# Does regional variation in wage levels identify the effects of a national
minimum wage?

Daniel Haanwinckel Department of Economics, UCLA and NBER. Contact
information<EMAIL_ADDRESS>
This paper benefited from conversations with Martha Bailey, Adriana Lleras-
Muney, Maurizio Mazzocco, Yotam Shem-tov, and many others at UCLA, as well as
with participants of the 2023 Columbia Junior Micro/Macro Labor Conference,
the 2023 NBER Labor Studies Fall Meeting, and the 2024 SOLE Meeting. Filipe
Fiedler provided excellent research assistance.

(May 2024)

###### Abstract

I evaluate the performance of estimators that exploit regional variation in
wage levels to identify the employment and wage effects of national minimum
wage laws. For the "effective minimum wage" design, I show that the
identification assumptions in Lee (1999) are difficult to satisfy in settings
without regional minimum wages. For the "fraction affected" design, I show
that economic factors such as skill-biased technical change or regional
convergence may cause parallel trends violations and should be investigated
using pre-treatment data. I also show that this design is subject to
misspecification biases that are not easily solved with changes in
specification.

##  1 Introduction

One approach to measuring the effects of minimum wage regulations involves
leveraging variation in wage levels across different regions within a country.
For instance, the influence of the US federal minimum wage is more pronounced
in states like Mississippi or Arkansas than in Texas or Georgia due to the
higher median wages in the latter states. Thus, one can use wage distributions
to construct treatment intensity measures for a “differences-in-differences”
analysis. Classic applications using cross-state variation from the US are
Card (1992), who introduced the “fraction affected” design, and Lee (1999),
who introduced the “effective minimum wage” design. This strategy is
particularly convenient in countries with little or no spatial variation in
minimum wage laws. Examples of such applications published by leading economic
journals include Mexico (Bosch and Manacorda, 2010), South Africa (Dinkelman
and Ranchhod, 2012), Germany (Dustmann et al., 2021), the US in the 1960s and
1970s (Bailey, DiNardo and Stuart, 2021), and Brazil (Engbom and Moser, 2022).

This paper examines the identification assumptions underlying those
econometric approaches. Specifically, it uses a combination of economic theory
and simulation exercises to investigate whether fraction affected and
effective minimum wage designs can accurately capture the causal effects of
the minimum wage on employment and wages, focusing on applications where the
minimum wage is set at the national level.

For the effective minimum wage design, I show that two identification
assumptions emphasized by Lee (1999) are crucial for unbiased estimation of
causal effects but very difficult to satisfy in applications where there is no
regional variation in minimum wage laws. The first assumption is that observed
median log wages are a good proxy for the centrality of the latent log wage
distribution—that is, the distribution of log wages that would prevail with no
minimum wage in place. I show that employment effects lead to violations of
that assumption by introducing correlated measurement errors in the design.
The ensuing biases can be significant regardless of whether the average
employment effects are negative, zero, or positive, and even if the
correlation between the unobserved centrality of the latent wage distribution
and the observed median log wage is above 0.99.

The second assumption in Lee (1999) is that overall wage levels should be
uncorrelated with the dispersion of latent log wages across regions. This
assumption may be violated if, for example, regions differ in the share of
skilled workers, and there is more latent wage dispersion for skilled workers
(Lemieux, 2006). I show that correlations as small as 0.07—no larger than what
one would infer from US data—can introduce large biases.

I discuss the effectiveness of potential solutions to those problems with the
effective minimum wage design. They include 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 or trends included in the regression. The general message is
that the design is unlikely to recover true causal effects unless there exists
an economic factor that (i) changes the “bite” of the minimum wage in some
regions compared to others; (ii) is otherwise unrelated to employment outcomes
and the shape of the log wage distribution, conditional on the fixed effects
and controls included in the regression; and (iii) has sufficient residual
variance. If the data includes a variable that is a good candidate for being
that economic factor, then an instrumental variables approach that directly
exploits it is more likely to be successful.

Next, I discuss fraction affected and gap designs implemented at the regional
level (this paper does not address estimators that compare firms within the
same region based on the firm-level share of affected workers, e.g. Card and
Krueger, 1994; Harasztosi and Lindner, 2019). The key identification
assumption is transparent in those designs: in a counterfactual scenario
without an increase in the national minimum wage, trends in outcomes would be
orthogonal to the treatment intensity variable used, conditional on the
controls and fixed effects used in the design. My analysis points out three
potential issues. First, I show that structural factors that may seem
unproblematic, such as a common trend in the dispersion of latent log wages
affecting all regions in the same way (due, e.g., to skill-biased technical
change), may cause violations of the parallel trends assumption. Second, the
design is subject to bias from regression to the mean in regional wage
statistics, originating from sampling variation when constructing those
statistics and from region-specific productivity shocks. Third, the design is
sensitive to the functional form chosen for the treatment intensity variable,
with misspecification biases being more significant when the minimum wage is
more binding or when it causes an increase in the number of workers earning a
bit more than the minimum wage.

I also discuss possible diagnostics and solutions to those issues. I show that
tests for differential pre-trends may effectively detect the first two issues
when the data includes a “pre-treatment” period without significant changes in
minimum wage laws, provided that the econometrician is careful when
implementing the test. Similarly, if sufficient pre-treatment data is
available, regression to the mean can be controlled using the procedure
discussed in Dustmann et al. (2021). For the misspecification issue, I
consider three potential solutions: binary treatment intensity measures,
quadratic specifications, and using one treatment intensity variable as an
instrument for the other. Neither of these approaches solves the problem, and
the binary version typically displays larger biases than the continuous
fraction affected or gap specifications.

The paper is structured as follows. Section 2 introduces the setup, clarifies
the scope of my analysis, and shows how it relates to other papers on the
econometrics of minimum wage. Sections 3 and 4 analyze the effective minimum
wage and the fraction affected estimators, respectively. In Section 5, I show
that simulation results shown in Sections 3 and 4 also hold when using
alternative economic models as the data-generating processes. The final
section concludes with a summary of recommendations for researchers interested
in measuring the effects of national minimum wage changes in contexts without
regional variation in minimum wage laws.

##  2 Setup and relationship to literature

I consider two-period ($t\in\\{0,1\\}$) data-generating processes (DGP) of the
following form:

$\displaystyle\bm{y}_{r,t}$
$\displaystyle=f\left(mw_{t},\bm{\theta}_{r,t}\right)$ (1)
$\displaystyle[\bm{\theta}^{\prime}_{r,0},\bm{\theta}^{\prime}_{r,1}]^{\prime}$
$\displaystyle\sim G$

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, as determined by the distribution $G$, but they are
independent across regions.

I assume henceforth that $mw_{1}>mw_{0}$.111This assumption is without loss of
generality given the data-generating process above, as outcomes depend only on
current values of $mw_{t}$ and $\bm{\theta}_{r,t}$. It would not be without
loss of generality in a model that accounted for nominal rigidities or other
dynamic concerns. See the discussion at the end of this section. Given the
data-generating process described above, there are two natural ways to define
the ceteris paribus causal effects of the rise in the national minimum wage:

$\displaystyle\bm{ATE}_{0}$
$\displaystyle=\mathbb{E}\left[f\left(mw_{1},\theta_{r,0}\right)-f\left(mw_{0},\theta_{r,0}\right)\right]$
$\displaystyle=\mathbb{E}\left[f\left(mw_{1},\theta_{r,0}\right)\right]-E_{G}\left[\bm{y}_{r,0}\right]$
$\displaystyle\bm{ATE}_{1}$
$\displaystyle=\mathbb{E}\left[f\left(mw_{1},\theta_{r,1}\right)-f\left(mw_{0},\theta_{r,1}\right)\right]$
$\displaystyle=\mathbb{E}\left[\bm{y}_{r,1}\right]-E_{G}\left[f\left(mw_{0},\theta_{r,1}\right)\right]$

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 $G$ distribution is time-invariant, an assumption that holds
for only some of the DGPs I study. For consistency throughout the paper, I use
the average of these two definitions as the object of interest to be recovered
by the econometric designs:

$\bm{ATE}=\frac{\bm{ATE}_{0}+\bm{ATE}_{1}}{2}$

All exercises in the paper 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, though I explore idiosyncratic, mean-zero 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 (1) 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 the fact that minimum
wage effects may take some time to materialize. Using Canadian data, Baker,
Benjamin and Stanger (1999) document that employment negatively responds to
low-frequency minimum wage variation. In contrast, the response to the high-
frequency component of the variation is positive (though statistically
insignificant). Sorkin (2015) builds a putty-clay model where firms cannot
easily adjust labor inputs in the short run. Using this model, he argues that
short- and long-run effects of the minimum wage can be very different and that
it may be impossible to identify long-run effects through econometric designs
if the data does not contain permanent shocks to the real minimum wage (which
is often the case, as minimum wage increases are eroded by inflation over a
few years). Meer and West (2016) argue that, in many economic models
(including the putty-clay model of Sorkin, 2015), minimum wages cause changes
in employment growth rates, not levels. Then, standard difference-in-
differences designs may fail to capture the actual effects of the minimum wage
on employment, especially if the designs include region-specific time trends.
Vogel (2023) documents that minimum wage effects are concentrated on the
bottom of the wage distribution on impact but “trickle up” over the next 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 (1) does not include measurement error, meaning that the issues I
discuss in this paper are also different from the mechanical bias issue
discussed in Autor, Manning and Smith (2016). These authors propose an
instrumental variables estimator as an improvement over the effective minimum
wage design of Lee (1999). Later in the paper, I will show that the estimator
of Autor, Manning and Smith (2016) can also solve some of the issues I
document. However, that strategy is only feasible with alternative data-
generating processes that include 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. See Huang (2019),
Chapter 2 for an investigation of whether such spillovers create biases in the
context of the US.

Finally, the issues I discuss complement recent papers studying the
econometrics of difference-in-differences models. The restriction to a two-
period model means that issues related to staggered treatment and treatment
effect heterogeneity across groups do not apply; see de Chaisemartin and
D’Haultfœuille (2020) and Roth et al. (2023). Callaway, Goodman-Bacon and
Sant’Anna (2024) study difference-in-differences models with continuous
treatment intensity variables, a group that includes the fraction affected-
style regressions I study in Section 4. Part of my results can be interpreted
as showing that, under a range of economic models, the fraction affected
design and related estimators do not satisfy the “strong parallel trends”
assumption defined by Callaway, Goodman-Bacon and Sant’Anna (2024), causing
biases that may be large in some cases and whose sign is difficult to predict.

##  3 The effective minimum wage design

###  3.1 Definition

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:

$y_{i,r,t}=\alpha_{i,r}+\delta_{i,t}+\beta_{i}\left[mw_{t}-w_{0.5,r,t}\right]\\\
+\gamma_{i}\left[mw_{t}-w_{0.5,r,t}\right]^{2}+\epsilon_{i,r,t},$ (2)

where $i$ indexes the particular 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. It measures how binding the national minimum
wage is in a particular region and time, using the observed median wage as the
benchmark. This baseline specification includes region and time fixed effects,
following the bulk of the literature. I discuss alternative specifications
later.

To calculate the predicted treatment effects of the minimum wage increase in
each region $r$, 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 the estimated average treatment effects of the
national minimum wage increase following the definition from Section 2.222That
is, the predicted average treatment effects of the minimum wage are:
$\widehat{ATE}_{q}=\frac{1}{R}\sum_{r}\left\\{\hat{\beta}_{q}\left[\left(mw_{1}-w_{0.5,r,1}\right)-\left(mw_{0}-w_{0.5,r,0}\right)\right]+\hat{\gamma}_{q}\left[\left(mw_{1}-w_{0.5,r,1}\right)^{2}-\left(mw_{0}-w_{0.5,r,0}\right)^{2}\right]\right\\}$
The econometrician could be interested not in the average change in region-
specific quantile gaps but in the change in quantile gaps based on the
national log wage distribution. I focus on the former definition because it is
more closely linked to the regression model.

This regression model was first introduced by Lee (1999), who focused on
quantile gaps as the outcomes of interest. The design was later used to
estimate employment effects as well; Engbom and Moser (2022) is one
example.333Such 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 Neumark and Wascher (1992), 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
notation from Lee (1999). Suppose 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. Assume that the cumulative distribution
function for those latent log wages has the form:

$F_{t}\left(\frac{w-\mu_{r,t}}{\sigma_{r,t}}\right)$

where $\mu_{r,t}$ and $\sigma_{r,t}$ are the centrality (or location) and
dispersion parameters, respectively.

Using this notation, Lee (1999) 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 (2)—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 included in the regression.

The central message of my analysis below is that those assumptions are
essential but unlikely to hold in practice for economic reasons. Later, I will
also show that the issues are less severe in contexts with regional variation
in the minimum wage, and in those contexts, the instrumental variables
estimator of Autor, Manning and Smith (2016) is preferable.

###  3.2 Issue #1: Imperfect measurement of latent centrality

The effective minimum wage design is predicated upon minimum wage effects
being stronger where it bites more into the latent wage distribution.
Following that logic, 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 even
minor deviations between $w_{0.5,r,t}$ and $\mu_{r,t}$ are enough to introduce
economically significant biases.

####  3.2.1 Good and bad variation

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 1
illustrates this scenario.

Panel A: Good variation arising from a shock to location, $\Delta\mu_{B}$

Panel B: Bad variation arising from differences in dispersion,
$\sigma_{A}>\sigma_{B}$

Panel C: Bad variation arising from a shock to dispersion, $\Delta\sigma_{B}$

Figure 1: Good and Bad Variation in the Effective Minimum Wage Design

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 $\mu_{r,t}$.444One 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 Footnote
2 that after estimating Equation (2), 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 will 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 1. The minimum wage has no bite in that region because latent
wages tightly 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 1 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, it would only be possible
to know the direction of the bias by making stronger assumptions about the
data-generating process. 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 have
a markdown parameter of $m=0.7$.

Figure 2 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.

Figure 2: Minimum wage effects in the baseline simulation model

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
$t=1$).555For 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 A.1 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 and how heterogeneous they are
across regions.

Table 1: Effective minimum wage design: good vs. bad variation

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
Panel A: Regions differ only in location
True average causal effect | -0.010 | 0.019 | 0.006 | -0.004
Effective min. wage | -0.010 | 0.020 | 0.006 | -0.004
| (0.000) | (0.001) | (0.000) | (0.000)
Panel B: Regions differ in location and dispersion
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.007 | 0.034 | 0.015 | -0.023
| (0.002) | (0.011) | (0.007) | (0.014)
Panel C: As above, but larger increase in min. wage
True average causal effect | -0.032 | 0.078 | 0.017 | -0.012
Effective min. wage | -0.014 | 0.117 | 0.046 | -0.080
| (0.007) | (0.020) | (0.012) | (0.026)
Panel D: St. dev. of dispersion is 50% larger
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.003 | 0.050 | 0.025 | -0.047
| (0.003) | (0.017) | (0.010) | (0.021)

Notes: This table summarizes simulation results with 200 regions and two
periods. The top row in each panel reports the average of 1,000 simulations of
the true $ATE_{i}$ for different outcomes $i$, corresponding to different
columns. The second row shows estimated average treatment effects for each
outcome based on the effective minimum wage regressions, averaged over the
same 1,000 simulations. The third row shows the average over simulations of
the corresponding standard errors, which are clustered at the region level in
each simulation. The data-generating process includes truncation and censoring
effects of the minimum wage. Each panel corresponds to different assumptions
on the data-generating process. In Panel A, regions differ only in the
location parameter $\mu_{r,t}$, with a correlation between the initial and
final location of 0.89. Panel B includes differences in the dispersion
parameter $\sigma_{r,t}$, with a correlation between initial and final
dispersion of 0.46. Panel C is like Panel B but with an increase of the log
minimum wage of 0.4 instead of 0.2. Panel D increases the between-region
standard deviation of the $\sigma_{r,t}$ parameters by 50%. See Appendix A.1
for details on the calibration of the model.

The first panel in Table 1 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 very 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 Lee (1999):
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 A1 in Appendix A shows that their correlation is 0.999.
Still, the estimator displays economically meaningful 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.666The direction of the biases suggests that the
bias coming from shocks to dispersion parameters $\mu_{r,t}$ are the main
problem in the simulations, instead of the permanent differences in
dispersion. Among the problems illustrated in Figure 1, the one in Panel C
seems the most relevant. In unreported simulations, I confirm this intuition,
noting 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 more significant 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 A1), measured employment effects are only 40% as large in magnitude
as the true causal effects. At the same time, the lower-tail spillovers are
amplified by 50%. 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 $\mu_{r,t}$. Thus, a larger minimum wage shock
does not help reduce bias. On the contrary, it amplifies biases because the
effects on observed median wages become stronger.

Panel D highlights how differences in the dispersion of latent log wages
between regions constitute the source of the bias. Making those differences
50% larger in magnitude while keeping the other parameters constant is enough
to essentially 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 2. Such effects could emerge from
increased search effort by workers (see Adams, Meer and Sloan, 2022, for an
empirical evaluation of this hypothesis in the context of the US). The details
on that model are provided in Appendix A.1.3.

Results are reported in Appendix Tables A3 and A4. The simulations are
analogous to those shown in Table 1, but the average employment effects of the
minimum wage are zero (in Table A3) or positive at around one percentage point
(in Table A4). These alternative models also display economically significant
biases, though they are smaller than in Table 1. This exercise clarifies that
the econometric problem arises from idiosyncratic, rather than average,
differences between $\mu_{r,t}$ and $w_{r,t}$.

###  3.3 Issue #2: Correlation between location and dispersion parameters

The second assumption emphasized by Lee (1999) 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.

Below, I show through simulations that even a mild contemporaneous correlation
between location and dispersion parameters can introduce significant biases in
the effective minimum wage design. Next, I argue that there are plausible
economic reasons why we should expect such correlations to occur.

Table 2: Correlation between location and dispersion parameters

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
Panel A: No correlation between location and dispersion
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.007 | 0.033 | 0.014 | -0.022
| (0.002) | (0.011) | (0.007) | (0.014)
Panel B: Contemporaneous correlation of 0.076
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.002 | 0.077 | 0.040 | -0.076
| (0.002) | (0.010) | (0.006) | (0.013)
Panel C: Full correlation matrix in US data
True average causal effect | -0.010 | 0.019 | 0.006 | -0.004
Effective min. wage | -0.014 | -0.006 | -0.009 | 0.027
| (0.002) | (0.011) | (0.006) | (0.014)

Notes: Each panel displays average results for 1,000 simulations, each with
200 regions and two periods, for different assumptions on the data-generating
process (see the notes below Table 1 for an explanation of the table’s
structure). Panel A is identical to Panel B in Table 1: regions differ in
location ($\mu_{r,t}$) and dispersion ($\sigma_{r,t}$) parameters, but they
are orthogonal to each other. Panel B introduces a correlation of 0.076
between location and dispersion parameters within each period. Panel C uses
the full set of correlations between
$[\mu_{r,0},\sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$ observed in US data. See
Appendix A.1 for details on the calibration of the model.

Panel A in Table 2 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.

The US data also displays intertemporal correlations between location and
dispersion.777Specifically, initial location has a significant correlation
with final dispersion, and initial dispersion has a mild correlation with
final location. Panel C includes those correlations in the simulated model.
The biases now have the opposite sign. That result shows that it may be
difficult to predict the direction of the bias in empirical applications.

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 Lemieux (2006), 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
(Ferreira, Firpo and Messina, 2017). Then, if regions differ in workforce
composition, the correlation we discussed above may follow.

Education is not the only economic factor that can introduce problems for the
effective minimum wage design. Regional differences in endowments, leading to
heterogeneity in industrial composition, may also generate a correlation
between location and dispersion parameters. That is because industries—or
clusters of connected industries that tend to co-locate—may differ in wage
premiums and the breadth of occupations and skill levels used in production.

###  3.4 Fixed effects, trends, controls, and confounders

The baseline specification in Lee (1999) 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 3.2, 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 3 illustrates Lee’s argument through simulations. The data-generating
process for those simulations is the same as reported in Panel B of Table 1. 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. One example would be
that unregistered employment, not visible in the data, is more relevant in
low-wage areas, generating a spurious negative correlation between measured
employment-to-population and the effective minimum wage. Another is regional
differences in the supply of skills, corresponding latent log wage
distributions with different shapes (and thus different quantile gaps). In
this paper, I focus on the version with region fixed effects because those
concerns make it more popular in recent literature.

Indeed, the specifications in papers such as Bosch and Manacorda (2010),
Autor, Manning and Smith (2016), and Engbom and Moser (2022) 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. One example is changes in educational
composition, which, as discussed in the previous subsection, may affect both
the location and the dispersion of latent log wages. Another example is
demand-side shocks such as “the China syndrome” (Autor, Dorn and Hanson,
2013), whose wage effects are not uniform over the distribution and whose
employment effects may generate mechanical shifts in the median wage.

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
discussion on including 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).

Table 3: Effective minimum wage: alternative fixed effects specifications

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.007 | 0.033 | 0.014 | -0.022
| (0.002) | (0.011) | (0.007) | (0.014)
Effective min. wage, no region FE | -0.010 | 0.022 | 0.007 | -0.007
| (0.001) | (0.004) | (0.002) | (0.005)
Effective min. wage, no time FE | -0.007 | 0.052 | 0.024 | -0.041
| (0.001) | (0.003) | (0.002) | (0.004)

Notes: Each panel displays average results for 1,000 simulations with 200
regions and two periods. See the notes below Table 1 for an explanation of the
table’s structure. The data-generating process corresponds to Panel B from
Table 1; see Appendix A.1 for details. The table reports predicted average
treatment effects for three estimators: the baseline effective minimum wage
design and the same design without region effects or without time effects.

Based on this discussion, one may wonder whether dropping the time effects
from the design would add more “good” variation. Lee (1999) explains that this
choice is only wise if the econometrician believes that latent log wages’
shape and average dispersion do not change over time. It is not warranted in
the presence of secular wage trends coming from technical change or
international trade, for example.

Table 3 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.

###  3.5 Does using a higher quantile as the deflator help?

In some applications, the econometrician may have a strong 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,
Bosch and Manacorda (2010) use quantile 0.7 as the deflator in a study of
Mexico, and Engbom and Moser (2022) use quantile 0.9 when studying Brazil.

Table 4: Effective minimum wage using percentile 90 as the deflator

| Outcome
---|---
| Emp. | p10 - p90 | p25 - p90 | p50 - p90
True average causal effect | -0.010 | 0.024 | 0.009 | 0.000
Effective min. wage, p90 | 0.009 | 0.219 | 0.176 | 0.000
| (0.002) | (0.013) | (0.011) | (0.000)

Notes: This table has the same structure as Table 3, but reports regression
results where the effective minimum wage is calculated based on percentile 90
of the observed log wage distribution.

Lee (1999) 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. 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 4 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 3). 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.

###  3.6 Is the standard diagnostic test effective?

Lee (1999) 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. Autor, Manning and Smith (2016) use the same
specification test to validate their instrumental variables implementation of
the effective minimum wage design.

However, the econometrician must know that such a test is subject to both
false positives and false negatives. False positives—detecting a problem where
none exists—may arise because many plausible mechanisms could lead to minimum
wage spillovers that extend beyond the median wage. Engbom and Moser (2022)
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 Haanwinckel (2023) 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 Engbom and Moser (2022). Thus, an
econometrician with a strict rejection rule based on effects in the upper tail
may reject a valid model.

There are two concerns regarding false negatives. One is that the estimator
may be biased in the lower tail but not in the upper tail. This may happen if,
for example, the negative upper-tail bias illustrated in Table 1 is combined
with positive bias arising from measurement error, as discussed by Autor,
Manning and Smith (2016). Second, even if spurious upper tail spillovers are
detected, the econometrician may still interpret those results as not
indicating a problem if they have the prior that such spillovers are
economically plausible in the specific application.

###  3.7 State-level minimum wages and instrumental variables

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.

One may consider an instrumental variables (IV) estimator that isolates that
source of good variation. One approach is to use the prevailing institutional
minimum wage (and its square) as an instrument for the effective minimum wage
(and its square). In their pursuit of an effective minimum wage estimator
robust to measurement error, Autor, Manning and Smith (2016, henceforth AMS)
propose an IV estimator along those lines but include a third instrument: the
interaction of the log minimum wage with the average median wage in each
region. Because it uses observed median wages in its construction, this third
instrument may be subject to some of the abovementioned concerns.

Table 5: State-level minimum wages and instrumental variables approaches

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
Panel A: No regional variation in minimum wage.
True average causal effect | -0.010 | 0.020 | 0.006 | -0.004
Effective min. wage | -0.002 | 0.076 | 0.040 | -0.075
| (0.002) | (0.010) | (0.006) | (0.013)
Panel B: 20% of regions with local min. wage
True average causal effect | -0.015 | 0.035 | 0.008 | -0.005
Effective min. wage | -0.015 | 0.051 | 0.015 | -0.018
| (0.001) | (0.005) | (0.003) | (0.006)
Two instruments | -0.015 | 0.036 | 0.009 | -0.006
| (0.002) | (0.006) | (0.004) | (0.008)
Three instruments (AMS) | -0.017 | 0.042 | 0.009 | -0.005
| (0.002) | (0.005) | (0.003) | (0.006)
Panel C: 40% of regions with local min. wage
True average causal effect | -0.020 | 0.052 | 0.011 | -0.007
Effective min. wage | -0.019 | 0.059 | 0.015 | -0.016
| (0.001) | (0.004) | (0.002) | (0.005)
Two instruments | -0.020 | 0.051 | 0.011 | -0.007
| (0.002) | (0.004) | (0.002) | (0.006)
Three instruments (AMS) | -0.020 | 0.052 | 0.011 | -0.007
| (0.001) | (0.004) | (0.002) | (0.005)

Notes: Each panel displays average results for 1,000 simulations, each with
200 regions and two periods, for different assumptions on the data-generating
process (see the notes below Table 1 for an explanation of the table’s
structure). Models in all panels are similar to those from Panel B in Table 2,
where there is a small intra-temporal correlation between location
($\mu_{r,t}$) and dispersion ($\sigma_{r,t}$) parameters. Panels B and C
introduce region-specific minimum wages. They differ in the share of regions
with a local minimum wage higher than the national minimum wage. “Two
instruments” corresponds to regressions that employ the nominal minimum wage
and its square as instruments for the effective minimum wage and its square.
“Three instruments (AMS)” adds a third instrument following Autor, Manning and
Smith (2016). See Appendix A.1.4 for details.

Table 5 presents the outcomes of simulations that incorporate region-specific
minimum wages and implement alternative instrumental variables estimators. As
with the previous simulations, the parameters of the data-generating process
are tailored to mirror the US context; for more details, refer to Appendix
A.1.4. Panel A showcases the baseline model, where there is a slight
correlation between location and dispersion parameters (as in Table 2). Panels
B and C introduce region-specific minimum wages that surpass the national
minimum wage. The distinction between the panels is the proportion of regions
with local minimum wages exceeding the national minimum wage. In Panels B and
C, I present results not only for the regular effective minimum wage design
but also for instrumental variables specifications, with either two or three
instruments.

From the table, three key findings emerge. First, the more variation derived
from state-level minimum wages, the smaller the biases, even when using the
ordinary least squares estimator. This is evident when comparing the
“Effective min. wage” rows across panels, which gradually align with the
corresponding “Mean causal effect” rows. However, some bias persists. Second,
the use of instrumental variables approaches significantly mitigates this
bias. Third, the biases are least pronounced when employing the estimator with
two instruments, albeit at the expense of precision.

Therefore, the issues discussed in this section are an additional reason to
adopt instrumental variables regressions in the style of AMS when the data
includes regional-level variation in minimum wage laws. Such estimators
circumvent previously discussed biases by eschewing the potentially endogenous
variation from median wages. This section also provides a rationale for
avoiding the “interaction” instrument in AMS if the minimum wage instruments
alone offer sufficient identifying variation.

###  3.8 Taking stock

The core message of this discussion is that to evaluate whether the effective
minimum wage strategy is likely to be successful, the econometrician should
have a clear sense of what constitutes the identifying variation. Exogenous
changes in state-level minimum wages are the clearest example of such
variation. Because the effective minimum wage estimator might be biased even
when that variation is available, instrumental variables approaches such as
that in Autor, Manning and Smith (2016) are recommended.

If the data includes little or no variation in state-level minimum wage laws,
does the variation in wage levels identify the effects of the national minimum
wage? My analysis shows that this is only the case if, conditional on the
fixed effects (or trends) used in the regression, differences in median wages
come from an underlying structural factor that shifts the location of latent
log wage distributions but has no independent effects on shape, dispersion, or
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.

The existence of this structural factor is necessary but insufficient for
unbiased estimation. That’s because observed median wages are not a perfect
proxy for the centrality of the latent log wage distribution, which gives rise
to biases originating from correlated measurement error. Ideally, this
underlying shifter of minimum wage bindingness should be observable, in which
case the econometrician may use it as an instrument.

##  4 Fraction Affected and Gap estimators

###  4.1 Definition

Now, I study a difference-in-differences model with a time-invariant,
continuous measure of treatment intensity based on the initial distribution of
wages:

$y_{i,r,t}=\alpha_{i,r}+\delta_{i,t}+\beta_{i}FA_{r}\cdot\bm{1}\\{t=1\\}+\epsilon_{i,r,t}$
(3)

where the subscript $i$ indexes a specific equilibrium outcome, such that
different $o$ 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 $mw_{1}$.888When 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 (see e.g. Bailey, DiNardo and Stuart, 2021). In all simulations below,
there is perfect compliance, so both approaches are equivalent. The
regressions include region and time fixed effects.

Given the linearity of this model, the estimated average treatment effect of
the national minimum wage increase on outcome $i$ is given by the product of
the average of $FA_{r}$ and the estimated $\hat{\beta}_{i}$ parameter.

Card (1992) 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
emphasizes that much of the identifying variation in his application comes
from significant heterogeneity in the bindingness of state-level minimum wages
in the preceding years. 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
(Ahlfeldt, Roth and Seidel, 2018; Fedorets and Shupe, 2021).

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:”

$\displaystyle y_{i,r,t}=$
$\displaystyle\alpha^{Gap}_{i,r}+\delta^{Gap}_{i,t}+\beta^{Gap}_{i}Gap_{r}\cdot\bm{1}\\{t=1\\}+\epsilon^{Gap}_{i,r,t}$
$\displaystyle Gap_{r}=$
$\displaystyle\frac{\sum_{j=1}^{J_{r}}\max\\{\exp(mw_{1})-\exp(w_{j,0}),0\\}}{\sum_{j=1}^{J_{r}}\exp(w_{j,0})}$

where $j\in\\{1,\dots,J_{r}\\}$ indexes workers in the initial period and
$w_{j,0}$ is their log wage in that period. Card and Krueger (1994) 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. Dustmann et al.
(2021) provides an example, again in the context of Germany. It would
correspond to the resulting relative increase in the average wage in each
region if all low-wage workers 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.

###  4.2 Issue #1: Sensitivity to functional form assumptions

The fraction affected design and its variations rely on specific functional
forms that determine the sensitivity of individual regions to the national
change in the minimum wage. If that sensitivity measure is misspecified,
estimates of average treatment effects based on that estimator may be biased.

Table 6: Misspecification biases in the Fraction Affected and Gap designs

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Small initial min. wage, truncation/censoring only
True average causal effect | -0.006 | 0.016 | 0.008 | 0.004 | 0.002
Fraction affected | -0.008 | 0.020 | 0.010 | 0.006 | 0.003
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Gap measure | -0.006 | 0.015 | 0.007 | 0.004 | 0.002
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Panel B: Large initial min. wage, truncation/censoring only
True average causal effect | -0.031 | 0.118 | 0.036 | 0.020 | 0.010
Fraction affected | -0.039 | 0.185 | 0.044 | 0.026 | 0.013
| (0.000) | (0.009) | (0.001) | (0.001) | (0.001)
Gap measure | -0.028 | 0.127 | 0.031 | 0.019 | 0.009
| (0.000) | (0.009) | (0.001) | (0.001) | (0.001)
Panel C: Small initial min. wage, positive emp. effects
True average causal effect | 0.010 | -0.002 | -0.012 | -0.006 | -0.003
Fraction affected | 0.002 | 0.067 | -0.002 | -0.001 | -0.000
| (0.000) | (0.003) | (0.001) | (0.001) | (0.001)
Gap measure | 0.001 | 0.052 | -0.001 | -0.001 | -0.000
| (0.000) | (0.002) | (0.001) | (0.001) | (0.001)
Panel D: Large initial min. wage, positive emp. effects
True average causal effect | -0.003 | 0.149 | 0.039 | 0.002 | 0.001
Fraction affected | -0.038 | 0.132 | 0.143 | 0.026 | 0.012
| (0.001) | (0.007) | (0.004) | (0.002) | (0.001)
Gap measure | -0.028 | 0.090 | 0.102 | 0.019 | 0.008
| (0.001) | (0.006) | (0.003) | (0.001) | (0.001)

Notes: In all panels, the national minimum wage increases by 20 log points
from the first period to the second. Regions differ only in the time-invariant
location parameter $\mu_{r}\sim\mathcal{N}(0,0.2^{2})$. Each panel displays
average results for 1,000 simulations, each with 200 regions. For each
outcome, the numbers correspond to the mean true ATE across simulations, the
mean estimates of causal effects based on the regressions listed on the left,
and the average standard error associated with the estimates (in parentheses,
clustered at the region level).

To investigate this possibility, I run simulations based on the same model
used in the previous section: latent log wages are Normal in each region, and
the minimum wage causes truncation, censoring, and potentially positive
employment effects for workers a bit above the minimum wage (see Appendix A.1
for details). I consider four data-generating processes encompassing different
values for the initial minimum wage and whether it causes positive employment
effects. In the scenarios with the higher initial minimum wage, it bites more
into the latent distribution after the increase of 20 log points, such that
the disemployment and censoring effects become more significant.

To focus on the role of functional form assumptions, I designed those
simulations 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 their time-invariant location parameter
$\mu_{r}$. 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 6 shows 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, and 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 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 this particular data-generating process, 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.

In the remainder of this subsection, I consider three alternative econometric
specifications and show that neither of them adequately solves the
misspecification biases.

Binary design: Recent papers have explored a binary version of the fraction
affected intensity measure (Derenoncourt et al., 2021; Parente, 2024). One may
wonder whether such a strategy can help with misspecification issues.
Callaway, Goodman-Bacon and Sant’Anna (2024) argue that, in difference-in-
differences designs with continuous treatment variables, a binary design has a
more straightforward interpretation when the control group in the binary
design is composed of entirely untreated units. However, this is not the case
for the data-generating processes studied here.

In Appendix Table A5, I show that a binary definition of treatment suffers
from more substantial biases than the continuous versions. 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 Callaway,
Goodman-Bacon and Sant’Anna (2024), 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 precision loss, and a significant bias
remains.

Instrumental variables approach: Another potential solution for the
misspecification biases discussed above 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 A6 reports the results of using
such strategies. They have no impact on the estimates compared to the basic
OLS estimator.

Quadratic specification: One potential source of misspecification biases is
failing to account for the fact that minimum wage effects can be very
heterogeneous depending on how binding it is. Appendix Table A7 reports
results for models that include a quadratic term in the regressions to account
for such heterogeneity. Biases become smaller for some outcomes but larger for
others. Thus, this approach is not a practical solution for the
misspecification problems either.

###  4.3 Issue #2: Regression to the mean at the regional level

The fraction affected and gap measures are constructed based on extreme wage
observations, in the sense that individual workers only contribute to those
measures if their wages are below some threshold. Thus, these estimators may
be subject to bias emerging from regression to the mean. This issue is well-
known in minimum wage studies at the individual worker or firm levels. For
example, in their worker-level analysis, Dustmann et al. (2021) use data from
before the minimum wage was implemented to control for regression to the mean.
However, that issue is typically not discussed in regional-level studies.

One potential source of regression to the mean at the regional level is
sampling error. A region may have a high fraction affected because of an
"unlucky" draw of workers in the survey in the year used to construct the
treatment intensity variable. The ensuing bias is likely to be negligible with
large sample sizes. However, one must be aware of this possibility in studies
that define regions at a fine geographical level, especially if regions with
the smallest samples also have low average wages.

Time-varying structural factors that determine regional wages may also
introduce reversion to the mean, even when samples are large. Caliendo et al.
(2017) document that regional-level productivity shocks are quantitatively
significant in the United States. Gennaioli et al. (2014) collect time-series
data on regional GDP for 83 countries and document within-country regional
convergence. Their results mean that, in general, regions that have
particularly low GDP per capita in a given period are likely to have stronger
growth ex-post. Since these regional productivity shocks may affect both wages
and employment, potential biases are not limited to regressions where wages
are the dependent variable. That kind of regression to the mean will likely be
more consequential for longer-run specifications.

Table 7: Sensitivity of the Gap design

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Only permanent differences in location
True average causal effect | -0.010 | 0.026 | 0.012 | 0.007 | 0.003
Gap measure | -0.009 | 0.027 | 0.011 | 0.006 | 0.003
| (0.000) | (0.002) | (0.001) | (0.001) | (0.001)
Panel B: Adding location shocks, stable distributions
True average causal effect | -0.010 | 0.026 | 0.012 | 0.007 | 0.003
Gap measure | -0.007 | 0.043 | 0.030 | 0.026 | 0.023
| (0.001) | (0.005) | (0.006) | (0.006) | (0.006)
Panel C: Adding dispersion differences and shocks, stable distributions
True average causal effect | -0.010 | 0.026 | 0.013 | 0.007 | 0.003
Gap measure | -0.007 | 0.052 | 0.034 | 0.024 | 0.009
| (0.001) | (0.006) | (0.006) | (0.006) | (0.009)
Panel D: Average dispersion falls over time
True average causal effect | -0.010 | 0.027 | 0.013 | 0.007 | 0.003
Gap measure | -0.004 | 0.046 | 0.032 | 0.023 | 0.009
| (0.001) | (0.006) | (0.006) | (0.006) | (0.009)

Notes: All panels illustrate scenarios where the only time-varying factor is
an increase in the national minimum wage of 20 log points. Each panel displays
average results for 1,000 simulations with 200 regions and two periods. For
each outcome, the numbers correspond to the mean true ATE across simulations,
the mean estimates of causal effects based on the regressions listed on the
left, and the average standard error associated with the estimates (in
parentheses, clustered at the region level). See the notes for Tables 1 and 2
for a description of Panels A and B.

The comparison between Panels A and B in Table A8 illustrates this issue in
the context of the Normal-markdown model previously used in this paper. It
reports results for the Gap design; Table A8 in Appendix B shows similar
results for the Fraction Affected design. In Panel A, regions only differ in a
time-invariant location parameter $\mu_{r}$. Panel B introduces time-specific
location parameters $\mu_{r,t}$, with each region’s initial and final
parameters being jointly Normal with a correlation of 0.894 (equal to the
correlation between state-level mean log wages in the US for 1989 and 2004).
As expected, the estimated wage effects become substantially more positive.

Panel C further to explore this issue by including time-varying heterogeneity
in dispersion parameters $\sigma_{r,t}$ between regions. Those parameters are
assumed to be independent of the location parameters but have an
autocorrelation of 0.456 between periods. That magnifies the positive bias in
the lower tail, though it reduces it in the upper tail.

Fortunately, biases arising from regression to the mean can be detected with
tests for differential pre-trends if the context allows such tests. Appendix
Table A9 illustrates this concept using a placebo exercise that parallels
Table 7. The Gap measure is calculated as if the national minimum wage would
increase by 0.2, but there is no increase from period 0 to 1. That placebo
exercise shows positive employment effects that have about the same size as
the biases discussed above.

For the pre-trends test to work well in detecting that issue, the
econometrician needs to be careful in how to define the treatment intensity
variable. As an example, consider Dustmann et al. (2021), who calculate the
regional Gap measure for each year in the period before the new minimum wage
and then use the average for all of those years as the main regressor. Such a
definition may help with precision and reduce regression to the mean
originating from sampling error. However, it does not solve regression to the
mean originating from region-level shocks. More importantly, constructing the
treatment variable in that way prevents regression to the mean from being
detected in tests for differential pre-trends. Thus, using a single pre-period
year is preferable for diagnostics purposes. Ideally, if sufficient pre-
treatment data is available, the econometrician should directly control for
regression to the mean; see the individual-level design in Dustmann et al.
(2021) for details.

Another implication of regression to the mean is that using region-specific
linear trends to control for deviations from the parallel trends assumption
may not be a valid strategy. The reason is that, in an “event study” graph,
regression to the mean implies a V-shaped pattern with the bottom located in
the year used to construct the treatment variable. Thus, if one were to
extrapolate the pre-trends into the post-period and use it as the
counterfactual, one would increase the bias in the estimated treatment effects
instead of attenuate it.999The recommendation to avoid region-specific trends
was first given by Meer and West (2016), based on the fact that the minimum
wage may cause changes in employment trends rather than a step change in
employment levels.

###  4.4 Issue #3: Trends in the dispersion of latent wages

Suppose that, in all regions, there is a change in the dispersion of latent
log wages occurring at the same time as the increase in the national minimum
wage. Since that structural shock is the same in all regions, one may expect
the time effects included in the regression to account for it. This subsection
shows that this may not be the case when the minimum wage is not zero in the
initial period.

Panel A: Fraction affected is larger at the top region

Panel B: Observed employment rises in the top region

Figure 3: A truncation model with a secular decline in latent log wage
dispersion

Consider the model illustrated in Figure 3. Regions differ in a location
parameter $\mu_{r}$ constant over time. The shape parameter $\sigma_{t}$ is
the same in all regions and decreases over time. All else equal, regions with
lower $\mu_{r}$ have a higher fraction affected and a higher causal effect of
the minimum wage. However, with the fall in $\sigma_{t}$, these regions may
see a relative increase in employment. That is because the truncation effects
of the minimum wage may decrease 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 despite the causal
effects being negative for all observations in all periods.

The comparison between Panels C and D in Table 7 illustrates this issue. Panel
C was discussed in the previous subsection: it features idiosyncratic shocks
to both location and dispersion parameters, but the distributions of those
parameters are stable over time. Panel D differs from it by reducing the
average dispersion parameter $\sigma_{r,t}$, from 0.54 in $t=0$ to 0.51 in
$t=1$.101010Those numbers are based on the previously discussed calibration
based on state-level statistics from CPS data for 1989 and 2004. In the
context of the US, the decline in dispersion could be partly explained by the
higher prevalence of state-level minimum wages in 2004 compared to 1989, in
which case it should not be interpreted as a change in latent log wages. But
as mentioned before, this exercise aims to illustrate an econometric issue
using reasonable numbers, not evaluate the effects of the national minimum
wage in the US. This small change is enough to reduce the measured employment
effects by almost half, making them statistically different from the true
causal effects on average.

Tests for differential pre-trends can detect this issue if the trend in
dispersion starts before the minimum wage change and has a roughly constant
effect on employment over time. The placebo regressions reported in Appendix
Table A9 illustrate that possibility. The comments made in the previous
subsection apply: for those tests to be most effective, the treatment
intensity variable should be calculated based on a single year instead of
averaged over all pre-treatment years. In addition, one must be aware of the
possibility that pre-trends look flat because they combine the effect of
regression to the mean (discussed in the previous subsection) and other
secular trends that have a stronger effect on treated regions.

###  4.5 Taking stock

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 fundamentally rely on the
functional form used to construct the treatment variable, they are sensitive
to model misspecification. Practitioners are encouraged to report predicted
employment and wage effects from both the Fraction Affected and Gap designs to
assess the possibility of misspecification biases. However, readers should be
made aware that there is no theoretical guarantee that the true average
treatment effect lies between those estimates. In addition, I showed that
binarized versions of those estimators should be avoided as they display more
significant misspecification biases.

##  5 Robustness to alternative data generating processes

One may wonder whether the conclusions from the previous sections were
sensitive to the data-generating process used in the simulations. In this
section, I evaluate the performance of the estimators using two alternative
frameworks: the canonical model of labor demand and the task-based
monopsonistic model with firm heterogeneity from Haanwinckel (2023).

###  5.1 The canonical model of labor demand

Consider a competitive economy with two types of labor: low- or high-skilled.
Each worker is characterized not only by their type but also by their amount
of efficiency units of labor. The skill wage premium is pinned down by the
ratio of marginal products of labor between the two labor types. Marginal
products of labor are, in turn, determined by a representative production
function with constant returns and constant elasticity of substitution. This
setup mirrors the classic model of Katz and Murphy (1992).

Now, consider the implications of a binding minimum wage in that model. The
representative firm will not employ workers whose productivity is below the
minimum wage. Thus, one can think of this model as one where each worker has a
latent log wage given by the sum of the log price of the efficiency unit of
their type (low- or high-skill) and the log of their amount of efficiency
units. The observed wage distribution is the truncated version of the latent
one. See Appendix A.2 for details on the model.

The critical difference between this model and the one presented before is
that the shape of the latent log wage distribution responds to the minimum
wage. As the minimum wage causes more disemployment for low-skill workers,
returns to skill are expected to fall. These price responses generate wage
spillovers for other low-skill workers and attenuate the minimum wage’s
disemployment effects.

Appendix Table A10 reports simulations that assess the effectiveness of the
Fraction Affected and Gap designs using the Canonical model as the data-
generating process. In those simulations, regions only differ in the initial
share of workers in the high-skill group, and the only time-varying factor is
the minimum wage. In that sense, they parallel Table 6 above, illustrating
ideal scenarios for the Fraction Affected and Gap designs. Also, similar to
that previously discussed table, I report results for different model
specifications, varying the bindingness of the initial minimum wage and the
elasticity of substitution between low- and high-skill workers.

The takeaways from that exercise are the same as those from Table 6. The
predicted average causal effects estimated using those approaches do not
always equal the true ones, and those misspecification biases are more
significant when the minimum wage is more binding.

Appendix Table A11 evaluates the effective minimum wage design using the same
simulations. The baseline effective minimum wage specification displays severe
biases when both region and time fixed effects are included. That is because
the model does not include region-specific wage shocks, the source of “good”
variation in that design. For that reason, I also report estimates for the
model without region fixed effects (the baseline specification in Lee, 1999),
which exploit differences in wage levels instead of changes. The model is
almost ideal for that design in that there are no sources of “endogeneity”
(like systematically lower latent employment levels in low-wage regions) and
also in that the dispersion of efficiency units for both low- and high-skill
workers are the same (which reduces the correlation between “location” and
“dispersion” parameters of latent log wages). Even so, that model displays
significant biases in some specifications.

###  5.2 A task-based, monopsonistic model with firm heterogeneity

Table 8: Task-based, monopsonistic model with two-sided heterogeneity

Panel A: Fraction Affected and Gap Designs

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
True average causal effect | -0.046 | 0.318 | 0.197 | 0.120 | 0.107
Fraction affected | -0.021 | 0.232 | 0.278 | 0.325 | 0.175
| (0.007) | (0.012) | (0.015) | (0.015) | (0.026)
Gap measure | -0.012 | 0.103 | 0.123 | 0.153 | 0.085
| (0.004) | (0.017) | (0.019) | (0.021) | (0.014)
Binary measure, 50% treated | -0.003 | 0.092 | 0.113 | 0.135 | 0.072
| (0.003) | (0.006) | (0.008) | (0.009) | (0.012)
Binary measure, 90% treated | 0.002 | 0.227 | 0.260 | 0.271 | 0.102
| (0.008) | (0.021) | (0.025) | (0.029) | (0.032)

Panel B: Effective Minimum Wage Designs

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
True average causal effect | -0.046 | 0.198 | 0.077 | -0.013
Effective min. wage | -0.015 | 0.218 | 0.122 | 0.070
| (0.012) | (0.011) | (0.013) | (0.037)
Effective min. wage, no region FE | -0.073 | 0.196 | 0.088 | -0.016
| (0.006) | (0.005) | (0.006) | (0.014)
Effective min. wage, no time FE | 0.113 | 0.212 | 0.121 | -0.139
| (0.004) | (0.003) | (0.004) | (0.012)
AMS, no time FE | 0.125 | 0.211 | 0.121 | -0.159
| (0.005) | (0.003) | (0.003) | (0.011)

Panel C: Effective Minimum Wage based on percentile 90

| Outcome
---|---
| Emp. | p10 - p90 | p25 - p90 | p50 - p90
True average causal effect | -0.046 | 0.211 | 0.090 | 0.000
Effective min. wage, p90 | 0.025 | 0.384 | 0.306 | 0.000
| (0.012) | (0.015) | (0.021) | (0.000)

Notes: This table compares the true average causal effects of the minimum wage
from the model of Haanwinckel (2023) to the predicted average treatment
effects from different estimators. The model has 151 regions (corresponding to
microregions in Brazil) and two time periods (corresponding to 1998 and 2012).
Estimated standard errors (clustered at the microregion level) are shown in
parentheses.

The previous simulation exercises have two important limitations. First, the
causal channels from the minimum wage to employment and wages are somewhat
limited, excluding margins that have been shown to matter, such as the
reallocation of workers from low- to high-wage jobs (Dustmann et al., 2021).
Second, they relied on simplistic—and thus potentially inaccurate—functional
form assumptions on the shape of the underlying worker productivity
distribution and how it varies across regions.

To address those limitations, I use the model estimated in Haanwinckel (2023)
as the data-generating process. In that model, workers belong to one of ten
comparative advantage types and differ in the number of efficiency units of
labor within groups. The quantity of workers of each type is associated with
the educational composition of the region at each period. Firms differ in
their demand for skills because they use task-based production technologies,
and some firms may have a higher demand for high-complexity tasks that skilled
workers perform better. Firms also differ in the wages they pay to each worker
type; that is, the model features firm wage premiums. Minimum wages have a
range of effects, including disemployment (truncation), mechanical wage
increases (censoring), between-firm reallocation, changes in returns to skill
within firms, in the distribution of firm types (that is, the share of high-
wage, high-skill firms versus low-wage, low skill ones), in participation
decisions of workers, and in prices for goods. Regions differ not only in
educational composition but also in total factor productivity (TFP), three
types of labor demand parameters (which may correlate with TFP and initial
educational composition), and some parameters regulating employment rates
(which capture unobserved heterogeneity related to the Brazilian informal
sector).

Haanwinckel (2023) estimates that model using Brazilian matched employer-
employee data, targeting a large set of moments at the region-time levels.
There are 151 regions (corresponding to microregions as defined by the
Brazilian Statistical Bureau, IBGE) and two time periods (corresponding to
1998 and 2012). Appendix Table A12 shows that the estimated model fits several
dimensions of the data pretty well: inequality measures between and within
educational groups, the contribution of firm wage premiums to inequality
(measured using the methodology developed by Kline, Saggio and Sølvsten,
2018), the share of workers earning up to 30 log points of the minimum wage,
and the minimum wage “spike” (measured as the share of workers earning up to 5
log points of the minimum wage).

Table 8 evaluates the performance of the estimators discussed in this paper
when the fitted model of Haanwinckel (2023) is used as data. Each row labeled
as an estimator in that table reports the predicted average treatment effects
on a range of outcomes using the results of that single regression. The table
also shows the corresponding standard errors in parenthesis. At the top row of
each panel, I report the true average causal effect of the national minimum
wage predicted by the fitted model, following the definition from Section 2.

The overall result is that all of the estimators display significant biases.
With one exception, they all underestimate disemployment effects by at least
50%. Some versions of the effective minimum wage design predict positive,
instead of negative, employment effects. Most estimators also fail to
accurately capture the spillover effects on the wage distribution, though the
effective minimum wage estimator without region fixed effects performs
surprisingly well in that regard.

##  6 Conclusion

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.

The main takeaway of my analysis is that 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)
construct the treatment intensity variable using only one pre-treatment year
rather than averaging over years; (ii) check for differential pre-trends;
(iii) control for regression to the mean using the procedure in Dustmann et
al. (2021); (iv) check for trends in the dispersion of log wages in the pre-
treatment period, which may cause bias if the initial minimum wage is not
zero; and (v) do not use a binary version of the treatment intensity variable.
Even if the econometrician takes such precautions, the estimator may be
subject to significant misspecification biases arising from functional form
assumptions, especially when the minimum wage increase is significant or
generates positive employment effects. At a minimum, econometricians may
consider reporting results of both the “fraction affected” and “gap” designs
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, one may 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 one includes both region and time fixed effects
and constructs 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 the
econometrician believes 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.

In contexts that are not ideal for the fraction affected/gap designs due to a
lack of a pre-period with stable minimum wages and where there is no quasi-
experimental shifter of minimum wage bindingness at the regional level, the
econometrician may consider two alternative strategies. One is to use within-
region difference-in-differences designs where the unit of analysis is either
firms (Harasztosi and Lindner, 2019) or workers (Dustmann et al., 2021), and
treatment and control groups are defined based on initial wages. In that case,
the econometrician should be mindful that some concerns discussed above, such
as regression to the mean, may still apply. In addition, careful
interpretation of results is warranted; potential effects on firm or worker
entry may be undetectable, and the control group may also be affected by the
shock even if general equilibrium effects are believed to be small (for
example, high-wage firms may be affected by the minimum wage if it induces
worker reallocation, as in Dustmann et al., 2021).

The second alternative strategy is to estimate a parametric economic model;
see, for example, Engbom and Moser (2022) and Haanwinckel (2023) in the
Brazilian context. Such models can use information from the data to quantify
and correct 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 adequately controls for data features
that are problematic for the designs studied in this paper.

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Appendix

##  Appendix A Details on data-generating processes for simulations

This appendix lists the parameters used in all simulations. Every simulation
exercise is repeated 1,000 times. The increase in the log minimum wage is
always 0.2, except when otherwise noted.

###  A.1 Normal-markdown model

####  A.1.1 Model description

Each region $r$ and each time $t$ has a Normal distribution of latent log
wages
$G^{*}_{r,t}(w^{*})=\Phi\left(\frac{w^{*}-\mu_{r,t}}{\sigma_{r,t}}\right)$.
The employment-to-population ratio and the distribution of observed wages
depend on latent wages, the level of the national minimum wage, and a
“markdown” parameter $m\in(0,1]$ as follows:

$\displaystyle emp_{r,t}=$ $\displaystyle 1-\Phi\left(\frac{mw_{t}-\log
m-\mu_{r,t}}{\sigma_{r,t}}\right)$ $\displaystyle G_{r,t}(w)=$
$\displaystyle\frac{\Phi\left(\frac{w-\mu_{r,t}}{\sigma_{r,t}}\right)-\Phi\left(\frac{mw_{t}-\log
m-\mu_{r,t}}{\sigma_{r,t}}\right)}{1-\Phi\left(\frac{mw_{t}-\log
m-\mu_{r,t}}{\sigma_{r,t}}\right)}\qquad\text{for }w\geq mw_{t}$

This model generates both truncation and censoring of the latent wage
distribution. Workers whose latent log wages are below the minimum minus the
log markdown become disemployed. For those with latent log wages above the log
minimum wage, the observed wage is equal to the latent wage. Finally, those
who remain employed but have latent log wages below the log minimum wage see a
mechanical increase in their wage. The latter group corresponds to the minimum
wage “spike” in the log wage distribution.

The model can be understood as reflecting an economy with an inelastic labor
supply, exogenous worker productivities, and identical monopsonistic firms
paying wages that are below the marginal products of labor unless mandated to
pay higher wages via the minimum wage. When the markdown $m$ is low,
disemployment effects are smaller, and positive effects on wages are bigger.
Unless otherwise noted, I use $m=0.7$.

####  A.1.2 Calibration

The meta-parameters governing the distribution of region-specific parameters
$[\mu_{r,0},\sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$ are based on data from the
US Current Population Survey for 1989 (corresponding to period $t=1$) and 2004
(corresponding to $t=0$). I chose those years because the national minimum
wage was small and approximately the same, in real terms, in both years and
the unemployment rate was also approximately equal.

The data was processed using the same procedures as in Lemieux (2006). The
sample is restricted to workers between 16 and 64 years of age, with positive
potential experience, and whose wages and worked hours are reported by the
respondent instead of inferred. Top-coded earnings are adjusted by a factor of
1.4.

Table A1: Simulation meta-parameters: Normal-markdown model, effective minimum
wage design

|  | Min. Wage | Means | Std. Dev. | Correlations | Corr.
---|---|---|---|---|---|---
Model | $m$ | $mw_{0}$ | $mw_{1}$ | $\sigma_{r,0}$ | $\sigma_{r,1}$ | $\mu_{0}$ | $\sigma_{0}$ | $\mu_{1}$ | $\sigma_{1}$ | $\mu_{0},\sigma_{0}$ | $\mu_{0},\mu_{1}$ | $\mu_{0},\sigma_{1}$ | $\sigma_{0},\mu_{1}$ | $\sigma_{0},\sigma_{1}$ | $\mu_{1},\sigma_{1}$ | $\mu_{r,t},w_{0.5,r,t}$
Tables 1, A3, and A4
Panel A | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.000 | 0.112 | 0.000 | 0.000 | 0.894 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
Panel B | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.999
Panel C | 0.7 | -1.0 | -0.6 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.994
Panel D | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.039 | 0.112 | 0.074 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.998
Table 2
Panel A | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.999
Panel B | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.076 | 0.894 | 0.000 | 0.000 | 0.456 | 0.076 | 0.999
Panel C | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.076 | 0.894 | 0.366 | 0.063 | 0.456 | 0.264 | 0.999
Tables 3 and 4
All panels | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.999
Table 5
All panels | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.123 | 0.026 | 0.112 | 0.049 | 0.076 | 0.894 | 0.000 | 0.000 | 0.456 | 0.076 | 0.999

Notes: This table shows meta-parameters used to draw simulation samples for
the Normal-markdown model. See Appendix A.1 for details.

Table A2: Simulation meta-parameters: Normal-markdown model, Fraction
Affected/Gap designs

|  | Min. Wage | Means | Std. Dev. | Correlations | Corr.
---|---|---|---|---|---|---
Model | $m$ | $mw_{0}$ | $mw_{1}$ | $\sigma_{r,0}$ | $\sigma_{r,1}$ | $\mu_{0}$ | $\sigma_{0}$ | $\mu_{1}$ | $\sigma_{1}$ | $\mu_{0},\sigma_{0}$ | $\mu_{0},\mu_{1}$ | $\mu_{0},\sigma_{1}$ | $\sigma_{0},\mu_{1}$ | $\sigma_{0},\sigma_{1}$ | $\mu_{1},\sigma_{1}$ | $\mu_{r,t},w_{0.5,r,t}$
Tables 6, A5, A6, and A7
Panel A | 0.7 | -1.1 | -0.9 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
Panel B | 0.7 | -0.7 | -0.5 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 0.994
Panel C | 0.7 | -1.1 | -0.9 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
Panel D | 0.7 | -0.7 | -0.5 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999
Tables 7 and A8
Panel A | 0.7 | -1.0 | -0.8 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999
Panel B | 0.7 | -1.0 | -0.8 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.894 | 0.000 | 0.000 | 0.000 | 0.000 | 0.999
Panel C | 0.7 | -1.0 | -0.8 | 0.526 | 0.526 | 0.118 | 0.038 | 0.118 | 0.038 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.999
Panel D | 0.7 | -1.0 | -0.8 | 0.542 | 0.510 | 0.118 | 0.038 | 0.118 | 0.038 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 0.999
Table A9
Panel A | 0.7 | -1.0 | -1.0 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.999 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
Panel B | 0.7 | -1.0 | -1.0 | 0.526 | 0.526 | 0.118 | 0.000 | 0.118 | 0.000 | 0.000 | 0.894 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000
Panel C | 0.7 | -1.0 | -1.0 | 0.526 | 0.526 | 0.118 | 0.038 | 0.118 | 0.038 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 1.000
Panel D | 0.7 | -1.0 | -1.0 | 0.542 | 0.510 | 0.118 | 0.038 | 0.118 | 0.038 | 0.000 | 0.894 | 0.000 | 0.000 | 0.456 | 0.000 | 1.000

Notes: This table shows meta-parameters used to draw simulation samples for
the Normal-markdown model. See Appendix A.1 for details.

Using this sample, I calculate the mean and standard deviation of log wages in
each combination of state and year, weighting by the CPS sampling weights and
worker hours. Then, I de-mean the $\mu_{r,t}$ elements using simple averages
within the period so that the $\mu_{r,t}$ are mean zero in both periods. I
treat those statistics as corresponding to the
$[\mu_{r,0},\sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$ vector for each state. Thus,
I calculate the corresponding covariance matrix of that vector and use it to
calibrate the simulation models.

Finally, I calibrate the simulations using the estimated vector of means and
covariance matrix. As stated in the main text, in each simulation, the vectors
$[\mu_{r,0},\sigma_{r,0},\mu_{r,1},\sigma_{r,1}]$ for each region $r$ are
drawn from a Multivariate Normal distribution. The parameters for that meta-
distribution are created by either “shutting down” some of the correlations in
the estimated covariance matrix, eliminating differences in dispersion
parameters, increasing the correlation between some initial and final region
parameters to one (to impose that those parameters are time-invariant), or
averaging some meta-parameters between both periods so that the distributions
are stable over time. Tables A1 and A2 report the meta-parameters used in
every simulation exercise with the Normal-markdown model.

####  A.1.3 Positive employment effects

For some simulation exercises, I augment the Normal-markdown model to include
the possibility of positive employment effects. I add two parameters to the
model: $P_{base}$ and $P_{height}$. The total employment mass added to the
model is equal to
$\frac{P_{base}P_{height}}{2}\phi\left(\frac{mw_{t}-\mu_{r,t}}{\sigma_{r,t}}\right)$,
where the latter term corresponds to the density of the latent log wage
distribution evaluated at the point where the minimum wage binds. The wage
distribution for that extra mass is triangular, with support
$[mw_{t},mw_{t}+P_{base}]$ and peak at the left extreme of the support.

Intuitively, that model corresponds to one where the minimum wage increases
labor force participation of individuals with potential wages just above the
minimum wage, in the interval $[mw_{t},mw_{t}+P_{base}]$. This effect’s
overall intensity is assumed to be proportional to the density of latent log
wages evaluated at the minimum wage level and to the $P_{height}$ parameter.
In this model, a small minimum wage is likely to have positive employment
effects, which are initially increasing. However, at some point, the effects
of disemployment start to become more significant. Eventually, the effects of
minimum wage on employment will become negative.

####  A.1.4 The Normal-markdown model with state-level minimum wages

For the exercise shown in Table 5, I augment the Normal-markdown model to
include the possibility of state-specific minimum wages that surpass the
national minimum wage. I first choose the share of regions that, in each
period, are selected to have a higher local minimum wage. Those shares are 0.2
or 0.4, depending on the Panel in Table 5. For reference, the share of states
in the US that had local minimum wages at least 5 log points above the federal
minimum wage was 0.23 in both 1989 and 2004.

When simulating the model for the initial period, I randomly draw the subset
of regions with higher local minimum wages. Given the shares chosen above,
those subsets have the same size in all simulations. Then, I draw a number
from a Normal distribution with a mean of 0.25 and a standard deviation of
0.075. I assign a local log minimum wage that is equal to the federal minimum
wage plus that number (or the federal minimum wage plus 0.05, whatever is
higher). The numbers 0.25 and 0.075 above are chosen to match the mean and
standard deviation of the log gap between local minimum wages and the federal
minimum wage in 2004 for the subset of states for which the minimum wage is
higher than the federal one (the corresponding numbers are 0.15 and 0.052 for
1989).

In the second period simulation, I follow the same procedure, except that I do
not allow for reductions of local minimum wages between periods. So, the local
minimum wage is either calculated from the procedure above or observed in the
first period, whichever is higher.

The definition of the average treatment effect to be estimated is updated in
the following way to account for the possibility of local minimum wages:

$\displaystyle\bm{ATE}_{0}$
$\displaystyle=\mathbb{E}\left[f\left(mw_{r,1},\theta_{r,0}\right)-f\left(mw_{r,0},\theta_{r,0}\right)\right]$
$\displaystyle=\mathbb{E}\left[f\left(mw_{r,1},\theta_{r,0}\right)-\bm{y}_{r,0}\right]$
$\displaystyle\bm{ATE}_{1}$
$\displaystyle=\mathbb{E}\left[f\left(mw_{r,1},\theta_{r,1}\right)-f\left(mw_{r,0},\theta_{r,1}\right)\right]$
$\displaystyle=\mathbb{E}\left[\bm{y}_{r,1}-f\left(mw_{r,0},\theta_{r,1}\right)\right]$
$\displaystyle\bm{ATE}$ $\displaystyle=\frac{\bm{ATE}_{0}+\bm{ATE}_{1}}{2}$

The only difference is that the counterfactuals being considered correspond to
state-specific changes in the minimum wage induced by the minimum wage, caused
either by the increase in the federal minimum wage or by a random draw of a
higher local minimum wage in the latter period. Calculating the estimated
average treatment effects is the same, using region-specific changes in the
observed effective minimum wage. That is consistent with the updated
definition, as changes in the effective local minimum wage reflect both
national and local minimum wage changes.

###  A.2 The canonical model of labor demand

####  A.2.1 Model description

Consider a competitive economy where the only production factors are skilled
($i=1$) and skilled ($i=2$) labor, both of which have inelastic supply. A
representative firm produces the numeraire good in the economy using a
constant elasticity of substitution (CES) production function:

$F(L_{1},L_{2})=\left[\alpha{L_{1}}^{\frac{E-1}{E}}+(1-\alpha){L_{2}}^{\frac{E-1}{E}}\right]^{\frac{E}{E-1}}$

The measure of workers is normalized to one, and the region-specific share of
skilled workers is $s_{r}$. Each worker supplies $\exp(e)$ efficiency units of
labor, where $e$ has a Normal distribution with mean zero and standard
deviation $D$. Workers whose log marginal product of labor falls below the log
minimum wage $mw_{t}$ are not employed by the representative firm. That is,
the minimum wage truncates the worker productivity distribution. The
equilibrium log prices per efficiency unit of labor, $p_{i}$, are then given
by the solution to a system of two equations:

$\displaystyle p_{i,r,t}$ $\displaystyle=\log
F_{i}\left(s_{r}E(p_{1,r,t},mw_{t}),(1-s_{r})E(p_{2,r,t},mw_{t})\right)\qquad
i\in\\{1,2\\}$ $\displaystyle\text{where }E(p,mw)$
$\displaystyle=\int_{mw_{t}-p}^{\infty}\exp(e)\phi\left(\frac{e}{D}\right)de$
$\displaystyle\text{and }F_{i}(L_{1},L_{2})$
$\displaystyle=\frac{dF(L_{1},L_{2})}{dL_{i}}$

In the expressions above, $\phi$ denotes the density of a standard Normal
distribution. The function $E(\cdot)$ calculates the average amount of
efficiency units supplied by workers of a given type, taking into account the
disemployment effects of the minimum wage.

The resulting employment-to-population ratio in a given region and period is
given by:

$emp_{r,t}=s_{r}\left[1-\Phi\left(\frac{mw_{t}-p_{1,r,t}}{D}\right)\right]+(1-s_{r})\left[1-\Phi\left(\frac{mw_{t}-p_{2,r,t}}{D}\right)\right],$

and the corresponding cumulative distribution function for log wages is:

$G_{r,t}\left(w\right)=s_{r}\frac{\Phi\left(\frac{w-p_{1,r,t}}{D}\right)-\Phi\left(\frac{mw_{t}-p_{1,r,t}}{D}\right)}{1-\Phi\left(\frac{mw_{t}-p_{1,r,t}}{D}\right)}+(1-s_{r})\frac{\Phi\left(\frac{w-p_{2,r,t}}{D}\right)-\Phi\left(\frac{mw_{t}-p_{2,r,t}}{D}\right)}{1-\Phi\left(\frac{mw_{t}-p_{2,r,t}}{D}\right)}\qquad\text{for
}w\geq mw_{t}$

where $\Phi$ is the cumulative distribution function of a standard Normal.

####  A.2.2 Calibration

I also use US Current Population Survey data to calibrate the simulations.
Using the same sample restrictions described in the previous subsection and
data for 1989, I define a worker as belonging to the skilled group $i=1$ if
they have at least four years of college education. Then, I calculate the mean
and standard deviation of log wages by skill group for each state and the
share of workers in each group.

On the labor supply side, the (unweighted) average of the share of skilled
workers across states is 0.224, and the standard deviation is 0.047. Then, in
the simulations, I draw the share of skilled workers in each region from a
Normal distribution with the corresponding mean and standard deviation,
trimming the results so that the share of each worker type can never be below
0.01. The average standard deviation of log wages within states is close to
0.5 for both educational groups. Thus, I set $D=0.5$.

On the demand side, the mean log wage gap between skilled and unskilled
workers is also close to 0.5. Thus, I choose the $\alpha$ parameter such that
the skill premium $p_{2,r,t}/p_{1,r,t}$ is 0.5 in an equilibrium of the model
with the share of skilled workers equal to the cross-state average, and at the
lowest initial value of the minimum wage used (see below). That corresponds to
$\alpha=0.563$ when the elasticity of substitution used in the simulation is
$E=3$, and $\alpha=0.493$ for $E=1.4$.

The simulations are run for six scenarios. They combine the two values for the
elasticity of substitution in production and three initial values of the
minimum wage: -2.2, -1.8, and -1.5. The corresponding initial employment-to-
population ratios given the average share of skilled workers are around 0.995,
0.966, and 0.896, respectively.

##  Appendix B Additional tables

Table A3: Effective minimum wage design with zero employment effects

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
Panel A: Regions differ only in location
True average causal effect | -0.000 | 0.020 | 0.000 | -0.000
Effective min. wage | -0.000 | 0.020 | 0.000 | -0.000
| (0.000) | (0.003) | (0.000) | (0.000)
Panel B: Regions differ in location and dispersion
True average causal effect | -0.000 | 0.021 | 0.000 | -0.000
Effective min. wage | 0.001 | 0.025 | 0.004 | -0.008
| (0.001) | (0.012) | (0.007) | (0.015)
Panel C: As above, but larger increase in min. wage
True average causal effect | -0.008 | 0.106 | 0.009 | -0.003
Effective min. wage | 0.001 | 0.123 | 0.028 | -0.043
| (0.006) | (0.019) | (0.013) | (0.028)
Panel D: St. dev. of dispersion is 50% larger
True average causal effect | -0.001 | 0.022 | 0.001 | -0.000
Effective min. wage | 0.002 | 0.031 | 0.009 | -0.019
| (0.002) | (0.017) | (0.011) | (0.022)

Notes: This table is analogous to Table 1, except that the markdown parameter
is reduced from 0.7 to 0.65, and the data-generating process includes positive
employment effects. The size of those effects is calibrated such that the
average impact of a 20 log point increase in the minimum wage is zero in Panel
A, corresponding to $P_{height}=0.5$ and $P_{base}=0.25$. See Appendix A.1.3
for details.

Table A4: Effective minimum wage design with zero employment effects

| Outcome
---|---
| Emp. | p10 - p50 | p25 - p50 | p90 - p50
Panel A: Regions differ only in location
True average causal effect | 0.009 | 0.026 | -0.005 | 0.003
Effective min. wage | 0.009 | 0.027 | -0.005 | 0.003
| (0.000) | (0.003) | (0.001) | (0.000)
Panel B: Regions differ in location and dispersion
True average causal effect | 0.009 | 0.028 | -0.005 | 0.003
Effective min. wage | 0.009 | 0.023 | -0.007 | 0.007
| (0.001) | (0.012) | (0.008) | (0.015)
Panel C: As above, but larger increase in min. wage
True average causal effect | 0.015 | 0.137 | 0.004 | 0.005
Effective min. wage | 0.020 | 0.140 | 0.019 | -0.011
| (0.005) | (0.019) | (0.015) | (0.029)
Panel D: St. dev. of dispersion is 50% larger
True average causal effect | 0.008 | 0.030 | -0.004 | 0.003
Effective min. wage | 0.009 | 0.019 | -0.008 | 0.009
| (0.001) | (0.018) | (0.012) | (0.022)

Notes: This table is analogous to Table 1, except that the markdown parameter
is reduced from 0.7 to 0.6, and the data-generating process includes positive
employment effects. The size of those effects is calibrated such that the
average impact of a 20 log point increase in the minimum wage is close to one
percentage point in Panel A, corresponding to $P_{height}=1.0$ and
$P_{base}=0.25$. See Appendix A.1.3 for details.

Table A5: Difference-in-differences with binary treatment

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Small initial min. wage, truncation/censoring only
True average causal effect | -0.006 | 0.016 | 0.008 | 0.004 | 0.002
Binary measure, 50% treated | -0.003 | 0.007 | 0.003 | 0.002 | 0.001
| (0.000) | (0.001) | (0.000) | (0.000) | (0.000)
Binary measure, 90% treated | -0.004 | 0.011 | 0.006 | 0.003 | 0.002
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Panel B: Large initial min. wage, truncation/censoring only
True average causal effect | -0.031 | 0.118 | 0.036 | 0.020 | 0.010
Binary measure, 50% treated | -0.009 | 0.053 | 0.010 | 0.006 | 0.003
| (0.001) | (0.002) | (0.001) | (0.000) | (0.000)
Binary measure, 90% treated | -0.017 | 0.084 | 0.020 | 0.012 | 0.006
| (0.001) | (0.004) | (0.001) | (0.001) | (0.001)
Panel C: Small initial min. wage, positive emp. effects
True average causal effect | 0.010 | -0.002 | -0.012 | -0.006 | -0.003
Binary measure, 50% treated | 0.001 | 0.021 | -0.001 | -0.001 | -0.000
| (0.000) | (0.002) | (0.000) | (0.000) | (0.000)
Binary measure, 90% treated | 0.003 | 0.017 | -0.004 | -0.002 | -0.001
| (0.000) | (0.003) | (0.001) | (0.001) | (0.001)
Panel D: Large initial min. wage, positive emp. effects
True average causal effect | -0.003 | 0.149 | 0.039 | 0.002 | 0.001
Binary measure, 50% treated | -0.008 | 0.033 | 0.035 | 0.006 | 0.003
| (0.001) | (0.002) | (0.002) | (0.001) | (0.000)
Binary measure, 90% treated | -0.013 | 0.078 | 0.051 | 0.009 | 0.004
| (0.001) | (0.005) | (0.003) | (0.001) | (0.001)

Notes: This table is analogous to Table 6, except that it reports results for
a difference-in-differences estimator based on a binary version of treatment.
Treated status is based on initial median wages being below some simulation-
specific threshold, chosen such that the share of treated units corresponds to
the desired level.

Table A6: Difference-in-differences with instrumental variables

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Small initial min. wage, truncation/censoring only
True average causal effect | -0.006 | 0.016 | 0.008 | 0.004 | 0.002
FA instrumended by GAP | -0.008 | 0.021 | 0.010 | 0.006 | 0.003
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
GAP instrumended by FA | -0.006 | 0.015 | 0.007 | 0.004 | 0.002
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Panel B: Large initial min. wage, truncation/censoring only
True average causal effect | -0.031 | 0.118 | 0.036 | 0.020 | 0.010
FA instrumended by GAP | -0.040 | 0.182 | 0.044 | 0.027 | 0.013
| (0.000) | (0.010) | (0.001) | (0.001) | (0.001)
GAP instrumended by FA | -0.028 | 0.131 | 0.031 | 0.019 | 0.009
| (0.000) | (0.008) | (0.001) | (0.001) | (0.001)
Panel C: Small initial min. wage, positive emp. effects
True average causal effect | 0.010 | -0.002 | -0.012 | -0.006 | -0.003
FA instrumended by GAP | 0.002 | 0.069 | -0.001 | -0.001 | -0.000
| (0.000) | (0.003) | (0.001) | (0.001) | (0.001)
GAP instrumended by FA | 0.001 | 0.051 | -0.001 | -0.001 | -0.000
| (0.000) | (0.002) | (0.001) | (0.001) | (0.001)
Panel D: Large initial min. wage, positive emp. effects
True average causal effect | -0.002 | 0.149 | 0.039 | 0.002 | 0.001
FA instrumended by GAP | -0.039 | 0.128 | 0.144 | 0.027 | 0.012
| (0.002) | (0.007) | (0.004) | (0.002) | (0.001)
GAP instrumended by FA | -0.027 | 0.094 | 0.102 | 0.019 | 0.008
| (0.001) | (0.006) | (0.003) | (0.001) | (0.001)

Notes: This table is analogous to Table 6, except that it reports results for
a difference-in-differences estimator where the main regressor (the
interaction between one treatment intensity variable and an indicator for the
post period) is instrumented with an alternative treatment intensity variable
interacted with the dummy for the post period.

Table A7: Difference-in-differences with quadratic treatment intensity

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Small initial min. wage, truncation/censoring only
True average causal effect | -0.006 | 0.016 | 0.008 | 0.004 | 0.002
Quadratic on FA | -0.007 | 0.017 | 0.009 | 0.005 | 0.002
| (0.000) | (0.002) | (0.002) | (0.002) | (0.002)
Quadratic on GAP | -0.006 | 0.015 | 0.008 | 0.004 | 0.002
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Panel B: Large initial min. wage, truncation/censoring only
True average causal effect | -0.031 | 0.118 | 0.036 | 0.020 | 0.010
Quadratic on FA | -0.034 | 0.281 | 0.038 | 0.023 | 0.011
| (0.001) | (0.019) | (0.003) | (0.003) | (0.003)
Quadratic on GAP | -0.030 | 0.211 | 0.033 | 0.020 | 0.010
| (0.000) | (0.008) | (0.002) | (0.002) | (0.002)
Panel C: Small initial min. wage, positive emp. effects
True average causal effect | 0.010 | -0.002 | -0.012 | -0.006 | -0.003
Quadratic on FA | 0.007 | 0.029 | -0.010 | -0.004 | -0.002
| (0.000) | (0.007) | (0.002) | (0.002) | (0.002)
Quadratic on GAP | 0.004 | 0.040 | -0.006 | -0.002 | -0.001
| (0.000) | (0.004) | (0.001) | (0.001) | (0.001)
Panel D: Large initial min. wage, positive emp. effects
True average causal effect | -0.003 | 0.149 | 0.039 | 0.002 | 0.001
Quadratic on FA | -0.013 | 0.246 | 0.126 | 0.007 | 0.004
| (0.001) | (0.004) | (0.014) | (0.004) | (0.004)
Quadratic on GAP | -0.019 | 0.167 | 0.118 | 0.012 | 0.006
| (0.001) | (0.005) | (0.006) | (0.002) | (0.002)

Notes: This table is analogous to Table 6, except that it reports results for
a difference-in-differences that allows for treatment effects to vary with the
treatment intensity through a quadratic functional form.

Table A8: Sensitivity of the Fraction Affected design

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Only permanent differences in location
True average causal effect | -0.010 | 0.026 | 0.012 | 0.007 | 0.003
Fraction affected | -0.013 | 0.036 | 0.015 | 0.009 | 0.004
| (0.000) | (0.003) | (0.001) | (0.001) | (0.001)
Panel B: Adding location shocks, stable distributions
True average causal effect | -0.010 | 0.026 | 0.012 | 0.007 | 0.003
Fraction affected | -0.010 | 0.059 | 0.042 | 0.036 | 0.033
| (0.001) | (0.007) | (0.008) | (0.008) | (0.009)
Panel C: Adding dispersion differences and shocks, stable distributions
True average causal effect | -0.010 | 0.026 | 0.012 | 0.007 | 0.003
Fraction affected | -0.008 | 0.072 | 0.047 | 0.031 | 0.005
| (0.002) | (0.008) | (0.008) | (0.008) | (0.012)
Panel D: Average dispersion falls over time
True average causal effect | -0.010 | 0.027 | 0.013 | 0.007 | 0.003
Fraction affected | -0.005 | 0.065 | 0.044 | 0.030 | 0.005
| (0.001) | (0.008) | (0.008) | (0.008) | (0.012)

Notes: This table is analogous to Table 7, except that it shows results for
the Gap design instead of the Fraction Affected design.

Table A9: Sensitivity of the Gap design: placebo

| Outcome
---|---
| Emp. | p10 | p25 | p50 | p90
Panel A: Only permanent differences in location
True average causal effect | -0.000 | 0.000 | 0.000 | 0.000 | 0.000
Gap measure | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
| (0.000) | (0.001) | (0.001) | (0.001) | (0.001)
Panel B: Adding location shocks, stable distributions
True average causal effect | -0.000 | 0.000 | 0.000 | 0.000 | 0.000
Gap measure | 0.001 | 0.017 | 0.018 | 0.019 | 0.019
| (0.000) | (0.005) | (0.006) | (0.006) | (0.006)
Panel C: Adding dispersion differences and shocks, stable distributions
True average causal effect | -0.000 | 0.000 | 0.000 | 0.000 | 0.000
Gap measure | 0.002 | 0.026 | 0.022 | 0.016 | 0.004
| (0.001) | (0.007) | (0.006) | (0.006) | (0.008)
Panel D: Average dispersion falls over time
True average causal effect | -0.000 | 0.000 | 0.000 | 0.000 | 0.000
Gap measure | 0.004 | 0.022 | 0.021 | 0.016 | 0.004
| (0.001) | (0.007) | (0.006) | (0.006) | (0.009)

Notes: This table is analogous to Table 7, but it reports a placebo scenario
with no increase in the national minimum wage. The Gap measure, however, is
calculated as if the national log minimum wage would increase by 0.2 between
periods (as is the case in Table 7).