Make slide with no dummies, add time dummy, then state dummy, then interaction term

Difference in Difference Model with two states and two time periods. The estimating equation is: y st = α 0 + α 1 Treatment s +α 2 Post t + + δ Treatment s *Post t + ε st Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy α0α0 α 0+ α 1 α2α2 α 2+ δ

Difference in Difference Model with two states and two time periods. The estimating equation is: y st = α 0 + α 1 Treatment s +α 2 Post t + + δ Treatment s *Post t + ε st Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy α0α0 α 0+ α 1 α2α2 δ α2α2

Difference in Difference Model with two states and two time periods. The estimating equation is: y st = α 0 + α 1 Treatment s +α 2 Post t + + δ Treatment s *Post t + ε st Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy. α2α2 α2α2 δ

This estimating equation is exactly the same as: y ist = α 0 + α 1 dCO s +α 2 d2 t + + δ Policy st + ε ist dCO is dummy for CO; d2 is dummy for year 2; Policy is dummy =1 in CO after policy is passed Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy. α2α2 α2α2 δ

Traffic Fatality Rate time Colorado Nebraska. α2α2 δ

Traffic Fatality Rate time Colorado Nebraska. δ

When we have lots of states and years, an author typically writes y st = β 0 + δPolicy st + v s + z t +ε st And then the author might say that the equation is estimated including state and year fixed effects Start with state fixed effects, common time trend

If we’re using state-level data, then each state contributes one observation per year. The estimating equation with a common time trend is y ist = α 0 + α 1 dCO s +α 2 t + δ Policy st + ε ist Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy. α0α0 α 0 + α 1

Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy.

Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy. New Mexico New Mexico Implements Policy. With multiple states δ will be the mean vertical change y st = α 0 + α 2 t + δPolicy st + v s + ε st

If we’re using repeated cross-sectional data at the individual level, then each state contributes multiple observations per year. The estimating equation is: y ist = α 0 + δ Policy st + α 2 t + v s + ε ist Traffic Fatality Rate time Colorado Nebraska Colorado Implements Policy... New Mexico New Mexico Implements Policy.

Now look at time shocks. What if all states experience a common shock in a given year? What if the means varies by time as well as by state?

z3z3 Crime Rate time Barber Jefferson Barber legalizes by-the-drink sales Common time fixed effects-- δ will still be the average vertical change from before and after the policy Violent Crime ct = π 0 + δ Wet Law ct + X ‘ ct π 2 + v c + z t + ε ct. Jefferson legalizes by-the-drink sales Positive shock to crime Negative shock to crime z3z3 z8z8 z8z8

Crime Rate time Barber Jefferson Barber legalizes by-the-drink sales But note if you look at it, Barber is increasing faster than Jefferson: Violent Crime ct = π 0 + π 1 Wet Law ct + X ‘ ct π 2 + v c + z t + ε ct. Jefferson legalizes by-the-drink sales Positive shock to crime Negative shock to crime

Crime Rate time Barber Jefferson Barber legalizes by-the-drink sales Franklin Franklin legalizes by-the- drink sales Adding state specific time trends Violent Crime ct = π 0 + δ Wet Law ct + X ‘ ct π 2 + v c + z t + Θ c ∙ t + ε ct. Jefferson legalizes by-the-drink sales Positive shock to crime Negative shock to crime

17 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

18 Typical benefits schedule Min( pY,C) P=percent replacement Y = earnings C = cap e.g., 65% of earnings up to $400/month

19 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

20 Y i = X i β + αR i + ε i Y (duration) R (replacement rate) Expect α > 0 Expect Cov(R i, ε i ) Higher wage workers have lower R and higher duration (understate) Higher wage states have longer duration and longer R (overstate)

21 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

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24 Model Y it = duration of spell on WC A it = period after benefits hike H it = high earnings group (Income>E 3 ) Y it = β 0 + β 1 H it + β 2 A it + β 3 A it H it + β 4 X it ’ + ε it Diff-in-diff estimate is β 3

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26 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 β 3 ? Do the authors provide any evidence supporting these assumptions?

27 Almond et al. Neonatal mortality, dies in first 28 days Infant mortality, died in first year Babies born w/ low birth weight(< 2500 grams) are more prone to Die early in life Have health problems later in life Educational difficulties generated from cross-sectional regressions 6% of babies in US are low weight Highest rate in the developed world

28 Let Y it be outcome for baby t from mother I e.g., mortality Y it = α + bw it β + X i γ + α i + ε it bw is birth weight (grams) X i observed characteristics of moms α i unobserved characteristics of moms

29 Cross sectional model is of the form Y it = α + bw it β + X i γ + u it where u it =α i + ε it Many observed factors that might explain health (Y) of an infant Prenatal care, substance abuse, smoking, weight gain (of lack of it) Some unobserved as well Quality of diet, exercise, generic predisposition α i not included in model Cov(bw it,u it ) < 0

30 Solution: Twins Possess same mother, same environmental characterisitics Y i1 = α + bw i1 β + X i γ + α i + ε i1 Y i2 = α + bw i2 β + X i γ + α i + ε i2 ΔY = Y i2 -Y i1 = (bw i2 -bw i1 ) β + (ε i2 - ε i1 )

31 Questions to consider? What are the conditions under which this will generate unbiased estimate of β? What impact (treatment effect) does the model identify?

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33 Large change In R2 Big Drop in Coefficient on Birth weight

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35 More general model Many within group estimators that do not have the nice discrete treatments outlined above are also called difference in difference models Cook and Tauchen. Examine impact of alcohol taxes on heavy drinking States tax alcohol Examine impact on consumption and results of heavy consumption death due to liver cirrhosis

36 Y it = β 0 + β 1 INC it + β 2 INC it-1 + β 1 TAX it + β 2 TAX it-1 + u i + v t + ε it i is state, t is year Y it is per capita alcohol consumption INC is per capita income TAX is tax paid per gallon of alcohol

37 Model requires that untreated groups provide estimate of baseline trend would have been in the absence of intervention Key – find adequate comparisons If trends are not aligned, cov(T it A it,ε it ) ≠0 Omitted variables bias How do you know you have adequate comparison sample?

38 Concern: suppose that the intervention is more likely in a state with a different trend Do the pre-treatment samples look similar? Tricky. D-in-D model does not require means match – only trends. If means match, no guarantee trends will However, if means differ, aren’t you suspicious that trends will as well?

39 Add state specific time trends Y it = β 0 + β 1 INC it + β 2 INC it-1 + β 1 TAX it + β 2 TAX it-1 + β i T + u i + v t + ε it i is state, t is year Y it is per capita alcohol consumption INC is per capita income TAX is tax paid per gallon of alcohol β i gives state specific time trend

First-Stage Estimates: Wet Laws and On-Premises Alcohol Licenses, On-Premises Licenses Wet Law.177*** (.025).143*** (.022) Mean of the dependent variable.617 N3,352 R2R F-Statistic Year FEsYes County FEsYes CovariatesYes County linear trendsNoYes Notes: Regressions are weighted by county population and standard errors are corrected for clustering at the county level. The dependent variable is equal to the number of active on-premises liquor licenses per 1,000 population in county c and year t. The years 1995, 1996, and 1999 are excluded because of missing crime data.

On-Premises Alcohol Licenses and Violent Crime, OLS Violent Crime OLS Violent Crime 2SLS Violent Crime 2SLS Violent Crime On-Premises Licenses.853* (.500) 1.24* (.667) 4.31** (1.77) 5.00*** (1.87) N3,352 R2R Year FEsYes County FEsYes CovariatesYes County linear trends NoYesNoYes Notes: Regressions are weighted by county population and standard errors are corrected for clustering at the county level. The dependent variable is equal to the number of violent crimes per 1,000 population in county c and year t. The years 1995, 1996, and 1999 are excluded because of missing crime data.

42 Falsification tests Add “leads” to the model for the treatment Intervention should not change outcomes before it appears If it does, then suspicious that covariance between trends and intervention Y it = β 0 + β 3 A it + α 1 A it-1 + α 2 A it-2 + α 3 A it-3 + u i + v t + ε it Three “leads” Test null: H o : α 1 =α 2 =α 3 =0

43 Pick control groups that have similar pre-treatment trends Most studies pick all untreated data as controls Example: Some states raise cigarette taxes. Use states that do not change taxes as controls Example: Some states adopt welfare reform prior to TANF. Use all non- reform states as controls Can also use econometric procedure to pick controls Appealing if interventions are discrete and few in number Easy to identify pre-post