 2.2e-16 *** as.factor(muniid) e e e+07 < 2.2e-16 *** as.factor(muniid) e e e as.factor(muniid) e e e+02 < 2.2e-16 *** as.factor(muniid) e e e+02 < 2.2e-16 *** as.factor(muniid) e+00 NA NA NA as.factor(muniid) e e e+01 < 2.2e-16 ***. Jonathan Mummolo (Stanford) 150C Causal Inference 21 / 59

24 Fixed Effects Regression Applying Fixed Effects We can use fixed effects for other data structures to restrict comparisons to within unit variation Matched pairs Twin fixed effects to control for unobserved effects of family background Cluster fixed effects in hierarchical data School fixed effects to control for unobserved effects of school Jonathan Mummolo (Stanford) 150C Causal Inference 22 / 59

25 Fixed Effects Regression Problems that (even) fixed effects do not solve y it = x it β + c i + ε it, t = 1, 2,..., T Where y it is murder rate and x it is police spending per capita What happens when we regress y on x and city fixed effects? Jonathan Mummolo (Stanford) 150C Causal Inference 23 / 59

26 Fixed Effects Regression Problems that (even) fixed effects do not solve y it = x it β + c i + ε it, t = 1, 2,..., T Where y it is murder rate and x it is police spending per capita What happens when we regress y on x and city fixed effects? ˆβ FE inconsistent unless strict exogeneity conditional on c i holds E[ε it x i1, x i2,..., x it, c i ] = 0, t = 1, 2,..., T implies ε it uncorrelated with past, current, and future regressors Most common violations: 1 Time-varying omitted variables economic boom leads to more police spending and less murders can include time-varying controls, but avoid post-treatment bias 2 Simultaneity if city adjusts police based on past murder rate, then spending t+1 is correlated with ε t (since higher ε t leads to higher murder rate at t) strictly exogenous x cannot react to what happens to y in the past or the future! Fixed effects do not obviate need for good research design! Jonathan Mummolo (Stanford) 150C Causal Inference 23 / 59

27 Outline Modeling Time 1 Panel Setup 2 Unobserved Effects Model and Pooled OLS 3 Fixed Effects Regression 4 Modeling Time 5 Modeling Dynamic Effects Jonathan Mummolo (Stanford) 150C Causal Inference 24 / 59

28 Common Shocks or Causation? Jonathan Mummolo (Stanford) 150C Causal Inference 25 / 59

29 Naturalization Rate Over Time Mean Nat Rate Year Jonathan Mummolo (Stanford) 150C Causal Inference 26 / 59

30 Representative Democracy Over Time representative democracy (0/1) Year Jonathan Mummolo (Stanford) 150C Causal Inference 27 / 59

31 Adding Time Effects Reconsider our unobserved effects model: y it = x it β + c i + ε it, t = 1, 2,..., T Fixed effects assumption: E[ε it x i, c i ] = 0, t = 1, 2,..., T : regressors are strictly exogenous conditional on the unobserved effect Typical violation: Common shocks that affect all units in the same way and are correlated with x it. Trends in farming technology or climate affect productivity Trends in immigration inflows affect naturalization rates We can allow for such common shocks by including time effects into the model Jonathan Mummolo (Stanford) 150C Causal Inference 28 / 59

32 Fixed Effects Regression: Adding Time Effects Linear time trend: y it = x it β + c i + t + ε it, t = 1, 2,..., T Linear time trend common to all units Time fixed effects: y it = x it β + c i + t t + ε it, t = 1, 2,..., T Common shock in each time period Generalized difference-in-differences model Unit specific linear time trends: y it = x it β + c i + g i t + t t + ε it, t = 1, 2,..., T Linear time trends that vary by unit Jonathan Mummolo (Stanford) 150C Causal Inference 29 / 59

33 Modeling Time Effects Jonathan Mummolo (Stanford) 150C Causal Inference 30 / 59

34 Modeling Time Effects > d$time <- as.numeric(d$year) > d[1:21,c("muniid","muni_name","year","time")] muniid muni_name year time 1 1 Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Aeugst A.A Affoltern A.A Affoltern A.A Jonathan Mummolo (Stanford) 150C Causal Inference 31 / 59

35 Fixed Effects Regression: Linear Time Trend > mod_fe <- plm(nat_rate~repdem+time,data=d,model="within") > coeftest(mod_fe, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) repdem ** time < 2.2e-16 *** Jonathan Mummolo (Stanford) 150C Causal Inference 32 / 59

36 Fixed Effects Regression: Year Fixed Effects > mod_fe <- plm(nat_rate~repdem+year,data=d,model="within") > coeftest(mod_fe, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) repdem e-05 *** year * year year e-05 *** year e-05 *** year e-06 *** year e-07 *** year e-08 *** year e-07 *** year < 2.2e-16 *** year e-12 *** year < 2.2e-16 *** year < 2.2e-16 *** year < 2.2e-16 *** year < 2.2e-16 *** year < 2.2e-16 *** year e-14 *** year < 2.2e-16 *** year e-07 *** --- Jonathan Mummolo (Stanford) 150C Causal Inference 33 / 59

37 ~ Modeling Time 'I( Unit Specific Time Trends240 Often Chapter 5 Eliminate Results ", AOt" TABLE Estimated effects of labor regulation on the performance of firms in Indian states Labor regulation (lagged) (.064) Log development expenditure per capita Log installed electricity capacity per capita (1) (2) (3) (4) (.051).240 (.128).089 (.061) Log state population.720 (.96) (.039).184 (.119).082 (.054) (1.192).0002 (.020).241 (.106).023 (.033) (2.326) Fixed Effe the estimates in colum specific covariates, suc and state population. addition of state-level the minimum wage stu Besley and Burgess es specific trends kills the in column 4. Apparen in states where output trend therefore drives Congress majority We've la beled the two time because this is th (.01) (.010) econometrics. But the Hard left majority (.017) (.009) of states, the subscrip some of which are af Janata majority For example, Kugler, J (.026) (.033) effects of age-specific e Regional majority Likewise, instead of t other types ofcharacte (.009) (.023) (1999), who studied State-specific trends Adjusted R2 No.93 No.93 No.94 Yes.95 laws on teen pregnan birth. Regardless of t Notes: Adapted from Besley and Burgess (2004), table IV. The table reports regression DD estimates of the effects of labor regulation on productivity. The dependent variable is log manufacturing output per capita. All models include always set up an impl question of whether careful consideration. state and year effects. Robust standard errors clustered at the state level are One potential pitfa reported parentheses. State amendments to the Industrial Disputes Act are labor regulationcoded increased 1 = pro-worker, in0 states = neutral, -1 where = pro-employer output and then was cumulated declining anyway position of the treatm over the period to generate the labor regulation measure. Log of installed result of treatment. electrical capacity is measured in kilowatts, and log development expenditure is real per capita state spending on social and economic services. Congress, Jonathan Mummolo (Stanford) 150C Causal Inference hard left, Janata, and regional majority are counts of the number of years and time comparisons of the generosity 34 / of 59 pu

38 Fixed Effects Regression: Unit Specific Time Trends > mod_fe <- plm(nat_rate~ repdem+muniid*time+year,data=d,model="within") > coeftest(mod_fe, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) repdem e-05 *** muniid2:time e e e+07 < 2.2e-16 *** muniid3:time e e e+07 < 2.2e-16 ***. Jonathan Mummolo (Stanford) 150C Causal Inference 35 / 59

39 Unit Specific Quadratic Time Trends > d$time2 <- d$time^2 > mod_fe <- plm(nat_rate~repdem+ muniid*time+muniid*time^2+year,data=d,model="within") > coeftest(mod_fe, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) repdem e-05 *** muniid2:time e < 3.4e-16 *** muniid2:time e e+07 < 5.6e-16 ***. Jonathan Mummolo (Stanford) 150C Causal Inference 36 / 59

40 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

41 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

42 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

43 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Important to consider realistic counterfactuals when interpreting results, i.e. a one-unit or one SD increase in x may no longer be of interest because x never shifts this much within a unit over time Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

44 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Important to consider realistic counterfactuals when interpreting results, i.e. a one-unit or one SD increase in x may no longer be of interest because x never shifts this much within a unit over time Could end up extrapolating beyond the coverage in the data, yielding unreliable counterfactual estimates Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

45 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Important to consider realistic counterfactuals when interpreting results, i.e. a one-unit or one SD increase in x may no longer be of interest because x never shifts this much within a unit over time Could end up extrapolating beyond the coverage in the data, yielding unreliable counterfactual estimates More important when treatment is continuous Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

46 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Important to consider realistic counterfactuals when interpreting results, i.e. a one-unit or one SD increase in x may no longer be of interest because x never shifts this much within a unit over time Could end up extrapolating beyond the coverage in the data, yielding unreliable counterfactual estimates More important when treatment is continuous Useful first step: examine (plot, summarize) the demeaned data Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

47 Interpreting FE Results Sensibly Using fixed effects often removes large amounts of variation from the data With unit fixed effects, we are left explaining changes in y within units over time, after removing all variation between units With unit and time fixed effects, we are left explaining changes in y within units over time net of common time shocks Important to consider realistic counterfactuals when interpreting results, i.e. a one-unit or one SD increase in x may no longer be of interest because x never shifts this much within a unit over time Could end up extrapolating beyond the coverage in the data, yielding unreliable counterfactual estimates More important when treatment is continuous Useful first step: examine (plot, summarize) the demeaned data Also a good way to check functional form assumptions: is the bivariate relationship of interest still linear after conditioning on covariates? Jonathan Mummolo (Stanford) 150C Causal Inference 37 / 59

48 Pooled Data Modeling Time 7 6 SD(Economic Output) for all data = 346 Range(Economic Output) for all data = 1,034 Country 5 Democracy (1 7 Scale) Country 2 Country 3 Country 4 2 Country Economic Output ($ billions) Jonathan Mummolo (Stanford) 150C Causal Inference 38 / 59

49 Reduction in Variation 3 SD(Economic Output) for demeaned data = 24.6 Range(Economic Output) for demeaned data = Mean Deviated Democracy 2 1 One SD Increase in Economic Output Min. to Max. Increase in Economic Output Mean Deviated Economic Output ($ billions) Jonathan Mummolo (Stanford) 150C Causal Inference 39 / 59

50 Example: Healy and Malhotra (2009) Estimates effect of federal disaster relief ($) on presidential vote share Jonathan Mummolo (Stanford) 150C Causal Inference 40 / 59

51 Example: Healy and Malhotra (2009) Estimates effect of federal disaster relief ($) on presidential vote share Observations on all U.S. counties over several election years Jonathan Mummolo (Stanford) 150C Causal Inference 40 / 59

52 Example: Healy and Malhotra (2009) Estimates effect of federal disaster relief ($) on presidential vote share Observations on all U.S. counties over several election years > head(d[,c("name_state","county","year", "all_current_relief","incum_vote")]) name_state county year all_current_relief incum_vote 1 AL AUTAUGA AL AUTAUGA AL AUTAUGA AL AUTAUGA AL AUTAUGA AL BALDWIN Jonathan Mummolo (Stanford) 150C Causal Inference 40 / 59

53 Example: Healy and Malhotra (2009) Pooled OLS Result > summary(lm(incum_vote~all_current_relief, data=d)) Call: lm(formula = incum_vote ~ all_current_relief, data = d) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** all_current_relief <2e-16 *** Jonathan Mummolo (Stanford) 150C Causal Inference 41 / 59

54 Bivariate Plot Pooled Ln(Disaster Relief per Capita) Incumbent Vote % OLS fit LOESS fit Jonathan Mummolo (Stanford) 150C Causal Inference 42 / 59

55 Example: Healy and Malhotra (2009) County Fixed Effects > summary(m1<-lm(incum_vote~ all_current_relief +factor(fips), data=d))$coefficients[1:2,] Estimate Std. Error t value Pr(> t ) (Intercept) e-16 all_current_relief e+00 Jonathan Mummolo (Stanford) 150C Causal Inference 43 / 59

56 County FE Pooled Ln(Disaster Relief per Capita) Incumbent Vote % County FE Ln(Disaster Relief per Capita) Within Counties Incumbent Vote % within Counties OLS fit LOESS fit Jonathan Mummolo (Stanford) 150C Causal Inference 44 / 59

57 What About Multi-Way Fixed Effects? How do we visualize data with multi-way fixed effects? Jonathan Mummolo (Stanford) 150C Causal Inference 45 / 59

58 What About Multi-Way Fixed Effects? How do we visualize data with multi-way fixed effects? Can residualize the data and then plot Jonathan Mummolo (Stanford) 150C Causal Inference 45 / 59

59 What About Multi-Way Fixed Effects? How do we visualize data with multi-way fixed effects? Can residualize the data and then plot Useful exercise with all multivariate regressions Jonathan Mummolo (Stanford) 150C Causal Inference 45 / 59

60 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

61 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

62 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) 2. Regress y on an intercept and z 1 -z k and store the residuals, r yz = (y ŷ yz ) Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

63 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) 2. Regress y on an intercept and z 1 -z k and store the residuals, r yz = (y ŷ yz ) 3. Estimate r yz = θ + β 1 r xz + v Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

64 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) 2. Regress y on an intercept and z 1 -z k and store the residuals, r yz = (y ŷ yz ) 3. Estimate r yz = θ + β 1 r xz + v Then ˆβ 1 = ˆβ 1 Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

65 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) 2. Regress y on an intercept and z 1 -z k and store the residuals, r yz = (y ŷ yz ) 3. Estimate r yz = θ + β 1 r xz + v Then ˆβ 1 = ˆβ 1 (Intuition: Remove all variation explained by z 1 z k that is shared with x. What relationship between x and y remains?) Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

66 Frisch-Waugh-Lovell Theorem Suppose we are interested in estimating β 1 in the following model: y = α + β 1 x + δ 2 z 1 + β 3 z β k z k + ε where x is the treatment and z 1 -z k are a set of control variables. β 1 can also be estimated through the following multi-step process. 1. Regress x on an intercept and z 1 -z k and store the residuals, r xz = (x ˆx xz ) 2. Regress y on an intercept and z 1 -z k and store the residuals, r yz = (y ŷ yz ) 3. Estimate r yz = θ + β 1 r xz + v Then ˆβ 1 = ˆβ 1 (Intuition: Remove all variation explained by z 1 z k that is shared with x. What relationship between x and y remains?) In one-way FE only, equivalent to demeaning the data within each cross-sectional unit Jonathan Mummolo (Stanford) 150C Causal Inference 46 / 59

67 Healy and Malhotra (2009) 1. Regress Disaster Relief on an intercept and county and year dummies, and store the residuals, r xz = (x ˆx xz ) Jonathan Mummolo (Stanford) 150C Causal Inference 47 / 59

68 Healy and Malhotra (2009) 1. Regress Disaster Relief on an intercept and county and year dummies, and store the residuals, r xz = (x ˆx xz ) 2. Regress Vote Share on an intercept and county and year dummies and store the residuals, r yz = (y ŷ yz ) Jonathan Mummolo (Stanford) 150C Causal Inference 47 / 59

69 Healy and Malhotra (2009) 1. Regress Disaster Relief on an intercept and county and year dummies, and store the residuals, r xz = (x ˆx xz ) 2. Regress Vote Share on an intercept and county and year dummies and store the residuals, r yz = (y ŷ yz ) 3. Estimate r yz = θ + β 1 r xz + v Jonathan Mummolo (Stanford) 150C Causal Inference 47 / 59

70 County and Year FE Dummy Model: > m1b<-lm(incum_vote~ all_current_relief+ factor(fips)+factor(year), data=d) > summary(m1b)$coefficients[1:5,] Estimate Std. Error t value Pr(> t ) (Intercept) e-30 all_current_relief e-119 factor(fips) e-01 factor(fips) e-01 factor(fips) e-01 Jonathan Mummolo (Stanford) 150C Causal Inference 48 / 59

71 County and Year FE Residualized Model: m3b<-lm(all_current_relief~factor(fips)+factor(year), data=d) m4b<-lm(incum_vote ~factor(fips)+factor(year), data=d) > summary(lm(m4b$residuals~m3b$residuals))$coefficients Estimate Std. Error t value Pr(> t ) (Intercept) e e e+00 m3b$residuals e e e-148 Jonathan Mummolo (Stanford) 150C Causal Inference 49 / 59

72 County and Year FE Pooled Ln(Disaster Relief per Capita) Incumbent Vote % SD(X)= County FE Ln(Disaster Relief per Capita) Within Counties Incumbent Vote % Within Counties SD(X*)= County and Year FE Ln(Disaster Relief per Capita) Within Counties/Years Incumbent Vote % within Counties SD(X**)=1.004 OLS fit LOESS fit Jonathan Mummolo (Stanford) 150C Causal Inference 50 / 59

73 Interpreting Results Modeling Time Dummy Model: > m1b<-lm(incum_vote~ all_current_relief+ factor(fips)+factor(year), data=d) > summary(m1b)$coefficients[1:5,] Estimate Std. Error t value Pr(> t ) (Intercept) e-30 all_current_relief e-119 With county and year FE, more reasonable to discuss a one-sd shift in X = disaster relief after residualizing with respect to county and year dummies * 2.17 = 2.26 additional points for incumbent using SD of X * 2.17 = 3.97 additional points for incumbent using SD of X =.569 meaning the substantive impact is cut in half once we consider a plausible shift in X. Jonathan Mummolo (Stanford) 150C Causal Inference 51 / 59

74 Outline Modeling Dynamic Effects 1 Panel Setup 2 Unobserved Effects Model and Pooled OLS 3 Fixed Effects Regression 4 Modeling Time 5 Modeling Dynamic Effects Jonathan Mummolo (Stanford) 150C Causal Inference 52 / 59

75 Modeling Dynamic Effects Leads and Lags Let D it be a binary indicator coded 1 if unit i switched from control to treatment between t and t 1; 0 otherwise Lags: D it 1 : unit switched between t 1 and t 2 Leads: D it+1 : unit switches between t + 1 and t Include lags and leads into the fixed effects model: y it = D it+2 β 2 + D it+1 β 1 + D it β 0 + D it 1 β 1 + D it 2 β 2 + c i + ε it Interpretation of coefficients: Leads β 1, β 2, etc. test for anticipation effects (should be zero!) Switch β 0 tests for immediate effect Lags β 1, β 2, etc. test for long-run effects highest lag dummy can be coded 1 for all post-switch years Jonathan Mummolo (Stanford) 150C Causal Inference 53 / 59

76 Modeling Dynamic Effects Lags and Leads of Switch to Representative Democracy > d[970:989,c(1:3,5,12:ncol(d))] muniid year muni_name repdem switcht lag1 lag2 lag3 lead1 lead2 lead3 lead4 lead Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Hagenbuch Pfungen Jonathan Mummolo (Stanford) 150C Causal Inference 54 / 59

77 Modeling Dynamic Effects Dynamic Effect of Switching to Representative Democracy > mod_all <- plm(nat_rate~lag3+lag2+lag1+switcht+ lead1+lead2+lead3+lead4+lead5+ + year,data=d,model="within") > coeftest(mod_all, vcov=function(x) vcovhc(x, cluster="group", type="hc1")) t test of coefficients: Estimate Std. Error t value Pr(> t ) lag * lag ** lag *** switcht lead lead lead lead lead year * Jonathan Mummolo (Stanford) 150C Causal Inference 55 / 59

78 Modeling Dynamic Effects Dynamic Effect of Switching to Representative Democracy Dynamic Panel Estimates Jonathan Mummolo (Stanford) 150C Causal Inference 56 / 59

79 Switching Plot Modeling Dynamic Effects Naturalization Rate (%) Direct Democracy Representative Democracy Years Before / After Change from Direct to Representative Democracy Jonathan Mummolo (Stanford) 150C Causal Inference 57 / 59

80 Modeling Dynamic Effects Heterogeneous Treatment Effects So far we have assumed that the treatment effect is constant across units Can allow for heterogeneous treatment effects by including interaction of treatment with other regressors y it = treat it α 0 + (treat it x it )α 1 + x it β + c i + t + ε it, t = 1, 2,..., T Often the treatment is interacted with a time-invariant regressor: y it = treat it α 0 + (treat it x i )α 1 + c i + t + ε it, t = 1, 2,..., T Note: The lower order term on the time-invariant x i is collinear with the fixed effects and drops out Jonathan Mummolo (Stanford) 150C Causal Inference 58 / 59

81 Modeling Dynamic Effects Heterogeneous Effect of Direct Democracy Effect: Direct to Representative Democracy 95 % Confidence Interval Change in Naturalization Rate (%) Quartiles SVP Vote Share Municipality Level SVP Vote Share (%) Jonathan Mummolo (Stanford) 150C Causal Inference 59 / 59