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2cmDoes regional variation in wage levels identify the effects of a national minimum wage?
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Keywords: minimum wage, employment, wage distribution, Brazil, Kaitz index, fraction affected
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One approach to measuring the labor market effects of minimum wages is to exploit between-region differences in wage levels within a country. For instance, Mississippi and South Carolina should be more strongly affected by an increase in the US federal minimum wage than Texas or Georgia because wages in the latter states are higher. Based on this idea, the econometrician estimates panel regressions where the treatment intensity variable is a function of region-specific wage distributions. Classic examples are the “fraction affected” design of Card1992 and the “effective minimum wage” design of Lee1999.
Both Card1992 and Lee1999 used data from the US, where much of the identifying variation came from state-specific minimum wage laws. Since then, researchers have developed more modern estimators that directly exploit that source of variation Cengiz2018,Ghanem2023. However, these newer estimators are infeasible in contexts without regional minimum wages, explaining why variations of the effective minimum wage and fraction affected designs are still in use. Recent applications published by leading journals include Mexico Bosch2010, South Africa Dinkelman2012, Germany Dustmann2021, the US in the 1960s and 1970s Bailey2021, Brazil Engbom2022, and the UK Giupponi2024.
This paper examines the validity of the fraction affected and effective minimum wage designs in such contexts. Specifically, it uses a combination of economic theory, simulation exercises, and original data analysis to investigate whether these techniques can accurately capture the causal effects of a national minimum wage on regional employment levels and wage distributions.
I begin by discussing the sources of identification for each estimator and documenting identification threats that have not been recognized in the existing literature. For the effective minimum wage design, I show that even when the assumptions emphasized by Lee1999 are satisfied, the design suffers from correlated measurement error, which can lead to economically significant biases. In addition, I demonstrate that even small deviations from the central identifying assumption in Lee1999—namely, the absence of correlation between the mean and standard deviation of latent log wage distributions across regions—can generate large biases. This finding is particularly important because there are economic reasons to expect such correlations, and empirical evidence from both the U.S. and Brazil suggests that they are not negligible.
The fraction affected design suffers from analogous issues. Even when the estimator is expected to perform well under the “parallel trends” assumption, functional form misspecification can lead to substantial biases. These biases are more likely to arise when the minimum wage has nontrivial employment effects—whether positive or negative. I also show that the parallel trends assumption can be violated by seemingly benign factors that affect all regions, such as changes in the dispersion of latent log wages driven by skill-biased technical change.
After presenting the econometric issues, I evaluate potential diagnostics tests and alternative regression specifications proposed in academic literature. For the effective minimum wage design, I discuss testing for spillovers in the upper tail as a diagnostics tool, using higher quantiles of the wage distribution to construct the effective minimum wage, or changing the set of fixed effects, trends, and controls included in the regression. The leading diagnostics test for the fraction-affected design is the well-known check for parallel pre-trends. As for alternative specifications, I consider other treatment intensity measures (such as the Gap measure or binary indicators), instrumental variables designs, and the estimator of Giupponi2024.
I employ two approaches to quantify the practical relevance of these issues and the effectiveness of potential solutions. The first is a battery of simulation exercises using different economic models as data-generating processes. They include scenarios where the minimum wage causes disemployment in the lower tail of the wage distribution; others where the minimum wage causes no employment effects but significantly increases wages in the lower tail, perhaps due to the reallocation of workers from low-wage to high-wage firms; others where both wages and employment increase, corresponding to scenarios with significant monopsony power; and still others where minimum wages can affect the returns to skill in a “canonical” model in the style of Katz1992.
The second approach is a detailed analysis of the rise of the Brazilian federal minimum wage starting in 1995. Several published and working papers use the effective minimum wage and fraction affected designs to estimate minimum wage effects in this context. I start by reviewing that literature and showing that it contains significant discrepancies in estimation results. Next, I conduct original data analysis to investigate the sensitivity of those estimators to specification choices. I also benchmark those estimates against the minimum wage effects predicted by a structural model that is estimated on the same data but does not suffer from the same econometric problems.
Both exercises conclude that when those estimators are used in contexts where the minimum wage is set at the national level, economically significant biases may be the norm rather than the exception. The effective minimum wage design requires the existence of shocks that change the bindingness of minimum wages in particular regions but are otherwise unrelated to both the shape of the wage distribution and employment levels. The only plausible shock of that kind explicitly mentioned in the literature is region-specific minimum wage regulations. Even if those shocks exist, misspecification biases still apply. In those cases, an instrumental variables approach in the style of Autor2016 can perform better than the ordinary least squares estimator.
Out of the two estimators, the fraction affected design is more appealing, especially because tests based on pre-treatment data can detect many of the issues I discuss in the paper (if properly implemented). However, those tests are only feasible when a sufficiently long pre-treatment period with constant minimum wages exists. Even with parallel pre-trends, practitioners should investigate the possibility of functional form misspecification. Longer-term studies should also consider regression to the mean and regional convergence issues.
The paper is structured as follows. Section 2 clarifies the scope of my analysis and relates it to existing literature. Sections 3 and 4 analyze the effective minimum wage and the fraction affected estimators, respectively. Section 5 presents additional simulation results to assess the generality of previously discussed issues. Section 6 analyzes the rise of the Brazilian minimum wage. The final section concludes with a summary of recommendations for researchers.
I consider two-period ($t\in\{0,1\}$) data-generating processes (DGP) of the following form:
where $r\in\{1,\dots,R\}$ indexes regions, $\bm{y}_{r,t}$ is a vector of equilibrium outcomes (such as employment to population ratio or quantiles of the log wage distribution), $mw_{t}$ denotes the logarithm of the national minimum wage, $\bm{\theta}_{r,t}$ is a vector of region-time-specific determinants of the outcomes of interest that differ across regions, and $f$ is a function that outputs the equilibrium outcomes of a particular economic model (which I will later specify in the simulation exercises). The variables that compose the $\bm{\theta}_{r,t}$ vectors may display correlations across periods within regions but are independent across regions.
I assume that $mw_1>mw_0$. There is no loss of generality in the DGP from Equation (ref), as outcomes depend only on current values of $mw_{t}$ and $\bm{\theta}_{r,t}$. There would be loss of generality in models with nominal rigidities or other dynamic concerns, but I abstract from such issues in this paper.
There are two natural ways to define the ceteris paribus causal effects of the rise in the national minimum wage:
where the expectation is taken with respect to the distribution of the $\theta_{r,t}$ variables. The first formulation, $\bm{ATE}_0$, requires evaluating a counterfactual where the minimum wage rises from $mw_0$ to $mw_1$ but other characteristics remain at their $t=0$ levels. The second formulation compares the outcomes as of $t=1$ to a counterfactual scenario where the minimum wage remained at the $t=0$ level. The two definitions are identical if the $\theta_{r,t}$ distribution is time-invariant. I use the average of these two definitions as the object of interest to be recovered by the econometric designs:
Does this definition really correspond to the central object of interest in papers using the effective minimum wage and fraction affected designs? I claim that the answer is yes. Appendix (ref) substantiates this point by showing that all of the recently published papers cited early in the introduction seek to identify an average (or aggregate) effect of this type.
I impose an additional restriction on the data-generating process: there are no trends in overall wage levels. Suppose the minimum wage change is simultaneous with an unobserved shock to total factor productivity (TFP) affecting all regions. In that case, it is only possible to separately identify the average effects of the minimum wage by imposing further assumptions. To abstract from this “missing intercept” issue, I rule out common TFP shocks. In practice, econometricians should interpret estimates coming from these regressions as the impact of the minimum wage net of common TFP shocks.
Comparison to existing literature: Equation (ref) is fairly general, but it imposes important constraints that limit the scope of my analysis. Discussing these limitations helps pinpoint how my findings differ from and complement existing literature.
Many papers discuss econometric challenges arising from minimum wage effects taking some time to materialize. Using Canadian data, Baker1999 document that employment responds differently to low- or high-frequency minimum wage variation. Sorkin2015 argues that short- and long-run effects of the minimum wage can be different, and that it may be impossible to identify long-run effects with regressions in some contexts. Meer2016 discusses econometric challenges that arise if minimum wages change employment growth rates instead of employment levels. Vogel2023 documents that minimum wage effects “trickle up” the wage distribution over a few years. Because I study a two-period model without dynamics, all of the issues I document in this paper are separate from those discussed above.
Equation (ref) does not include measurement error, meaning that the issues in this paper are also different from the mechanical bias discussed in Autor2016. These authors propose an instrumental variables estimator as an improvement over the effective minimum wage design of Lee1999. Later in the paper, I will show that the estimator of Autor2016 can also solve some of the issues I document. However, that strategy is only feasible if the data includes region-specific changes in the minimum wage.
My model also imposes complete independence between regions, ruling out spillover effects coming from, e.g., migration responses Cadena2014,Huang2019.
The issues I discuss complement recent work on the econometrics of difference-in-differences models. Because the analysis is restricted to a two-period setting, concerns related to staggered treatment timing do not apply; see Chaisemartin2020 and Roth2023. However, two other issues from this literature are closely related to my results: violations of the “strong parallel trends” assumption in models with continuous treatment intensity Callaway2024, and biases arising from heterogeneous treatment effects (see, e.g., deChaisemartin2018; and Dube2024 for a discussion of how the issues raised in Sun2022 apply in the minimum wage context).
Relative to these papers, my contribution is to show that such problems arise naturally from economic theory in minimum wage studies; that the resulting biases can be economically meaningful; and that general-purpose solutions proposed in the broader literature may not be effective in this setting. I elaborate further on these connections after describing my results.
Finally, my paper focuses on designs where regions are the unit of analysis. For this reason, I do not evaluate “fraction affected” designs where the unit of analysis is workers Dustmann2021 or firms Harasztosi2018. These estimators are not designed to capture labor market-level treatment effects, as defined in Equation (ref), but rather how outcomes for different groups of firms or workers evolve in response to the minimum wage. Even if the control group in such designs is not directly affected by minimum wage regulations, the resulting estimands generally do not correspond to market-level effects. For example, the reallocation effects documented in Dustmann2021 imply that high-wage (control) firms may still be indirectly affected by an inflow of low-skilled labor, which could alter internal task allocations, as in Haanwinckel2023. Alternatively, minimum wages may affect firm entry and exit Aaronson2018 or worker search effort Adams2022, in which case individual-level designs would not capture the impact of compositional changes on the aggregate labor market. Because the estimands differ in important ways, a detailed econometric analysis of these methods does not fit neatly within the framework of this paper and is left to future work.
Let $w_{q,r,t}$ denote quantile $q$ of the log wage distribution in region $r$ at time $t$. Now suppose that the econometrician is interested in two types of endogenous outcomes $y_{i,r,t}$: quantile gaps of the form $w_{q,r,t} - w_{0.5,r,t}$ and an employment measure such as the employment-to-population ratio. The effective minimum wage design uses the following ordinary least squares regression to estimate the effects of the national minimum wage increase:
where $i$ indexes the outcome of interest, such that each outcome corresponds to a separate regression. The term $mw_t - w_{0.5,r,t}$ is the (log) effective minimum wage, a measure of the bindingness of the national minimum wage in a particular region-time. This baseline specification includes region and time fixed effects. I discuss alternative specifications later.
To calculate the predicted treatment effects of the minimum wage increase in each region, the econometrician multiplies the changes in the effective minimum wage (and its square) by the estimated $\hat{\beta}$ and $\hat{\gamma}$ parameters. Those products can then be added up and averaged across regions, yielding an estimate of average treatment effects as defined in Equation (ref): $$\widehat{ATE}_i = \frac{1}{R}\sum_r \left\{\hat{\beta}_i \left[\left(mw_1 - w_{0.5,r,1}\right)-\left(mw_0 - w_{0.5,r,0}\right) \right] + \hat{\gamma}_i \left[\left(mw_1 - w_{0.5,r,1}\right)^2-\left(mw_0 - w_{0.5,r,0}\right)^2 \right]\right\} $$ This regression model was first introduced by Lee1999, who focused on quantile gaps as the outcomes of interest. The design was later used to estimate employment effects as well; Engbom2022 is one example.\footnote{Such regressions follow in the tradition of earlier papers that used variation in wage levels to measure employment effects of minimum wages. A well-known example is Neumark1992, who use as the treatment variable the nominal minimum wage in a state-year multiplied by the state-specific minimum wage coverage and divided by the state-specific average wage. My analysis focuses on the quantile-based effective minimum wage because it is more common in recent work.}
In order to discuss identifying assumptions, it is helpful to introduce the semiparametric conceptual framework of Lee1999. Each region has a latent distribution of log wage in each period---that is, the distribution of log wages that would prevail with no minimum wage regulation. The cumulative distribution function for those latent log wages is: $$F_t\left(\frac{w-\mu_{r,t}}{\sigma_{r,t}}\right)$$ where $\mu_{r,t}$ and $\sigma_{r,t}$ are the centrality (or location) and dispersion parameters, respectively.
Using this notation, Lee1999 emphasizes two identification assumptions. First, the deflator used to construct the effective minimum wage---that is, the median wage $w_{0.5,r,t}$ in Equation (ref)---should provide a good approximation for the centrality parameter $\mu_{r,t}$. Second, the location and dispersion parameters should be uncorrelated across regions conditional on $t$. When employment is the outcome of interest, one must also assume that latent employment is uncorrelated with location parameters conditional on the fixed effects and controls included in the regression.
The central message of the analysis below is that, even when those assumptions are satisfied, the effective minimum wage estimator may not perform well, and that minor deviations from the second assumption can introduce large biases.
The effective minimum wage design is predicated upon minimum wage effects being stronger where it bites more into the latent wage distribution. The econometrician would ideally use $mw_t-\mu_{r,t}$ as the key regressor. But because $\mu_{r,t}$ is not observed, $mw_t-w_{0.5,r,t}$ is used instead. In this subsection, I argue that the minimum wage introduces deviations between $w_{0.5,r,t}$ and $\mu_{r,t}$ that can cause significant biases even if they are small in magnitude, because these deviations correlate with minimum wage effects on the outcomes of interest.
To understand why even minor deviations between $w_{0.5,r,t}$ and $\mu_{r,t}$ can be problematic, consider a simple model with only two regions, $A$ and $B$. In this model, a more binding minimum wage has small but positive effects on the median wage. These effects at the median can arise from strong spillovers (e.g., workers moving from low- to high-wage firms) or because the minimum wage causes disemployment effects in the lower tail, such that all quantiles of the log wage distribution mechanically move to the right.
I start by discussing the ideal source of variation for the effective minimum wage designs: random shocks to the location parameters $\mu_{r,t}$. In period $t=0$, both regions have identical distributions of latent log wages. In period $t=1$, two things happen. First, the national minimum wage rises. Second, the location parameter in Region $B$ increases--- for simplicity, assume that it increases by the same amount as the minimum wage. Panel A in Figure (ref) illustrates this scenario.
In this scenario, Region $A$ is the “treatment group” while $B$ is the “control.” In $B$, the minimum wage binds as much in period $t=1$ as it did in $t=0$. Thus, we should not expect any changes in the effective minimum wage or outcomes of interest. Thus, comparing $A$ and $B$ provides a valid quasi-experiment from which we can recover the causal effects of the minimum wage, even though the change in the national minimum wage was the same everywhere. The broader point is that, in the effective minimum wage design with region and time fixed effects, the ideal identifying variation comes from idiosyncratic shocks to the location parameters $\mu_{r,t}$.\footnote{One may wonder whether the small spillover effects in Region $A$ generate bias, since the change in the effective minimum wage is smaller than the change in the national minimum wage. To see why this is not a problem, recall from Subsection (ref) that after estimating Equation (ref), the predicted treatment effects are calculated by multiplying the coefficients and changes in the effective minimum wage, not changes in the national minimum wage.}
Next, I show how heterogeneity in dispersion parameters $\sigma_{r,t}$ can introduce “bad variation:” a spurious empirical link between the effective minimum wage and the outcomes of interest. Again, we explore a scenario where Region $A$ is affected by the increase in the national minimum wage, while Region $B$ is not. But now assume that neither location nor dispersion parameters change over time; the reason why $B$ is the “control” is because the time-invariant dispersion $\sigma_{B}$ is minimal, as illustrated in Panel B of Figure (ref). The minimum wage has no bite in that region because latent wages concentrate around the median.
What would the regression recover in this scenario? The effective minimum wage rises in both regions as the minimum wage gets closer to the median wage. However, the relative increase is higher in Region $B$ due to spillover effects on the median wage in Region A. Thus, if those permanent differences in the dispersion of latent wages are the only source of variation, then the predicted treatment effects would have the opposite sign compared to the actual causal effects of the minimum wage.
Shocks to location parameters $\sigma_{r,t}$ can also introduce bad variation. Panel C in Figure (ref) shows a scenario where the only change over time is a fall in the dispersion parameter in Region $A$. That shock reduces inequality, moving percentile 25 of the log wage distribution closer to the median. Moreover, because the minimum wage becomes less binding, the median log wage falls, such that the effective minimum wage rises. Thus, comparing changes between Region $A$ and Region $B$, the estimator will estimate a positive relationship between the effective minimum wage and the log wage gap $w_{0.25,r,t}-w_{0.5,r,t}$. However, the magnitude of this link is likely to be significantly overstated compared to the causal minimum wage effects. That is because the regression entirely attributes the inequality-reducing effects of the $\sigma_{r,t}$ change to that small change in the effective minimum wage induced by the spillovers at the median.
I finish this discussion with three remarks. First, since the scenarios in Panels B and C imply biases in opposite directions, one needs stronger assumptions about the DGP to infer the sign of the total bias. Second, to see how this is fundamentally a measurement error issue, note that the issues I described above would not exist if the econometrician could observe $\mu_{r,t}$ and use it to construct the effective minimum wage. If that were the case, only the scenario shown in Panel A would generate identifying variation for the effective minimum wage design. Third, in all of those examples, there is no systematic relationship between location and dispersion parameters in all of those scenarios. I discuss problems arising from such correlations in the following subsection.
Simulations. To investigate the potential magnitude of the bias in empirical applications, I perform simulation exercises with parameters calibrated based on state-level data from the US Current Population Survey. I assume that latent log wages are Normally distributed in every region. There is a “markdown” parameter $m\in[0,1]$ such that the latent distribution is truncated at $mw_{t}+\log m$ and censored at $mw_t$. That is, workers who would earn less than the minimum wage times the markdown become disemployed, and those with latent wages above that cutoff but below the minimum earn exactly the minimum wage (that is, they create a minimum wage “spike” in the simulated log wage distribution). Unless otherwise noted, all simulations impose $m=0.7$.
Figure (ref) illustrates the minimum wage effects in these simulations for two regions that differ in minimum wage bindingness. The truncation and censoring effects above correspond to the red and orange areas. That figure also includes the possibility of positive employment effects slightly above the minimum wage, illustrated in green. The first set of simulation results reported below does not include those positive employment effects; I will return to this point at the end of this section.
Each region is described by a vector $[\mu_{r,0}, \sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$, drawn from a multivariate Normal distribution. The parameters for that multivariate Normal are calibrated based on state-level data from the US Current Population survey, based on the years 1989 (corresponding to $t=0$) and 2004 (corresponding to $t=1$).\footnote{For each state, I calculate the mean log wage and the and standard deviation of log wages for each year. Next, I calculate the means, variances, and pairwise correlations for this four-element vector across states. I use those summary statistics to calibrate the simulations. I use 1989 and 2004 because the real federal minimum wage bottomed out in those years. In addition, unemployment rates are also similar in both years. Thus, the summary statistics based on these two years provide a reasonable approximation for how the latent distribution of log wages varies between states and over time.} As explained below, each particular simulation exercise makes different assumptions about that meta-distribution of parameters across regions. In all exercises shown, the data contains 200 regions. See Appendix (ref) for details.
Before showing the results, I make an important note. Since I ignore state-level minimum wage regulations and select particular years in the analysis, these exercises do not constitute an evaluation of the effective minimum wage design in the US context. Instead, I use the US data to argue that the econometric issues I describe could be significant in contexts similar to the US regarding latent log wage distributions.
The first panel in Table (ref) shows a case where all regions have the same dispersion parameters $\sigma_{r,t}=\sigma_t$, but differ in location parameters---which are subject to changes over time. This model corresponds to the ideal scenario with only “good” variation. Correspondingly, the estimator performs very well. The true average causal effects from the model are nearly identical to the predicted effects from the regressions, averaged over 1,000 simulations (shown in the second row of each panel of the table). In addition, the confidence intervals are tight, as implied by the standard errors reported in the third row (which are also averaged over the 1,000 simulations).
Panel B introduces differences in dispersion parameters. The data-generating process still satisfies the structural assumptions emphasized by Lee1999: distributions only differ in location and dispersion parameters, not shape, and the location and dispersion parameters have zero correlation. In addition, the median wage is an excellent proxy for the latent $\mu_{r,t}$ location parameter: Table (ref) in Appendix (ref) shows that their correlation is 0.999. Still, the estimator displays biases. They arise because, though the differences between $w_{0.5,r,t}$ and $\mu_{r,t}$ constitute a tiny share of the variation, they are systematically correlated with the outcomes of interest.\footnote{The direction of the biases suggests that the issue illustrated in Panel C of Figure (ref) is more serious than the one in Panel B. I confirm this intuition by verifying that a model that maintains cross-sectional dispersion in $\sigma_{r,t}$ but makes it invariant over time is almost unbiased.}
Panel C shows that biases are larger when the simulated increase in the federal minimum wage is 40 log points instead of 20 log points, even though the correlation between $\mu_{r,t}$ and $w_{r,t}$ remains above 0.99 (see Table (ref)). This exercise reinforces the idea that the “good” identifying variation comes not from the change in the minimum wage itself but from idiosyncratic shocks to $\mu_{r,t}$.
Panel D highlights how differences in the dispersion of latent log wages between regions constitute the source of the bias. Increasing those differences by 50% is enough to double the average bias in the regressions.
Are there biases if average employment effects are zero or positive? To answer this question, I simulate an alternative model where the minimum wage can increase employment levels for individuals with latent wages just above it, corresponding to the green areas in Figure (ref) (see Appendix (ref) for details). Such effects could emerge from, e.g., increased search effort by workers Adams2022,Piqueras2024. The results (available upon request) show that biases exist even when employment effects are zero or positive on average. That's because the correlated measurement error issue arises from heterogeneity in employment effects across regions, not from its average effects. In Section (ref), I will come back to this issue by quantifying biases a large number of alternative DGPs, many of which have zero or positive employment effects of the minimum wage.
Connection to previous econometric literature: The dispersion in shape parameters $\sigma_{r,t}$, combined with the non-linearity of minimum wage effects in standard economic models, implies that the effective minimum wage design may be subject to econometric concerns related to heterogeneous treatment effects in two-way fixed effects settings Blundell2009,deChaisemartin2018,Sun2022,Callaway2024. As a result, these regressions may yield biased estimates of the average treatment effect of the national minimum wage, even if the econometrician observes the latent centrality parameter $\mu_{r,t}$. Accordingly, part of the bias observed in my simulation exercises stems from this treatment effect heterogeneity.
That said, the econometric concerns discussed in this section go beyond those highlighted in the existing literature. In particular, prior work does not consider the endogeneity of the treatment intensity variable in the effective minimum wage design, which is the central focus of my analysis.
The second assumption emphasized by Lee1999 is independence between the location and dispersion parameters, $\mu_{r,t}$ and $\sigma_{r,t}$, conditional on $t$. For an intuition of why this assumption is essential for measuring spillover effects, consider again the “good variation” example from the previous subsection. In that example, Region $A$ was “treated” by the minimum wage because its location parameters $\mu_{A,t}$ are constant over time, while Region $B$ is the “control” because $\mu_{B,1}-\mu_{B,0}=mw_1-mw_0$. Now, suppose that, along with the increase in location, the dispersion parameter also increases for Region $B$. That would increase all quantile gaps $w_{q,B,t}-w_{0.5,B,t}$ in the control region. Thus, comparing changes in treatment versus control regions would no longer provide a valid estimate of the causal effects of the minimum wage.
A correlation between location and dispersion parameters is also problematic if the outcome is employment. The reason is that changes in dispersion parameters can make the minimum wage bind more or less in some regions, causing independent effects on the median wage in the presence of a minimum wage. For example, rising dispersion can add more probability mass in the lower tail of the latent log wage distribution, increasing the amount of truncation and, thus, the mechanical effects of the minimum wage on the median wage. That effect would magnify the correlated measurement error issues discussed in the previous subsection.
Panel A in Table (ref) shows a baseline scenario where regions differ in dispersion parameters, but dispersion and correlation parameters are uncorrelated. Panel B introduces a within-period correlation of 0.076, the value I find in US data for 1989 (the correlation for 2004 is 0.264). That mild correlation is enough to bring the estimated employment effects to almost zero and make estimated spillover effects much larger than the true ones. Note that this correlation does not significantly affect estimated standard errors; if anything, the estimates become more precise.
Correlations between the location and dispersion of latent log wages may emerge from economic reasons. One is the observation that if workers are split into education-age groups, higher-wage groups tend to display more within-group inequality. This fact is discussed in detail by Lemieux2006, who argues that much of the increase in inequality observed in the US from 1973 to 2003 is a compositional effect deriving from increased educational achievement. The same result has been found in other contexts, such as Brazil Ferreira2017. Then, if regions differ in workforce composition, the correlation we discussed above may follow. Another economic factor that could generate this correlation would be regional differences in endowments that could affect industrial composition, which in turn would lead to non-random sorting of workers and firms tho different regions.
The baseline specification in Lee1999 does not include region fixed effects. Concerning the inclusion of such fixed effects, he writes: “... the reduced identifying variation resulting from eliminating the "permanent" state effects may magnify biases due to misspecification, in the same way biases stemming from measurement error in the independent variable are magnified when true variation in the independent variable is reduced.” Using the language introduced in Subsection (ref), the estimator without region fixed effects has another source of “good” variation: within-period differences in the location parameters of latent log wage distributions (instead of simply differential shocks to location). That may significantly reduce the influence of “bad” variation coming from correlated measurement error in the centrality measure, reducing the amount of bias.
Table (ref) illustrates Lee's argument through simulations. The data-generating process for those simulations is the same as reported in Panel B of Table (ref). I report predicted treatment effects using the default effective minimum wage design and two alternative specifications, the first being the estimator without region fixed effects. The comparison of the fifth row to the third shows that, by using more “good” variation coming from level differences in $\mu_{r,t}$, the estimator without region fixed effects can indeed perform better.
Still, it is easy to contemplate omitted variable biases that could cause problems for estimators without region fixed effects. For example, a persistent shock reducing labor demand over a long period would introduce a negative correlation between the effective minimum wage and employment. Alternatively, regions may differ in the shape of their distribution of worker skills, leading to different baseline levels for quantile wage gaps. Such concerns make specifications with fixed effects overwhelmingly more popular in published literature.
Indeed, the specifications in papers such as Bosch2010, Autor2016, and Engbom2022 go beyond region fixed effects and include region-specific trends as well. These trends may absorb region-specific supply and demand shocks that affect the median wage and the outcomes of interest.
However, it is not evident that region-specific trends and controls reduce biases; they may instead amplify them. The reason is analogous to Lee's argument on region fixed effects. By including the trends, the econometrician may throw out the “good variation” with the bathwater. The fixed effects, region-specific trends, and controls may absorb much of the $\Delta \mu_{r,t}$ shocks, such that the measurement errors become a larger share of the residual variation in the effective minimum wage. In addition, once those controls are included, it may be difficult to interpret where the variation in the effective minimum wage is coming from. This lack of intuition is problematic; ideally, the econometrician should be able to defend the assumption that there exists an economic factor, separate from all trends and controls, that shifts wage levels but does not affect employment levels or the shape of the log wage distribution in any way (other than making the minimum wage more or less binding). In Section (ref), I discuss those issues in the Brazilian context.
Based on this discussion, one may wonder whether dropping the time effects from the design would also add more “good” variation. Lee1999 explains that this choice is unwise if latent log wages' shape and average dispersion change over time. It is not warranted in the presence of inequality trends coming from technical change or trade shocks, for example.
Table (ref) illustrates the sensitivity of that estimator to changes in the economic environment. The baseline data-generating process displays minor differences in the marginal distributions of $\mu_{r,t}$ and $\sigma_{r,t}$ between periods. The most salient differences are that average $\sigma_{r,t}$ falls from 0.54 to 0.51, and the standard deviation of $\sigma_{r,t}$ between regions increases from 0.026 to 0.049. Those small changes are enough to warrant the inclusion of time effects, as biases are much more prominent when the model does not include them.
Sometimes, the econometrician may have a prior that the minimum wage significantly impacts the median wage, making it a poor measure of centrality. In those cases, they may consider using a higher quantile of the wage distribution to construct the effective minimum wage. For example, Bosch2010 use quantile 0.7 as the deflator in a study of Mexico, and Engbom2022 use quantile 0.9 when studying Brazil.
Lee1999 argues that the deflator should be a good approximation for centrality $\mu_{r,t}$ instead of merely an overall measure of wages. Otherwise, the regression may yield non-zero estimates even when the observed log wage distribution is identical to the latent wage distribution.\footnote{See Lee's discussion around Equation (5) in page 996.} The discussion regarding correlated measurement error introduces another reason to be wary of choosing higher quantiles of the wage distribution. While it is true that those higher quantiles may be less affected by the minimum wage, the effects will still not be zero if the minimum wage has employment effects, positive or negative. In addition, higher quantiles are likely to be more sensitive to cross-region differences in the dispersion of latent log wages. Due to those two issues, the biases may be more significant when a quantile other than the median is used as the deflator.
Table (ref) evaluates the performance of an estimator based on quantile 90 of the log wage distribution using the baseline scenario with regional differences in location and dispersion parameters (the same from Table (ref)). The biases are significantly larger than those for other estimators previously discussed. In unreported simulations, I tested that estimator in a broader range of scenarios and found that it consistently underperforms relative to the estimator based on the median.
Lee1999 proposes estimating relative effects on high log wage quantiles $q>0.5$ to validate the model. The justification for that approach is that, in many applications (such as in the US), the econometrician may have a strong prior that the minimum wage should have minimal effects on the upper tail of the wage distribution. Autor2016 use the same specification test to validate their instrumental variables implementation of the effective minimum wage design.
As with any test, one should consider the possibility of false positives and false negatives. False positives may arise because many plausible mechanisms could lead to minimum wage spillovers that extend beyond the median wage. Engbom2022 develop and estimate an on-the-job search model where minimum wages cause spillovers that extend far into the upper tail of the wage distribution, primarily due to worker reallocation from low- to high-wage firms. The model in Haanwinckel2023 also includes endogenous changes in within-firm returns to skill in response to reallocation flows, firm entry responses, and price effects as mechanisms that can generate spillovers in the upper parts of the wage distribution. Those channels may be quantitatively important even when net disemployment effects are minor, as in Engbom2022. Thus, an econometrician with a strict rejection rule based on effects in the upper tail may reject a valid model.
False negatives may emerge from two reasons. The first are cases where the true model includes spillovers in the upper tail. The second possibility is that the estimator may be biased for lower tail outcomes, or for employment, while being unbiased for spillovers in the upper tail. This may happen if, for example, the negative upper-tail bias illustrated in Table (ref) is combined with positive bias arising from measurement error, as discussed by Autor2016.
In Section (ref), I will come back to the effectiveness of this diagnostics test when examining biases in a large number of DGPs.
In a previous subsection, I argued that adding region fixed effects, trends, and controls can absorb much of the “good” variation that could be exploited and thus magnify biases coming from misspecification. One exception to that logic is where the data includes changes in region-specific minimum wage laws. In that scenario, the estimator can exploit variation from those regulatory changes while using a battery of controls and region-specific trends to net out the influence of other factors. Still, the effective minimum wage estimator may remain biased, as it uses both the good variation from state-specific minimum wages and the bad variation induced by measurement error and the residual correlation between location and dispersion parameters.
In Appendix (ref), I investigate these scenarios. I show that, the more variation coming from regional minimum wages, the smaller the biases in the effective minimum wage design. In addition, I show that the remaining biases can be removed by using an instrumental variables approach inspired by Autor2016.
The main takeaway from the previous discussion is that the econometrician should have a clear sense of what constitutes the identifying variation behind the effective minimum wage design. Exogenous changes in state-level minimum wages are the clearest example of such variation. If the data includes little or no variation in state-level minimum wage laws, then identification relies on the existence of a latent residual shock that shifts the location of latent log wage distributions but has no independent effects on their shape, nor on employment. If the econometrician does not have an intuitive sense of what that structural factor could be, then the effective minimum wage design may not be warranted, as labor supply and demand shocks are typically biased towards some types of workers and affect employment levels in addition to wages.
Even if a plausible shifter of regional minimum wage bindingness exists, the estimator may still have economically significant biases arising from correlated measurement error. These biases can be more severe if the specification includes many controls, as they may diminish the residual variance of the “good” source of identification. If the bindingness shifter is observable, the econometrician may reduce misspecification biases by using an instrumental variables approach in the spirit of Autor2016.
If one must use this regression design anyway, my analysis confirms the advice from Lee1999. First, the effective minimum wage should be constructed using the median rather than other quantiles of the wage distribution. Second, time fixed effects should always be used. As for whether to include region fixed effects, regional trends, or other regional controls, econometric theory provides little guidance.
Some of the issues discussed in this section arise from the fact that the treatment intensity variable used in the effective minimum wage design may be endogenous to the minimum wage itself. One potential strategy to address this concern is to construct the treatment variable using pre-treatment information. In that case, identification would no longer rely on shocks to the location parameters $\mu_{r,t}$, but rather on their initial levels. This idea serves as the central motivation for the estimators introduced in the following section.
Now, I study a difference-in-differences model with a time-invariant, continuous measure of treatment intensity based on the initial distribution of wages:
where the subscript $i$ indexes a specific equilibrium outcome, such that different $i$ correspond to separate regressions. The treatment intensity variable $FA_{r}$ is the “fraction affected,” that is, the share of workers in the initial period earning less than $mw_1$.\footnote{When the data includes non-compliance with minimum wage regulations, researchers typically define the fraction affected as the share earning between $mw_0$ and $mw_1$, but not always Bailey2021. In all simulations below, there is perfect compliance, so both approaches are equivalent.} The regressions include region and time fixed effects. The estimated average treatment effect $ATE_i$ is obtained by multiplying the average of $FA_{r}$ by the $\hat{\beta}_i$ estimate.
Card1992 first introduced the fraction affected design in an analysis of the 1990 increase in the federal minimum wage in the US. In that paper, he emphasized that much of the identifying variation in his application comes from state-level minimum wage laws passed in the 1980s. Since that original application, that estimator has been applied in other contexts with no regional variation in nominal minimum wages, such as the introduction of a federal minimum wage in Germany in 2015 Ahlfeldt2018, Fedorets2021.
Identification comes from comparing the evolution of outcomes for “more treated” versus “less treated” units, where the treatment intensity variable only uses information from the initial period. This design is thus fundamentally different from the effective minimum wage one, which, as discussed in the previous section, relies on idiosyncratic shocks to the location parameter of latent log wage distributions (when the regression includes both region and time fixed effects). The core identification assumption is standard for differences-in-differences designs: absent the increase in the national minimum wage, outcomes in treatment and control regions would evolve similarly.
This design is ideal in scenarios where the minimum wage increases after at least a few years without adjustments. In those cases, the econometrician can use pre-treatment data to check for differential trends, which may provide support for the parallel trends identification assumption. In the next subsections, I will discuss the effectiveness of the parallel trends assumption in detecting each of the issues I highlight.
I also study other closely related designs. The main one is based on the “Gap measure:”
where $j\in\{1,\dots,J_r\}$ indexes workers in the initial period and $w_{j,0}$ is their log wage in that period. Card1994 introduces the gap measure in a firm-level econometric design. It has later been extended to region-level designs like the ones studied in this paper. Dustmann2021 provides an example, again in the context of Germany. The Gap measure corresponds to the increase in regional average wage if all low-wage workers in that region were to receive raises to comply with the new minimum. Other variations, such as transforming the treatment intensity variable into a binary indicator, are also explored as potential ways to solve issues discussed in the following subsection.
As in any regression, unbiasedness requires correct specification of the conditional expectation function. That is, the effects of the national minimum wage should be approximately linear in the continuous treatment intensity variable. If that sensitivity measure is misspecified, estimates of average treatment effects based on that estimator may be biased.
In this section, I show that the fraction affected estimator and its variants are sensitive to this issue, at least when the DGP is of the Normal-markdown class I introduced previously on the paper. Later, I show that the same is true when the DGP comes from the “canonical” model of labor demand from Katz1992 (and in that case, functional form misspecification issues are relevant for the effective minimum wage design as well).
To focus on the role of functional form assumptions, I design the simulations of the Normal-markdown model to be ideal applications for the fraction affected and gap designs. First, the national minimum wage increase is the only time-varying factor in the model, preventing violations of the parallel trends assumption. Second, regions only differ in $\mu_{r}$, the location parameter of their latent log wage distributions. Thus, to the extent that those simulations find misspecification issues, one may expect they would be even more severe if the data also includes heterogeneity in the dispersion and shape of latent log wage distributions.
Table (ref) reports results for four different DGPs. They differ in two dimensions. First, the starting minimum wage may be lower or higher, in which case the minimum wage increase has stronger effects (as the minimum bites more into the latent distribution). Second, in some DGPs, I allow for positive employment effects, as illustrated in the green areas of Figure (ref).
I find that even in this ideal scenario, biases arising from functional form misspecification may be statistically and economically significant. The direction of the bias may change depending on the model used; in unreported simulations, I find that it also changes with the markdown parameter $m$. The biases are more significant when the minimum wage is more binding or when it causes positive employment effects. For example, in Panel C, the minimum wage increases employment by one percentage point, but both the fraction affected and gap estimators yield precise estimates that are very close to zero.
Given these particular DGPs, the measured employment and wage effects tend to be smaller in magnitude when estimated with the Gap measure. If one is exclusively interested in the ratio of employment effects to wage effects in the lower tail (proxying for the employment elasticity with respect to the worker's wage), then the estimators are remarkably similar to each other across panels. However, the estimated ratio is generally different from the true one.
\paragraph{Relationship to previous literature and alternative specifications.} Misspecification biases arise because the effect of a marginal increase in the treatment intensity variable is not constant, either within or between regions. In this sense, the issue can be viewed as a deviation from the “strong parallel trends” assumption formalized in Callaway2024, a paper that studies difference-in-differences designs with continuous treatment variables. It is also related to unit-level heterogeneity in treatment effects emphasized by Sun2022 and recently discussed in the minimum wage context by Dube2024.\footnote{Previous work on similar econometric issues include Blundell2009,deChaisemartin2018,Chaisemartin2020,deChaisemartin2024.}
My contribution relative to those methodological papers is threefold. First, the previous results illustrate how these problems arise naturally in the minimum wage context from the basic mechanics of standard labor economics models. Second, I show that these misspecification biases can be large in general, by exploring a large range of possible data-generating processes (see Section (ref)). Third, as I discuss below, I show that possible solutions discussed in those methodological papers may not be effective in this context.
Callaway2024 argue that, if there is a control group composed of entirely untreated units, then a binary difference-in-differences specification can recover an average treatment effect on treated units. However, in the minimum wage context, it may be hard to argue that any single region is entirely untreated by a national minimum wage without imposing strong structural assumptions on the data-generating process. Still, some applied papers explore this idea by creating a binary version of treatment intensity. Recent examples in the Brazilian case, which will be further discussed in Section (ref), are Derenoncourt2021 and Parente2024.
In Appendix Table (ref), I show that rather than solving the problem, a binary treatment specification can suffer from more substantial biases than the baseline models. In this exercise, I split regions into treatment or control groups based on whether the initial median wages are below a given threshold. I choose thresholds such that either half or 90% of the regions are in the treatment group. Consistent with the logic of Callaway2024, biases are smaller when 90% of the sample is in the treatment group since it makes the “zero” group closer to being entirely untreated. However, there is a loss of precision, and significant biases remains.
Section 3.2.4 in Dube2024 discusses the fraction affected and related estimators. They emphasize the issue that treatment effect heterogeneity may lead to biases, as shown by Sun2022. These papers argue that, if treatment effects are constant within within fine observable groups (say, industries), it is relatively straightforward to adjust the econometric model to solve the misspecification bias. However, in the discussion above, I show that the problematic heterogeneity in treatment effects is not of the form discussed in Sun2022 and Dube2024. Rather, the fundamental problem is that with monopsony power or minimum wage-induce reallocation effects, minimum wage effects are likely to be nonlinear within fine groups. In that case, re-parameterizing the econometric model is much less straighforward, and can only be done if the econometrician is willing to make parametric structural assumptions on the channels through which minimum wages affect local labor markets.
One may wonder whether adding a quadratic term to the regression specification could effectively capture that kind of treatment effect heterogeneity. Appendix Table (ref) reports results for those models. Biases become smaller for some outcomes but larger for others. Thus, this approach is not a practical solution for the misspecification problems either.
One alternative specification discussed by Dube2024 is the bin-level estimator of Giupponi2024. It differs from the baseline fraction affected design in three ways. The first is that it uses a binary treatment intensity measure, with the regions in the top decile of wages being the control group and the other nine deciles being the treated group. Second, this split is based not on the averaged wages but rather location wage premiums estimated in a Mincerian regression, netting out the influence of worker's gender, education, occupation, industry, and even person effects in some specifications. Third, it measures impacts at the level of “skill” bins (with skill defined as wage purged from the location wage premium). That is, one can view their strategy as separate binary fraction affected designs, one for each bin.
Those alterations are interesting but do not fundamentally alter the conclusions from the previous specifications; the estimator is subject to the same concerns. It is unclear whether functional form misspecification biases should be larger or smaller using this estimator. As discussed above, binary versions of the fraction affected design are more likely to perform poorly. On the other hand, defining treatment based on regional wage premiums rather than raw wages, and estimating impacts at the skill bin level, could reduce omitted variable bias in the presence of skill-specific shocks correlated with minimum wage bindingness. Finally, the bin-specific approach relies strongly on the assumption that treatment and control groups differ only on the location of the latent log wage distributions, not their shape. Thus, like the effective minimum wage design, this estimator may be very sensitive to correlations between location and dispersion parameters across regions. The baseline effective minimum wage design does not share that sensitivity.
A final potential solution would be to use the regressor in the Fraction Affected design as an instrument for the regressor based on the Gap measure, or vice-versa. This strategy could be justified if each of those regressors were equal to an unobserved metric of propensity to be affected by the minimum wage plus some random noise and if those noise terms are uncorrelated with one another. Appendix Table (ref) reports the results of using such strategies. They have no impact on the estimates compared to the basic OLS estimator.
As with any difference-in-differences design, the validity of the fraction affected and gap estimators depends critically on the parallel trends assumption. In Appendix (ref), I discuss two potential threats to this assumption that have been raised in the prior literature. The first is regression to the mean: if regional wage levels contain a transitory component, then defining treatment intensity based on wages from a single period (or a small number of periods) can introduce a dynamic form of bias. The second, closely related concern is regional convergence driven by other factors, which can also result in violations of the parallel trends assumption. One key takeaway from that discussion is that, although such problems can often be detected using standard tests for pre-trends, these tests are only informative when the treatment intensity variable is constructed using a single pre-treatment period, rather than being averaged across multiple periods.
In this section, I highlight another potential threat to identification that, to my knowledge, has not been discussed in the existing literature. Suppose that, across all regions, there is a simultaneous change in the dispersion of latent log wages that coincides with an increase in the national minimum wage. Since this structural shock is common to all regions, one might expect the time fixed effects included in the regression to account for it. However, as I show below, this is not necessarily the case when the minimum wage is already binding in the initial period.
Consider the model illustrated in Figure (ref). Regions differ in a location parameter $\mu_{r}$ that is constant over time. The shape parameter $\sigma_t$ is common across regions and decreases over time. All else equal, regions with lower $\mu_{r}$ have a higher fraction affected and a larger (more negative) causal effect of the minimum wage. However, as $\sigma_t$ declines, these same regions may experience a relative increase in employment. This is because the truncation effects of the minimum wage may weaken as latent wages become more concentrated around the median. As a result, the estimator may recover a positive relationship between the fraction affected and employment, even though the causal effects are negative for all observations in all periods.
The comparison between Panels C and D in Appendix Table (ref) illustrates this issue. Panel C, discussed in Appendix (ref), features idiosyncratic shocks to both the location and dispersion parameters, but the distributions of these parameters remain stable over time. Panel D differs only in that the average dispersion parameter $\sigma_{r,t}$ declines from 0.54 in $t = 0$ to 0.51 in $t = 1$.\footnote{These values are based on the calibration discussed earlier, using state-level statistics from CPS data for 1989 and 2004. In the U.S. context, the decline in dispersion could partly reflect the increased prevalence of state-level minimum wages in 2004 compared to 1989, and should not necessarily be interpreted as a change in latent log wages. As noted before, the purpose of this exercise is to illustrate an econometric issue using empirically grounded parameters, not to evaluate the effects of the national minimum wage in the U.S.} Even this small change is enough to meaningfully increase the bias in the estimated employment effects.
In Appendix (ref), I also discuss how tests for parallel pre-trends may effectively detect biases arising from regression to the mean and regional convergence. Similarly, such tests may be able to detect the issue discussed in this section, provided that the trend in dispersion begins prior to the minimum wage change and has a relatively stable effect on employment over time. This is illustrated in the placebo regressions reported in Appendix Table (ref). An additional suggestion for practitioners is to check for the existence of trends in the variance of log wages before the implementation of the national minimum wage.
I showed that the Fraction Affected and Gap designs explore a fundamentally different and more transparent source of variation compared to the effective minimum wage design. It relies on the well-understood parallel trends assumption. Part of my contribution is showing that some factors that do not seem problematic, such as national trends in the dispersion of latent log wages or region-level productivity shocks, can imply violations of the parallel trends assumption and cause quantitatively significant biases. That makes it all the more important to report tests for differential pre-trends. Those tests are most effective when the treatment variable is constructed using only one pre-treatment period.
I also showed that because those estimators may be particularly sensitive to functional form misspecification. To assess the possibility of misspecification biases, practitioners are encouraged to report predicted employment and wage effects from both the Fraction Affected and Gap designs, and perhaps from quadratic specifications as well. In addition, I showed that binarized versions of those estimators should be avoided as they display more significant misspecification biases.
The previous sections presented a series of econometric issues that may invalidate the use of the effective minimum wage and fraction affected designs. The remainder of the paper is devoted to gauging their practical relevance. In this section, I use simulation exercises to estimate the likelihood of significant biases arising under a wide range of economic scenarios. Later in the paper, I explore the same question in a particular empirical setting.
In the main exercise of this section, I quantify biases for 500 different DGPs in the Normal-markdown class introduced in Subsection (ref). Each DGP corresponds to different assumptions on distribution of region-specific parameters $[\mu_{r,0}, \sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$, markdown parameter $m$, positive employment effect parameters $P_{base},P_{height}$, and levels of the national minimum wage $mw_0,mw_1$ (measured relative to the center of the latent log wage distribution). They run the gamut from models with significant disemployment effects, to positive employment effects, to no employment effects but significant wage spillovers. The specific parameter values are randomly drawn, with most distributions centered around the corresponding values observed for US states (used for previous simulations in the paper).
I make conservative choices regarding DGP elements that have been shown to be problematic. For example, the contemporaneous correlation between location and dispersion parameters is allowed to be positive, but its value is never larger than 0.20. That number is smaller than the correlation between regional mean log wages and regional standard deviation of log wages observed for the US in 2004 (0.26) or Brazil in 1998 (0.34). See Appendix (ref) for details.
For each DGP, I simulate 1,000 samples, each with 200 regions and two periods. For each sample, I estimate the causal effects of the national minimum wage using the effective minimum wage and fraction affected designs. Then, I average those estimates over the 1,000 samples to obtain the expected value of those estimators under that particular DGP, assuming a sample size of 200. I classify an estimator as being biased for a specific outcome, under that specific GDP, if the expected estimated effect is significantly different than the expected causal effect of the minimum wage on that outcome under that DGP.
What does “significantly different” mean in this context? For employment effects, the estimator is considered biased if the gap between expected estimate and expected causal effect is at least 0.005 (half a percentage point) and at least 25% of the true causal effect. For quantile gaps, the estimator is biased if the gap is at least 0.05 (five log points) and at least 25% of the true causal effect. For example, an estimator is considered biased for the p10-p50 quantile gap if the expected estimate is 0.04 while the expected causal effect is 0.09. However, it is not classified as biased if the expected estimate is 0.48 and the expected causal effect is 0.40.
Table (ref) reports the results of that exercise. To provide additional clarity on when each estimator is expected to perform better or worse, I split DGPs in three groups: positive employment effects (0.005 or more), negative employment effects (-0.005 or less), and small employment effects but significant wage effects.\footnote{To be included in the analysis, the DGP must be such that the minimum wage causes effects of magnitude at least half a percentage point on the employment to population ratio or 5 log points on at least one quantile gap. This choice rules “trivial” cases with non-binding minimum wages. I also exclude DGPs where the causal effects of the minimum wage on the employment to population ratio is of four percentage points or more, either positive or negative. This criterion ensures that the biases are not arising from atypical scenarios such as a large minimum wage hike in a country where latent log wages have low dispersion.} All numbers with two decimal places in the table correspond to the share of DGPs in a given group for which one of the estimators (rows) is positively or negatively biased for a given outcome (columns).
The broad message of the table is that, in all three categories of DGPs, both estimators are more likely than not to be biased for at least one outcome, and often for two or more. A careful examination reveals some interesting patterns. The effectively minimum wage design is more likely to be positively- than negatively-biased for employment effects, especially when the true effects are negative. For the fraction affected design, the opposite is true: it is more likely to be negatively biased for employment effects, especially when the true effects are positive. The effective minimum wage design is also more likely to overestimate, rather than underestimate, the degree to which minimum wages reduce inequality in the lower tail of the wage distribution.
Appendix Table (ref) repeats this exercise using a more stringent requirement for classifying an estimator as biased. Specifically, it requires gaps between expected estimates and expected causal effects that are twice as large as in the baseline exercise. It finds that large biases are common for all combinations of estimator and DGP group. Taken together, these results suggest that the reliability of the effective minimum wage and fraction affected designs should not be taken for granted if the data does not include regional variation in minimum wage laws.
In Subsection (ref), I discussed using estimates of wage effects in the upper tail of the wage distribution for model selection. Appendix Table (ref) replicates the exercise in Table (ref) restricting attention to DGPs where the effective minimum wage estimate of the p90-p50 quantile gap effect is less than 0.05 in magnitude. This procedure approximates a two-step process where the econometrician first measures the upper-tail spillovers and abandons the project if they find large effects.
The first takeaway from that table is that there are few DGPs without upper-tail spillovers: only 77 out of 500. This result is consistent with the fact that papers using the effective minimum wage design commonly detect upper-tail spillovers. Using US data, both Lee1999 and Autor2016 find evidence of such spillovers in some specifications and argue that they should be used for model validation. In the Brazilian context, Hinojosa2019 and Engbom2022 also detect upper-tail spillovers but interpret them as causal effects, rather than evidence of misspecification.
The second takeaway is that, while the incidence of biases is indeed reduced in that subsample, they are not completely eliminated. For example, there remains a 16% chance of positive biases on the p10-p50 effect when the true effect on employment is positive, and a 40% chance of positive bias on the employment effect when the true employment effect is negative.
In Appendix Table (ref), I implement a similar exercise for the fraction affected design: estimates are considered invalid if the estimator detects positive effects on the 90th quantile of the wage distribution. This procedure is less stringent than the one for the effective minimum wage, leaving 345 out of 500 DGPs. However, economically significant biases remain common. Thus, a lack of wage effects on the upper tail is no guarantee of unbiasedness for the fraction affected design.
All of the simulations performed so far used the Normal-markdown class of simulation models. That class is very flexible and can approximate a broad range of economic models. However, it imposes symmetry of the latent log wage distribution and implicitly assumes that the marginal returns to labor are invariant to the minimum wage. To address those limitations, I perform an additional simulation exercise based on an equilibrium labor market model very similar to that of Katz1992, with parameters calibrated to reflect US states as of 1989.
Appendix (ref) describes that exercise and presents the results. They mirror previous results for the fraction affected estimator regarding functional form issues, except that they now extend to the effective minimum wage design as well. Specifically, model misspecification introduces economically significant biases even in cases that, conceptually, should be ideal applications of those estimators. These results suggest that simulation results based on the Normal-markdown model may be conservative in the sense that misspecification biases can be larger in cases where minimum-wage-induced employment effects or reallocation introduce changes in marginal returns to labor.
This section discusses the limitations of the effective minimum wage and fraction affected estimators in a specific context: the rise of the Brazilian federal minimum wage beginning in 1995, illustrated in Figure (ref). Brazil underwent a series of macroeconomic reforms in 1993 and 1994 which successfully stabilized the economy after years of hyperinflation. Over the 22 years following stabilization, three different presidents promoted yearly adjustments that, in total, increased the minimum wage by 167 percent in real terms. Productivity also grew, but there is no question that the minimum wage became more binding. The red dash-dot line in Figure (ref) shows that the minimum wage grew substantially relative to median wages. Because the Brazilian workforce became more skilled over the same period, the relative change is more stark if one conditions on a specific worker group, such as young men with high school degrees (the blue dashed line).\footnote{Young, high school-educed men are a positively selected subsample earlier in the studied period because men earned significantly more than women, and because most workers had less than complete high school.}\footnote{Figure (ref) in the appendix displays alternative bindingness measures based on shares of workers earning close to the minimum wage. They reinforce the view that the “bite” of the minimum wage grew substantially before 2007.}
This setting is ideal for analyzing the performance of the effective minimum wage and fraction affected estimators for four reasons. First, Brazil is a large country with substantial variation across regional labor markets. Second, there is no regional variation in minimum wage laws, making those estimators more appealing.\footnote{Brazilian law allows for state- and occupation-specific wage floors. However, they exist in few states, apply to few occupations, and are imperfectly enforced Moura2008,Corseuil2015.} Third, there are several published papers and recent working papers using those estimators in the Brazilian context, allowing me to discuss how those strategies are implemented in practice and to evaluate their results. Fourth, in previous work Haanwinckel2023, I developed a structural model that predicts minimum wage impacts at the regional level and estimated it using a panel of Brazilian regions. This model yields region- and time-specific estimates of minimum wage effects that are directly comparable to those coming from the reduced-form designs, but that are not subject to the econometric issues discussed in this paper. Thus, they can be used as benchmark values for minimum wage effects in this context.
In the following, I first show that existing papers often come to conflicting conclusions about its impacts on employment and wages, and that econometric theory provides little guidance on which of them---if any---is correct. Next, I conduct an original data analysis based on Brazilian data to investigate the sensitivity of those estimators to arbitrary specification choices, and to benchmark the estimates I find against those coming from the structural model.
I focus on the six papers listed in Table (ref), which were selected based on a search on the IDEAS database.\footnote{The search was executed on 10/29/2024. I used the keywords “Brazil minimum wage” and included published papers from 2000 or later, or working papers from 2019 or later, that (i) used the methods studied in this paper; and (ii) used data from the post-estabilization period (i.e., after 1994). Derenoncourt2021 was not present in the IDEAS data set, but was included because it was known to me through participation in conferences.} The table summarizes the data sources for each paper, the preferred econometric specification, and the main results.
Even though all of those papers use regional variation in wage levels to identify the effects of the same national minimum wage increase, they often reach different conclusions. Parente2024 estimates an increase in informal employment relative for formal employment, while Derenoncourt2021 and Engbom2022 estimate zero effects on that margin. Parente2024 also finds that wage dispersion rises for all workers (formal and informal), while Hinojosa2019 finds the opposite. Neumark2006 finds negative employment effects, while most other papers looking at that outcome do not.\footnote{Using a different design (still based on regional variation in wage levels), Hinojosa2019 also finds negative and statistically significant effects on employment, though the magnitude of those effects is smaller than in Neumark2006. Studying the period from 1982 to 2000 with PNAD data, Lemos2009IntRev also reports that estimates of employment effects are sensitive to the choice of estimator, with some specifications leading to negative employment effects and others yielding zero effects.} Estimates of spillover effects are different in Hinojosa2019 relative to Engbom2022.
\defcitealias{Derenoncourt2021}{Derenoncourt, Gérard, Lagos, and Montialoux (2021)}
An even clearer illustration comes from the sensitivity analysis from Engbom2022. The central component of their masterful study of the Brazilian minimum wage is a structural labor market model with search frictions and two-sided heterogeneity. But in addition, their motivating descriptive analysis is thorough and exceptionally transparent. Figure B.14 in their Online Appendix, replicated as Appendix Figure (ref) here for the reader's convenience, shows that their effective minimum wage estimates vary dramatically depending on specification choices. Based on previous discussions in Subsections (ref) and (ref), it may be warranted to restrict attention to specifications that include time fixed effects and that use the median wage as the deflator, as opposed to the 90th percentile. Even then, estimates still differ substantially depending on the inclusion of region fixed effects or region-specific time trends.
In many cases, disagreements between papers studying the same empirical context can be informative. For example, estimates of returns to education may differ depending on whether an instrument is used or not, or which specific instrument is used. Researchers can rationalize those differences based on transparent discussions of identification assumptions: the presence or not of omitted variable bias, instrument validity, and who is the complier population for each instrument.
However, in the case of the papers in Table (ref), it is difficult to argue which specification should be preferred over the others because the source of identification is not as transparent. In the first three papers in that table (using effective minimum wage designs), identification relies on the existence an unobserved shock that shifts the location of the latent wage distribution, but is orthogonal to determinants of its shape and of latent employment levels. None of those papers specify what economic factor could play that role. The most relevant shocks affecting Brazilian labor markets between 1995 and 2018 are changes in educational achievement, trade liberalization, the commodities boom, import competition from China in the manufacturing sector, increased enforcement of labor regulations, the expansion of technologies such as broadband, computers, and numerical control machines, and social programs like conditional cash transfers FirpoPortella2019. All of those factors are biased towards either low-, mid-, or high-skilled workers (meaning that they are not simply a locational shift of the log wage distribution), and all are likely to have independent effects on employment.
Similar arguments can be made for latter three papers in Table (ref), which use difference-in-differences estimators analogous to the fraction affected design. The parallel trends assumption cannot be tested given the lack of a pre-treatment period with a stable minimum wage. In addition, the long-run estimates from the last two papers can be subtantially biased if there are other structural factors causing regional convergence Manzi2023,Abreha2024.
One may wonder whether controlling for all of the shocks known to affect the Brazilian economy in this period would be sufficient to ensure unbiased estimation. Indeed, this is one motivation behind the use of region-specific trends in the effective minimum wage papers.\footnote{Whether state-specific trends adequately capture the impact of those factors depends on whether their joint effect in each state is linear, or at least more linear than the impact of the minimum wage. I discuss this assumption in Appendix D.8.4 of Haanwinckel2023.} But using controls only tells us what is not being used as the source of identification. The estimators still rely on the existence of unobserved regional parameters satisfying some properties---except that now these parameters are defined as residuals, netting out the contribution of the included controls. Without knowing what that residual latent parameter is, it is impossible to evaluate the identification assumptions. In addition, the inclusion of controls can magnify biases arising from misspecification if they reduce the variance of identifying variation.
The previous discussion has two limitations. First, some of the conflicting results in existing literature may be coming from using different samples, different treatment of dynamic issues, or different ways to display the results (though Figure B.14 in Engbom2022 goes a long way in addressing this limitation). Second, one would ideally want to benchmark the potentially problematic numbers using estimates that are not subject to the biases discussed in this paper.
In this final analysis, I address those limitations with quantitative exercises based on Brazilian data. To keep the analysis comparable to the remainder of the paper, the data includes exactly two periods: 1998 and 2012, the same years used in Haanwinckel2023. This exercise aims at the long-run effects of the minimum wage, starting from a period a few years after the stabilization reforms and ending at a period where the economy has had a few years to adjust to the higher minimum wage levels (from Figures (ref) and (ref), bindingness measures are mostly stable following 2007). The outcomes of interest are formal employment and formal wages. They are measured using an administrative matched employer-employee data set (“Relação Anual de Informações Sociais,” henceforth RAIS).
Each spatial unit is a “microregion” from the Brazilian Statistical Bureau (IBGE). These units are analogous to commuting zones in the United States in terms of size, population, and in that less than 5% of workers live in one microregion and work in another. They are commonly used to determine the boundaries of labor markets in studies using Brazilian data Costa2016,DixCarneiro2017,Ponczek2021.\footnote{Microregions are grouped to ensure constant boundaries throughout the studied period. Following the sample selection criterion in Haanwinckel2023, I select regions with at least 15,000 workers in the RAIS data set in each year, and at least 1,000 in each educational group (less than high school, high school, and complete college or more). Those criterion select 151 out of 486 microregions that, together, account for 73% of the adult population in Brazil.}
The reference estimates of minimum wage effects come from Haanwinckel2023. In that paper, I develop an imperfectly competitive labor market model with worker and firm heterogeneity to study the impact of rising schooling achievement, labor demand shocks, and the federal minimum wage on Brazilian labor markets. The predicted impacts of the federal minimum wage incorporate several channels discussed in previous literature: disemployment of very low productivity workers (i.e., truncation of the latent log wage distribution); mechanical wage increases for some workers (i.e., censoring); increase in employment for some worker types due to the presence of monopsony power; reallocation of workers to firms with higher wage premiums (as in Dustmann2021); changes in returns to skill, both in the aggregate economy and differential changes within firms; effects on the composition of firms operating in the economy (as in Aaronson2018); and pass-through effects to consumer prices (as in Harasztosi2018).
The magnitudes of those mechanisms are regulated by parameters which, in turn, are estimated using a simultaneous equation nonlinear least squares procedure. The estimator targets several endogenous outcomes: employment rates, measures of wage inequality between and within educational groups, the contribution of firm wage premiums to inequality, and a measure of sorting of high-wage workers to high-wage firms Kline2018. To account for potential confounders on the labor supply side, the model directly incorporates the role of education in affecting labor market outcomes such as the distribution of, and returns to, skill. The model also includes three types of unobserved labor demand shifters at the regional level, which are partly determined by initial sectoral shares in agriculture and manufacturing. These labor demand shocks absorb the influence of important transformations mentioned above, such as trade liberalization and technical change.\footnote{Appendix Table (ref) shows that the estimated model fits the original data well. See Haanwinckel2023 for additional information on the model and identification.}
Table (ref) reports the average causal effects of the minimum wage predicted by that structural model, along with estimates of causal effects using different versions of the effective minimum wage and fraction affected estimators based on the same data. The structural model predicts sizable disemployment effects: a reduction of 3.3 percentage points in the share of the adult population (18 to 54 years old) employed in the formal sector. Both the baseline fraction affected design and the baseline effective minimum wage design also predict reductions in formal employment that are economically and statistically significant. However, the magnitudes differ a lot. For the fraction affected, the decline is smaller at 1.9 percentage point, while for the effective minimum wage it is larger at 6.6 percentage points.
The structural model predicts sizable compression of wages in the lower tail of the wage distribution, but no inequality-reducing effects on the upper tail.\footnote{Wage levels rise by about nine log points in that part of the distribution because the disemployment of low-skilled workers means that the same quantile of the wage distribution corresponds to more skilled workers following the minimum wage hike.} The effective minimum wage estimate is very close for the p10-p50 quantile gap, but overestimates the compression of the p25-p50 gap relative to the structural model. It also predicts an increase in inequality in the upper tail. The baseline fraction affected estimate predicts large wage effects that are mostly uniform over the wage distribution. To the extent that there is wage compression, it happens in the upper tail.
Table (ref) also reports estimates for alternative estimators. They encompass using binary versions of the fraction affected design, changing the fixed effects included in the effective minimum wage design, using an instrumental variables version of this estimator, or constructing the effective minimum wage using the 90th percentile as the deflator, instead of the median. Those specifications mirror the variety of implementations in existing literature, summarized in Table (ref). Each of those estimators finds different numbers for the causal effects of interest, helping explain the diversity of results in existing literature. If one takes the structural estimates as the correct benchmark, the baseline specifications of the effective minimum wage and fraction affected estimators seem to perform better than the alternative ones.
All in all, the analysis of the Brazilian case shows that when researchers try use regional variation in wage levels to identify the effects of a national minimum wage, the estimates may differ substantially based on specification choices for which economic theory provides little to no guidance. Based on these results, practioners are urged to pinpoint the source of identification in their empirical context, discuss threats to identification, and report results from several specifications to gauge the sensitivity to arbitrary choices.
In this paper, I analyzed the performance of two classes of estimators of the employment and wage effects of minimum wages in contexts where that policy is set at the national level. I discussed the key source of identifying variation for each of them, showed that identification assumptions required for unbiased estimation are stronger than what existing literature documents, and discussed potential solutions via adjustments of the estimation procedures.
If the data in a specific application includes several pre-treatment periods when the minimum wage was constant, then the “fraction affected” or “gap” designs should be the preferred choice. In that case, the specific recommendations coming from this paper are (i) to construct the treatment intensity variable using only one pre-treatment year rather than averaging over years; (ii) to check for differential pre-trends; (iii) to assess the possibility of regression to the mean/regional convergence and, if needed, account for that in the estimation model; (iv) to check for trends in the dispersion of log wages in the pre-treatment period, which may cause biases as discussed in Section (ref); and (v) not to use a binary version of the treatment intensity variable. Even if the econometrician takes such precautions, the estimator may be subject to economically significant misspecification biases arising from functional form assumptions, especially when the minimum wage increase is significant. Econometricians may consider reporting results of both the “fraction affected” and “gap” designs, and also a version of those estimators that included a quadratic term, to partially assess the relevance of this problem in a particular application.
If the empirical context does not feature a stable pre-treatment period, then the validity of estimates from the “fraction affected” and “gap” designs cannot be evaluated. In such cases, it may be tempting to consider the effective minimum wage design. However, my analysis shows that this estimator relies on assumptions unlikely to hold in any application without regional differences in minimum wage regulation. Among variations of the effective minimum wage design, the best-performing ones include time fixed effects and construct the effective minimum wage using the median wage (as opposed to a higher wage quantile). However, even in these cases, biases can be considerable. The effective minimum wage design should only be trusted if there exists an economic factor that increases the bite of the minimum wage in some regions compared to others but that does not affect the shape of the wage distribution or employment levels in any other way (conditional on the controls included in the design). If such a variable is available, the econometrician should consider an instrumental variables design that directly exploits it as an instrument, as explained in Appendix (ref).
In contexts that are not well-suited to either estimator, the econometrician may consider two alternative strategies. One is to employ within-region difference-in-differences designs, where the unit of analysis is either firms Harasztosi2018 or workers Dustmann2021, and treatment and control groups are defined based on initial wages. As discussed at the end of Section (ref), such designs do not directly identify the policy counterfactual analyzed in this paper; recovering those effects from the estimated coefficients requires additional structural assumptions. Moreover, the econometrician should be aware that some of the concerns discussed earlier—such as regression to the mean—may still apply. See Dustmann2021 for an example of how this issue can be addressed in a worker-level analysis.
The second alternative strategy is to estimate a parametric structural model; see, for example, Engbom2022 and Haanwinckel2023 in the Brazilian context. Such models can use information from the data to quantify and correct for potential sources of bias. Those solutions come at the cost of higher complexity and the need to pre-specify the causal pathways through which the minimum wage affects the economy. A promising direction for further research is developing an econometric model that is simple to implement and agnostic about economic channels, but that adequately controls for the issues studied in this paper.
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