Panel Data and Difference-in-Differences

Question 1

What assumption does DnD require?

Answer 1

Common trends assumption: requires assumption that T would have the same trend as C were it not treated.

Question 2

What are three magnitudes that will be key for a formal representation of diff-in-diff identification?

Answer 2

1. the group-level fixed effects of the T and C groups
2. the common trend (in the absence of treatment)
3. the causal effect (aka the difference-in-differences)

Question 3

Two formal assumptions of DnD

Answer 3

1. E(Y1 −Y0|s,t)=δ ⇒ conditional treatment effect is a constant
2. E(Y0 |s,t)=γs +λt ist ⇒ potential outcomes in the untreated state have an additive structure. Specifically, they must consist of

• a unit-specific fixed effect γs
• a period effect (or trend) λt that is common across all units

This assumption is violated if the potential outcomes are such that in the absence
of treatment their true modelling requires state-specific time effects, as e.g. when
the expected potential outcomes are given by E (Y 0 |s, t ) = γs + λt + κst .

Question 4

DnD as a Regression

Answer 4

Diff-in-diff estimates can also be implemented as a regression. Recall equation (1) above:

Yist = γs + λt + δDst + εist . It’s straight-forward to formulate a regression that estimates the key parameters as coefficients

Yist = α+β1INJ +β2TNov +δ(INJ ×TNov)+εist, where INJ and TNov are state and time dummies, respectively (IPA and TFeb are the reference categories for which no dummy is included). INJ × TNov is the same as Dst above. The coefficients contain the following magnitudes (as specified on previous slide), and can be interpreted accordingly:

• α=γPA+λFeb - β1=γNJ−γPA
• β2 =λNov −λFeb - δ=δ

Question 5

Advantage of DnD Regression?

Answer 5

• One key advantage of regression diff-in-diff is the possibility to include controls: Y =α+βI+βT+X′β+δ(I×T)+ε. ist 1 NJ 2 Nov ist 3 NJ Nov ist
• Another key advantage of regression diff-in-diff is the possibility to estimate effect of a non-binary treatment.

Question 6

Dnd Controls in Minimum Wage example?

Answer 6

• chain of restaurant (McD, KFC, Wendy’s, etc)
• ownership (company-owned vs franchise-taker)
• region within NJ and PA (to capture regional demand shocks)

Question 7

Robustness Check for DnD

Answer 7

If the number of time periods is at least > 2 (in practice it must be much larger), a good robustness check of diff-in-diff results is to estimate a model with unit-specific trends: Y = α+I′β +T′β +Istτ+X′ β +D δ+ε , ist s 1 t 2 ist 3 st ist where

• its interaction with the unit dummies Is allow each unit to have an individual linear trend
• τ are the trend coefficients to be estimated.
  ⇒ If δ remains largely unaffected by the inclusion of Is′ Tt τ , then this is strong support for trends being common.

Question 8

Interpret the general DnD Regression: Y = α+I′β +T′β +X′ β +D δ+ε ,

Answer 8

• Is and Tt are vectors of state and time dummies;
• Xist is a vector with variables that can differ across subjects, states, and time;
• Dst can either be the a binary indicator of treatment of unit s at time t, or it can be a continuous variable capturing treatment intensity of unit s at time t.

Question 9

Main requirement of DnD

Answer 9

• availability of panel data with at least two pre-treatment observations
• treated and untreated subjects
• common pre-treatment trends of the untreated and (later) treated
• constant treatment effect
• no shocks coinciding with treatment

Question 10

When is DnD most valuable to evaluate Policy?

Answer 10

• is implemented at an organizational level whose units contain many subjects (e.g. students within schools or classes, firms in regions, households in cities, etc), and
• is applied to some, though not all of the units (some but not all schools, regions, cities)
• staggered or (randomly) phased-in treatment can also be exploited (→ natural
  experiment)

Question 11

Policy areas for DnD

Answer 11

• education policy
• labour market policy
• but e.g. also reform effectiveness in organizations