Difference-in-Differences

Introduction

Previously: how to estimate “causal effects” when treatments are randomly offered to individuals.
Can we ever do this in non-experimental settings?
With additional assumptions, we can try.
We’ll now explore the method of difference-in-differences, which infers causality by making a particular modelling assumption for conditional means.

Consider the Medicaid Expansion introduced by the ACA
- Increased eligiblity of individuals for public health insurance
Some states adopted, some states did not.
Let $Y_{st}$ be all-cause mortality in state $s$ at time $t$.
Suppose we have data on two states $s\in\{1,2\}$ in $T=10$ periods.
Suppose that only state 2 adopts the expansion in period 5…

When covid hit, states adopted different policies with regards to school closures.
Let $Y_{pt}$ be average standardized math score for students in districts with policy $p$ in year $t$.
Suppose we compare districts that remained fully in person to those that went fully remote…

Let $j$ index treatment units $j\in\{1,2\}$
Let $t$ index periods $t\in\{1,2\}$
Let $Y(p)$ be the potential outcome of interest where $p\in\{0,1\}$ indicates treatment
Let $P(j,t)\in\{0,1\}$ indicate whether the policy is in place. Here: \[ P(1,1)=P(1,2) = 0\ \text{and}\ P(2,1)=0,\ P(2,2) = 1 \]
So policy introduced in unit 2 in period 2.

Recall that causal inference is all about making assumptions to construct a counterfactual
Here, that is given by the parallel trends assumption: that average differences in $Y(0)$ between the untreated ($j=1$) and treated ($j=2$) units are stable over time
Define $Y_{jt}(p) = \mathbb{E}[Y(p)|j,t]$
Math: \[ Y_{2t}(0) - Y_{1t}(0) = \text{constant for all}\ t \]

We want: \[ ATT = Y_{22}(1) - Y_{22}(0) \]

Use parallel trends to calculate the counterfactual:

\[ Y_{22}(0) = Y_{12}(0) + Y_{21}(0) - Y_{11}(0) \]

So: \[ \begin{multline} ATT = \mathbb{E}[Y|j=2,t=2] - \mathbb{E}[Y|j=1,t=2] \\ - (\mathbb{E}[Y|j=2,t=1]-\mathbb{E}[Y|j=1,t=1]) \end{multline} \]

Hence the name: difference-in-differences.

Do higher minimum wages decrease employment?
Card and Krueger (1994) consider impact of New Jersey’s 1992 minimum wage increase from $4.25 to $5.05 per hour.
Compare employment in 410 fast-food restaurants in New Jersey and eastern Pennsylvania before and after rise.
Use survey data on employment and wages from two waves:
- March 92 - one month before min wage increase
- Dec 92 - eight months after

Estimation is straightforward in the $2\times2$ case (exercise)
Let’s work out inference in two cases:
1. Each unit and time period has an independent sample
2. Individuals in each unit are observed in both periods

Card & Krueger find no significant effect of minimum wage on employment.
Are you satisfied with this evidence? Any potential issues?
This paper sparked a decades-long empirical debate. Still no resolution.
Depends who you read, what specification you use, which outcomes you look at, who you look at.
Most recent: Seattle minimum wage study - two groups of researchers finding conflicting results.
Krueger was Assistant Secretary of Treasury and chair of Obama’s CEA.