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Difference in Difference Models. Bill Evans Spring 2008. Difference in difference models. Maybe the most popular identification strategy in applied work today Attempts to mimic random assignment with treatment and “comparison” sample Application of two-way fixed effects model .

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difference in difference models

Difference in Difference Models

Bill Evans

Spring 2008

difference in difference models1
Difference in difference models
  • Maybe the most popular identification strategy in applied work today
  • Attempts to mimic random assignment with treatment and “comparison” sample
  • Application of two-way fixed effects model
problem set up
Problem set up
  • Cross-sectional and time series data
  • One group is ‘treated’ with intervention
  • Have pre-post data for group receiving intervention
  • Can examine time-series changes but, unsure how much of the change is due to secular changes
slide4

Y

True effect = Yt2-Yt1

Estimated effect = Yb-Ya

Yt1

Ya

Yb

Yt2

ti

t1

t2

time

slide5
Intervention occurs at time period t1
  • True effect of law
    • Ya – Yb
  • Only have data at t1 and t2
    • If using time series, estimate Yt1 – Yt2
  • Solution?
difference in difference models2
Difference in difference models
  • Basic two-way fixed effects model
    • Cross section and time fixed effects
  • Use time series of untreated group to establish what would have occurred in the absence of the intervention
  • Key concept: can control for the fact that the intervention is more likely in some types of states
three different presentations
Three different presentations
  • Tabular
  • Graphical
  • Regression equation
slide9

Y

Treatment effect=

(Yt2-Yt1) – (Yc2-Yc1)

Yc1

Yt1

Yc2

Yt2

control

treatment

t1

t2

time

key assumption
Key Assumption
  • Control group identifies the time path of outcomes that would have happened in the absence of the treatment
  • In this example, Y falls by Yc2-Yc1 even without the intervention
  • Note that underlying ‘levels’ of outcomes are not important (return to this in the regression equation)
slide11

Y

Yc1

Treatment effect=

(Yt2-Yt1) – (Yc2-Yc1)

Yc2

Yt1

control

Treatment

Effect

Yt2

treatment

t1

t2

time

slide12
In contrast, what is key is that the time trends in the absence of the intervention are the same in both groups
  • If the intervention occurs in an area with a different trend, will under/over state the treatment effect
  • In this example, suppose intervention occurs in area with faster falling Y
slide13

Y

Estimated treatment

Yc1

Yt1

Yc2

control

True treatment effect

Yt2

True

Treatment

Effect

treatment

t1

t2

time

basic econometric model
Basic Econometric Model
  • Data varies by
    • state (i)
    • time (t)
    • Outcome is Yit
  • Only two periods
  • Intervention will occur in a group of observations (e.g. states, firms, etc.)
slide15
Three key variables
    • Tit =1 if obs i belongs in the state that will eventually be treated
    • Ait =1 in the periods when treatment occurs
    • TitAit -- interaction term, treatment states after the intervention
  • Yit = β0 + β1Tit + β2Ait + β3TitAit + εit
more general model
More general model
  • Data varies by
    • state (i)
    • time (t)
    • Outcome is Yit
  • Many periods
  • Intervention will occur in a group of states but at a variety of times
slide18
ui is a state effect
  • vt is a complete set of year (time) effects
  • Analysis of covariance model
  • Yit = β0 + β3 TitAit + ui + λt + εit
what is nice about the model
What is nice about the model
  • Suppose interventions are not random but systematic
    • Occur in states with higher or lower average Y
    • Occur in time periods with different Y’s
  • This is captured by the inclusion of the state/time effects – allows covariance between
    • ui and TitAit
    • λt and TitAit
slide20
Group effects
    • Capture differences across groups that are constant over time
  • Year effects
    • Capture differences over time that are common to all groups
meyer et al
Meyer et al.
  • Workers’ compensation
    • State run insurance program
    • Compensate workers for medical expenses and lost work due to on the job accident
  • Premiums
    • Paid by firms
    • Function of previous claims and wages paid
  • Benefits -- % of income w/ cap
slide22
Typical benefits schedule
    • Min( pY,C)
    • P=percent replacement
    • Y = earnings
    • C = cap
    • e.g., 65% of earnings up to $400/month
slide23
Concern:
    • Moral hazard. Benefits will discourage return to work
  • Empirical question: duration/benefits gradient
  • Previous estimates
    • Regress duration (y) on replaced wages (x)
  • Problem:
    • given progressive nature of benefits, replaced wages reveal a lot about the workers
    • Replacement rates higher in higher wage states
slide24
Yi = Xiβ + αRi + εi
  • Y (duration)
  • R (replacement rate)
  • Expect α > 0
  • Expect Cov(Ri, εi)
    • Higher wage workers have lower R and higher duration (understate)
    • Higher wage states have longer duration and longer R (overstate)
solution
Solution
  • Quasi experiment in KY and MI
  • Increased the earnings cap
    • Increased benefit for high-wage workers
      • (Treatment)
    • Did nothing to those already below original cap (comparison)
  • Compare change in duration of spell before and after change for these two groups
model
Model
  • Yit = duration of spell on WC
  • Ait = period after benefits hike
  • Hit = high earnings group (Income>E3)
  • Yit = β0 + β1Hit + β2Ait + β3AitHit + β4Xit’ + εit
  • Diff-in-diff estimate is β3
questions to ask
Questions to ask?
  • What parameter is identified by the quasi-experiment? Is this an economically meaningful parameter?
  • What assumptions must be true in order for the model to provide and unbiased estimate of β3?
  • Do the authors provide any evidence supporting these assumptions?
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