Ann Reg Sci (2010) 45:313–329DOI 10.1007/s00168-009-0304-9

ORIGINAL PAPER

Measuring intra-distribution income dynamics:an application to the European regions

Adolfo Maza · María Hierro · José Villaverde

Received: 25 November 2008 / Accepted: 20 April 2009 / Published online: 7 May 2009© Springer-Verlag 2009

Abstract This paper addresses the issue of intra-distribution income dynamics at theEuropean regional level for the periods 1980–1993 and 1993–2005. In order to assesswhether the results depend on the estimation method, the paper applies and comparestwo relatively novel techniques: the highest conditional density region approach anda new mobility measure based on a discrete-time Markov chain approach. An inter-esting conclusion and a methodological lesson have been obtained. The conclusion isthat, whatever the technique used, the degree of intra-distribution mobility—which hasfavoured convergence—has been much higher in the first than in the second period.The lesson is that the use of different approaches is highly recommended not only forthe sake of robustness but also because they offer additional insights into intra-distri-bution dynamics.

JEL Classification O15 · O18 · C21

1 Introduction

In the last two decades there has been a huge amount of literature devoted to the studyof the topic of per capita income (or productivity) disparities between economies; thisis especially true for the case of the European regions, where the interest has beenfostered by concern about the ongoing process of economic integration. As a resultof this, it is virtually impossible to acknowledge all researchers that have at sometime dealt with this issue. Just as a short reference to some of the most interesting

A. Maza (B) · M. Hierro · J. VillaverdeDepartment of Economics, University of Cantabria, Avda. de los Castros,s/n, 39005 Santander, Spaine-mail: adolfo.maza@unican.es; mazaaj@unican.es

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works, it is worth mentioning the old but excellent survey edited by Armstrong andVickerman (1995). More recently, the books edited by Cuadrado-Roura and Parellada(2002) and Fingleton (2003), reviews as Magrini (2004), and papers such as those ofLópez-Bazo et al. (1999), Badinger et al. (2004), Miller and Genc (2005), Fingletonand López-Bazo (2006), Meliciani (2006), Arbia et al. (2008), Fischer and Stumpner(2008), and Mora (2008) are very good references.

Most of the articles analysing the evolution of disparities between economies haveapplied the classical σ and β convergence approaches [see the seminal articles byBarro and Sala-i-Martin (1991, 1992)]. Although being quite illustrative, these twocomplementary approaches not only present several econometric problems but alsofail to capture some potentially interesting features of the distribution dynamics, asthey devote their attention to the individual (or representative) unit and not to the entiredistribution. As regards the evolution of the distribution over time, at least two impor-tant drawbacks arise from these approaches: they are informative neither about thelocation and shape of the distribution nor about its internal changes (Quah 1996a,b).

Leaving aside the study of the shape dynamics, this paper concentrates on the anal-ysis of intra-distribution dynamics. Our interest in this issue has two main motivations:one is theoretical, in which, as emphasized by Quah (1996a), movements within thedistribution might be even more important than changes in its external shape; the otheris methodological, as the debate regarding the most suitable technique to evaluate thesemovements is still open, because the use of different techniques sometimes leads toquite different conclusions.

In order to address this topic, most researchers use the so-called stochastic kernelapproach, which provides some interesting insights on intra-distribution dynamics.However, this is not the approach we adopt here, as we consider there are more usefulways to deal with the study of transitional dynamics; in particular, the main innova-tion of this paper lies in the application of two relatively novel techniques that offeradditional features to the traditional stochastic kernel approach: the highest condi-tional density region approach and a new mobility measure based on the discrete-timeMarkov chain approach.1

On one hand, the highest conditional density region approach provides, among otherstatistical advantages, a much clearer graphical visualization and a more straightfor-ward interpretation of the results than the conventional stochastic kernel technique.On the other hand, the main contribution of our mobility measure—computed usinga discrete-time Markov chain approach—is that it enables us to quantify the mobilitydegree within the distribution far more accurately than with other conventional mea-sures, such as, for example, those proposed by Shorrocks (1978) and Bartholomew(1996). Furthermore, the employment of these two techniques allows us to test for therobustness of the results about the mobility of the regions up and down the incomeladder, a critical issue to assess the soundness and suitability of the EU cohesion policy.

This paper uses Europe as the case of study for the analysis of intra-distribu-tion dynamics. Our observation units are European regions, namely EU-15 NUTS-2 regions; the relevant variable of the analysis is relative per capita income (per

1 In so doing we follow the advice of Schluter (1997) of using different approaches in a complementaryfashion.

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Measuring intra-distribution income dynamics: an application to the European regions 315

Table 1 EU per capita income: regional growth rates

Annual average rates of growth 1980–1993 1993–2005

EU-15 1.67 2.21

Ten highest regions 4.06 4.25

Ten lowest regions −0.99 0.54

Standard deviation 1.00 0.86

capita GAV in PPS)2 and the sample period goes from 1980 to 2005. For the sakeof comparison, we have split our sample into two periods, 1980–1993 and 1993–2005. These periods have been selected because, being of similar length, they areunmistakably distinct as the rates of growth (Table 1) have been much higher in thesecond (2.21%) than in the first (1.67%).

Regarding this type of data, and arising not only from measurement problems butalso mainly from interactions across regions, a batch of studies has focused in thelast few years on the fact that regional economic data may be spatially dependent [forrecent papers see Fischer and Stumpner (2008), and Maza and Villaverde (2009)].Therefore, to ascertain whether our data are affected or not by spatial dependence, wecarry out an Exploratory Spatial Data Analysis (ESDA) and proceed accordingly.

The remainder of the paper is organized as follows. To begin with, Sect. 2 testsfor spatial dependence and, eventually, removes it by filtering the data. After that, theremainder sections look at the intra-distribution dynamics. Section 3 provides the basictheory behind the highest conditional density region approach (developed by Hyndmanet al. 1996) and applies it to our filtered data set. Next, Sect. 4, using a discrete-timeMarkov chain approach, proposes a new aggregated mobility index that provides anumeric support for the conclusions previously obtained. Finally, Sect. 5 concludes.

2 An analysis of spatial dependence

As mentioned above, there is a high probability that our EU regional income data arespatially autocorrelated. In order to confirm or rule out this intuition, we carried out anESDA analysis, for which the most representative statistic is the well-known standard-ized Moran’s I. For any variable Y , this statistic is expressed as follows (Anselin 1988):

I = n∑

i∑

j wi j

∑i∑

j wi j (yi − y)(y j − y

)

∑i (yi − y)2 (1)

where yi (y j ) is the per capita income of region i ( j), y the European average percapita income, wi j an element of the distance matrix between each pair of regions,

2 Data are drawn from the Cambridge Econometrics regional database. The entire list of European regionsconsidered in this paper is shown in the Appendix. For the computing of regional PPS only national valueshave been applied. Although we are conscious that this may produce some autocorrelation problems, it is notpossible to deal with them as there is no data bank offering homogeneous information about European-wideregional differences in prices. The same applies for the ECU/EUR conversion.

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Table 2 Spatial dependence

Years Moran’s I Z -value p-value

1980 0.255 33.90 0.00

1981 0.255 33.81 0.00

1982 0.253 33.52 0.00

1983 0.235 31.23 0.00

1984 0.230 30.61 0.00

1985 0.223 29.74 0.00

1986 0.214 28.57 0.00

1987 0.203 27.05 0.00

1988 0.208 27.68 0.00

1989 0.207 27.62 0.00

1990 0.163 21.87 0.00

1991 0.208 27.75 0.00

1992 0.198 26.48 0.00

1993 0.154 20.75 0.00

1994 0.151 20.26 0.00

1995 0.130 17.63 0.00

1996 0.119 16.09 0.00

1997 0.115 15.61 0.00

1998 0.122 16.59 0.00

1999 0.120 16.32 0.00

2000 0.117 15.81 0.00

2001 0.113 15.35 0.00

2002 0.110 14.97 0.00

2003 0.108 14.74 0.00

2004 0.107 14.51 0.00

2005 0.105 14.30 0.00

∑i∑

j wi j a standardization factor that corresponds to the sum of all the weights,and n is the number of regions.

The raw and standardized (Z -value) results obtained for Moran’s I, using the inverseof the standardized distance as a distance matrix, reveal the existence of statisticallysignificant positive spatial dependence (Table 2), thus proving that regions are not iso-lated economies.3 Furthermore, it is also shown that spatial autocorrelation decreasesconsiderably during the sample period.

3 Although we also tried other distance matrices (such as the “8 nearest neighbour”, “10 nearest neigh-bour” and “12 nearest neighbour” matrices), the results obtained were roughly the same. Additionally, it isconvenient to note that the significance level of the Moran’s I statistic was computed by assuming that thestandardized statistic follows a normal distribution; in any case, and for the sake of robustness, we also usedother approaches—the randomization and permutation approaches—and the results obtained were quitesimilar.

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Measuring intra-distribution income dynamics: an application to the European regions 317

Accordingly, spatial dependence should be removed from data because the viola-tion of the independence assumption may lead to misleading conclusions. To do that,we opt for the filtering methodology proposed by Getis (1995), based on a spatialstatistic developed by Getis and Ord (1992).4 This procedure is designed to convertspatially dependent variables (Y ) into spatially independent ones (Y F ); thus, the dif-ference between these two variables is a new variable representing the spatial effectsembedded in Y . Hence, the filtered variables should be interpreted as that part of theregional per capita income not explained by the spillover effects from the remainingregions.

The filtered per capita income of region i is defined as follows:

yFi = yi

∑j wi j (δ)

(n − 1)Gi (δ)(2)

where

Gi (δ) =∑

j wi j (δ)y j∑

j y j(3)

for j �= i, δ being a distance parameter indicating the extent to which further distantobservations are down-weighted. To apply this filter, the inverse of the standardizeddistance is again used as a distance matrix. Therefore, we assume thatwi j (δ) = (γ i j )

−δ

with δ = 1 and γi j the distance between the capitals of regions i and j.Once this filter has been applied, in the rest of the paper we examine intra-distribu-

tion mobility across European regions using only filtered data; this is done because,as previously mentioned, ‘if we are to evaluate growth and convergence dynamicsacross regions correctly the use of spatially filtered data is pretty much essential toavoid misleading interpretations’ (Fischer and Stumpner 2008, p. 130).5

3 Intra-distribution dynamics: the highest conditional density region approach

One of the techniques most commonly used in the analysis of the dynamics within adistribution involves the calculation of stochastic kernels (see Quah 1996a; Durlaufand Quah 1999).6 This approach is based on the estimation of the conditional densityof a variable Y given a variable X ; in this paper and from here on, variables Y and

4 An alternative way of dealing with spatial dependence when it comes to analyse intra-distribution dynam-ics is the so-called “spatial distribution dynamics approach”. This approach has been directly employed inthe context of Markov chains (Rey 2001; Le Gallo 2004). However, as this paper applies not only discretebut also continuous approaches to the study of intra-distribution dynamics, we have opted for the filteringapproach because, although it provides less insights on local mobility, it offers a more straightforward,simpler and clearer way of looking at intra-distribution global mobility.5 Anyway, we have also examined intra-distribution dynamics using actual data. The results, which areavailable upon request, show that, at large, mobility is lower with filtered than actual data, meaning thatfactors behind spatial dependence have a positive effect on intra-distribution mobility.6 For more recent papers see, for instance, Meliciani (2006), Pittau and Zelli (2006), Ezcurra et al. (2007),Villaverde and Maza (2008) and Maza and Villaverde (2009).

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X refer, respectively, to the filtered relative per capita income in periods t + τ and t,where t + τ and t are the final and initial years of each sample period.

The traditional stochastic kernel estimator (see, among others, Arbia et al. 2006) isdefined as:

fτ ( y| x) = gτ (x, y)

hτ (x), (4)

where the estimator for the joint density gτ (x, y) is given by:7

gτ (x, y) = 1

nab

n∑

i=1

K

(‖x − Xi‖x

a

) (‖y − Yi‖y

b

)

, (5)

and the estimator for the marginal density hτ (x) is:

hτ (x) = 1

na

n∑

i=1

K

(‖x − Xi‖x

a

)

, (6)

being ‖.‖x and ‖.‖y the Euclidean distance metrics on the space of X and Y respec-tively, a and b the smoothing or bandwidths parameters, and K (.) the function kernel.

An equivalent expression for Eq. (4) is the following:

fτ (y |x ) = 1

b

n∑

i=1

ωi (x)K

(‖y − Yi‖y

b

)

, (7)

where

ωi (x) = K

(‖x − Xi‖x

a

)/n∑

j=1

K

(∥∥x − X j

∥∥

x

a

)

. (8)

As can be seen, this estimator shows that a conditional density can be obtained by thesum of n kernel functions in Y space weighted by the ωi (x) in the X space.

Following this approach, Hyndman et al. (1996) have developed a new techniquewhich offers additional insights with respect to the former (Basile 2006; Fischer andStumpner 2008; Hierro and Maza 2009, and Maza et al. 2009). Two are the mainadvantages of this alternative conditional density estimator: first, it offers better sta-tistical properties than the traditional stochastic kernel estimator; second, it providesa much powerful visualization tool, this allowing an easier and direct interpretation ofthe results. The estimator proposed by Hyndman et al. (1996) is given by:

f ∗τ ( y| x) = 1

b

n∑

i=1

ωi (x)K

(∥∥y − Y ∗

i (x)∥∥

y

b

)

, (9)

7 In the bivariate case, the product kernel is only slightly less efficient than other multivariate kernels.

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Measuring intra-distribution income dynamics: an application to the European regions 319

where Y ∗i (x) = ei + r(x) − l(x), being r(x) the estimator of the conditional mean

function r(x) = E[Y |X = x], ei = yi −r(xi ), and l(x) the mean of the estimated con-ditional density of e|X = x . It can be proved that when r(x) = m(x) = ∑n

i=1 wi (x)Yi ,then f ∗

τ (y|x) = fτ (y|x), but the mean function f ∗τ (y|x) has better bias properties than

the traditional kernel regression, as well as a smaller integrated mean square error.A fundamental factor in the estimation of stochastic kernels—both the traditional

estimator [Eq. (7)] and the one employed in this paper [Eq. (9)]—is the choice of thebandwidths; the function of these bandwidths is to put less weight on observations thatare further from the point being evaluated.8 Specifically, we use optimal bandwidthsin the two directions x and y following the Bashtannyk and Hyndman (2001) rules.9

As regards the kernel function, another relevant factor in the computation of stochastickernels, we use the Gaussian kernel given by:

K (x) =(√

2π)−1

exp

(

−1

2x2

)

. (10)

Apart from this new estimator, Hyndman et al. (1996) also propose, as it was mentionedbefore, new ways to visualize the conditional density: the so-called stacked densityplot and the highest conditional density region plot. The stacked density plot showsa number of densities plotted side by side in a perspective graph; as a result of that,it highlights the conditioning inside the distribution. The highest conditional densityregion plot represents, on the other hand, the so-called highest density regions; withoutentering into details, the highest density region is defined ‘as the smallest region ofthe sample space containing a given probability’ (Hyndman et al. 1996, p. 327).

The results of this new approach, applied to the periods 1980–1993 and 1993–2005, are displayed in Fig. 1: the stacked density plot on its left side and the highestconditional density region plot on its right side. With reference to the stacked densityplots, they allow us to see the changes in the shape of the relative per capita incomedistribution for a given per capita income value in the initial year. According to theresults, a striking difference between the two sample periods is evident: while, in thesecond one, the probability mass and most of the peaks tend to be clustered alongthe main diagonal, in the first period there are some apparent deviations from thisdiagonal, mainly at high rates of relative income. This simply means that the mobilitydegree in the European regional income distribution was much higher during the firstthan during the second period.

However, a more informative way to represent the changes occurring in a distri-bution is based on the highest conditional density region plot. Each vertical strip onthe right hand side of Fig. 1 represents the conditional density for a relative per capitaincome level in the initial year. In particular, this figure shows the highest density

8 The bandwidth election gives a trade-off between bias and variance. Small bandwidths produce smallbias and large variance, while large bandwidths yield large bias and small variance.9 These authors proposed a three-step strategy for bandwidth selection: first, bandwidth selection withthe traditional rule suggested by Silverman (1986); second, a bootstrap bandwidth selection approach forestimating conditional distribution functions (Hall et al. 1999); third, a regression-based bandwidth selector(Fan et al. 1996).

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Fig. 1 Stacked conditional density and highest conditional density region plots. Notes: From darkto light, the shadings represent 25, 50, 75 and 90% of the total probability. Bullets indicate the mode.Both the stacked conditional density plot and the high conditional density region plot were estimated at50 points

regions for a probability of 25, 50, 75 and 90% (as it passes from a darker to a lessdark area). In addition, it illustrates, as a bullet (•), the mode (value of relative percapita income in the final year where the density function takes on its maximum value)for each value in the initial year.

With respect to the first period (right side of Fig. 1a), the position of the modes indi-cates that, generally speaking, the poorest (richest) regions have improved (worsened)their relative per capita income levels; this process of catching-up is crystal-clear asthe modes of the lower (higher) tail of the distribution are above (below) the maindiagonal. On the other hand, if we observe the mass of probability (dark areas), we seethat, in both the lower and higher extremes of the distribution, the area representinga probability of 25 and 50% (and even 75 and 90% in some cases) does not cross thediagonal; this reveals again the existence of both a catching-up process, thus givingsupport to the EU cohesion/regional policy, and a high degree of mobility within the

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Measuring intra-distribution income dynamics: an application to the European regions 321

per capita income distribution (particularly evident in income range above 200 of theaverage).

Relative to the second period under study, it is important to notice that, as a resultof the convergence process previously mentioned, the ratio between extreme valuesof the income distribution is much lower than in the first. Additionally, the right-handside of Fig. 1b distinctively shows that the mobility degree was much lower (or thepersistence much higher) than in the first one; as can be seen, the modes are nownearer to the diagonal and the dark areas representing a probability of 25% cross it inmore cases than in the first period. In particular, this figure shows that mobility hasbeen mainly confined to regions in the income range of 110–140 and 160–190 of theaverage; on the contrary to what happened during the first period, these results seemsto call into question the effectiveness of EU regional policy between 1993 and 2005.

4 Intra-distribution dynamics: the estimation of mobility measures basedon a Markov chain approach

The conclusions drawn from the previous approach are relevant in light of the polit-ical debate about the merits of the EU cohesion policy as an instrument to reducedisparities across regions. Indeed, its main results are, on the one hand, that mobil-ity within the distribution fostered convergence and, on the other, that the degree ofintra-distributional dynamics was much lower in the second than in the first period.

The highest conditional density region approach fails, however, to provide a quan-titative value of the mobility degree. This shortcoming can be overcome using theMarkov chain approach in a discrete state space set up. From a chronological pointof view this approach was, in fact, the first methodology used in modelling intra-dis-tribution dynamics; although popularized by Quah (1996a) in the analysis of incomedistribution, it goes back, at least, to Prais (1955). The appeal of this approach liesin the fact that it provides useful representations of dynamic processes through theestimation of the well-known transition matrices. Recent papers employing this meth-odology are, for instance, Le Gallo (2004) and Tortosa et al. (2005).

In order to give a simple definition of the concept of transition matrix, it is nec-essary to consider that regions are classified into exhaustive and mutually exclusivestates according to their relative per capita income at times t0 and t1, p(t0) and p(t1)being the initial and final income distribution. The link between these distributionsis given by p(t1) = p(t0)P , where operator P represents the transition probabilitymatrix with elements pi j . This expression describes the time evolution of p(t0), bymapping p(t0) onto p(t1), that is, the changes in the income distribution between t0and t1. Thus, the interpretation of a transition matrix is really intuitive: each one of itscells pi j indicates the probability of moving from a per capita income state i to a percapita income state j .

As mentioned before, the main advantage of this technique is that scalar summaryindices of mobility can be derived (Shorrocks 1978; Sommers and Conlisk 1978;Bartholomew 1996; Parker and Rougier 2001). However, the conventional Markovchain approach is not without limitations. In fact, its main drawback—implying thatsome authors avoid to using it—is the arbitrariness when defining the states. Actually,

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the results critically depend on this choice and there is no any theoretical method toachieve an appropriate partition of the distribution.10

Although we are well aware of this limitation, in this paper we have opted for includ-ing this approach because we propose a revised version of the traditionalBartholomew (1996) measure on mobility. As is well known, this has been one ofthe most extensively used measures in the literature on social mobility because itintroduces distance into the quantification of mobility [see, for instance, Parker andRougier 2001; Ezcurra et al. 2006; Hierro 2007]; additionally, and unlike other mea-sures, such as that proposed by Shorrocks (1978), Bartholomew’s general class ofmobility measures is concerned with all transitions between states.

In this paper, we suggest an extension of Bartholomew (1996) measure. Our mobil-ity index is given by the expression:

d(P) =∑

i

∑

j

1

kipi pi j di j , (11)

where pi is the initial proportion of regions in i’s income state (that is, the size of eachincome state), pi j denotes, as we have already indicated, the probability of movingfrom the relative per capita income state i to the relative per capita income state j, di j isa distance measure between income states i and j (defined below) and, finally, ki is anelement introduced into the expression in order to normalize the mobility index. Thus,three elements are considered in the quantification of mobility: size (pi ), transitions(pi j ) and income distance (di j ).

In addition, mobility indices for each state can be obtained, which allows immobileand mobile states to be identified; in consequence, a separate analysis of the perfor-mance of each income state can be carried out. This is because the preceding aggregatemeasure can be expressed as the weighted sum of corresponding state-by-state mobil-ity measures, d(P i ),

d(P i ) =∑

j

1

kipi j di j , (12)

that is,

d(P) =∑

i

pi d(P i ) (13)

This index, d(P), differs from Bartholomew’s measure (1996) in three importantaspects. First, it considers an alternative definition of distance for situations of state-hierarchy, which is, in our opinion, especially suitable for income contexts.11 This

10 When it comes to the choice of the number of states, the researcher should weigh up the clarity of theresults and the number of observations in each state. For some interesting references on this issue see,among others, Magrini (1999) and Bulli (2001).11 When there is a natural ordering of the states, Bartholomew proposes di j = |i − j | as the distancemeasure.

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Measuring intra-distribution income dynamics: an application to the European regions 323

distance is defined as the absolute difference between the average per capita income(y) of the states under consideration, that is, di j = | y j − yi |. Therefore, accordingto this definition, when dealing with changes within each income state, the distancemoved, dii , is 0. On the other hand, transitions between income states will give positivevalues of di j ; as is evident, the distance is higher when the differences in the averageincome of the states considered are larger.

Second, another difference of our index with the original Bartholomew mobilitymeasure is that, instead of the ergodic distribution, it includes, as a weighting factor,the initial income distribution, given by the elements pi . This decision responds to thefact that, with non-stationary processes (as is usual in income contexts), the ergodicdistribution does not provide a faithful picture of the ultimate consequences of the cur-rent income distribution because intra-distribution patterns do not remain unchanged.On the contrary, the initial income distribution allows us to capture the relative size ofeach income state at the time the mobility degree is being analysed.

Finally, the last difference consists of the introduction of the element ki , definedas the largest value in the i‘s row of D (matrix of distance with generic elementsdi j ). This element is included to obtain a normalized measure in the interval [ 0, 1 ];accordingly, the closer is the value to 1, the higher is the mobility degree.

As should be obvious, the definition of distance conditions the meaning of max-imum mobility. In our particular case, a situation of maximum mobility, given byd(P) = 1, arises when all transitions occur towards the more distanced income state,regardless of being an upward or a downward transition. However, considering thata value of the index close to unity is impossible from an economic point of view, wehave carried out several simulations to discriminate between high, medium and lowmobility degrees. On the basis of the results of these simulations and to ease interpre-tation, we adopt the following criteria: we label a situation as “high mobility degree”if d(P) is over 0.15 (obtained when 40% of the regions transit to a contiguous state);“medium mobility degree” if d(P) is between 0.15 and 0.06 (20% of the regions moveto a contiguous state); and “low mobility degree” if d(P) is below 0.06.

Bearing these considerations in mind, it is important to notice again that, in orderto apply these indices, the grouping for discretization is necessarily arbitrary and hasto be established beforehand. This being so, and to get an straightforward economicinterpretation of the results, we have initially considered—for our two sample peri-ods—five exhaustive and mutually exclusive income states distinguishing betweenregions with low, middle-low, intermediate, middle-high and high levels of per capitaincome relative to the European average: [0, 75), [75, 90), [90, 110), [110, 125),[125, +∞), respectively; this grouping coincides with that usually employed by theEuropean Union.12 However, before computing the indices we carry out two statisticaltests. On the one hand, we test for the existence of Markovian dependence in eachone of the two sample periods using the χ2-test suggested by Anderson and Goodman(1957); the results lead us to reject the null hypothesis of non-Markovian dependence

12 In order to compare different values of the index, the definition of income states must be assumed tobe the same in the two sample periods. Thus, the application of other discretization proposals based on theproperties of the sample [see, for example, Magrini (1999)] would be problematic as the states obtaineddiffer for both periods.

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324 A. Maza et al.

Table 3 Aggregated and state-by-state mobility indices

States [0, 75) [75, 90) [90, 110) [110, 125) [125, +∞)

(a) 1980–1993

[0, 75) 0.60 0.19 0.21 0.00 0.00

[75, 90) 0.05 0.40 0.49 0.07 0.00

[90, 110) 0.00 0.10 0.79 0.10 0.00

[110, 125) 0.00 0.06 0.67 0.22 0.06

[125,+∞) 0.00 0.00 0.33 0.28 0.40

pi 0.32 0.22 0.15 0.09 0.22

d(P i ) 0.16 0.18 0.07 0.21 0.26

d(P) 0.18

(b) 1993–2005

[0, 75) 0.80 0.18 0.03 0.00 0.00

[75, 90) 0.09 0.48 0.39 0.00 0.03

[90, 110) 0.00 0.16 0.77 0.07 0.00

[110, 125) 0.00 0.00 0.64 0.18 0.18

[125,+∞) 0.00 0.00 0.00 0.17 0.83

pi 0.20 0.17 0.42 0.11 0.09

d(P i ) 0.06 0.14 0.06 0.33 0.07

d(P) 0.10

at the 0.05 significance level (both p-values are 0.000). On the other hand, we performthe χ2-test of stationarity also proposed by Anderson and Goodman (1957) in order toevaluate if differences between periods are significant; the result rejects the hypothesisof stationary transition probabilities at the 0.05 significance level (p-value is 0.000).

Thus, after having shown the validity of the Markovian property and on the basisof the existence of clear differences in the dynamics across periods, we compute ourmobility indices. The results reported in Table 3 are of particular interest since theyprovide a numeric support to the conclusions drawn from the highest conditional den-sity region approach. In particular, it is shown that the aggregate mobility measured(P) reaches a much higher value for the first period (0.18) than for the second (0.10).This means that mobility in the European income distribution has fallen more than40%, implying, in compliance with our criteria, the change from high to mediummobility degree.

A first look into the transition matrices and the state-by-state indices allows us tolook more deeply into this result. Specifically, a feature of note is that the mobilitydegree is considerably higher in the period 1980–1993 than over the period 1993–2005 not only for the whole distribution but also for most of the income states. Thisfinding comes from the existence of higher transition probabilities to contiguous andeven non-contiguous states, along with the fact that relative distances between themare greater over this period (see Table 4). Besides, it is worth highlighting the perfor-mance of the upper income state of the distribution, [125, +∞), which reaches the

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Measuring intra-distribution income dynamics: an application to the European regions 325

Table 4 Absolute distances between states

States [0, 75) [75, 90) [90, 110) [110, 125) [125, +∞)

(a) 1980–1993

[0, 75) 0.00 36.05 55.30 72.23 111.60

[75, 90) 36.05 0.00 19.25 36.18 75.55

[90, 110) 55.30 19.25 0.00 16.93 56.30

[110, 125) 72.23 36.18 16.93 0.00 39.37

[125,+∞) 111.60 75.55 56.30 39.37 0.00

(b) 1993–2005

[0, 75) 0.00 23.58 37.54 55.67 91.95

[75, 90) 23.58 0.00 13.96 32.09 68.37

[90, 110) 37.54 13.96 0.00 18.13 54.41

[110, 125) 55.67 32.09 18.13 0.00 36.28

[125,+∞) 91.95 68.37 54.41 36.28 0.00

highest mobility (0.26); this, together with its relatively large size in the distribution(0.22), contributes to its effect on the aggregate mobility being eventually noteworthy.

As regards the second period, the higher probabilities on the main diagonal pointout to a lower mobility; this is especially remarkable in the tails of the distribution.Actually, most income states exhibit low values of the index. The only exception refersto the middle-high income state [110, 125), for which the state-by-state mobility indexis 0.33; however, its small size (0.11) makes its contribution to the aggregate mobilityindex quite small.

Finally, in order to check the robustness of the results just discussed, we performthe analysis for an additional set of three alternative discretizations of the incomedistribution. Therefore we have four cases: (A) the previous (benchmark) case, (B)the case of five income states but with a different grid than in the benchmark, (C) thecase of four income states and (D) the case of six income states. The results, displayedin Table 5, show that the differences among these four cases are not very significant,especially in the first period. In fact, this outcome highlights a desirable property ofthis index; by construction, its results do not depend to a great extent on the choice ofthe number and length of states.

5 Conclusions

Intra-distribution dynamics has become one of the most heated topics in the literatureon regional per capita income distribution; however, no definitive conclusion has beenreached as there is still an open debate regarding which is the most suitable techniqueto undertake this topic. This paper tries to contribute to this debate by analysing theinternal changes of the relative per capita income distribution between the Europeanregions in two periods: 1980–1993 and 1993–2005. After removing spatial depen-dence from the original data, the paper adds to literature by employing and comparing

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Table 5 Robustness of the results

Periods (1980–1993) (1993–2005)

Cases A B C D A B C D

pi 0.32 0.21 0.21 0.21 0.20 0.09 0.09 0.09

0.22 0.27 0.21 0.11 0.17 0.18 0.13 0.12

0.15 0.23 0.33 0.22 0.42 0.56 0.66 0.17

0.09 0.16 0.25 0.15 0.11 0.11 0.12 0.42

0.22 0.13 0.12 0.09 0.06 0.12

0.19 0.08

d(P i ) 0.16 0.18 0.21 0.18 0.06 0.06 0.06 0.05

0.18 0.18 0.28 0.23 0.14 0.08 0.07 0.09

0.07 0.02 0.03 0.16 0.06 0.08 0.08 0.13

0.21 0.25 0.26 0.06 0.33 0.25 0.14 0.11

0.26 0.25 0.21 0.07 0.06 0.20

0.26 0.05

d(P) 0.18 0.17 0.18 0.18 0.10 0.09 0.08 0.11

A: 5 states -[0, 75), [75, 90), [90, 110), [110, 125), [125, +∞) -; B: 5 states - [0, 60), [60, 85), [85, 115),[115, 140), [140, +∞)-; C: 4 states - [0, 60), [60, 80), [80, 120), [120, +∞)-; D: 6 states - [0, 60), [60,75),[75, 90), [90, 110), [110, 130), [130, +∞) -

two relatively novel techniques: the highest conditional density region approach anda new mobility measure based on a discrete-time Markov chain approach.

To begin with, the highest conditional density region approach, a variant of the tra-ditional kernel approach, is used. The results reveal that, especially in the first period,the relative per capita income of some European regions has changed in a significantway, this implying a high degree of mobility. In particular, it is convenient to note thatmobility in the lower end of the distribution during the period 1980–1993 was mainlycaused by the improvement of per capita income in Greek, Portuguese and Spanishregions; as these three countries joined the EU in the 1980s, this result is a reflection ofthe immediate positive impact effect of becoming a member of the EU. Besides this,the results show that mobility has been higher in the first than in the second period,and that it has favoured the convergence process between the European regions, aspoorest (richest) regions have improved (worsened) their relative income; this seemsto show that the higher the rate of growth, the higher the probability of disparitiesbeing reduced.

Subsequently, a new mobility measure based on an adaptation for income contextsof the discrete-time Markov chain approach is proposed. When it is applied to theEuropean regions, this measure not only confirms that intra-distributional mobilityhas been much higher in the first than in the second period, therefore showing therobustness of the previous results, but also quantifies this difference; intra-distributionmobility has been, according to our index, 1.8 times as high between 1980 and 1993as between 1993 and 2005. In other words, the degree of persistence has increasedover time.

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Measuring intra-distribution income dynamics: an application to the European regions 327

According to the previous findings, two main conclusions emerge. The first one isof a methodological nature. The use of different approaches to analyse the internalmobility of a given distribution enables us to gain more insight into its dynamics.In fact, what this paper has shown is that the two approaches employed to deal withintra-distribution dynamics should be considered complementary, as each one offersadditional views of the issue at stake.

The second conclusion has a more political flavour. Considering that regional mobil-ity is good in itself and that this has been large in the EU between 1980 and 1993,it seems that European policies (mainly regional policy) were successful. However,as European regional mobility has decreased in the last decade, it is no wonder thatthe effectiveness of these policies has been called into question (Dall’erba and LeGallo 2008). Despite this or precisely because of this, what seems to be clear is that arenewed, more business-oriented European regional policy is today even more neces-sary than it was in the past. Although it is a fact that a new strategy has been adoptedby the European cohesion policy for the period 2007–2013, we are afraid that its ratherlow budget might be at odds with the pursuing of additional regional convergence.

Acknowledgments The authors wish to acknowledge the editor Börje Johansson and three anonymousreferees for helpful comments and suggestions.

Appendix: list of European regions

The complete list of regions employed in this paper is the following: BELGIUM(10 regions): Antwerpen, Limburg, Oost-Vlaanderen, Vlaams Brabant, West-Vla-anderen, Brabant Wallon, Hainaut, Liege, Luxembourg, Namur. DENMARK (1):Denmark.GERMANY(31):Stuttgart,Karlsruhe,Freiburg,Tübingen,Oberbayern,Nie-derbayern, Oberpfalz, Oberfranken, Mittelfranken, Unterfranken, Schwaben, Berlin,Bremen, Hamburg, Darmstadt, Giessen, Kassel, Braunschweig, Hannover, Lüne-burg, Weser-Ems, Düsseldorf, Köln, Münster, Detmold, Arnsberg, Koblenz, Trier,Rheinhessen-Pfalz, Saarland, Schleswig-Holstein. GREECE (13): Anat. Mak., Kent.Makedonia., Dytiki Makedonia, Tesalia, Ipeiros, Ionia Nisia, Dytiki Ellada, StereaEllada, Peloponnisos, Attiki, Voreio Aigaio, Notio Aigaio, Kriti. SPAIN (17): Galicia,Asturias, Cantabria, Pais Vasco, Navarra, Rioja, Aragon, Madrid, Castilla-Leon,Castilla-la Mancha, Extremadura, Cataluña, Com. Valenciana, Baleares, Andalucia,Murcia, Canarias. FRANCE (22): Ile de France, Champagne-Ard., Picardie, Haute-Normandie, Centre, Basse-Normandie, Bourgogne, Nord-Pas de Calais, Lorraine,Alsace, Franche-Comte, Pays de la Loire, Bretagne, Poitou-Charentes, Aquitaine,Midi-Pyrenees, Limousin, Rhone-Alpes, Auvergne, Languedoc-Rouss., Prov-Alpes-Cote d’Azur, Corse. IRELAND (2): Border, Southern and Eastern. ITALY (20):Piemonte, Valle d’Aosta, Liguria, Lombardia, Trentino-Alto Adige, Veneto, Fr.-Vene-zia Giulia, Emilia-Romagna, Toscaza, Umbria, Marche, Lazio, Abruzzi, Molise,Campania, Puglia, Basilicata, Calabria, Sicilia, Sardegna. LUXEMBOURG (1):Luxembourg. NETHERLANDS (12): Groningen, Friesland, Drenthe, Overijssel,Gelderland, Flevoland, Utrecht, Noord-Holland, Zuid-Holland, Zeeland, Noord-Brabant, Limburg. AUSTRIA (9): Burgenland, Niederosterreich, Wien, Kärnten,

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Steiermark, Oberösterreich, Salzburg, Tirol, Vorarlberg. PORTUGAL (7): Norte, Cen-tro, Lisboa e V.do Tejo, Alentejo, Algarve, Açores, Madeira. FINLAND (6): Ita-Suomi,Vali-Suomi, Pohjois-Suomi, Uusimaa, Etela-Suomi, Aland. SWEDEN (8): Stockholm,OstraMellansverige,Sydsverige,NorraMellansverige,MellerstaNorrland,OvreNorr-land, Smaland med oarna, Vastsverige. UK (37): Tees Valley and Durham, Northumb.et al., Cumbria, Cheshire, Greater Manchester, Lancashire, Merseyside, East Riding,North Yorkshire, South Yorkshire, West Yorkshire, Derbyshire, Leics., Lincolnshire,Hereford et al., Shrops., West Midlands (county), East Anglia, Bedfordshire, Essex,Inner London, Outer London, Berkshire et al., Surrey, Hants., Kent, Avon et al., Dorset,Cornwall, Devon, West Wales, East Wales, North East Scot., Eastern Scotland, SouthWest Scot., Highlands and Islands, Northern Ireland.

References

Anderson TW, Goodman LA (1957) Statistical inference about Markov chains. Ann Math Stat 28:89–109.doi:10.1214/aoms/1177707039

Anselin L (1988) Spatial econometrics: methods and models. Kluwer, DordrechtArbia G, Basile R, Piras G (2006) Analyzing intra-distribution dynamics: a reappraisal. ERSA Conference

Papers, Volos (Greece)Arbia G, Le Gallo J, Piras G (2008) Does evidence on regional economic convergence depend on the estima-

tion strategy? Outcomes from the analysis of a set of NUT2 EU regions. Spat Econ Anal 3(2):209–224.doi:10.1080/17421770801996664

Armstrong HW, Vickerman RW (eds) (1995) Convergence and divergence among European regions. PionLtd, London

Badinger H, Müller WG, Tondl G (2004) Regional convergence in the European Union, 1985–1999: aspatial dynamic panel analysis. Reg Stud 38(3):241–253. doi:10.1080/003434042000211105

Barro R, Sala-i-Martin X (1991) Convergence across states and regions. Brook Pap Econ Act 1:107–182.doi:10.2307/2534639

Barro R, Sala-i-Martin X (1992) Convergence. J Polit Econ 100(21):223–251. doi:10.1086/261816Bartholomew DJ (1996) The statistical approach to social measurement. Academic Press, LondonBashtannyk DM, Hyndman RJ (2001) Bandwidth selection for kernel conditional density estimation. Com-

put Stat Data Anal 36:279–298. doi:10.1016/S0167-9473(00)00046-3Basile R (2006) Intra-distribution dynamics of regional per-capita income in Europe: evidence from alter-

native conditional density estimators. Paper presented at the 53rd North-American Meetings of theRSAI, Toronto

Bulli S (2001) Distribution dynamics and cross-country convergence: a new approach. Scott J Polit Econ48(2):226–243. doi:10.1111/1467-9485.00196

Cuadrado-Roura JR, Parellada M (eds) (2002) Regional convergence in the european union: facts, prospectsand policies. Springer, Berlin

Dall’erba S, Le Gallo J (2008) Regional convergence and the impact of European structural funds over1989–1999: a spatial econometric analysis. Pap Reg Sci 87(2):219–244. doi:10.1111/j.1435-5957.2008.00184.x

Durlauf SN, Quah D (1999) The new empirics for economic growth. In: Taylor J, Woodford M (eds)Handbook of macroeconomics, vol. I.. North-Holland, Amsterdam pp 235–308

Ezcurra R, Gil C, Pascual P (2006) Regional mobility in the European Union. Appl Econ 38(19):2237–2253.doi:10.1080/00036840500427494

Ezcurra R, Pascual P, Rapún M (2007) Spatial disparities in the European Union: an analysis of regionalpolarization. Ann Reg Sci 41(2):401–429. doi:10.1007/s00168-006-0111-5

Fan J, Yao Q, Tong H (1996) Estimation of conditional densities and sensitivity measures in nonlineardynamical systems. Biometrika 83(1):189–206. doi:10.1093/biomet/83.1.189

Fingleton B (ed) (2003) European regional growth. Springer, BerlinFingleton B, López-Bazo E (2006) Empirical growth models with spatial effects. Pap Reg Sci 85:177–198.

doi:10.1111/j.1435-5957.2006.00074.x

123

Measuring intra-distribution income dynamics: an application to the European regions 329

Fischer MM, Stumpner P (2008) Income distribution dynamics and cross-region convergence in Europe.Spatial filtering and novel stochastic kernel representations. J Geogr Syst 10(2):109–140. doi:10.1007/s10109-008-0060-x

Getis A (1995) Spatial filtering in a regression framework: examples using data on urban crime, regionalinequality, and government expenditures. In: Anselin L, Florax RJGM (eds) New directions in spatialeconometrics. Springer, Berlin, pp 172–185

Getis A, Ord JK (1992) The analysis of spatial association by use of distance statistics. Geogr Anal 24:189–206

Hall P, Wolff RC, Yao Q (1999) Methods for estimating a conditional density distribution. J Am Stat Assoc94(455):154–163. doi:10.2307/2669691

Hierro M (2007) The effect of foreign-born residents on migratory patterns of natives in Spain. Econ Bull10(3):1–6

Hierro M, Maza A (2009) Per capita income convergence and internal migration in Spain: Are foreign-bornmigrants playing an important role? Pap Reg Sci (forthcoming)

Hyndman RJ, Bashtannyk DM, Grunwald GK (1996) Estimating and visualizing conditional densities.J Comput Graph Stat 5(4):315–336. doi:10.2307/1390887

Le Gallo J (2004) Space-time analysis of GDP disparities among European regions: A Markov chainapproach. Int Reg Sci Rev 27:138–163. doi:10.1177/0160017603262402

López-Bazo E, Vayá E, Mora A, Suriñach J (1999) Regional economic dynamics and convergence in theEuropean Union. Ann Reg Sci 33(3):343–370. doi:10.1007/s001680050109

Magrini S (1999) The evolution of income disparities across the regions of the European Union. Reg SciUrban Econ 29:257–281. doi:10.1016/S0166-0462(98)00039-8

Magrini S (2004) Regional di(convergence). In: Henderson J, Thisse J (eds) Handbook of regional andUrban economics. Elsevier, Armsterdam, pp 2741–2796

Maza A, Villaverde J (2009) Provincial wages in Spain: convergence and flexibility. Urban Stud (forth-coming)

Maza A, Villaverde J, Hierro M (2009) Regional productivity distribution in the EU: which factors arebehind? Eur Plann Stud 17(1):149–159. doi:10.1080/09654310802514052

Meliciani V (2006) Income and employment disparities across European regions: The role of national andspatial factors. Reg Stud 40(1):75–91. doi:10.1080/00343400500450075

Miller JR, Genc I (2005) Alternative regional specification and convergence of US regional growth rates.Ann Reg Sci 39(2):241–252. doi:10.1007/s00168-004-0216-7

Mora T (2008) Factors conditioning the formation of European regional convergence clubs. Ann Reg Sci42(4):911–927. doi:10.1007/s00168-007-0191-x

Parker SC, Rougier J (2001) Measuring social mobility as unpredictability. Economica 68:63–76. doi:10.1111/1468-0335.00233

Pittau MG, Zelli R (2006) Empirical evidence of income dynamics across EU regions. J Appl Econ 21:605–628. doi:10.1002/jae.855

Prais SJ (1955) The formal theory of social mobility. Popul Stud 9:72–81. doi:10.2307/2172343Quah D (1996a) Twin peaks: Growth and convergence in models of distribution dynamics. Econ J 106:1045–

1055. doi:10.2307/2235377Quah D (1996b) Empirics for economic growth and convergence. Eur Econ Rev 40:1353–1375. doi:10.

1016/0014-2921(95)00051-8Quah D (1997) Empirics for growth and distribution: stratification, polarization, and convergence clubs.

J Econ Growth 2:27–59. doi:10.1023/A:1009781613339Rey S (2001) Spatial empirics for economic growth and convergence. Geogr Anal 33:195–214Schluter C (1997) On the non-stationary of German income mobility. Discussion Study No. DARP/30,

Toyota Center LSE, LondonShorrocks AF (1978) The measurement of mobility. Econom 46:1013–1024. doi:10.2307/1911433Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, LondonSommers PM, Conlisk J (1978) Eigenvalue inmobility measures for Markov chains. J Math Sociol 6:253–

276Tortosa E, Pérez F, Mas M, Goerlich FJ (2005) Growth and convergence profiles in the Spanish provinces

(1965–1997). J Reg Sci 45(1):147–182. doi:10.1111/j.0022-4146.2005.00367.xVillaverde J, Maza A (2008) Productivity convergence in the European regions, 1980–2003: a sectoral and

spatial approach. Appl Econ 40(10):1299–1313. doi:10.1080/00036840600771361

123