Minimum-Wage Evaluation with Contaminated Difference-in-Differences
A labor ministry is evaluating a municipal minimum-wage increase. Six of fourteen matched cities raised their minimum from 12.00 to 14.50 per hour in January 2023; eight kept it at 12.00. The evaluation uses quarterly administrative payroll data for 2019Q1-2024Q4 on teen employment (thousands) and average hourly earnings. A two-way fixed-effects difference-in-differences regression of log teen employment on a treated-city times post-2023 indicator yields -0.032 (cluster-robust standard error 0.014), while the earnings equation yields +0.081 (0.022). Treated cities are larger and had faster employment growth in 2019-2022 (1.9 percentage points per year) because of a semiconductor investment. Two treated cities simultaneously introduced a childcare subsidy. Eleven percent of affected firms report the new wage but not hours, and 6% of teen workers live in one city and work in another. The ministry must decide whether to cite the -3.2% estimate in a national policy brief.
1. Interpret the two coefficients economically. What wage elasticity do they imply, and what caveats attach to that calculation?
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Point estimates. Teen employment in treated cities is estimated to be 3.2% lower, and average hourly earnings 8.1% higher, than the untreated counterfactual after the increase. With a nominal wage change of 20.8% (12.00 to 14.50), the implied employment elasticity is about -0.032/0.208 = -0.15, at the low end of the published range.
Caveats. (1) The earnings coefficient is not the treatment intensity: the minimum binds only for workers below 14.50, so the effective wage change for affected workers is larger and for the average teen smaller; 8.1% is an average over all teen employees and is contaminated by composition - if low-wage jobs are lost, average earnings rise mechanically even with no wage change for anyone (selection). (2) 11% of firms report the new wage but not hours, so the earnings series is more complete than the hours series, and part of the employment effect may reflect hours cuts rather than headcount. (3) The elasticity assumes the earnings change is policy-induced; the semiconductor boom could be raising wages independently in treated cities. (4) With six treated clusters, cluster-robust inference is fragile: the effective number of clusters is 14, and the t-statistic of -2.3 should be checked with wild-cluster bootstrap or randomisation inference before being described as significant.
2. Assess the parallel-trends assumption, the threats to identification, and the diagnostics you would require before publication.
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Parallel trends. The raw fact that treated cities grew 1.9 points per year faster in 2019-2022 is a direct warning: the DiD estimate compares post-period changes to pre-period changes, and if that differential trend continues, it biases the estimate upward (less negative). The true disemployment effect could be larger than -3.2%, not smaller. Equally, the childcare subsidy in two treated cities raises employment and biases upward; both threats push the same way, so -3.2% is plausibly a lower bound on harm.
Diagnostics required. (1) Event-study specification with leads and lags: coefficients on pre-2023 leads should be near zero and jointly insignificant; a trend in the leads invalidates the design. (2) Sensitivity to city-specific linear trends, reported alongside the baseline because trends can absorb genuine dynamic effects. (3) Synthetic control or reweighting on pre-period employment and wages, so the comparison group matches treated cities on growth. (4) Border-pair design using contiguous city pairs, which handles regional shocks better than distant controls. (5) Explicit control or exclusion for the childcare cities. (6) Wild-cluster bootstrap and randomisation inference for the six treated clusters.
Spillovers. 6% of teen workers cross city boundaries: controls next to treated cities may also see wage rises, violating SUTVA and attenuating the estimate. A border-discontinuity sample, or dropping controls adjacent to treated cities, tests this.
3. Draft the conclusion the ministry should publish, including the specific wording of the estimate and its limitations.
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Recommended conclusion. "The evaluation finds that the municipal increase from 12.00 to 14.50 is associated with teen employment about 3.2% lower than the comparison cities, with a 95% confidence interval of roughly -6.0% to -0.4% under the baseline specification, alongside 8.1% higher average teen earnings. These estimates are not yet suitable for extrapolation to a national increase."
Why this wording. (1) "Associated with" rather than "caused" reflects the pre-trend and the co-occurring childcare reform; the direction of the bias is upward, so the point estimate is likely conservative for disemployment, but that is an argument, not a proof. (2) The interval must be shown: with six treated clusters and a t-statistic near 2.3, the result is fragile to inference method, and wild-bootstrap intervals may cross zero. (3) External validity: six larger cities experiencing a semiconductor boom are not representative of the national labor market, and a 20.8% increase is far above the national proposal. (4) The earnings effect cannot be read as a pure wage gain because composition changed.
Before national use. Require the event study, synthetic control and border-pair estimates to agree in sign; if they do, a national brief can say that a large increase probably reduces teen employment modestly while raising earnings for those retained, and that the magnitude for a smaller increase is uncertain. If they disagree, the ministry should publish the range and defer the national claim.