Replication Lab: Autor (2003) Wrongful Discharge Laws and Outsourcing

Overview

David Autor's 2003 paper "Outsourcing at Will: The Contribution of Unjust Dismissal Doctrine to the Growth of Temporary Employment" (Journal of Labor Economics, 21(1), 1–42; DOI: 10.1086/344122) examines how the adoption of wrongful discharge laws (specifically, the implied-contract exception to employment-at-will) affected the use of temporary help services across U.S. states. The paper uses the staggered adoption of these laws across states between the 1970s and 1990s as a natural experiment, estimating event study models to trace out the dynamic effects.

Key findings:

• Temporary help employment increases significantly after adoption of wrongful discharge laws
• The pre-adoption coefficients are close to zero, supporting the parallel trends assumption
• The effect builds gradually over several years post-adoption
• The magnitude implies that wrongful discharge laws account for a meaningful share of the rise in temporary employment

What you will learn:

• How to construct event-time (relative time) indicators for a staggered adoption design
• How to estimate a two-way fixed effects (TWFE) event study regression
• How to create and interpret the event study plot
• How to formally test for pre-trends
• How to implement the Sun and Abraham (2021) heterogeneity-robust estimator
• Why staggered adoption creates problems for TWFE and how to address them

Prerequisites: Difference-in-differences, fixed effects (see the DiD and FE labs).

Step 1: Generate the Simulated Panel Dataset

1library(fixest)
2library(modelsummary)
3
4# Simulate state-year panel with staggered adoption
5set.seed(42)
6n_states <- 50
7n_years <- 25
8years <- 1975:1999
9
10# Staggered adoption
11adopt_year <- rep(NA, n_states)
12treated_states <- sample(1:n_states, 35)
13for (s in treated_states) {
14adopt_year[s] <- sample(1980:1994, 1)
15}
16
17state_fe <- rnorm(n_states, 0, 0.5)
18year_fe <- seq(0, 1.5, length.out = n_years)
19
20df <- expand.grid(state = 1:n_states, year = years)
21df$adopt_year <- adopt_year[df$state]
22df$ever_treated <- as.integer(!is.na(df$adopt_year))
23df$event_time <- ifelse(df$ever_treated == 1, df$year - df$adopt_year, NA)
24df$treated_post <- as.integer(!is.na(df$adopt_year) & df$year >= df$adopt_year)
25
26# Dynamic treatment effect
27df$tau <- ifelse(df$treated_post == 1 & df$event_time >= 0,
28               0.05 * pmin(df$event_time, 8) + 0.01 * df$event_time, 0)
29
30df$y <- 2.0 + state_fe[df$state] + year_fe[df$year - 1974] +
31      df$tau + rnorm(nrow(df), 0, 0.15)
32
33cat("Panel:", n_states, "states x", n_years, "years =", nrow(df), "obs\n")
34cat("Treated:", sum(!is.na(adopt_year)), "  Never-treated:", sum(is.na(adopt_year)), "\n")

Expected output:

Summary statistics:

Step 2: Construct Event-Time Dummies

1# Create event-time dummies for TWFE
2# Bin endpoints, reference period = -1
3df$et <- ifelse(df$ever_treated == 1, df$event_time, NA)
4df$et_binned <- pmax(pmin(df$et, 10), -6)  # Bin at -6 and +10
5
6# For fixest, we can use the i() function directly
7# But let us also create manual dummies for transparency
8for (k in -6:10) {
9if (k == -1) next  # reference period
10varname <- paste0("et_", ifelse(k < 0, paste0("m", abs(k)), k))
11if (k == -6) {
12  df[[varname]] <- as.integer(df$ever_treated == 1 & df$et <= k)
13} else if (k == 10) {
14  df[[varname]] <- as.integer(df$ever_treated == 1 & df$et >= k)
15} else {
16  df[[varname]] <- as.integer(df$ever_treated == 1 & df$et == k)
17}
18}
19
20cat("Event-time dummies created. Reference period: t = -1\n")

Expected output:

Step 3: Estimate the TWFE Event Study

1# TWFE Event Study using fixest
2# fixest handles event studies elegantly with i() syntax
3es_model <- feols(y ~ i(et_binned, ever_treated, ref = -1) |
4                state + year, data = df,
5                cluster = ~state)
6
7summary(es_model)
8
9# The coefficients are automatically named by event time

Expected output:

The DGP treatment effect is tau = 0.05 * min(k, 8) + 0.01 * k. Post-treatment coefficients should rise steadily, leveling off around k = 8 where the 0.05-per-year component caps.

Step 4: Plot the Event Study Coefficients

1# Event study plot using fixest
2iplot(es_model,
3    main = "Event Study: Wrongful Discharge Laws and Temp Employment",
4    xlab = "Event Time (years relative to adoption)",
5    ylab = "Coefficient (relative to t = -1)")
6abline(v = -0.5, col = "red", lty = 2)
7
8# Alternative: manual plot
9coefs <- coef(es_model)
10ses <- se(es_model)
11# Extract event times from coefficient names
12# (fixest names them automatically)

Step 5: Formal Pre-Trends Test

1# Joint test of pre-trends using fixest::wald
2# Test that all pre-treatment coefficients are jointly zero
3pre_test <- wald(es_model, "et_binned::-[2-6]")
4print(pre_test)
5
6# Alternative: manual F-test
7# Get pre-treatment coefficient names
8pre_names <- grep("et_binned::-[2-6]", names(coef(es_model)), value = TRUE)
9if (length(pre_names) > 0) {
10linearHypothesis_result <- car::linearHypothesis(es_model, pre_names)
11print(linearHypothesis_result)
12}

Expected output:

All individual pre-treatment coefficients are small and statistically insignificant (|t| < 1.96).

Step 6: Sun-Abraham Heterogeneity-Robust Estimator

Recent research by Sun and Abraham (2021) shows that TWFE event study estimates can be biased under treatment effect heterogeneity across cohorts. The Sun-Abraham estimator provides a heterogeneity-robust alternative.

1# Sun-Abraham estimator using fixest
2# fixest implements this with sunab()
3df$cohort <- ifelse(is.na(df$adopt_year), 10000, df$adopt_year)  # Never-treated = Inf
4
5sa_model <- feols(y ~ sunab(cohort, year) | state + year,
6                data = df[df$cohort != 10000 | df$ever_treated == 0, ],
7                cluster = ~state)
8
9summary(sa_model)
10
11# Compare TWFE and Sun-Abraham plots
12par(mfrow = c(1, 2))
13iplot(es_model, main = "TWFE Event Study")
14iplot(sa_model, main = "Sun-Abraham Event Study")
15
16# Aggregate Sun-Abraham estimates
17summary(sa_model, agg = "att")

Expected output:

With the relatively homogeneous treatment effects in this DGP, the TWFE and Sun-Abraham estimates should be very similar. With real data exhibiting treatment effect heterogeneity across cohorts, larger differences would emerge.

Step 7: Compare with Published Results

Key comparisons with Autor (2003):

The central finding is that wrongful discharge laws cause an increase in temporary employment, with the effect building gradually over several years. The pre-treatment coefficients should be close to zero, supporting the parallel trends assumption.

Extension Exercises

1. Callaway-Sant'Anna estimator. Implement the Callaway and Sant'Anna (2021) estimator using the did package in R or csdid in Stata. Compare the group-time ATT estimates with the TWFE results.

2. Treatment effect heterogeneity. Modify the DGP so that early adopters have larger treatment effects than late adopters. Re-estimate the TWFE event study and the Sun-Abraham estimator. How much does the bias in TWFE increase?

3. Anticipation effects. Add anticipation effects to the DGP (effects begin 1-2 years before formal adoption, as states may announce laws before enacting them). How does this affect the pre-trends test and the estimated treatment effects?

4. Bacon decomposition. Implement the Goodman-Bacon (2021) decomposition to understand the weights TWFE places on different 2x2 DiD comparisons. Which comparisons receive negative weights?

5. Wild cluster bootstrap. With only 50 clusters, asymptotic cluster-robust inference may be unreliable. Implement the wild cluster bootstrap ((Cameron et al., 2008)) and compare confidence intervals with the conventional clustered SEs.

Summary

In this replication lab you learned:

• Event study designs trace out dynamic treatment effects and provide a visual test of parallel trends
• The reference period (t = -1) is conventional; all coefficients are relative to the period just before treatment
• Flat pre-treatment coefficients are necessary (but not sufficient) for the parallel trends assumption
• Standard TWFE event studies can be biased under staggered adoption with heterogeneous treatment effects
• The Sun and Abraham (2021) estimator avoids "forbidden comparisons" by estimating cohort-specific effects and aggregating them with proper weights
• Our simulated results reproduce the key pattern from Autor (2003): flat pre-trends followed by a gradual increase in temporary employment after wrongful discharge law adoption
• It is recommended to report the event study plot, pre-trends test, and sensitivity to alternative estimators