The Minimum Wage and the Distribution of

Wages: Analysing the Spillover Effects of the

National Minimum Wage in the UK ∗

Panagiotis Nanos†

University of Southampton

March 2011

Abstract

This paper investigates the impact of the National Minimum Wage on the distribu-

tion of wages in the UK. Past research suggests that minimum wage spillover effects are

rather small and insignificant in the UK as opposed to the USA. I address this puzzle

using an empirical model employed in previous studies (Lee 1999, Manning 2003) and

demonstrate that the magnitude of the estimated spillover effects is dependent on the

initial assumptions about the counterfactual wage distribution. This finding implies

that the emergence of the puzzle is due to the lack of a unified estimation framework

capable of producing comparable results. My estimations are suggestive of a positive

minimum wage spillover effect in the UK.

JEL Classification: E24, J3, J31

Keywords: wage structure, minimum wage, inequality

∗Part of this work was completed while a doctoral student at the Department of Economics, University ofOxford, and was supported by the Greek State Scholarships Foundation. I am grateful to Margaret Stevensfor her continual advice that substantially improved this paper. I am also indebted to Steve Bond, AlanManning, Patricia Rice, and seminar participants at Oxford and AUEB for their constructive comments andsuggestions. I remain solely responsible for all errors, omissions, and interpretations.†Email: P.Nanos@soton.ac.uk, http://www.personal.soton.ac.uk/pn1c09/

1 Introduction

The introduction of the National Minimum Wage (NMW) in the UK in April 1999 and its

subsequent upratings prompted many scholars to investigate its impact on labour market

outcomes.1 Although there is a large literature addressing the effects of the National Mini-

mum Wage on employment, the gender pay gap, and the number of hours worked,2 very few

studies have been advanced analysing the impact of this policy on the distribution of wages

and wage inequality. Two prominent exceptions to this rule are the studies by Dickens and

Manning (2004a, 2004b), which conclude that the NMW, apart from the creation of a spike

at the level of the minimum wage, has had no effect on the wage distribution indicating

that it has not significantly reduced wage inequality.

These findings are in sharp contrast to the results reported in Lee (1999) and Manning

(2003) on the corresponding effects of the US minimum wage. Lee (1999) suggests that

the declining level of the US real minimum wage throughout the 1980’s contributed to the

significant increase in wage inequality in the USA. His analysis indicates that changes in the

level of the minimum wage impact on a large part of the wage distribution, thus affecting

wage inequality. Although Lee’s study does not focus on the estimation of minimum wage

effects on the pay of workers not directly affected, what is often referred to as spillover

effect, the results presented indicate the existence of significant and positive spillovers. In

the same spirit, Manning (2003) uses Lee’s (1999) empirical model to estimate minimum

wage spillover effects in the USA from the late 1970’s to the late 1990’s. The results of

his estimations signal the existence of significant spillover effects, which are indicative of a

non-negligeable effect of the minimum wage on wage inequality.

The differential impact of the minimum wage on the wage distribution across the At-

lantic, as documented in these studies, has been a major source of confusion among scholars:

the labour markets of the UK and the USA exhibit considerable similarities, which would

imply a homogeneous minimum wage effect in these countries. In this sense, the existence

of such an apparent difference constitutes a puzzle that remains unanswered.

The objective of this paper is to address this puzzle by estimating the model of Lee

(1999) for the UK, and thus, contribute to the understanding of the NMW effects on the

bottom half of the wage distribution.

The estimation of minimum wage effects on the wage distribution is subject to the

following methodological problem: one has to control for other changes occurring simulta-

neously with minimum wage upratings. The most obvious way to correct for this problem is

to find an appropriate counterfactual wage distribution, that is, the distribution of wages in

the absence of the minimum wage. Prior knowledge of the counterfactual wage distribution

would simplify matters, since one could calculate changes in wage inequality and spillovers

effects by comparing the pre- and post-change distributions. However, the underlying wage

distribution is unknown, since minimum wage effects on observed wages cannot be sepa-

1The National Minimum Wage was introduced at £3.60 per hour and since 2000 it is uprated everyOctober.

2Seminal studies include among others Stewart (2002, 2004) and Connolly and Gregory (2002).

1

rated from the influence of other factors, such as demographic change, annual wage growth,

and so on.

Lee’s (1999) model relies on such an assumption about the counterfactual wage distri-

bution: he assumes that the counterfactual has the same shape but differs in its centrality

and scale parameters across geographical regions. In this way, Lee employs geographical

variation to estimate a measure of counterfactual wage inequality, which is then utilised to

assess the impact of the minimum wage on post-change wage inequality. Although Man-

ning (2003) and Dickens and Manning (2004a) estimate the same model, their assumptions

about the counterfactual wage distribution are different: Manning (2003) assumes that the

shape of the counterfactual wage distribution is given by the lognormal distribution, while

Dickens and Manning (2004a) assume that it is given by the observed wage distribution in

a period prior to the minimum wage introduction.

I believe that the results obtained from the estimation of this model are sensitive to

changes in the assumptions about the counterfactual. For that reason, I fit the model to

the UK wage distribution employing the “normality” approach used in Manning (2003),

the “empirical counterfactual” approach presented in Dickens and Manning (2004a), and a

modified version of the “non-parametric” approach introduced by Lee (1999). I estimate the

model on the bottom-half and the bottom-quarter percentiles of the wage distribution and

explain that the comparison of parameter estimates corresponding to different percentile

ranges could be used as a specification test. For example, the estimation of model param-

eters that do not differ significantly across the wage distribution would indicate that the

model performs well in explaining the data.

Using this specification test, I demonstrate that the “non-parametric” approach outper-

forms the “parametric” and “empirical counterfactual” approaches. The implication of this

finding is that the same model under different assumptions about the counterfactual can

lead to different results. More importantly, the robustness of the results depends on the

plausibility of the assumptions about the counterfactual distribution. In this sense, I argue

that the puzzle is due to the lack of a unified estimation framework capable of producing

comparable results. The estimates obtained using the non-parametric approach indicate

the existence of positive and significant spillover effects of the NMW in the UK.

This paper is organized as follows. Section 2 reviews the empirical literature on spillovers.

I provide an extensive discussion of Lee (1999), Manning (2003), and Dickens and Manning

(2004a) to highlight the pivotal role of these studies in motivating my methodology. In

Section 3, I present the analytic strategy and the estimation approach. Section 4 provides

a summary of the dataset and presents the results of my estimations. Section 5 concludes

with a discussion of my findings. Finally, in the Appendix I discuss the properties of the

empirical model estimated in this paper.

2

2 Spillover Effects in Context

In this Section, I review past empirical research on minimum wage spillover effects. The

main purpose of this survey is to provide a detailed presentation of the methodological

approaches that motivated this paper, highlight the assumptions they employ, compare

their findings with respect to spillover effects, and discuss possible explanations for the

“puzzles” they have identified.

Any attempt to estimate the effects of policy measures, such as the minimum wage, on

the wage distribution over time is subject to the following methodological problem: one

has to control for other changes occurring simultaneously with minimum wage upratings.

In fact, one has to separate the minimum wage effects from the effects of other factors

on the wage distribution. An example is wage growth that occurs even in the absence

of a legislated wage floor. The most obvious way to correct for this problem is to find

an appropriate counterfactual wage distribution, that is, the distribution of wages in the

absence of the minimum wage. One of the most important features of the methodologies

developed to estimate minimum wage spillover effects is the technique employed to find the

counterfactual wage distribution.

Researchers have addressed this issue in a variety of ways. Card and Krueger (1995)

argue that a method of finding an appropriate counterfactual is to identify a subset of

workers, who do not experience a change in the minimum wage, but are similar in all other

aspects to the subset of workers that experience a change in their minimum wage; this

is conditional on having observations for both subsets over the same time period. They

examine the impact of the 1990 and 1991 federal minimum wage changes on the lower

percentiles of the wage distribution in US states with a high concentration of minimum

wage workers among teenagers; their counterfactual is the corresponding impact of the

minimum wage on US states with low concentration of workers near the minimum. Their

findings suggest that there exists a spillover effect at least up to the 10th percentile: given a

27 percent increase in the minimum wage during this period, the 5th percentile experiences,

on average, a 10 percent increase, while the 10th percentile experiences, on average, a 6

percent increase.3

A primary limitation of this methodology is that it does not control for factors that are

traditionally considered important in wage determination, such as demographic character-

istics. A study that accounts for this problem including controls for demographic trends

is Neumark, Schweitzer, and Wascher (2004). The basic approach of this study is to use

individually-matched data from 1979 to 1997 to regress individual workers’ wage growth on

changes in the minimum wage. In essence, Neumark et al. (2004) do not use a counter-

factual wage distribution, but rather a counterfactual wage profile. To capture the impact

of the minimum wage on different locations of the wage distribution, they regress the per-

centage difference of an individual’s wage on: a number of state-year indicator variables,

3Pollin, Brenner, and Wicks-Lim (2004) estimate a similar model for the period from 1991 to 2000 andconclude that the spillover effect extends no further than the 15th percentile of the wage distribution.

3

controls for race, sex, education, and experience, the percentage change in the minimum

wage, a measure of proximity of the individual’s wage to the minimum wage, as well as

interaction terms of these controls. In other words, instead of estimating the effect of the

minimum wage on different percentiles of the state wage distributions, they estimate the

effect on the earnings of different groups of individuals, which are constructed depending

on the distance of each individual’s first period earnings from the then prevailing minimum

wage. Their findings suggest a positive spillover effect, which is strongest near the minimum

wage level and dissipates quickly as one moves to higher wage bands.

The measure they use to identify the location of the distribution where the minimum

wage impacts depends crucially on the initial wage of individuals. A limitation of this

technique is that individuals initially classified into a specific wage band may experience an

idiosyncratic positive shock in their earned wages, which would result in an overstatement

of the estimated spillover effect for this particular location of the wage distribution.

An alternative approach to estimating the counterfactual density based on micro-level

data is presented in the seminal study by DiNardo, Fortin, and Lemieux (1996), who use

a semiparametric model to analyse the effects of institutional and labour market factors

on US wage inequality. This study addresses the issue of minimum wage spillover effects

in a somewhat implicit way. The methodology employed is very much in the spirit of

the Oaxaca (1973) decomposition with the difference that instead of focusing on means

alone, the authors focus on the entire density of wages. Their objective is to examine

whether the increase in wage inequality in the USA between 1979 and 1988 is explained

by the decline in the real value of the minimum wage. To address this question, they

estimate the wage densities of a homogeneous group of workers in 1979 and in 1988. The

counterfactual 1988 density is given if the minimum wage is raised to its 1979 real value.

Using appropriately weighted sections of the 1979 wage density, they simulate the 1988

counterfactual density. They compare the simulated counterfactual density and the kernel

estimated density for 1988 and argue that the decline in the real value of the minimum wage

explains a substantial proportion of the increase in wage inequality in the USA throughout

the 1980’s. Visual comparison of the densities indicates a considerable difference in the

area immediately to the right of the 1979 minimum wage, which signifies the existence of

positive minimum wage spillover effects.

One study that takes a comprehensive approach to modelling changes in the distri-

bution of wages associated with minimum wage upratings is Lee (1999). Because of the

important role this study plays in motivating my methodology, I provide a relatively in-

depth discussion of Lee’s approach and discuss the significance of his results. Moreover, I

present two subsequent studies (Manning 2003, Dickens and Manning 2004) that extend

Lee’s methodology and estimate spillover effects in an explicit way.

4

2.1 Lee (1999)

Lee examines the rising trend in US wage inequality during the 1980’s and identifies two

contributing factors: a rise in latent wage dispersion or a fall in the real minimum wage.4

Since the individual contributions of each one of these two factors to the rise in wage

inequality cannot be quantified using aggregate time series data, Lee uses regional variation

in the real or relative minimum wage across US states in order to decompose changes in

wage inequality into an average across states minimum wage effect and an average trend in

latent wage dispersion. The real or relative minimum wage is a measure of the differential

impact of the minimum wage on states’ wage distributions. To generate variability in the

relative minimum wage, Lee deflates the level of the legislated minimum using the median

of each geographical region’s wage distribution. It is evident that the variability of the

relative minimum wage arises due to the fact that the federal minimum wage has a sizeable

impact on low-wage states and a minimal impact on high wage states.

Apart from the relative minimum, Lee uses a measure of wage inequality given by the

differential between some lower-tail percentile and the median of the wage distribution in

every region. Similarly, latent wage inequality, that is, wage dispersion in the absence of

a minimum wage, is given by the differential between some lower-tail percentile and the

median of the latent wage distribution in every region.

Under the assumption that there is no stochastic variation in latent wage dispersion

across states, which implies that the shape of the latent wage distribution is identical in

all states up to a particular percentile, Lee argues that the relationship between the mini-

mum wage and wage dispersion is described by the following three scenarios: censoring -no

spillovers, no disemployment; spillovers, no disemployment; and truncation -no spillovers,

full disemployment. The effect of spillovers and disemployment on the observed wage dis-

tribution cannot be distinguished empirically. Lee admits that a hybrid case with some

disemployments and some spillover effects would be more realistic, but also very difficult

to implement empirically. Therefore, he assumes away disemployment effects and acknowl-

edges that any estimated effect of the relative minimum wage on wage dispersion would

overstate true spillover effects.

To account for stochastic variation in latent wage dispersion, Lee assumes that, condi-

tional on the year, the centrality measure of the state wage distribution is not systematically

correlated with latent wage dispersion across states. Lee employs this assumption to correct

for the mechanical correlation that could potentially arise due to the usage of the median

wage in both his dependent variable (the differential between a lower tail percentile and the

median of the regional wage distributions) and independent variable (the minimum wage

deflated by the regional median wage). If his identifying assumption does not hold, then the

empirical relation between these two quantities may exaggerate the impact of the minimum

wage. For example, if there is relatively greater variability in the median wage than in the

particular percentile of interest, then a region with a large median wage would have both

4The latent wage distribution is defined as the distribution of wages in the absence of the minimum wage.

5

a lower relative minimum, as well as a larger measure of wage dispersion, independently of

any minimum wage effect. Therefore, if there is a systematic relation between latent wage

dispersion and the location of the distribution (as measured by the median), Lee’s model is

not correctly specified.

A convenient way to test whether overall latent wage dispersion is empirically related

to wage levels is the identification of such a correlation at a point on the wage distribution

where the minimum wage is not likely to be a factor, e.g. the upper tail of the wage

distribution. If there is no significant association between the relative minimum and upper

tail measures of dispersion, then the relationship between the relative minimum and wage

dispersion in the lower part of the wage distribution (i.e. in the vicinity of the minimum

wage) reflects the actual impact of the minimum wage.

Lee presents the following specification, which captures the non-linear nature of the

relationship between observed wages, latent wages, and the minimum wage:

wpjt = wp∗jt +minwaget − wp∗jt

1− exp[−β(minwaget − wp∗jt

)] , (1)

where wpjt and wp∗jt are the observed and latent pth percentile of the log-wage distribution

in region j at time t, minwaget is the nominal value of the minimum wage in period t, and

β > 0 is a curvature parameter. If the minimum wage has a pure censoring effect, then

percentile p of the observed wage distribution is equal to percentile p of the latent wage

distribution or the minimum wage (whichever is higher). The above specification reduces

to the pure censoring case if the value of the curvature parameter β tends to infinity. For

finite values of β, the minimum wage has spillover and/or disemployment effects.5

Assuming that the shape of the latent log-wage distribution in year t, denoted Ft (.),

is the same in all regions, Lee expresses percentile p of the latent log-wage distribution in

state j at time t as follows:

wp∗jt = µjt + σjtF−1t (p), (2)

where µjt and σjt are centrality and scale parameters. Suppose the median wage is a

sufficient measure of centrality and is unaffected by the minimum wage, then µjt = w50∗jt =

w50jt . Under these assumptions, it is possible to express the average across regions difference

between percentile p and the median of the latent wage distribution in period t as follows:

αpt = E(wp∗jt − w

50jt |t)

= E (σjt|t)F−1t (p) = σtF−1t (p), (3)

where σt is the average across regions standard deviation of latent wages at t.

The combination of equations (3), (2), and (1) gives the non linear specification esti-

5The properties of this specification are analysed in the Appendix, where I illustrate the effects of theminimum wage for different values of β.

6

mated in Lee’s study:

E(wpjt − w

50jt |rmwjt, t

)=

rmwjt − αpt1− exp (−β(rmwjt − αpt ))

+ αpt , (4)

where rmwjt is the relative minimum wage(rmwjt = minwaget − w50

jt

). Clearly, the esti-

mated values of αpt provide a quantitative measure of the average across regions latent wage

dispersion and can be used to decompose wage changes into changes attributable to latent

wage growth and changes attributable to the minimum wage. The estimated value of the

curvature parameter β provides qualitative information about the relationship between the

minimum wage and percentile p of the log-wage distribution across states; this parameter

can also be employed to estimate the magnitude of minimum wage spillover effects. An-

other important point is that, besides the identifying assumptions presented above, one can

estimate the model without prior knowledge of the counterfactual/latent wage distribution.

If these assumptions hold, Lee’s model can produce a good estimate of the minimum wage

spillover effect and of the counterfactual wage percentile differentials, αpt ’s.

To investigate how changes in US minimum wages affected wage inequality, Lee uses

microdata from the Current Population Survey (CPS) Outgoing Rotation Group Earnings

Files. For workers who are not paid on an hourly basis, usual weekly earnings and usual

weekly hours are used to construct an average hourly wage. The annual frequency and

the large sample size of this dataset allows Lee to explore the effects of the minimum

wage throughout the 1980’s using regional variation in the level of the relative minimum

wage. Each observation of the dependent variable in Lee’s panel dataset corresponds to the

differential between percentile p and the median of the log hourly wage distribution in a

particular state in a particular year,(wpjt − w50

jt

); in most cases the 10 − 50 differential is

examined. The explanatory variable used is the relative minimum wage, that is, the gap

between the federal, or state legislated minimum wage, whichever is higher, and the median

of the corresponding (state-year) log hourly wage distribution.

Lee’s findings suggest that after accounting for the decline in the relative minimum wage,

average growth in wage dispersion in the lower tail of states’ wage distributions is quite

modest throughout the 1980’s. His estimates for men, women and the combined sample

imply that almost all of the growth in the wage gap between the 10th and 50th percentiles

is attributable to the erosion of the real value of the federal minimum wage during the

period 1979-1988. Estimation of the same model using variation in legislated changes in

minimum wages (arising from an interaction between the 1990-1991 federal increases and

preexisting variability in state-specific minimum wage laws) produces quite similar results

to those generated by cross-state variability in the relative minimum.

Although Lee’s paper focuses on how minimum wages “sweep up” workers in the lower

tail of the wage distribution, as opposed to an analysis of the effects on the part of the wage

distribution above but near the new minimum wage, his estimates are indicative of positive

spillovers in this region of the distribution; the estimated curvature parameter was β = 9.4,

which implies that the maximum minimum wage spillover effect on the 10th percentile of

7

the regional log wage distributions was approximately 1β = 0.106 log points.6

Apart from the results on the effects of the minimum wage on wage inequality, an

important contribution of this study is the detailed description and graphical illustration of

the various effects of the minimum wage on the wage distribution as well as the introduction

of an empirical specification (equation [4]) capable of capturing these effects. Lee’s empirical

model provides a non-structural methodology for the estimation of the minimum wage

spillover effects: their magnitude can be calculated from the estimated value of parameter

β. Another advantage of Lee’s approach is that, as long as the identifying assumptions

hold, the researcher can estimate the impact of the minimum wage on each percentile of the

wage distribution being agnostic about the shape of the latent wage distribution. In this

way, one need not make strong assumptions about the counterfactual distribution of wages

a priori. This approach has been very influential in the literature focusing on the spillover

effects of the minimum wage. In the remainder of this Section, I review two studies that

extend and implement empirically this model for the USA and the UK.

2.2 Manning (2003)

Contrary to Lee, Manning focuses on the impact of the minimum wage further up the wage

distribution, and thus, adjusts model (4) to explicitly capture the spillover effects of the

minimum wage. The objective of Manning’s estimations is the measurement of minimum

wage spillover effects on the bottom half of the hourly wage distribution (the estimation

and interpretation of parameter β). To this end, he adjusts Lee’s model appropriately and

adopts a parametric approach making an assumption about the shape of the latent wage

distribution. He assumes that the latent log wage distribution is normal and has constant

variance, that is,

wp∗jt = w50jt + σΦ−1 (p) , (5)

where w50 is the log of the median wage, σ is the average across states and over time

standard deviation of the log wage distribution and Φ−1 () is the inverse of the standard

normal distribution function. In this way, the counterfactual log wage distribution can be

estimated using the median of each state’s log wage distribution, the average across states

and over time standard deviation of the log wage distribution, and the values of the inverse

standard normal distribution.

The empirical model used in Manning’s estimations is:

wpjt =[w50jt + σΦ−1 (p)

]+

minwaget −[w50jt + σΦ−1 (p)

]1− exp

[−β(minwaget −

[w50jt + σΦ−1 (p)

])] . (6)

The parameters to be estimated are β and σ. The estimate of the average standard deviation

of the latent log-wage distribution, σ, is used to calculate latent log-wages; the estimated

6Spillovers effects are maximized at the point where the minimum wage binds. The magnitude of themaximum spillover effect is equal to the inverse of the curvature parameter β. In the Appendix, I analyseand illustrate this point.

8

latent wages can then be used along with the estimate of the curvature parameter, β, to

calculate the magnitude of the minimum wage spillover effects.

The spillover effect at percentile p is given by the difference between the total effect (the

difference between the observed wage and the latent wage at this location) and the direct

effect (the increase in latent wages required for compliance). In the Appendix, I demonstrate

that spillover effects are determined by the gap between the minimum wage and the latent

wage, as well as the curvature parameter β.

Manning estimates Lee’s model by non linear least squares on US data from 1979 to

2000.7 The model in (6) is fitted to all percentiles below the median, where each observation

is a percentile of the log hourly wage distribution in a particular state in a particular year.

The percentiles that are below the minimum are excluded from the analysis. The estimation

for the entire 22 year period gives a standard deviation of log wages σ = 0.6 and a spillover

parameter β = 8.8, which implies that the maximum size of the spillover effect is 1β = 0.11

log points. Manning concludes that all of the observed variation in wage inequality at

the bottom half of the wage distribution can be attributed to variation in the level of the

minimum wage.

Hence, assuming that the latent log wage distribution is normal, Manning estimates

the average across regions and overtime standard deviation of the log wage distribution, σ,

and the value of parameter β. Conditional on the estimated value of σ, one can estimate

the latent wage distribution, and then, conditional on the value of β and the latent wage

distribution, one can compute the predicted spillover effect at any percentile of the log wage

distribution.

To give some idea of how well the model fits the data, Manning also calculates the

actual spillover effects, given by the difference between the actual wage and the predicted

direct effect. Both the actual and predicted spillover effects are evaluated conditional on

the estimated standard deviation of the latent wage distribution.

Manning presents an elaborate methodology for the explicit estimation of minimum

wage spillover effects. A key feature of his estimation approach is the assumption that

latent log wages are normally distributed. The plausibility of the “normality assumption”

is discussed in the following Sections.

2.3 Dickens and Manning (2004a)

Dickens and Manning (2004a) use model (1) to estimate the spillover effects of the intro-

duction of the National Minimum Wage on the wage distribution of care homes in the UK.

They use data from a postal survey of workers in residential homes for the elderly. The

sample was created by collecting monthly data from one-ninth of the total population of

UK care homes for each of the nine months before and for each of the nine months after

the introduction of the National Minimum Wage.

To estimate model (1), the authors assume that the latent wage distribution is given by

7Manning uses CPS microdata. For workers not paid on an hourly basis, usual weekly earningns andusual weekly working hours are used to construct a measure of average hourly earnigns (same as Lee).

9

the observed wage distribution in the period prior to the introduction of the minimum wage.

The estimate of the spillover parameter is β ' 17, which implies a maximum predicted

spillover effect of 0.060. This result can be interpreted as follows: workers initially paid

£3.60 had their pay raised by 0.06 log points, an estimate that is not enormous but is not

small either. Their finding suggests that the direct effect of the NMW introduction was to

raise the average log wage by 5.1% but the total effect is for it to rise by 7.2% implying

that spillover effects increase the total effect by 2.1%, i.e. about 30% of the total.

They explain that the very simple model they use overstates the size of the spillover

effects as it ascribes any wage growth observed at the higher percentiles of the wage distri-

bution to spillover effects. Hence, they assume that in the absence of the minimum wage,

log wages grow by the same amount at all points of the wage distribution and estimate the

model again. Their estimates show that the estimated spillover effects are reduced (β ' 42

and the maximum predicted spillover effect is 0.024 log points). The authors conclude that

after allowing for general wage growth, the direct and spillover effects of the minimum wage

are significantly reduced.

2.4 Discussion

The studies by Manning (2003) and Dickens and Manning (2004a) suggest that there exists

a significant difference between the impact of the minimum wage in the USA and in the

UK. An important point to note is that the assumptions employed in the two studies are

not the same. Manning (2003) assumes that the shape of the latent log wage distribution in

the USA is normal and calculates the counterfactual distribution using the medians of the

state-year log wage distributions and the estimated value of average standard deviation.

In Dickens and Manning (2004a), the counterfactual wage distribution is given by the

observed wages in the period prior to the introduction of the minimum wage. Another

difference between these two studies, which is by and large attributable to the nature of the

datasets used, is that Manning (2003) estimates spillover effects using the bottom half of

the wage distributions in many different states, while Dickens and Manning (2004a) do not

exploit regional variation -they pool the bottom 90 percentiles of the wage distribution. In

the following section, I discuss the sensitivity of the results to changes in the assumptions

employed in each estimation approach.

3 Methodology

The basic analytic strategy builds on the model by Lee (1999) and its modification by

Manning (2003) that addresses the estimation of spillovers explicitly. As described in the

previous Section, past research employed a prima facie unified framework for the estimation

of the minimum wage spillover effect in the USA (Lee 1999, Manning 2003) and the UK

(Dickens and Manning 2004), and pointed out the differential impact of the minimum wage

across the Atlantic. One would have no reason to suspect that the approaches differ in any

way, since the empirical models in these studies are algebraically equivalent (equations [4],

10

[6], and [1], respectively) and the parameters estimated are similar, the curvature parameter

(β) and some measure of latent wage dispersion (αpt or σ). Notwithstanding the similarities

in the empirical approach, I believe that the results presented in these three studies are not

comparable.

A possible explanation for the non-comparability of these studies is their differential ori-

entation, which necessitated the adoption of different assumptions about the counterfactual

wage distribution. In the remainder of this Section, I outline the assumptions employed in

each case, examine the conditions under which these assumptions are satisfied, and analyse

how I use these approaches to estimate the spillover effects of the minimum wage. I also

present a simple specification test that can be used to evaluate the performance of each

approach in estimating spillover effects.

Non-parametric Approach Lee uses regional variation in the relative minimum wage

and assumes that the shape of the regional wage distributions is the same in all regions; he

also assumes that, conditional on the time period, the medians of these regional distributions

are not systematically correlated with latent wage dispersion across regions. Therefore, Lee

estimates the model being agnostic about the actual shape of the regional wage distributions

and allowing the distributions to differ in their centrality and scale parameters. These

minimalistic assumptions lead to an estimate of the average across regions latent wage

dispersion (αpt ); the curvature parameter (β) is estimated as a by-product.

The main virtue of this approach is that it allows the researcher to estimate the effects

of minimum wage changes on wage inequality without specifying the shape of the latent

wage distribution. Nevertheless, there are reasons to be skeptical of this approach when

it comes to the estimation of minimum wage spillover effects: it only focuses on changes

at a single percentile of the wage distribution, thereby providing a fragmentary picture of

minimum wage spillover effects, as opposed to a more comprehensive approach that would

estimate the impact of the minimum wage on the bottom half, or the bottom quarter of the

wage distribution.

I present a simple alternative that accounts for this limitation. Instead of relying on an

across regions estimation of the minimum wage spillover effect at a particular percentile,

I pool all percentiles below the median of all regional wage distributions and estimate the

following specification:

wpjt − w50jt =

rmwjt − αpt Ipt

1− exp (−β(rmwjt − αpt Ipt ))

+ αpt Ipt + upjt, (7)

where(rmwjt = minwaget − w50

jt

)is the relative minimum wage, Ipt is a dummy variable

indicating the percentile (p) of the regional latent wage distributions, and upjt is the error

term; the parameters to be estimated are the curvature parameter (β) and an average across

regions measure of the differential between the median and percentile p of the latent wage

distribution (αpt ∀p < 50).

The estimation of this specification relies on the existence of sufficient variation in the

11

level of the relative minimum wage. Stratifying the aggregate wage distribution by region

and exploiting the fact that the impact of the minimum wage is more pronounced in low-

wage regions than in high-wage regions, I can estimate the parameters of interest without

making a parametric assumption about the shape of the latent wage distribution. The

conditions that need to be satisfied are Lee’s identifying assumptions: (a) the shape of the

latent wage distribution is the same across regions, (b) the observed median wage is equal to

the latent median wage in every region (w50jt = w50∗

jt ), and (c) conditional on the time period,

the medians of all regional wage distributions are uncorrelated with latent wage dispersion

across regions.8 In addition, I make the simplifying assumption that the minimum wage

has no disemployment effects, since such effects cannot be empirically distinguished from

spillovers. This implies that the estimated spillover effects would be inflated estimates of

true spillovers if in reality the minimum wage causes job losses; however, there is substantial

evidence to the contrary.9

This model can also account for a minimum wage that varies over time as long as the

relative minimum is not correlated with changes in latent wage inequality across regions.

In this paper, however, I only consider variation in the relative minimum arising due to

variation in the median of the regional wage distributions; therefore, for every different

level of the nominal minimum wage, I obtain different estimates of the model’s parameters.

This strategy allows me to estimate the latent wage distribution at different points in time,

and thus, analyse separately the effects of each change in the level of the nominal minimum

wage.

Using the estimates of average latent wage dispersion (αpt ) for all percentiles below the

median, I am able to calculate the bottom half of the aggregate latent wage distribution.

Note that αpt gives an average across regions estimate of the gap between percentile p and

the median of the latent wage distribution, so it does not capture regional differences in

latent wage dispersion. The estimated αpt ’s could only be used to calculate regional latent

wage-percentiles if latent wage dispersion did not vary stochastically across regions, that is,

if αpjt = αpt for any region j. It should be emphasized that the parameters of specification (7)

are identified when latent wage dispersion varies stochastically across regions; the absence of

stochastic variation in latent wage dispersion is only required for the estimation of regional

latent wage-percentiles. This can be clarified by means of an example.

Suppose latent wage dispersion varies stochastically across regions. The residuals pro-

duced by estimating specification (7) are given by:

upjt =[wpjt −

(w50jt + αpt I

pt

)]− rmwjt − αpt I

pt

1− exp(−β(rmwjt − αpt I

pt ))

︸ ︷︷ ︸predicted spillover

. (8)

8If (c) is violated in the data, the model is not correctly specified. In a recent study, Autor, Manning,and Smith (2010) address the issue of misspecification caused by the violation of this assumption.

9Card and Krueger (1995) report a lack of adverse employment effects of the minimum wage in the USA.Stewart (2002, 2004a, and 2004b) and Stewart and Swaffield (2002) argue that the NMW had similar effectson employment in the UK.

12

One would be tempted to interpret upjt’s as the difference between the actual spillover

effects and the predicted spillover effects of the minimum wage. However, a more careful

examination of (8) indicates that the term in the square brackets on the right-hand side

would only correspond to the actual spillover effect of the minimum wage if latent wage

dispersion were constant across regions, that is, αpjt = αpt for any j. If αpjt 6= αpt , then the

“economic” interpretation of the residuals in this context is not straightforward: the model

does not allow for the calculation of regional latent wage percentiles, so actual spillover

effects cannot be inferred. The interpretation of the residuals will be further discussed in

the context of my empirical results.

Fitting specification (7) to a range of percentiles of regional wage distributions, I obtain

estimates of the curvature parameter (β) and average latent wage dispersion (αpt ’s), which

are utilised to describe the spillover effects of the minimum wage. In the following Section,

this modification of Lee’s model is used to analyse the impact of the minimum wage on the

bottom half percentiles of the wage distribution. Henceforth, this approach is referred to

as the “non-parametric” approach.

Parametric Approach The primary objective of Manning’s study is the estimation of

minimum wage spillover effects across the wage distribution. His analysis is predicated

on the assumption that latent log-wages are normally distributed. If this assumption is

satisfied, the model produces accurate estimates of spillover effects. However, it is likely

that the latent wage distribution has a different shape.10 If latent wages are not log-normally

distributed, this approach may lead to unreliable estimates of spillover effects.

Manning adopts this parametric assumption to examine how the minimum wage affects

wages in the USA, where many states have legislated wage floors higher than the federal

minimum. Given the nature of his dataset, Manning assumes that the distribution of

latent wages is normal in every state and has constant variance across states. If the labour

market does not exhibit variation in the nominal minimum wage across regions (e.g. the

UK labour market, considered in this paper), the parametric assumption ensures that the

parameters of interest are identified without the need to stratify the data by region; under

the parametric assumption, the model can be estimated using percentiles of the aggregate

wage distribution. Pooling the bottom 50 percentiles of the aggregate wage distribution, I

estimate the following specification

wpt =[w50t + σΦ−1 (p)

]+

minwaget −[w50t + σΦ−1 (p)

]1− exp

[−β(minwaget −

[w50t + σΦ−1 (p)

])] + upt , (9)

where wpt corresponds to percentile p of the aggregate wage distribution at time t, Φ−1 () is

the inverse of the standard normal distribution, and upt is the error term; the parameters to

be estimated are the curvature parameter (β) and the standard deviation of the distribution

of latent wages (σ). Armed with these estimates, I can then calculate the bottom half of

10In the following Section, I analyse my dataset and provide an example of a wage measure that mightnot be log-normally distributed.

13

the aggregate latent wage distribution and infer the actual and predicted spillover effects of

the minimum wage. In what follows, I refer to this approach as the “parametric approach”

without regional variation.

I also explore whether the combination of the parametric assumption about the latent

wage distribution with variation in the relative minimum wage across regions, as employed

by Manning, can lead to more informative conclusions. Specifically, I stratify the data by

region and make the following assumptions: (a) the latent wage distribution is log-normal

in every region, (b) the observed median wage is equal to the latent median wage in every

region (w50jt = w50∗

jt ), and (c) the variance of the latent wage distribution is constant across

regions. I then fit all percentiles below the median of the regional wage distributions to

specification (6) and estimate the curvature parameter (β) and the average across regions

standard deviation of the latent wage distribution (σt). As in the non-parametric case,

this model can be used to estimate the impact of the minimum wage over time; however,

I choose to estimate the model separately for every period t corresponding to a different

level of the nominal minimum wage. Using the estimates of the model parameters and

the normality assumption, I form the latent wage distribution and compute the actual and

predicted spillover effects of the minimum wage. Henceforth, this approach is referred to as

the “parametric approach” with regional variation.

Empirical Latent Approach Dickens and Manning make the assumption that the latent

wage distribution is the distribution of accepted wages just before the introduction of the

minimum wage. This approach is ideal for the calculation of direct and indirect effects

in response to the introduction of a wage floor. However, if this approach is employed

to investigate the effects of a minimum wage uprating, it produces estimates that are very

different in magnitude. The wage distribution prior to a minimum wage uprating is distorted

by the existing wage floor; using this approach one calculates the incremental effects of the

minimum wage uprating on an already deformed distribution, so the estimated spillover

effects are bound to be smaller. Such underestimation may also be the result of wage

adjustments in anticipation of a minimum wage introduction.

To examine how this assumption of Dickens and Manning regarding the latent wage

distribution affects the estimates, I fit percentiles of the aggregate wage distribution to the

following specification

wpt = wpt−1 +minwaget − wpt−1

1− exp[−β(minwaget − wpt−1

)] , (10)

where wpt−1 is percentile p of the aggregate wage distribution in the period before the

minimum wage change; the parameter to be estimated is the curvature parameter β. In the

remainder of this paper, I refer to this approach as the “empirical latent approach”.

Specification Test The estimation approaches presented above can produce accurate

estimates of spillover effects under the conditions specified by their identifying assumptions.

14

If all these assumptions are satisfied, then one would expect consistency in the estimated

parameters across approaches. The existence of statistically significant variation in the

estimates across approaches signals that some identifying assumption is violated in the

data. The salient differences across the three estimation strategies are their assumptions

regarding the latent wage distribution. Since these assumptions are not testable, how can

I demonstrate which approach produces reliable estimates?

Notwithstanding the differences in the assumptions employed, the principal objective

of the three empirical approaches analysed above is to infer the spillover effects of the

minimum wage using the estimated spillover parameter (β). To estimate each specification,

I use the bottom half percentiles of regional or aggregate wage distributions. According

to the theoretical analysis of the previous subsection, the impact of a given change in

the minimum wage level on the wage distribution is characterized by the value of the

spillover parameter, which should be the same across the entire wage distribution. In other

words, if the models are correctly specified, the estimated curvature parameter β should be

independent of the part of the wage distribution used in its estimation. A simple strategy to

evaluate the plausibility of the assumptions employed in each one of the three specifications

is to test the consistency of the estimates they produce across the wage distribution. Since

the minimum wage is more likely to affect the lower half of the wage distribution, I estimate

each specification on the bottom half percentiles and on the bottom quarter percentiles of

the wage distribution and examine whether their estimates (β’s) are sensitive to changes

in the dataset used. I argue that this comparison can serve as a specification test of the

three approaches: if β’s differ significantly across the wage distribution, the approach is

misspecified. Using this convenient method, I am able to identify the assumptions that

are not supported in the data, and thus, determine which model estimates minimum wage

spillover effects in a reliable and accurate way.

In the remainder of this paper, I apply the framework developed so far to the estimation

of minimum wage spillover effects in the UK.

4 Dataset and Empirical Findings

4.1 Data

The analysis in this study utilises microdata from the Quarterly Labour Force Survey

(LFS), which is the largest representative UK survey with wage information. The National

Minimum Wage (NMW) was introduced in the UK in April 1999 and from the year 2000

onwards it is uprated every October. The dataset I use covers a period that ranges from

the first quarter of 1999 to the fourth quarter of 2005, that is, I investigate the effects of

the introduction and six subsequent upratings of the NMW −the inaugural rate and the

upratings included in my sample are presented in Table 1. In particular, I consider the

quarters after the introduction or the uprating of the minimum wage: second quarter of

15

1999 and fourth quarters of 2000-2005.11

Table 1: The National Minimum Wage, 1999-2005.

Adult Ratefor workers aged 22+

1 April 1999 3.601 October 2000 3.701 October 2001 4.101 October 2002 4.201 October 2003 4.501 October 2004 4.851 October 2005 5.05

Given that the NMW sets a minimum hourly wage for workers, any assessment of

its effect should focus on changes in hourly wages. The LFS includes two basic hourly

wage measures, the variables hourly rate and hourly pay, henceforth hrrate and hourpay,

respectively. The former is the main job hourly wage as reported by workers, who are paid

on an fixed hourly rate, whereas the latter is a derived variable constructed by dividing

the main job gross weekly earnings measure by the measure of usual paid hours in the

main job. Clearly, the variable hrrate is more suitable for the analysis of the impact of

the NMW, as it provides a better measure of the “true” hourly wage than the variable

hourpay, but the problem is that, while hourpay is available for all those in work, hrrate

is only available for those paid by the hour −salaried workers do not answer this question.

As a result, hrrate is available for less than 40% of workers included in the LFS. Moreover,

hrrate only started being available in March 1999, one month before the introduction of

the NMW. Past research has addressed these weaknesses of the LFS data by suggesting

methods to impute the missing observations of the hrrate variable.12 However, the various

methodologies developed for the correction of the missing data problem rely on identifying

assumptions that are usually violated in the data, which casts doubt on the validity of their

predictions.

Lee (1999) and Manning (2003) have addressed this problem by replacing the missing

observations of the CPS hourly rate variable by the corresponding CPS average hourly

wage, which is equivalent to the LFS hourpay variable. Given the questionable results

of the imputation methodologies, I adopt Lee’s and Manning’s strategy and combine the

hrrate and hourpay variables to construct a new hourly earnings variable, denoted hearn.

This new variable includes all hrrate observations and replaces its missing values by hourpay

observations. In cases where an individual has both values, I use the hrrate. In this way, I

create a UK dataset that is qualitatively similar to the US datasets used in Lee (1999) and

11Note that the LFS quarters used are seasonal, rather than calendar quarters. In LFS terminology, theNMW was introduced in the first quarter of 1999 (hence, I use the second quarter of 1999), and uprated inthe third quarter of 2000-2005 (hence, I use the fourth quarters of 2000-2005).

12Dickens and Manning (2002) review this literature and present an alternative methodology, the “mea-surement error approach”.

16

Manning (2003).

However, scholars have expressed concerns about the large measurement error in the

hourpay variable,13 which indicate that the hearn variable, though comparable to the mea-

sures used by Lee and Manning, may produce unreliable results. The two Panels of Figure

1 depict kernel density estimates of the distribution of log-wages for the quarters before and

after the NMW introduction (fourth quarter of 1998 and second quarter of 1999).14

1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

Panel A

Log(Wage)

Den

sity

hrrate99b

hourpay99b

hearn99b

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

Panel B

Log(Wage)

Den

sity

hourpay98d

hourpay99b

Figure 1: Pre- and post-NMW introduction kernel density estimates of hourpay, hrrate andhearn.

Unfortunately, hrrate started being reported in March 1999, which means that hourpay

is the only measure available in both quarters. Hence, the lack of data does not allow us to

compare the densities of all three wage measures before and after the NMW introduction.

In Panel A, I present kernel estimates of the log-wage densities for hourly paid workers

(hrrate), salaried workers (hourpay), and the combined sample (hearn), based on the 1999

second quarter LFS −the quarter after the NMW introduction. The vertical line denotes the

inaugural NMW rate. The existence of a large spike in the hrrate density at the minimum

wage is the most striking impression from Panel A of Figure 1 −approximately 10% of hourly

paid workers receive the minimum wage. On the contrary, the hourpay density exhibits no

apparent spike, less than 1% of salaried workers receive the minimum, which implies that

the minimum wage is not an important feature of the lower part of the wage distribution.

The visual comparison of the hrrate and hourpay densities in Panel A provides a possible

13See Dickens and Manning (2002, 2004a).14All kerned density estimates are constructed using the Epanechnikov kernel and an “optimal” kernel

bandwidth.

17

explanation for this difference: it signifies that the hrrate measure oversamples low-wage

workers, while the hourpay measure is more representative of the working population. The

increased mass of the hearn density compared to the hourpay density in the area imme-

diately to the right of the minimum wage level reflects the inclusion of low-wage workers

in the salaried workers sample. Panel B shows that the introduction of the minimum wage

had virtually no effect on the distribution of wages of salaried workers (hourpay) and in

part corroborates the measurement error concerns expressed in the literature.

The visual comparison of the three variables suggests that hearn is a more representative

−hence more appropriate− measure than hourpay, though its accuracy compared to hrrate

is still in question. For that reason, apart from the hearn sample, I estimate the impact of

the minimum wage on the sub-sample of hourly paid workers.

Table 2: Summary StatisticsYEAR HEARN HRRATE MIN

Mean Sd Median N Mean Sd Median N WAGE

1999b 8.45 5.28 6.93 16,073 5.42 2.42 4.75 4,192 3.602000d 9.15 5.73 7.47 14,808 6.42 3.19 5.50 5,909 3.702001d 9.60 5.87 7.89 15,032 6.70 3.21 5.72 5,826 4.102002d 9.88 5.98 8.03 14,713 6.82 3.19 5.92 5,554 4.202003d 10.32 6.21 8.43 14,025 7.24 3.50 6.00 5,212 4.502004d 10.77 6.37 8.84 13,430 7.63 3.76 6.43 4,936 4.852005d 11.21 6.58 9.20 12,896 7.82 3.69 6.69 4,550 5.05

I only examine workers covered by the “adult” minimum wage rate (those aged 22

or older). The sample excludes the earnings of self-employed individuals, as well as all

second job earnings information. Observed wages below the minimum are evidence of

non-compliance, which is an existing but not widespread phenomenon, therefore, I choose

to include them in the sample. It is very likely though that the underlying reason for the

observation of wages below the minimum is measurement error, rather than non-compliance.

To alleviate this problem I exclude wage values below £2 per hour from the sample. Extreme

wage values at the other end of the wage distribution, that is, wages above £50 per hour are

also excluded from the sample. The resulting sample characteristics for the hearn and for

the hrrate measures are summarized in Table 2. The considerable gap between the mean

values of the two variables along with the differences in their standard deviations confirm

my claim that low-wage workers are overrepresented in the hrrate sample, while the hearn

sample is more representative of the entire working population. Despite these differences,

there is a rising trend in average hourly earnings throughout the period under study. In

what follows, I explore whether this increase in average hourly earnings can be attributed

to the rising level of the NMW.

In order to estimate versions of the model that require regional variation in the relative

minimum wage, I divide the aggregate sample to 21 regional sub-samples using the LFS

variable “region of place of work” (gorwkr).15 Data frequencies in every region are used

15In fact, the “region of place of work variable” includes workplaces outside the UK as a 22nd “region.”

18

Table 3: Summary Statistics by Region, fourth quarter 2005REGION HEARN HRRATE

Mean Sd Median N Mean Sd Median N

Tyne and Wear 9.80 5.06 8.30 331 7.23 2.53 6.20 119rest of NE 9.46 5.06 7.72 292 7.11 2.93 6.00 140

greater Manch. 10.95 6.64 8.97 477 7.69 4.38 6.22 155Merseyside 10.81 6.02 8.97 249 8.29 4.70 6.90 85rest of NW 10.55 6.17 8.50 576 7.46 3.39 6.25 222

South Yorkshire 9.92 5.39 8.43 310 8.02 3.99 6.72 116West Yorkshire 10.64 5.88 8.80 567 7.89 4.58 7.00 217

rest of Yorkshire 9.80 5.91 7.92 397 7.18 2.85 6.10 183East Midlands 10.32 5.98 8.34 1,077 7.53 3.57 6.50 446West Midlands 11.24 6.22 9.50 569 7.83 3.50 6.88 200

rest of West-Mid 10.27 5.79 8.50 528 7.43 3.07 6.50 206East of England 11.42 6.66 9.37 1,083 7.98 4.06 6.70 354Central London 18.64 9.66 16.35 399 11.28 6.53 9.75 52Inner London 15.18 7.86 13.49 348 9.80 4.10 9.35 56Outer London 12.64 6.39 11.41 603 8.61 3.93 7.20 155

South East 11.84 6.84 10.00 1,859 8.08 3.73 7.00 559South West 10.74 6.49 8.64 1,135 7.96 3.53 7.00 439

Wales 9.82 4.98 8.10 550 7.62 2.89 6.60 222Strathclyde 10.82 6.30 8.71 448 7.71 2.99 6.80 171

rest of Scotland 10.63 6.04 8.63 756 7.67 3.30 6.50 326Northern Ireland 9.90 5.01 8.22 342 7.47 3.27 6.60 127

Total 11.21 6.58 9.20 12,896 7.82 3.69 6.69 4,550

19

to construct sampling weights, so as the sample to be nationally representative. Table 3

summarizes all regional sub-samples for the fourth quarter of 2005.

Finally, in order to estimate the “empirical latent” version of the model, I relax the

definition of the latent wage distribution: instead of defining the latent as the distribution

of wages in the absence of a minimum wage, I define it as the distribution of wages before

the introduction or the uprating of the minimum wage. The underlying reason for this

modification is that the hrrate measure is reported in the LFS since the first quarter of

1999 (when the NMW was actually introduced), therefore, I cannot examine the impact of

the NMW introduction, as hrrate (or hearn) observations are not available for the fourth

quarter of 1998. This means that I can only evaluate the impact of the 2000-2005 upratings.

For that purpose, I use accepted wages included in the LFS quarter that preceded the 2000-

2005 upratings of the minimum wage.

4.2 Results

In the preceding Sections, I presented the empirical approaches to the estimation of Lee’s

specification (equation [1]) and claimed that they may produce different estimates of spillovers

depending on the dataset and the assumptions about the latent wage distribution. This

prediction is now tested in the UK labour market. To be more specific, I fit the para-

metric approach (with and without regional variation), the non-parametric approach, and

the empirical latent approach to two versions of the UK hourly wage distribution (namely,

the hrrate and hearn distributions) and compare the magnitude of the estimated spillover

parameter across methodologies. To test the robustness of each methodology’s results, I

estimate the various specifications on the bottom 25 and 50 percentiles of the wage distribu-

tion and check whether the parameter values estimated on the two parts of the distribution

differ significantly. For the sake of exposition, instead of reporting the standard error of the

estimated parameter value, all tables provide its 95% confidence interval.

4.2.1 Parametric Approach

First, I estimate spillover effects using the parametric approach without regional variation.

I fit model (9) to the bottom quarter and to the bottom half percentiles of the aggregate log-

wage distribution, assuming that latent log-wages are normally distributed, and estimate the

spillover parameter (β) and the standard deviation of log-wages (σ). All percentiles below

the minimum wage are dropped. The results of my estimations for the two alternative

measures of hourly wages are reported in Tables 4 and 5.

Table 4 indicates that this specification captures the most important aspects of the min-

imum wage impact on the hearn distribution in a satisfactory way: the spillover parameter

(β) and standard deviation (σ) estimates on the bottom quarter and on the bottom half

of the distribution do not differ significantly −the only exception is the standard deviation

estimates in the post NMW introduction period (second quarter of 1999).

Since the NMW only affects wage earners in the UK, earnings information on labour activity outside the

20

Table 4: Parametric Approach without Regional Variation −Variable HEARN

year β 25pctiles σ 25pctiles β 50pctiles σ 50pctiles

1999b 13.9362 0.5119195 9.899265 0.55368911.22 -16.66 0.497 - 0.527 8.33 - 11.47 0.533 -0.574

2000d 8.633641 0.5251541 7.088272 0.56868846.29 - 10.98 0.486 - 0.565 5.40 - 8.78 0.520 - 0.617

2001d 10.5962 0.5243181 8.70017 0.5577967.77 - 13.43 0.492 - 0.557 7.15 - 10.25 0.529 - 0.587

2002d 8.98442 0.5180127 7.715268 0.54994836.34 - 11.63 0.472 - 0.563 6.17 - 9.26 0.510 - 0.589

2003d 11.30819 0.5086666 8.400503 0.56211539.09 - 13.53 0.486 - 0.531 6.83 - 9.97 0.527 - 0.597

2004d 12.35581 0.5163078 9.744876 0.55396339.94 - 14.77 0.494 - 0.538 8.28 - 11.21 0.530 - 0.577

2005d 13.32879 0.5036298 11.19805 0.525501911.05 - 15.61 0.487 - 0.520 9.81 - 12.59 0.511 -0.540

Notes: Estimates and 95% confidence intervals of the parametric model fitted bynon-linear least squares to the bottom 25 and 50 percentiles of the aggregate wagedistribution.

Table 5: Parametric Approach without Regional Variation −Variable HRRATE

year β 25pctiles σ 25pctiles β 50pctiles σ 50pctiles

1999b 330.7593 0.2330185 39.90389 0.2517496-931 - 1593 0.229 - 0.236 25.94 - 53.86 0.243 - 0.26

2000d 20.80543 0.2939002 16.38736 0.312083214.12 - 27.48 0.278 - 0.310 11.39 - 21.38 0.29 - 0.33

2001d 44.48491 0.275269 23.54164 0.300328731.90 - 57.06 0.270 - 0.280 16.29 - 30.78 0.284 - 0.316

2002d 42.34545 0.255109 15.34914 0.315744928.66 - 56.02 0.250 - 0.261 7.77 - 22.925 0.264 - 0.367

2003d 52.50301 0.228456 9.193729 0.393864620.918 - 84.088 0.219 - 0.238 7.78 - 10.61 0.343 - 0.445

2004d 18.95961 0.3242217 20.90203 0.31245719.683 - 28.24 0.263 - 0.385 14.10 - 27.70 0.283 - 0.341

2005d 28.31414 0.2807519 17.29742 0.336618520.18 - 36.45 0.263 - 0.298 10.79 - 23.80 0.288 - 0.385

Notes: Estimates and 95% confidence intervals of the parametric model fitted bynon-linear least squares to the bottom 25 and 50 percentiles of the aggregate wagedistribution.

21

By contrast, the evidence in Table 5 shows that my estimates take different values on

different ranges of the hrrate distribution: the bottom 25 and bottom 50 estimates of β

and σ are very different in magnitude and in most periods these differences are statistically

significant. My results suggest that the parametric approach without regional variation is

suitable for the estimation of minimum wage effects on the hearn distribution, but mis-

specified for the hrrate distribution. The different performance of this specification across

the two variables signals that its assumptions are satisfied in the hearn data but violated

in the hrrate data.

Given that the model is misspecified, any conclusion about the minimum wage effect

on the wage distribution based on the hrrate estimates is liable to criticism. As a result,

I discard the hrrate results and focus on the analysis of the hearn estimates reported in

Table 4. Column 4 suggests that the impact of the NMW introduction and upratings on the

distribution of hourly wages has been rather pronounced: the estimated spillover parameter

ranges from 7.08 in the fourth quarter of 2000 to 11.19 in the fourth quarter of 2005, which

means that the maximum spillover effect throughout this period is approximately 0.10 log

points. Moreover, according to Column 5, the estimated variance of the underlying wage

distribution is relatively constant indicating that wage variation at the bottom end of the

wage distribution can be accounted for by changes in the level of the minimum wage.

Figure 2: Actual and predicted spillover effects of the NMW, hearn, fourth quarter 2005.

Figure 2 depicts the predicted and actual spillover effects against the bottom 50 per-

centiles of the wage distribution for the fourth quarter of 2005 and demonstrates that the

model describes the data quite well. As we can see, the spillover effect peaks at the 12th

percentile, where the minimum wage equals the latent wage, and takes positive values until

the 30th percentile of the wage distribution.

UK is excluded from the sample.

22

The parametric approach without regional variation produces reliable estimates of the

minimum wage effect on the hearn distribution, but does not describe equally well its ef-

fects on the hrrate distribution. In the discussion above, I claim that this difference is a

sign of some assumption of the model being violated in the hrrate data, while satisfied in

the hearn data. Before adopting an alternative assumption about the latent wage distribu-

tion, I estimate Manning’s (2003) specification, i.e. the parametric approach with regional

variation. I divide the aggregate UK labour market into 21 regional markets using the LFS

variable “region of place of work,” gorwkr, and assume that latent log-wages are normally

distributed in every region. Given the small size of UK regional labour markets relative to

the US states’ markets, one might express concerns about the plausibility of this estimation

approach. However, the differential impact of the minimum wage on high- and low-wage

regions could provide additional information and potentially produce accurate estimates for

both measures of hourly earnings used in this paper.16

For my estimations, I fit equation (6) to the bottom 25 and bottom 50 percentiles of

all regional wage distributions −excluding all percentiles below the minimum wage. Tables

6 and 7 report the results of my estimations of the parametric approach with regional

variation for the hearn and hrrate variables, respectively.

Table 6: Parametric Approach with Regional Variation −Variable HEARN

year β 25pctiles σ 25pctiles β 50pctiles σ 50pctiles

1999b 7.994916 0.5651175 7.956199 0.56439787.64 - 8.34 0.557 - 0 .5732 7.71 - 8.21 0.559 - 0.570

2000d 7.66869 0.5153801 7.649574 0.52172057.27 - 8.06 0.507 - 0.523 7.33 - 7.97 0.515 - 0.528

2001d 9.987522 0.5057648 9.56925 0.51507149.46 - 10.50 0.499 - 0.512 9.19 - 9.94 0.510 - 0.520

2002d 7.212537 0.5457771 7.277142 0.54831416.85 - 7.57 0.535 - 0.556 7.00 - 7.55 0.540 - 0.556

2003d 9.126266 0.5182733 9.156064 0.52032858.66 - 9.58 0.510 - 0.526 8.82 - 9.49 0.515 - 0.526

2004d 9.569676 0.5308347 9.540305 0.5313479.14 - 10.00 0.524 - 0.538 9.19 - 9.89 0.525 - 0.536

2005d 9.224078 0.5378865 8.769254 0.54223488.74 - 9.70 0.529 - 0.547 8.44 - 9.10 0.535 - 0.549

Notes: Estimates and 95% confidence intervals of the parametric model fittedby non-linear least squares to the bottom 25 and 50 percentiles of regional wagedistributions.

The estimates in Table 6 corroborate my claim that the parametric approach does a

16Stewart (2002) uses geographical wage variation to investigate the impact of the NMW introductionon employment. His analysis is conducted at local authority level −the UK is divided into 140 local areas.Although local area samples can be used to assess the employment effect of the NMW, they are not suit-able for my purposes, since the resulting wage distributions are not representative of the aggregate wagedistribution.

23

Table 7: Parametric Approach with Regional Variation −Variable HRRATE

year β 25pctiles σ 25pctiles β 50pctiles σ 50pctiles

1999b 22.15841 0.2727005 19.79743 0.279229719.17 - 25.14 0.262 - 0.283 18.01 - 21.58 0.271 - 0.287

2000d 9.206892 0.36747 9.98696 0.3527638.16 - 10.26 0.344 - 0.390 9.26 - 10.71 0.339 - 0.366

2001d 7.373601 0.5002185 9.731889 0.41951486.74 - 8.01 0.468 - 0.531 9.24 - 10.22 0.406 - 0.432

2002d 6.643993 0.4759995 10.01684 0.36676045.90 - 7.39 0.432 - 0.519 9.36 - 10.66 0.352 - 0.381

2003d 6.003069 0.5834728 9.729333 0.40749565.51 - 6.49 0.541 - 0.625 9.22 - 10.23 0.392 - 0.422

2004d 7.058857 0.5768402 9.483644 0.46176916.55 - 7.56 0.543 - 0.610 9.13 - 9.84 0.448 - 0.475

2005d 8.175177 0.5197085 10.65472 0.44654987.48 - 8.86 0.486 - 0.552 10.08 - 11.23 0.430 - 0.463

Notes: Estimates and 95% confidence intervals of the parametric model fittedby non-linear least squares to the bottom 25 and 50 percentiles of regional wagedistributions.

good job in explaining the hearn data: the values of the spillover parameter and standard

deviation estimated on the bottom quarter and bottom half percentiles of the hearn dis-

tribution are very close in magnitude for the entire period. The estimates presented in

Columns 2 and 4 of Table 6 range from 7.2 to 9.98, which suggests that the maximum

spillover effect of the NMW was approximately 0.12 log points −even greater than the pre-

diction based on the estimation of hearn without regional variation. It is also worth noting

that the estimated standard deviation of the latent wage distribution −Columns 3 and 5

of Table 6− is relatively constant throughout the period under study indicating that latent

wage dispersion at the bottom end of the hearn wage distribution did not change consider-

ably. Therefore, an important component of the variation in observed wage dispersion can

be attributed to the minimum wage introduction and upratings.

By contrast, Table 7 shows that regional variation cannot improve the robustness of

this specification’s estimates for the hrrate measure. Although the discrepancy between the

bottom quarter and bottom half estimates of the spillover parameter, or standard deviation,

is not as pronounced as in the respective case without regional variation, all estimates from

the fourth quarter of 2001 onwards are significantly different.

The above findings indicate that the parametric estimation with regional variation cap-

tures the effect of the NMW on the hearn distribution, but fails to provide a reliable

description of the NMW effect on the hrrate distribution. As in the parametric approach

without regional variation, the different performance of the model is a sign of some assump-

tion being violated in the hrrate data, while satisfied in the hearn data.

To illustrate this difference, I plot the actual and predicted spillover effects on the

24

Figure 3: Goodness of fit of the parametric model with regional variation, hearn, fourthquarter 2005.

Figure 4: Goodness of fit of the parametric model with regional variation, hrrate, fourthquarter 2005.

25

bottom half of all regional wage distributions against the gap between the minimum wage

and the latent wage. Figure 3 presents the predicted and actual spillovers of the 2005 NMW

uprating for the hearn measure and demonstrates that in this case the model performs very

well.

Figure 4 depicts the predicted and actual spillovers of the 2005 NMW uprating for

the hrrate measure. In this case, the model fails to describe the data in a satisfactory

way. Actual spillovers above the point where the minimum wage “bites” and below the

median take negative values, which clearly indicates that that the normality assumption

over-predicts latent hrrate percentiles just below the median. This accounts for the poor

fit of the parametric approach with regional variation to hrrate data.

4.2.2 Non-Parametric Approach

I now estimate the model using the non-parametric approach, which relies on the assumption

that regional wage distributions have the same shape but differ in their centrality and scale

parameters. I pool all percentiles below the median of all regional wage distributions and

use specification (7) to estimate the spillover parameter, β, as well as some measure of

the average across regions latent wage dispersion, αpt for every p < 50. Building on Lee’s

estimation strategy, I introduce dummy variables for every different percentile and exploit

regional variation to obtain αpt ’s, which I then use to infer the lower half of the latent wage

distribution. This specification is fitted to the bottom-half and -quarter of the hearn and

hrrate regional distributions.

Only wage percentiles that are weakly greater than the level of the minimum wage

are included in my sample. Percentiles 1-3 of regional wage distributions satisfying this

requirement are very few and limited to specific high wage regions. For that reason, I drop

all percentiles below 4 and estimate the model for the remaining 22 and 47 percentiles.

Tables 8 and 9 present the results of my estimations for the hearn and the hrrate

variables, respectively. The spillover parameter values estimated on the bottom half and

on the bottom quarter percentiles of the hearn distribution are very close in magnitude

indicating that the non-parametric approach performs very well in this case −Columns 2

and 3 in Table 8. The estimates range from 7.14 to 9.95 indicating that the maximum

spillover effect was approximately 0.12 log points.

Table 9 demonstrates that the non-parametric approach can account for the effects of the

minimum wage on the hrrate distribution in a satisfactory way. Columns 2 and 3 of Table 9

suggest that in most cases the value of the spillover parameter does not change significantly

with the range of percentiles on which it is estimated −the estimates corresponding to the

2003 minimum wage uprating are an exception. The introduction of the NMW in 1999 does

not appear to have caused significant spillover effects on the hrrate distribution: the value

of the spillover parameter estimated on the bottom half percentiles of the distribution is

approximately 17.5 and implies a maximum spillover effect of 0.057 log points. By contrast,

it seems that the impact of all subsequent upratings has been more pronounced: the value

of the spillover parameter is approximately 8, which implies a maximum spillover effect of

26

Table 8: Non-Parametric Approach −Variable HEARN

year β 25pctiles β 50pctiles

1999b 7.980105 7.8601177.64 - 8.32 7.61 - 8.11

2000d 7.624271 7.7103047.23 - 8.02 7.39 - 8.04

2001d 9.959924 9.8118449.45 - 10.47 9.42 - 10.20

2002d 7.230356 7.1473326.87 - 7.59 6.88 - 7.42

2003d 9.158034 9.2208948.70 - 9.61 8.88 - 9.57

2004d 9.617772 9.5146599.19 - 10.05 9.16 - 9.88

2005d 9.235057 8.6534088.75 - 9.72 8.32 - 8.98

Notes: Estimates and 95% confidence intervalsof the non-parametric model fitted by non-linearleast squares to the bottom 25 and 50 percentilesof regional wage distributions.

Table 9: Non-Parametric Approach −Variable HRRATE

year β 25pctiles β 50pctiles

1999b 19.71162 17.4992417.03 - 22.38 15.92 - 19.08

2000d 8.988694 8.0667717.95 - 10.03 7.46 - 8.69

2001d 7.961075 8.2093367.25 - 8.67 7.78 - 8.64

2002d 6.828002 7.586726.03 - 7.63 7.07 - 8.11

2003d 6.631105 8.5547466.09 - 7.17 8.01 - 9.10

2004d 7.452498 8.3304446.92 - 7.99 7.95 - 8.71

2005d 8.302095 8.0670187.58 - 9.03 7.60 - 8.53

Notes: Estimates and 95% confidence intervalsof the non-parametric model fitted by non-linearleast squares to the bottom 25 and 50 percentilesof regional wage distributions.

27

approximately 0.125 log points. This difference might reflect the fact that employers were

anticipating the introduction and had raised wages in advance.

The estimates seem to be robust for both measures indicating that the non-parametric

approach performs very well in explaining the data. The value of the spillover parameter

is suggestive of a significant spillover effect of the NMW on the hearn and the hrrate

distributions.

To further establish my claim that dropping the normality assumption and estimating

the effects of the minimum wage on the wage distribution by means of the non-parametric

approach improves the fit of the model, I plot the actual and predicted spillover effects of

the 2005 NMW uprating against the gap between the minimum wage and the estimated

latent wage for both the hearn and hrrate measures; these results are depicted in Figures

5 and 6, respectively.

Before analyzing the fit of the non-parametric approach to the data, I should emphasize

that the calculation of the actual spillover effects plotted in these two Figures is conditional

on the absence of stochastic variation in latent wage dispersion across regions. The discus-

sion in the previous Section demonstrates that the non-parametric specification (7) accounts

for stochastic elements in latent wage dispersion across states, so this additional assumption

is not necessary for the estimation of this approach. In other words, the estimates reported

in Tables 8 and 9 are not conditional on latent wage dispersion being constant across re-

gions. However, abstracting from stochastic elements in regional latent wage distributions

aids in describing the residuals of the non-parametric specification as the difference between

the actual and predicted spillover effects.

Under this additional assumption, I am able to compute the percentiles of regional latent

wage distributions as the sum of the estimated measures of latent wage dispersion (αpt ’s)

and the corresponding median wage; I then use these estimated latent wage percentiles to

infer the actual minimum wage spillover effect. In this way, I can assign an “economic”

meaning to the residuals of the non-parametric specification: as equation (8) suggests, they

are the difference between the actual spillover effect and the predicted spillover effect. Note

that assuming the absence of stochastic variation in latent wage dispersion is equivalent

to the assumption that the standard deviation of the latent wage distribution is constant

across regions, which was employed in the estimation of the parametric model with regional

variation. Although the adoption of this assumption in the non-parametric case is guided by

the need to interpret the residuals in an intuitive way, it also serves an additional purpose:

it unifies these two different approaches, and thus, facilitates the comparison of Figures 5

and 6 with Figures 3 and 4.

Figure 5 presents the actual and predicted spillover effects of the 2005 NMW for the

hearn measure and demonstrates that the non-parametric approach fits the data very well.

It is evident that the fit of the non-parametric approach to the hearn data is similar to the

fit of the parametric approach −Figures 3 and 5 are almost identical. This is not surprising,

as the hearn estimates of the spillover parameter (β) produced by the two approaches are

very close in magnitude. As discussed earlier, the good fit of the parametric approach to

28

Figure 5: Goodness of fit of the non-parametric model, hearn, fourth quarter 2005.

Figure 6: Goodness of fit of the non-parametric model, hrrate, fourth quarter 2005.

29

the hearn distribution indicates that the normality assumption is not violated in the data.

Figure 5 shows that the non-parametric approach, despite its less restrictive assumptions,

describes the effects of the NMW equally well.

The actual and predicted spillover effects of the 2005 NMW uprating for the hrrate

measure are depicted in Figure 6. The fit of the non-parametric model to the hrrate data

appears to be less satisfactory than its fit to hearn data −this specification seems to over-

predict or under-predict latent wage-percentiles just below the median. However, a close

examination of Figures 4 and 6 demonstrates that the fit of the non-parametric specification

is considerably better than the fit of the parametric approach with regional variation. The

residuals of the non-parametric approach, that is, the gap between actual and predicted

spillovers in Figure 6, are small in magnitude and zero on average. The corresponding

residuals of the parametric approach (Figure 4) seem to be different from zero, due to the

large negative values of the actual spillover at the percentiles below the median of the hrrate

regional distributions. These findings suggest that non-parametric approach corrects the

misspecification caused by the violation of the normality assumption in the hrrate data,

and thus, outperforms the parametric approach.

4.2.3 Empirical Latent Approach

I now estimate the same model assuming that the latent wage distribution is given by the

aggregate distribution of observed wages in the quarter prior to the introduction or the

uprating of the minimum wage. Given that the hrrate variable only became available in

the LFS in the first quarter of 1999, when the NMW was actually introduced, I do not

provide estimates of this approach for this period. I fit the bottom 25 and 50 percentiles of

the aggregate wage distribution to equation (1) and estimate the spillover parameter, β, for

the minimum wage upratings of 2000-2005. In fact, what I estimate using this method is the

effect of minimum wage upratings on a wage distribution that has already been distorted by

the existing minimum wage. My prediction is that in this case the results of my estimations

understate the effects of the minimum wage.

The results of the empirical latent approach for the hearn and the hrrate measures are

reported in Tables 10 and 11, respectively. Table 10 demonstrates that the estimates do

not change significantly with the range of the hearn distribution percentiles on which it is

estimated. The value of the spillover parameter ranges from 13 to 33, which signals that

the maximum spillover effect is between 0.03 and 0.08 log points.

The results reported in Table 11 suggest that the model produces robust estimates

when fitted to the hrrate data −the spillover parameter estimated on the bottom quarter

percentiles is not significantly different from the corresponding estimate on the bottom half

percentiles of the distribution. The spillover parameter estimated on the bottom half of the

hrrate distribution (Column 3) ranges from approximately 17 to 41, which implies that the

maximum spillover effect is between 0.024 and 0.06 log points.

The empirical approach produces spillover parameter estimates that are very large in

magnitude compared with the results of the other approaches. This signifies a rather small

30

Table 10: Empirical Latent Approach −Variable HEARN

year β 25pctiles β 50pctiles

2000 11.93862 13.228589.48 - 14.39 10.02 - 16.44

2001 19.48187 21.5781310.98 - 27.98 13.69 - 29.47

2002 22.08765 22.9283913.75 - 30.43 14.06 - 31.80

2003 24.89838 24.0241415.76 - 34.04 13.47 - 34.57

2004 24.37319 23.2877317.47- 31.27 14.71 - 31.87

2005 34.13582 33.8093420.20 - 48.07 14.21 - 53.41

Notes: Estimates and 95% confidence intervals ofthe empirical latent approach.

Table 11: Empirical Latent Approach −Variable HRRATE

year β 25pctiles β 50pctiles

2000 21.46772 17.9575315.19 - 27.74 14.18 - 21.73

2001 29.16244 33.6121220.61 - 37.71 23.90 - 43.32

2002 38.59058 40.2669224.24 - 52.94 22.38 - 58.15

2003 43.84405 41.4459833.67 - 54.02 29.79 - 53.10

2004 34.36508 27.5422726.99 - 41.74 21.87 - 33.22

2005 42.10171 37.7745931.41 - 52.79 28.82 - 46.72

Notes: Estimates and 95% confidence intervals ofthe empirical latent approach.

31

spillover effect, which corroborates my prediction that this specification captures the ef-

fects of the minimum wage on an already distorted wage distribution and results in the

underestimation of spillover effects.

4.2.4 Discussion

My results suggest that the non-parametric estimation of the Lee model does a very good

job in explaining the data for both the hearn and the hrrate measures. The parametric ap-

proach on the other hand performs well in the case of the hearn measure, but is misspecified

in the case of the hrrate measure.

Table 12: Comparison of NLS Estimates −Variable HEARN

Year Parametric No Reg Parametric Reg Nonparametric

1999b 9.899265 7.956199 7.8601178.33 - 11.47 7.71 - 8.21 7.61 - 8.11

2000d 7.088272 7.649574 7.7103045.40 - 8.78 7.33 - 7.97 7.39 - 8.04

2001d 8.70017 9.56925 9.8118447.15 - 10.25 9.19 - 9.94 9.42 - 10.20

2002d 7.715268 7.277142 7.1473326.17 - 9.26 7.00 - 7.55 6.88 - 7.42

2003d 8.400503 9.156064 9.2208946.83 - 9.97 8.82 - 9.49 8.88 - 9.57

2004d 9.744876 9.540305 9.5146598.28 - 11.21 9.19 - 9.89 9.16 - 9.87

2005d 11.19805 8.769254 8.6534089.81 - 12.59 8.44 - 9.10 8.32 - 8.98

To understand and interpret the difference I compare the estimates obtained using the

two approaches. Tables 12 and 13 report the spillover parameter estimated on the bottom

half percentiles of the hearn and hrrate distributions using the parametric approach (with

and without regional variation) and the non-parametric approach. Table 12 suggests that

the two approaches produce consistent results for the hearn variable: the magnitude of the

spillover parameter is similar across approaches.

By contrast, my estimates for the hrrate data reported in Table 13 seem to vary sig-

nificantly depending on the approach employed. A possible explanation for the significant

variation in the estimates of the spillover parameter across approaches is that the normality

assumption is violated in the hrrate data, so this variation is due to the misspecification

of the parametric approach. An alternative explanation for the differences in the estimates

pesented in Table 13 is that the hrrate oversamples low-wage workers, who are mostly af-

fected by changes in the minimum wage, so these differences are due to the inability of the

parametric and non-parametric specifications to appropriately account for the spike in the

32

Table 13: Comparison of NLS Estimates −Variable HRRATE

Year Parametric No Reg Parametric Reg Nonparametric

1999b 39.90389 19.79743 17.4992425.94 - 53.86 18.01 - 21.58 15.92 - 19.08

2000d 16.38736 9.98696 8.06677111.39 - 21.38 9.26 - 10.71 7.46 - 8.68

2001d 23.54164 9.731889 8.20933616.29 - 30.78 9.24 - 10.22 7.78 - 8.64

2002d 15.34914 10.01684 7.586727.77 - 22.93 9.36 - 10.66 7.07 - 8.11

2003d 9.193729 9.729333 8.5547467.78 - 10.61 9.22 - 10.23 8.01 - 9.10

2004d 20.90203 9.483644 8.33044414.10 - 27.70 9.13 - 9.84 7.95 - 8.71

2005d 17.29742 10.65472 8.06701810.79 - 23.80 10.08 - 11.23 7.60 - 8.53

hrrate distribution.

To examine whether my results are influenced by the spike in the hrrate distribution,

I repeat the estimation of the parametric approach with regional variation and the non-

parametric approach excluding the percentiles of regional wage distributions that are di-

rectly affected by the minimum wage. In some low-wage regions, the minimum wage “bite”

reaches up to the 16th percentile of the distribution, so I fit the two specifications on two

different percentile ranges: 17-38 and 17-50. The results of my estimations are reported in

Table 14 and suggest that my findings are not driven by the spike. The non-parametric

approach produces estimates that do not differ significantly across the wage distribution

and are very close in magnitude to the estimates of Table 9. By contrast, there is large and

significant variation in the estimates of the parametric approach, which supports my earlier

finding that the normality assumption is not satisfied in the hrrate data.

Figures 7 and 8 plot the actual and predicted spillovers of the 2005 NMW uprating on

the percentiles of the hrrate distribution above the spike. Simple inspection of the two

Figures indicates that eliminating the part of the distribution where the minimum wage

binds does not affect significantly the fit of the two models: compare Figures 7 and 8 with

Figures 4 and 6. The residuals of the parametric specification are large in magnitude and

seem to be different from zero on average; the non-parametric specification, on the other

hand, appears to fit the hrrate data sufficiently well.

The results of my estimations and the illustration of the actual and predicted spillover

effects suggest that the spike in the hrrate distribution is not the underlying reason for the

significant variation in the estimates of the curvature parameter across approaches. Exclud-

ing the spike does not alter my earlier findings that the normality assumption is violated in

the hrrate data, while the non-parametric specification produces robust estimates. There-

fore, non-parametric approach leads to the most reliable estimates of the minimum wage

33

Table 14: Robustness of Parametric & Non-Parametric Approach −HRRATEParametric Non-Parametric

Year Pcs 17-38 Pcs 17-50 Pcs 17-38 Pcs 17-50β σ β σ β β

1999 15.83382 0.291707 16.72615 0.286624 15.75815 15.9459513.97-17.70 0.27-0.31 15.18-18.27 0.28-0.30 13.87-17.64 14.42-17.47

2000 7.864656 0.399393 10.11989 0.348102 7.507934 7.6279877.11-8.62 0.37-0.43 9.27-10.97 0.33-0.36 6.72-8.30 7.00-8.25

2001 8.334528 0.447071 10.79571 0.385695 7.825738 8.5581477.78-8.90 0.43-0.47 10.14-11.45 0.37-0.40 7.29-8.36 8.07-9.05

2002 7.467218 0.446248 10.42457 0.358944 6.119393 7.4225036.80-8.13 0.41-0.48 9.63-11.22 0.34-0.38 5.56-6.68 6.88-7.97

2003 8.859379 0.421897 12.27439 0.346395 8.388912 10.141098.15-9.57 0.39-0.45 11.29-13.25 0.33-0.36 7.68-9.10 9.32-10.96

2004 7.976334 0.512580 10.11646 0.432897 7.905947 8.5972247.52-8.44 0.49-0.54 9.66-10.57 0.42-0.45 7.40-8.41 8.14-9.05

2005 8.04581 0.534804 11.04234 0.434120 7.141076 8.0921417.40-8.70 0.49-0.57 10.30-11.78 0.41-0.45 6.54-7.75 7.55-8.63

Notes: Estimates and 95% confidence intervals of the parametric and the non-parametric modelsfitted by non-linear least squares to percentiles 17-38 and 17-50 of regional wage distributions.

Figure 7: Goodness of fit of the parametric model above the spike, hrrate, fourth quarter2005.

34

Figure 8: Goodness of fit of the non-parametric model above the spike, hrrate, fourthquarter 2005.

effect on the hrrate distribution.

Let us now examine the implications of the estimates produced by the parametric and

non-parametric approaches for the effects of the minimum wage on wage inequality. Based

on the measures of wage dispersion estimated by these two specifications, it is possible

to infer the distribution of latent wages, which can then be used to analyse whether the

minimum wage has a significant impact on wage inequality. Both the parametric and non-

parametric approaches produce estimates of the average across regions wage dispersion (σ

and α). Under the assumption that latent wage dispersion is constant across regions, I

can use these estimates to infer the regional latent wage distributions. Figure 9 depicts

percentiles of the observed hrrate distribution in a low-wage and a high-wage region in

2005, as well as the corresponding percentiles of the latent distribution estimated by the

parametric and non-parametric approaches. The horizontal line denotes the minimum wage.

The assumption that latent wage dispersion is constant across regions implies that the

estimated latent distributions in low- and high-wage regions differ in their location (me-

dian wage), but are otherwise identical. In both the parametric and the non-parametric

cases, the high-wage region latent is a vertical translate of the low-wage latent distribu-

tion. However, it is evident that the predicted latent differs in shape between approaches.

First, I consider the differential impact of the minimum on the two regional wage distri-

butions ignoring any differences due to the specification used to estimate the latent. The

estimated effects of the minimum wage on the distribution of the low-wage region are rather

35

Figure 9: Observed and latent wage percentiles in a low-wage and a high-wage region,hrrate, fourth quarter 2005.

pronounced: the minimum bites somewhere between percentiles 25-30 (depending on the

approach) of the latent wage distribution and its spillover effects reach up to percentile 39.

In the high-wage region, the impact of the minimum wage is considerably smaller but still

noticeable: it bites between percentiles 9-12 and its spillover effects extend up to percentile

17. Consider now the differences in the shape of the latent estimated by the two approaches.

The non-parametric approach estimates lower latent wages than the parametric up to the

20th percentile of the distribution. This implies that, compared with the parametric spec-

ification, the non-parametric predicts a higher direct and spillover effect of the minimum

wage on the wage distribution, and thus, a more pronounced impact on wage inequality.

5 Conclusion

In light of these findings, it is possible to interpret the puzzle about the differential impact

of the minimum wage across the Atlantic. The puzzle was identified in the studies by

Lee (1999), Manning (2003), and Dickens and Manning (2004a), which employ the same

estimation framework, but rely on different assumptions about the latent wage distribution.

Using the same empirical model and a UK dataset, I demonstrate that the parameters of

the model are sensitive to changes in the assumptions about the latent wage distribution

and conclude that the results reported in these studies are not comparable. The underlying

reason for the emergence of this puzzle is methodological.

I present three alternative approaches to the estimation of spillover effects. All ap-

36

proaches use the empirical model of Lee (1999), but employ different assumptions about

the latent wage distribution. The dataset used consists of two different measures of the

UK hourly wage distribution: reported hourly wages (hrrate) and derived hourly earnings

(hearn). I fit the different approaches to each measure of hourly wages. To assess the

performance of each approach, I estimate the model parameters on the bottom half and the

bottom quarter percentiles of the wage distribution and explain that the comparison of the

two estimates can be used as a specification test.

My estimations of the parametric and non-parametric approaches for the two different

measures of hourly wages indicate the existence of a positive spillover effect of the National

Minimum Wage in the UK. When the model is not misspecified, the parametric and non-

parametric approaches produce an estimate of the spillover parameter, which is close in

magnitude to the corresponding estimate reported in Manning (2003) for the USA. By

contrast, the estimates obtained using the empirical latent approach are suggestive of a

rather small minimum wage spillover effect and are very close in magnitude to the estimates

reported in Dickens and Manning (2004a). In other words, the estimated value of the

spillover parameter estimated on the UK wage distribution takes different values depending

on the assumption about the latent wage distribution. Therefore, the emergence of the

puzzle can be attributed to the lack of a unified estimation framework.

While the contribution of this paper is mainly methodological, the estimates of positive

and significant minimum wage spillover effects suggest that the NMW has significantly con-

tributed to the reduction of wage inequality in the UK. In light of these findings, extending

the current framework to conduct a complete assessement of minimum wage effects on wage

inequality appears to be an attractive extension. However, this is by no means trivial, so I

choose to leave it to further research.

37

6 Appendix

This part examines the theoretical properties of the model (equation [1]) used in the em-

pirical studies presented in Section 2 for the estimation of minimum wage spillover effects.

To simplify notation, I omit subscripts and denote the nominal minimum wage rate wm

instead of minwaget.

6.1 Continuity, Monotonicity, and Extrema

For ease of exposition, I express the function in equation (1) using the following simplifica-

tion:

wp = wp∗ + f (wm − wp∗;β) ,

where

f (x;β) =x

1− e−βx∀β > 0.

Clearly, f (x;β) is not defined at x = 0, however, its limit exists at this point and it is given

by

limx→0

f (x;β) = limx→0

(x

1− e−βx

)=

1

β.

Therefore, it is possible to define a continuous function f (x;β) for any x ∈ R:

f (x;β) =

{ x1−e−βx , if x 6= 0

1β , if x = 0

}∀β > 0. (11)

In the special case where the curvature parameter tends to infinity, β →∞, f (x;β) simpli-

fies to

limβ→∞

f (x;β) = f∞ (x) =

x, if x > 0

0, if x = 0

0, if x < 0

. (12)

If x = (wm − wp∗), then the direct and total effects of the minimum wage on percentile

p of the wage distribution are given by equations (12) and (11), respectively.

Having discussed the continuity of f (x;β), I can now examine its monotonicity. The

derivative of f (x;β) with respect to x is given by

f ′ (x;β) =eβx

(eβx − 1− βx

)(eβx − 1)

2 > 0 if x 6= 0 & 0 < β <∞. (13)

Although f ′ (x;β) is not defined at x = 0, its limit exists and is

limx→0

f ′ (x;β) =1

2∀0 < β <∞.

Hence, the function’s derivative is continuous and positive for finite values of β.

From equation (12), it is evident that when the curvature parameter tends to infinity,

38

f∞ (x) is differentiable for any x 6= 0. Its first derivative is given by:

f ′∞ (x) =

{1, if x > 0

0, if x < 0

}.

Therefore, when β is finite f (x;β) is strictly increasing on (−∞,+∞) implying that

the total effect of the minimum wage on percentile p of the wage distribution is increasing

in (wm − wp∗). By contrast, in the pure censoring case, β → ∞, the direct effect of the

minimum wage, f∞ (x), is constant for (wm − wp∗) ∈ (−∞, 0) and linearly increasing for

(wm − wp∗) ∈ (0,+∞).

Let us now consider the behaviour of the function at the limits of its range. In the

limiting case where β →∞, there is nothing interesting to note: f∞ (x) coincides with the

45◦ line for positive values of the independent variable, and with the horizontal axis for

negative values of the independent variable. If 0 < β <∞, the limit of f (x;β) as x→ −∞is

limx→−∞

f (x;β) = limx→−∞

(x

1− e−βx

)= 0,

which means that if x takes extreme negative values, f (x;β) asymptotes to the horizontal

axis. Hence, for any value of β the function’s infimum is 0.

If 0 < β <∞ and x→ +∞,

limx→∞

f (x;β)

x= lim

x→∞

(1

1− e−βx

)= 1,

which implies that as x increases f (x;β) asymptotes to the 45◦ line.

Therefore, for any finite value of β, f (x;β) asymptotes to f∞ (x) at the limits of its

range. This result can be interpreted intuitively: when the minimum wage is very small

relative to the latent wage, it should not have any effect (the compliance and spillover effects

are both equal to zero); on the other hand, when the minimum wage is very large relative

to the latent wage, the spillover effect is so small in magnitude that total and compliance

effects are equalized.

6.2 Spillover Effects

In this part, I discuss how the magnitude of the spillover effect varies across the wage

distribution assuming that the nominal level of the minimum wage and the value of β are

constant.

The introduction or uprating of the minimum wage may affect workers’ earnings directly

(by increasing their pay to the level of the mandated wage floor) and indirectly (by causing

a ripple effect). The total effect of the minimum wage, defined as the difference between the

observed wage and the latent wage, can be decomposed into those two effects. Therefore,

the magnitude of the spillover effect can be calculated by subtracting the compliance effect

39

from the total effect. Algebraically,

s(p) = (wp − wp∗)−max {(wm − wp∗) , 0} .

Substituting the total and direct effects from equations (11) and (12), the spillover effect

at percentile p is given by

s(p) = f(wm − wp

∗;β)− f∞

(wm − wp

∗). (14)

The independent variable is the gap between the minimum wage and the latent wage. If

x = (wm − wp∗) < 0, the direct effect is zero, so the spillover effect is increasing in the

independent variable (see [13]). If x = (wm − wp∗) > 0,

∂s(p)

∂x= f ′ (x;β)− f ′∞ (x) =

eβx(eβx − 1− βx

)(eβx − 1)

2 − 1 < 0,

the function is decreasing in the independent variable. Therefore, the spillover effect is

maximum when x = 0 or wm = wp∗, and its magnitude at this point is

smax(p) = limx→0{f (x;β)− f∞ (x)} =

1

β.

Intuitively, the spillover effect is maximum for the workers, whose latent wage is equal to

the new minimum wage −these workers experience a zero compliance effect. At the lowest

percentiles of the wage distribution the minimum wage has a significant compliance effect

and a rather small spillover effect. As we move up the wage distribution, the compliance

effect decreases and the spillover effect increases until we reach the percentile where the

minimum wage “bites”. At this point, the compliance effect is zero and the spillover ef-

fect is maximum. From that point onwards, the spillover effect diminishes and gradually

disappears. These theoretical predictions are depicted in Figure 10: I plot the minimum

wage spillover effect against the bottom half percentiles of the wage distribution. In this

example, the minimum wage “bites” at the 10th percentile of the wage distribution. The

spillover effect reaches a maximum at percentile 10 and dissipates as one moves away from

this point.

6.3 Shape of the Function for Different Values of β

From the above analysis, it is clear that the behaviour of the function is by and large

determined by the value of parameter β. I now focus on the implications of changes in the

value of this parameter for the shape of the function. Given the latent wage distribution,

how does the function vary with β? Since the parameter can only take positive values, we

have:∂f (x, β)

∂β= −

[x

eβx − 1

]2eβx,

40

Figure 10: Predicted Spillover Effect.

which implies∂f (x, β)

∂β< 0, for x 6= 0 and β > 0.

This suggests that the impact of the minimum wage on the wage distribution decreases in

magnitude with β. The interpretation of a varying curvature parameter for the shape of

the function is facilitated by Figure 11. In the limiting case, β → ∞, the function is a

kinked line, which coincides with the 45◦ line for x = (wm − wp∗) > 0 and with the flat

line corresponding to the latent wage, wp∗, for x = (wp − wp∗) < 0. When β is small, the

function asymptotes to this kinked line at the limits of its range (β = 2 in the Figure). As

the magnitude of β increases, the function becomes less linear and coincides with the kinked

line on a bigger part of its range (β = 5, β = 10 in the Figure).

What is the practical meaning of these theoretical properties? Suppose we have a sample

of independent realizations of the wage distribution at the same point in time exhibiting

variation in the level of the minimum wage: for every level of the minimum wage, we have

a different realization of the wage distribution. The realization without a minimum wage

corresponds to the latent wage distribution. In this ideal case, if we wanted to examine how

a change in the minimum wage affects the wage distribution, we could fit the data (that is,

the percentiles of the corresponding distribution as well as the percentiles of the latent) to

equation (1) and estimate the value of the curvature parameter, β.

Figure 11 illustrates this relationship for different values of β. If the estimated value of

β is very large, that is, if β →∞, the minimum wage has a pure censoring effect: all workers

previously earning less than the minimum have their pay increased to the minimum wage

rate, while all workers earning more than the minimum are unaffected, wp = max {wp∗, wm}.In this case, the minimum wage only has the direct effect of increasing workers’ pay by the

amount needed to comply with the new mandated wage floor. Suppose the estimated value

41

Figure 11: Minimum Wage Effects and the Curvature Parameter β.

of β is finite. The relationship between the minimum wage and the observed wage is given

by a curve like the ones that asymptote to the kinked line in Figure 11 (the curvature being

determined by the magnitude of β). In this case, the minimum wage has two effects on the

wage distribution: a direct/compliance effect and an indirect/spillover effect.

At this point, it is worth emphasizing that, in theory, changes in the minimum wage

may also have adverse employment effects, which cannot be distinguished empirically from

spillover effects. So, in Figure 11, all curves corresponding to finite values of β could

illustrate a mixture of spillover and disemployment effects. For ease of exposition, I assume

that the minimum wage has no adverse employment effects; this assumption could lead to

inflated estimates of spillovers if the minimum wage does in fact cause job losses. However,

empirical evidence suggests that this is not the case.17

If the minimum wage does not affect employment, then its total effect on the wage

distribution is a combination of the direct and indirect effects and can be calculated by the

gap between the observed wage, wp, and the latent wage, wp∗. Graphically, the total effect

of the minimum wage is the distance between the curve fitted through the data and the

horizontal line wp = wp∗. The indirect effect, which corresponds to wage increases caused

by minimum wage changes for reasons other than compliance, is the difference between the

total effect and the direct effect (equation [14]), that is, the gap between the curve and the

kinked line in the Figure.

To recapitulate, the total minimum wage effect on the wage distribution diminishes as

the value of the curvature parameter increases −the curve shifts down as β becomes larger

and the gap between the observed wage and the latent wage decreases. For a particular

level of the minimum wage the compliance effect is constant (and given by the kinked line),

17See Card and Krueger (1995), Stewart (2002, 2004a, and 2004b), and Stewart and Swaffield (2002).

42

therefore a smaller total effect would imply a smaller spillover effect. In Figure 11, as β

increases the curve moves closer to the kinked line, so that in the limiting case (β → ∞)

the total effect reduces to a pure censoring effect.

43

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45