PUBL0050:AdvancedQuantitativeMethods Lecture5 ...πππ‘= + 1 π+ 2ππ+ 3( πβ ...
Transcript of PUBL0050:AdvancedQuantitativeMethods Lecture5 ...πππ‘= + 1 π+ 2ππ+ 3( πβ ...
PUBL0050: Advanced Quantitative Methods
Lecture 5: Panel Data and Difference-in-Differences
Jack Blumenau
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Course outline
1. Potential Outcomes and Causal Inference2. Randomized Experiments3. Selection on Observables (I)4. Selection on Observables (II)5. Panel Data and Difference-in-Differences6. Synthetic Control7. Instrumental Variables (I)8. Instrumental Variables (II)9. Regression Discontinuity Designs10. Overview and Review
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Was it βthe Sun wot won itβ?
Does the media influence vote choice?
β’ Randomized experiment?
β’ Hard to persuade newspapers to randomly endorse politicalcandidates
β’ Hard to randomly allocate citizens to read certain newspapers
β’ Selection on observables?
β’ The types of individual who read certain newspapers (i.e. The Sun)are likely different in many ways from those who read othernewspapers
β’ Difference-in-differences
β’ Collect data on vote choice before & after change in endorsementβ’ Did people who read The Sun change their vote choice more thanpeople who did not read The Sun?
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Lecture Outline
Difference-in-differences
Regression DD
Threats to Validity
Multiple periods
Conclusion
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Difference-in-differences setup
DefinitionTwo groups:
β’ π·π = 1 Treated units
β’ π·π = 0 Control units
Two periods:
β’ ππ = 0 Pre-Treatment period
β’ ππ = 1 Post-Treatment period
Potential outcome πππ(π‘)
β’ π1π(π‘) outcome of unit π in period π‘ when treated (at ππ = 1)β’ π0π(π‘) outcome of unit π in period π‘ when control (at ππ = 1)
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Difference-in-differences setup
DefinitionCausal effect for unit π at time π‘ is
β’ ππ(π‘) = π1π(π‘) β π0π(π‘)
For a given unit, in a given time period, the observed outcome ππ(π‘) is:
β’ ππ(π‘) = π1π(π‘) β π·π(π‘) + π0π(π‘) β (1 β π·π(π‘))β’ If treatment occurs only after π‘ = 0 we have:
ππ(1) = π0π(1) β (1 β π·π) + π1π(1) β π·π
β Fundamental problem of causal inference.
Estimand (ATT)πATT = πΈ[π1π(1) β π0π(1)|π·π = 1]
ProblemMissing potential outcome: πΈ[π0π(1)|π· = 1], ie. what is the average post-periodoutcome for the treated in the absence of the treatment?
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Illustration
Treated vs. control in post-treatment period
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
β
β
β
β
Problem: Missing potential outcome: πΈ[π0π(1)|π·π = 1]
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Illustration
Strategy: Treated vs. control in post-treatment period
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(1)|Di=0]
E[Yi(1)|Di=1]
DIGM
β
β
β
β
Assumption: No selection bias.
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Illustration
Strategy: Before vs. after for treatment units
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
Change over time in treatment group
β
β
β
β
Assumption: No effect of time independent of treatment.
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Illustration
Treated vs. control in post-treatment period
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
Change over time in control group
β
β
β
β
Treated vs. control in post-treatment period
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Illustration
Strategy: Difference-in-differences (1)
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(0)|Di=1]
E[Y0i(1)|Di=1]
β
β
β
β
β
β
Assumed change over time in treatment group
Assumption: Trend over time is the same for treatment and control.
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Illustration
Strategy: Difference-in-differences (1)
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Y0i(1)|Di=1]
β
β
β
β
β
β
Assumed change over time in treatment group
Difference in differences
Assumption: Trend over time is the same for treatment and control.
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Illustration
Strategy: Difference-in-differences (2)
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(0)|Di=0]
E[Yi(0)|Di=1]
Selection bias in preβtreatment period
β
β
β
β
Treated vs. control in post-treatment period
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Illustration
Strategy: Difference-in-differences (2)
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(1)|Di=0]
E[Y0i(1)|Di=1]
Assumed selection bias in postβtreatment period
β
β
β
β
β
β
Assumption: Selection bias is stable over time.
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Illustration
Strategy: Difference-in-differences (2)
t = 0 t=1
E[Yi(0)|Di=0]
E[Yi(1)|Di=0]
E[Yi(0)|Di=1]
E[Yi(1)|Di=1]
E[Yi(1)|Di=0]
E[Yi(1)|Di=1]
E[Y0i(1)|Di=1]
Assumed selection bias in postβtreatment period
β
β
β
β
β
β
Difference in differences
Assumption: Selection bias is stable over time.
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Identification with Difference-in-Differences
Two ways of stating the identifying assumption:
β’ Parellel trends
β’ If treated units did not receive the treatment, they would havefollowed the same trend as the control units
β’ No time-varying confounders
β’ Omitted variables related both to treatment and outcome must befixed over time
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Identification with Difference-in-Differences
Identification AssumptionπΈ[π0π(1) β π0π(0)|π·π = 1] = πΈ[π0π(1) β π0π(0)|π·π = 0] (parallel trends)
Identification ResultGiven parallel trends, πATT is identified as:
πΈ[π1π(1) β π0π(1)|π· = 1] = {πΈ[ππ(1)|π·π = 1] β πΈ[ππ(1)|π·π = 0]}
β {πΈ[ππ(0)|π·π = 1] β πΈ[ππ(0)|π· = 0]}
In other words:
πATT = {Difference in means in post-treatment period}
β {Difference in means in pre-treatment period}
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Difference-in-Differences: Estimators
Estimand (ATET)πΈ[π1(1) β π0(1)|π· = 1]
Estimator (Sample Means: Panel)
{ 1π1
βπ·π=1
ππ(1) β 1π0
βπ·π=0
ππ(1)}β{ 1π1
βπ·π=1
ππ(0) β 1π0
βπ·π=0
ππ(0)}
= { 1π1
βπ·π=1
{ππ(1) β ππ(0)} β 1π0
βπ·π=0
{ππ(1) β ππ(0)}} ,
where π1 is the number of treated individual and π0 is the number ofnon-treated individuals.
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Example
Persuasive Power of the News MediaDid the change in support for the Labour Party by the Sun newspaperincrease the number of people voting Labour? Ladd and Lenz (2009)use the British Election Panel Survey, which includes information onwhether individuals voted for Labour in 1992, whether they votedLabour in 1997, and which newspapers they read.
β’ Outcome (π , voted_lab): 1 if individual π voted Labour at time π‘
β’ Treatment (π·, reads_sun): 1 if individual π read the Sun (in 1992)
β’ Time (π , year): Election year (1992 or 1997)
β’ π = 1593, π1 = 211, π0 = 1382
β’ Note that this is panel data (repeated observations on the sameindividuals over time)
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Example
Typically useful to store data in βlongβ format (here, 2 rows per unit):str(sun)
## 'data.frame': 3186 obs. of 4 variables:## $ reads_sun: int 0 0 0 0 0 0 0 0 0 1 ...## $ voted_lab: int 1 1 0 1 1 1 1 1 1 1 ...## $ year : num 1992 1992 1992 1992 1992 ...## $ id : int 1 2 3 4 5 6 7 8 9 10 ...
table(sun$reads_sun, sun$year)
#### 1992 1997## 0 1382 1382## 1 211 211
Question: Which observations are βtreatedβ?
Answer: Sun readers in 1997.15 / 52
Example
# Untreated, pre-treatmenty_d0_t0 <- mean(sun$voted_lab[sun$reads_sun == 0 & sun$year == 1992])y_d0_t0
## [1] 0.3227207# Treated, pre-treatmenty_d1_t0 <- mean(sun$voted_lab[sun$reads_sun == 1 & sun$year == 1992])y_d1_t0
## [1] 0.3886256# Untreated, post-treatmenty_d0_t1 <- mean(sun$voted_lab[sun$reads_sun == 0 & sun$year == 1997])y_d0_t1
## [1] 0.4305355# Treated, post-treatmenty_d1_t1 <- mean(sun$voted_lab[sun$reads_sun == 1 & sun$year == 1997])y_d1_t1
## [1] 0.582938416 / 52
Example
0.30
0.35
0.40
0.45
0.50
0.55
0.60
% v
otin
g La
bour
1992 1997
Reads the SunDoesn't read the SunAssumed counterfactual trend
ATT
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Example
# Parallel trend calculation(y_d1_t1 - y_d1_t0) - (y_d0_t1 - y_d0_t0)
## [1] 0.08649803
# Stable selection bias calculation(y_d1_t1 - y_d0_t1) - (y_d1_t0 - y_d0_t0)
## [1] 0.08649803
Implication: The change in endorsement caused Labour support toincrease by 8.6 percentage points amongst readers of The Sun.
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Difference-in-Differences: Regression Estimator
Estimator (Regression 1)Alternatively, the same estimator can be obtained using regression techniques.
ππ = πΌ + π½1 β π·π + π½2 β ππ + πΏ β (π·π β ππ) + π,where πΈ[π|π·π, ππ] = 0. Then, it is easy to show that
πΈ[ππ|π·π, ππ] ππ = 0 ππ = 1 After - Beforeπ·π = 0 πΌ πΌ + π½2 π½2π·π = 1 πΌ + π½1 πΌ + π½1 + π½2 + πΏ π½2 + πΏTreated - Control π½1 π½1 + πΏ πΏ
Thus, the difference-in-differences estimate is given by:
πATT = (π½2 + πΏ) β π½2 = πΏ
Equivalently:πATT = (π½1 + πΏ) β π½1 = πΏ
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Example: Regression
dd_mod <- lm(voted_lab ~ reads_sun * as.factor(year),data = sun)
voted_lab
reads_sun 0.066β
(0.036)as.factor(year)1997 0.108βββ
(0.018)reads_sun:as.factor(year)1997 0.086β
(0.050)Constant 0.323βββ
(0.013)
Observations 3,186R2 0.022
β’ πΌ = Labour support amongstnon-Sun readers, 1992
β’ π½1 = difference between Sunand non-Sun readers, 1992
β’ π½1 + πΏ = difference betweenSun and non-Sun readers, 1997
β’ πΏ = ATT
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Example: Regression
dd_mod <- lm(voted_lab ~ reads_sun * as.factor(year),data = sun)
voted_lab
reads_sun 0.066β
(0.036)as.factor(year)1997 0.108βββ
(0.018)reads_sun:as.factor(year)1997 0.086β
(0.050)Constant 0.323βββ
(0.013)
Observations 3,186R2 0.022
β’ πΌ = Labour support amongstnon-Sun readers, 1992
β’ π½2 = 1992 to 1997 difference,amongst non-Sun readers
β’ π½2 + πΏ = 1992 to 1997difference, amongst Sunreaders
β’ πΏ = ATT
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Cross-sectional regression estimator
A nice feature diff-in-diff is it does not require panel data, i.e. repeatedobservations of the same units. Can also use repeated cross-sections:
β’ ππππ‘ where unit π is only measured at one π‘
β’ Units fall into treatment based on groups π
β’ Particularly useful as many βtreatmentsβ vary at some aggregatelevel:
β’ e.g. Law changes at the state/region level
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Cross-sectional regression estimator
Two options:
β’ Individual-level data:
ππππ‘ = πΌ + π½1π·π + π½2ππ + π½3(π·π β ππ) + ππππ‘
β’ Aggregated data:
πππ‘ = πΌ + π½1π·π + π½2ππ + π½3(π·π β ππ) + πππ‘
Both approaches will give the same result, as the treatment only varies atthe group level (so long as the aggregated version is weighted by cellsize).
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Regression estimator advantages
1. Easy to calculate standard errors (though be careful aboutclustering)
2. We can control for other variables
β’ Individual-level data, group-level treatment: controlling forindividual covariates may increase precision
β’ Time-varying covariates at the group-level may strengthen theparallel trends assumption, but beware of post-treatment bias
3. Simple to extend to multiple groups/periods (more on this later)
4. Can use multi-valued (not just binary) treatments
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Difference-in-Differences: First Differences Estimator
Estimator (Regression 2)With panel data we can use regression with first differences:
Ξππ = πΌ + πΏ β π·π + πβ²ππ½ + π’,
where Ξππ = ππ(1) β ππ(0), and π’ = Ξπ.
β’ With two periods gives identical result as other regressions
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Example: First-difference regression
1 row per unit:head(sun_diff)
## reads_sun voted_lab_92 voted_lab_97## 1 0 1 1## 2 0 1 0## 3 0 0 0## 4 0 1 1## 5 0 1 1## 6 0 1 1
sun_diff$diff <- sun_diff$voted_lab_97 - sun_diff$voted_lab_92head(sun_diff)
## reads_sun voted_lab_92 voted_lab_97 diff## 1 0 1 1 0## 2 0 1 0 -1## 3 0 0 0 0## 4 0 1 1 0## 5 0 1 1 0## 6 0 1 1 0
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Example: First-difference regression
first_diff_mod <- lm(diff ~ reads_sun, data = sun_diff)
Table 1: First difference model
diff
reads_sun 0.086βββ
(0.030)Constant 0.108βββ
(0.011)
Observations 1,593R2 0.005
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Non-parallel trends
Critical identification assumption: treatment units have similar trends tocontrol units in the absence of treatment.
Question: Why is this assumption untestable?
Answer: because of the FPOCI β we cannot observe potential outcomeunder the control condition for treated units in the post-treatmentperiod.
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Potential violations of parallel trends
β’ βAshenfelterβs Dipβ
β’ Participants in worker training programs may experience decreasedearnings before they enter the program (why are they participating?)
β’ If wages revert to the mean, comparing wages of participants andnon-participants leads to an upwardly biased estimate
β’ Targeting
β’ Policymakers may target units who are most improving
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Assessing (non-)parallel trends
What can we do?
β’ One treatment/control group
β’ Plot results and look at trends in periods before the treatmentβ’ Is the parallel trends assumption plausible?
β’ Multiple treatment/control comparisons
β’ Estimate treatment effects at different time points (i.e. placebo tests)β All estimated treatment effects before the treatment should be 0.
β’ Include unit-specific time trends β βrelaxβ parallel trendsassumption
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βGoodβ parallel trends example%
vot
ing
Labo
ur
1983 1987 1992 1997
Reads the SunDoesn't read the Sun
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βBadβ parallel trends example%
vot
ing
Labo
ur
1983 1987 1992 1997
Reads the SunDoesn't read the Sun
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Parallel and non-parallel trends
Parallel trends Treatment effect
Preβtreatment period Postβtreatment period
Nonβparallel trends No treatment effect
Preβtreatment period Postβtreatment period
Nonβparallel trends Treatment effect
Preβtreatment period Postβtreatment period
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Difference-in-Differences: Fixed-effect estimator
Estimator (Fixed-effect regression)We can generalise to multiple groups/time periods using unit and periodfixed-effects (βtwo-wayβ fixed-effect model):
πππ‘ = πΎπ + πΌπ‘ + πΏπ·ππ‘ + πππ‘
β’ πΎπ is a fixed-effect for groups (dummy for each group)
β’ πΌπ‘ if a fixed-effect for time periods (dummy for each time period)
β’ πΏ is the diff-in-diff estimate based on π·ππ‘, which is 1 for treatedunit-period observations, and 0 otherwise
Very flexible:
β’ can replace π·ππ‘ with almost any type of treatment (not only binary)β’ can extend easily to multiple periods (i.e. more than 2)β’ can have different units treated at different times
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Example: Fixed-effect regression
sun$treat <- sun$reads_sun == 1 & sun$year == 1997
fe_model <- lm(voted_lab ~ treat + as.factor(id) + as.factor(year),data = sun)
Table 2: Fixed-effect model
voted_lab
treat 0.086βββ
(0.030)
Observations 3,186R2 0.826
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Why does fixed-effect regression estimate the diff-in-diff?
β’ Unit/group FEs mean that we are only using within group variationin Y to calculate the effect of D
β’ Removes any omitted variable bias that is constant over time
β’ Time FEs means that we remove the effect of any changes to theoutcome variable that affect all units at the same time
β’ πΏ β πATTNote that unit dummies lead to smaller standard errors on our treatmenteffect. Why not always use unit dummies?
β’ Fine in panel data, as we have same units at several points in timeβ’ Not possible with repeated cross-section when we do not have thesame units in each time period
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Standard errors for the regression difference in differences
β’ Many papers using a DD strategy use data from many periods
β’ Treatments typically vary at the group level, while outcomesnormally measured at the individual level
β’ E.g. Minimum wage increases (state-level) and employment data(firm-level) in Cark and Krueger
β’ Will not bias treatment effect estimates, but will cause problems forvariance estimation when errors are serially correlated
β’ Implication: traditional standard errors will tend to be too small.
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Standard errors for the regression difference in differences
Solution (Bertrand et al (2004))Use cluster-robust standard errors where clusters are defined at thelevel of the treatment. If the number of groups is:
β’ β¦large (βͺ 30), use vcovCL in sandwichβ’ β¦small (βͺ 30), use block-bootstrap
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Example: Multiperiod diff-in-diff
Female role-models in politicsWhen women are promoted to high-office, do they motivate otherwomen to participate more in policymaking? Blumenau (2019) examinesthe participation of female MPs in parliamentary debates between 1997and 2018 as a proxy for policymaking involvement. If high-profilewomen act as role-models, when they are promoted they shouldencourage other women to participate more.
β’ Outcome (π , prop_words): Proportion of words spoken by women in debate π atin ministry π at time π‘
β’ Treatment (π·, female_minister): 1 if cabinet minister of ministry π at time π‘ isfemale
β’ Unit of analysis: Debates (clustered within about 30 ministries)
β’ π β 15,000 debates
β’ Time measured at the month level (i.e. February 2012, etc)
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Female role-models β model specification
πππππππππ ππππππ(ππ‘) = πΎπ +πΌπ‘ +πΏ βπΉππππππππππ π‘ππππ‘ +ππ(ππ‘)
β’ πΎπ β ministry fixed effect
β’ Controls for unobserved ministry characteristics that are timeinvariant
β’ πΌπ‘ β time (month-year) fixed effect
β’ Controls for changes in participation over time that are common toall ministries
β’ πΏ β average effect of switching from a male to a female minister based onwithin-ministry variation, amongst ministries that see a change in ministergender (i.e. πATT)
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R example
fe_mod <- lm(prop.women.words ~ minister.gender +as.factor(ministry) +
as.factor(yearmon),data= speeches)
screenreg(list(fe_mod),omit.coef = c(βministry|yearmonβ),digits = 3)
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R example
#### ===============================## Model 1## -------------------------------## (Intercept) 0.039## (0.044)## minister.genderF 0.041 ***## (0.005)## -------------------------------## R^2 0.106## Adj. R^2 0.091## Num. obs. 14320## RMSE 0.218## ===============================## *** p < 0.001, ** p < 0.01, * p < 0.05
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R example (clustered errors)
fe_mod_clustered <- coeftest(fe_mod,vcov = vcovCL, cluster = ~ ministry)
screenreg(list(fe_mod, fe_mod_clustered),omit.coef = c(βministry|yearmonβ),digits = 3)
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R example (clustered errors)
#### ===========================================## Model 1 Model 2## -------------------------------------------## (Intercept) 0.039 0.039## (0.044) (0.048)## minister.genderF 0.041 *** 0.041 ***## (0.005) (0.011)## -------------------------------------------## R^2 0.106## Adj. R^2 0.091## Num. obs. 14320## RMSE 0.218## ===========================================## *** p < 0.001, ** p < 0.01, * p < 0.05
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Female role-models β full results
Interpretation: The appointment of a female minister increasesparticipation of other women byβ 20% relative to male minister baseline
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Female role-models β parallel trends
Hard to provide a visual inspection of the parallel trends assumption astreatment switches on and off over time in different ministries.
Nevertheless, we are still assuming that treated/control ministries wouldhave evolved identically over time in absense of treatment.
One way forward, test for π βlagsβ and π βleadsβ of the treatment:
π¦π(ππ‘) = πΌ0 +π
βπ=βπ
πΏπ βπΉππππππππππ π‘πππ(π‘+π) +πΎπ0 +πΌπ‘ +ππ(ππ‘)
β’ πΏβ1, πΏβ2, ..., πΏβπ are the βleadβ effects which should all be 0
β’ πΏ1, πΏ2, ..., πΏπ are the βlaggedβ effects which may take differentvalues
Intuition: What is the DD estimate in the Sun example using only datafrom 1992 and 1996? 47 / 52
Data requirements for Diff-in-diff
Data structure:
β’ Panel data or repeated cross-sectionβ’ Single or multiple treatmentsβ’ Continuous or binary treatmentsβ’ Works both at individual/aggregate level
Does this require more data?
β’ Adding a time dimension can increase the amount of data you needβ’ No need to control for extensive covariates (so long as they are fixedwithin units over time) which might mean decreased data collection
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Examples of diff-in-diff designs
1. Card & Krueger, 1994β’ RQ: Do increases in the minimum wage reduce employment?β’ Outcome: Employment growth in fast-food restaurantsβ’ Treatment: Increased minimum wage in New Jersey; no change inPennsylvania
β’ Time: Before/after minimum wage changed
2. Dinas et al., 2019β’ RQ: What is the effect of refugee arrivals on support for the far right?β’ Outcome: Municipal support for far right partyβ’ Treatment: Refugee arrivals in Greek islandsβ’ Time: Elections before/after refugee crisis
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Examples of diff-in-diff designs
3. Bechtel & Heinmueller, 2011β’ RQ: What is the effect of good policy on government support?β’ Outcome: Support for the German SPD in parliamentaryconstituencies
β’ Treatment: Flooded German regions close to the River Elbeβ’ Time: Elections before/after 2002, when the Elbe flooded
4. Hainmueller & Hangartner, 2019β’ RQ: What is the effect of direct democracy on immigrantassimilation?
β’ Outcome: Naturalization rate of immigrants in Swiss municipalitiesβ’ Treatment: Whether municipality decides on naturalisation requestsvia expert or citizen councils
β’ Time: Decisions before/after legal changes to decision making inmunicipalities
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Conclusion
The DD design allows for a comparison over time in the treatment group,controlling for concurrent time trends using a control group.
DD requires data on multiple units in multiple periods, but can beapplied to panel data or repeated cross-sectional data.
DD is very widely used, as it is a powerful conditioning strategy thatdoesnβt require endless lists of covariates to strengthen the identifyingassumption.
The identification assumption β that treatment and control units wouldfollow parallel trends in the absense of treatment β should beinvestigated with every application!
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