In-Class Activity: DiD by Hand

Chapter 10 — Difference-in-Differences
Time: ~20 minutes

Setup

In January 2020, New Jersey increased its state minimum wage from $10.00 to $12.00 per hour. Neighboring Pennsylvania kept its minimum wage at $7.25. A researcher collected data on average employment at fast-food restaurants (measured as full-time equivalent workers per restaurant) in both states, before and after the policy change.

	Before (Nov 2019)	After (Mar 2020)
New Jersey (treatment)	20.4	21.0
Pennsylvania (control)	23.3	21.2

Questions

1. Calculate the simple before-and-after change in employment for New Jersey (the treatment group).

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2. A state legislator sees your answer to Question 1 and says, “See! Raising the minimum wage increased employment!” Explain why the simple before-after comparison is not a valid estimate of the causal effect of the minimum wage increase.

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3. Calculate the difference-in-differences (DiD) estimate of the effect of the minimum wage increase on fast-food employment.

Show your work. You can calculate this either as (a) the difference in before-after changes between the two groups, or (b) the difference in cross-state gaps between the two time periods.

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4. Interpret your DiD estimate from Question 3 in a complete sentence. Be specific about units and direction.

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5. State the parallel trends assumption in the context of this study. Be precise — what, specifically, must be true for the DiD estimate to be valid?

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6. Evaluate whether parallel trends is plausible here. Identify at least one reason it might hold and one reason it might fail.

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6b. A researcher suggests adding restaurant-level characteristics — franchise type, seating capacity, and years in operation — as control variables to the DiD regression. Explain two reasons why this might be useful: one about precision and one about the validity of the parallel trends assumption.

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7. (Bonus) Write the regression equation that would produce the DiD estimate from Question 3. Define your variables clearly and identify which coefficient gives the DiD estimate.

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INSTRUCTOR NOTES — DO NOT DISTRIBUTE

Answers

1. Simple before-after change for NJ:

$$\Delta_{NJ} = 21.0 - 20.4 = +0.6 \text{ FTE workers}$$

2. The before-after comparison confounds the effect of the minimum wage increase with any other changes happening over the same time period. PA employment fell by 2.1 FTE — if NJ would have followed the same trend absent the policy, the simple comparison overstates the true effect. March 2020 also coincides with the onset of COVID-19, making this a particularly sharp example of why common time trends must be accounted for.

3. DiD estimate (either approach):

Approach (a): Difference in before-after changes

$$\text{DiD} = \Delta_{NJ} - \Delta_{PA} = (21.0 - 20.4) - (21.2 - 23.3) = 0.6 - (-2.1) = +2.7$$

Approach (b): Difference in cross-state gaps

$$\text{DiD} = (21.0 - 21.2) - (20.4 - 23.3) = (-0.2) - (-2.9) = +2.7$$

Both give DiD = +2.7 FTE workers per restaurant.

4. “The minimum wage increase in New Jersey is estimated to have increased average fast-food restaurant employment by 2.7 full-time equivalent workers per restaurant, relative to what would have occurred absent the policy change.”

5. The parallel trends assumption requires that, in the absence of the minimum wage increase, average fast-food employment in New Jersey would have changed by the same amount as in Pennsylvania between November 2019 and March 2020. Any time trends affecting fast-food employment would have been the same in both states had NJ not raised its minimum wage.

Reasons it might hold:

NJ and PA are neighboring states with similar economies, labor markets, and demographics
Fast-food chains operate similarly across state lines (same companies, similar consumer bases)
Pre-treatment trends in fast-food employment might be similar

Reasons it might fail:

March 2020 is the start of the COVID-19 pandemic — differential pandemic impacts across states could violate parallel trends (this is the key one to discuss!)
Anticipation effects: NJ employers might have adjusted employment before formal implementation
Other state-level policies may have changed at the same time

6b.

Precision: Adding restaurant-level controls (franchise type, seating capacity, years in operation) reduces residual variance → tighter standard errors on the DiD estimate, even if parallel trends holds unconditionally.

Credibility: Parallel trends may only hold conditional on these covariates. If, for example, newer franchises were disproportionately opening in NJ during this period (anticipating higher wages), or if NJ and PA had systematically different franchise compositions, then observable restaurant characteristics could predict differential outcome trends. Controlling for them makes parallel trends more plausible. Omitting relevant covariates when they predict differential trends is OVB — the parallel trends assumption may only hold after conditioning.

7. Regression equation:

$$Employment_{st} = \beta_0 + \beta_1 \cdot NJ_s + \beta_2 \cdot After_t + \beta_3 \cdot (NJ_s \times After_t) + u_{st}$$

Where:

$NJ_s = 1$ if New Jersey, 0 if Pennsylvania
$After_t = 1$ if March 2020 (post-treatment), 0 if November 2019 (pre-treatment)
$\beta_3$ is the DiD estimate (= 2.7)

Interpretation of all coefficients:

$\beta_0 = 23.3$ (PA employment, before)
$\beta_1 = 20.4 - 23.3 = -2.9$ (NJ-PA gap, before)
$\beta_2 = 21.2 - 23.3 = -2.1$ (PA change over time)
$\beta_3 = 2.7$ (DiD: the additional change in NJ beyond the PA trend)

Teaching Notes

This activity is inspired by Card and Krueger (1994). Numbers here are illustrative and close to (but not identical to) the original study.
The COVID timing issue in Q6 is intentional — it creates a natural discussion point about threats to identification and how real-world events can compromise research designs.
Q6b connects to the “Adding Control Variables” section of the slides: stress the distinction between controls for precision vs. controls for identification (conditional parallel trends). This is different from RCTs where controls are only about precision.
For Q7, emphasize that the interaction term is the key — students often struggle to see how a 2×2 table maps to a regression with an interaction.
Common student error: confusing “parallel trends” with “equal levels.” Stress that the assumption is about changes (trends), not levels.