Testing for convergence: evidence from non-parametric multimodality tests

TESTING FOR CONVERGENCE: EVIDENCE FROMNON-PARAMETRIC MULTIMODALITY TESTS

MARCO BIANCHI�

Bank of England, Monetary Analysis, Threadneedle Street, London EC2R 8AH, UK

SUMMARY

In this paper, we test the convergence hypothesis in a cross-section of 119 countries by means of bootstrapmultimodality tests and nonparametric density estimation techniques. By looking at the density distributionof GDP across countries in 1970, 1980 and 1989, we ®nd low mobility patterns of intra-distributiondynamics and increasing evidence for bimodality. The ®ndings stand in sharp contrast with the convergenceprediction. # 1997 John Wiley & Sons, Ltd.

J. Appl. Econ., 12, 393±409 (1997)

No. of Figures: 7. No. of Tables: 3. No. of References: 23.

1. INTRODUCTION

There is a debate in growth economics as to whether less developed economies are catching upwith richer economiesÐwhat is commonly known as the `convergence hypothesis'. At the heartof the debate stands a fundamental controversy among researchers about whether or not a processof economic homogenisation (non-persistent inequality) takes place in the world economy astime passes.

Two measures of convergence have been proposed in the literature, called, respectively, b- ands-convergence (Barro and Sala-i-Martin, 1995). The former occurs when economies that startout poorer tend to display faster growth rates, in which case there is a negative correlationbetween the growth rate of income per capita and its initial level; the latter occurs when thedispersion of the real per capita income across economies tends to fall over time.

Although much empirical work has been devoted to support or question di�erent views aboutthe convergence hypothesis (and b-convergence in particular), no widely accepted conclusion hasbeen reached so far. Early empirical analyses employing cross-section, time-series, or panel datatechniques have found evidence to support the convergence hypothesis (see, among others,Sala-i-Martin, 1994, 1996, and references therein). More recently, however, evidence in favour oflack of convergence has been advocated in a number of studies, including Canova and Marcet(1995), Desdoigt (1994), Durlauf and Johnson (1995), Lee, Pesaran, and Smith (1997b), Paapand van Dijk (1994), Quah (1996a,b).

In the traditional approach to convergence analysis, the notion of b-convergence, that is,whether each country economy converges to its own equilibrium or steady state, has been mostwidely used. However, as the determinants of steady-state income may vary across economiesregardless how fast economies may be converging to their own steady states, b-convergence does

CCC 0883±7252/97/040393±17$17.50 Received 31 May 1995# 1997 John Wiley & Sons, Ltd. Revised 4 December 1996

JOURNAL OF APPLIED ECONOMETRICS, VOL. 12, 393±409 (1997)

� Correspondence to: M. Bianchi, Bank of England, Monetary Analysis, HO-4, Threadneedle Street, London EC2R8AH, UK. E-mail: [email protected]

not provide much information on how economies perform relative to each other. From theperspective of s-convergence, on the other hand, and particularly within our approach, theadvantage is that one looks at the evolution over time of the entire income distribution in a cross-section of countries, but with the shortcoming of failing to address directly the concept ofmobility patterns within the distribution (or intra-distribution dynamics). This concept playsinstead a crucial role in assessing persistence of income disparities over time (see Quah, 1996a,b).

Our preliminary step towards an analysis of the convergence hypothesis is based on the idea ofs-convergence, that a process of either economic homogenization or persistent inequality in theworld economy will be manifested in the shape of the density distribution of incomes. Inparticular, if we start with a bimodal density in a given point in time, indicating the presence oftwo groups in a population of countries (say a group of `poor' and a group of `rich' countries),convergence implies a tendency in the distribution to progressively move towards unimodalityover time.1 Such a prediction indicates the way the convergence hypothesis can be testedempirically: one can estimate the density of the frequency distribution of incomes across coun-tries at two (or more) distant points in time and evaluate whether and at what points in timeunimodality is most strongly rejected. As we shall discuss in the next section, we accomplish theformer task by means of non-parametric kernel density estimators; the latter, by bootstrapmultimodality tests.

A necessary condition must be satis®ed, however, for the above s-convergence type ofapproach to give more valuable insights, which is underlined by the following simple example.We consider the hypothetical situation where there are two groups of countries (say `rich' and`poor'), with 50% of countries in both groups. We then assume that there are dynamics over timeso that countries move from one group to the other until 50% of each group has transited to theother group at a `®nal' point in time. According to our s-convergence approach, the incomedistribution has not changed over time. Thus, s-convergence does not hold and there ispolarization (there are two modes). According to common sense, we would say instead that therehas been convergence among the two blocks. Although this is an extreme case which is veryunlikely to happen in reality, the example clari®es that the s-convergence analysis alone is notsu�cient to study convergence unless more information is gained on how units move within thedistribution. As we shall consider in the next section, this type of inference (i.e. inference on intra-distribution dynamics) can be done by a discriminant (or classi®cation) analysis based on theestimated density distributions.

Compared to other studies, our approach shares with Desdoigts (1994) the use of non-parametric density estimators to `let the data speak for themselves' and to ensure robustness ofthe results against possible misspeci®cation; but it also provides a formal test for the presence ofthose `convergence clubs' discussed by Quah (1996a,b), as in Paap and van Dijk (1994). Byestimating each distribution in isolation, in di�erent points in time, we can also easily compareincome distributions by plotting them together, on the same graph. Finally, we can makeinference on mobility patterns, such as rich countries becoming poor and vice versa, by adiscriminant analysis on the estimated densities.

A limit of our approach, on the other hand, is that it does not address the issue ofb-convergence.2 Moreover, although pertinent to testing for convergence, the method does not

1 Similarly, if we start with a unimodal density, the dispersion of this densityÐmeasured, for example, by the inter-quantile rangeÐwill tend to fall over time.2 For a clear exposition of panel data techniques for b-convergence empirics, see Lee, Pesaran, and Smith (1997a,b).

394 M. BIANCHI

J. Appl. Econ., 12, 393±409 (1997) # 1997 John Wiley & Sons, Ltd.

provide as detailed information about intra-distribution dynamics as, for example, Quah(1995a,b). Besides, representing a purely statistical investigation of stylized facts, our analysisdoes not provide theoretical justi®cations or economic rationales as to why convergence may ormay not occur.

Given the above limits, the results of our investigation present evidence supporting thefollowing four main facts: (1) only little mobility of countries between clubs has been observedduring the 1970s and the 1980s; (2) given that, there is evidence that countries do not appearconverging towards each other, but only within groups or `clubs'; (3) the gap between less andmore developed countries widened in the 1970s and in the 1980s; (4) a process of vanishing of themiddle class appears to have taken place in the 1980s, thus reverting a tendency of opposite signin the 1970s. These results, which are certainly consistent with some previous studies, in part-icular Paap and van Dijk (1994) and Quah (1996a,b), cast doubts on the validity of the converg-ence hypothesis within the world economy.The remainder of the paper is organized as follows. In Section 2 we brie¯y consider non-

parametric techniques for the estimation of the density distribution of incomes across countries.We also discuss non-parametric multimodality tests and discriminant analysis for intra-distribution dynamics. We report the empirical results in Section 3. Section 4 brie¯y summarizesand concludes.

2. THE STATISTICAL FRAMEWORK

2.1. Non-parametric Density Estimation and Multimodality Tests

Consider a random variable x with realizations xi, i � 1; . . . ; n. In our application, denoting by zithe GDP per capita (in US dollars at constant 1985 prices) in a cross-section of n countries, xirepresents some transformation of zi, such as, for example, each country's per capita incomerelative to the aggregate, i.e.

xi � zi=Xni�1

zi:3

Also denote by f(x) the density distribution of x.In the presence of m groups of countries (say, for example, m � 2 with a group of `rich' and a

group of `poor' countries), the density distribution of the data is a mixture of distributionsdescribed by

f �x� �Xmÿ1j�0

pj � gj�x;mj; s2j � pj 5 0 �1�

where pj's are mixing proportions with Xmÿ1j�0

pj � 1

and gj are densities with ®rst and second moments mj and s2j . We notice that if the gap in the �j'sis `large' relative to the s2j 's the modes in the density are said to be `well separated' and f(x) is

3 We will explain later the rationale for the transformation.

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# 1997 John Wiley & Sons, Ltd. J. Appl. Econ., 12, 393±409 (1997)

multimodal with m modes. If the distances in the means are `small' relative to the variances, themixture components in the density may not be well separated, that is, mixtures of distinctgroupings do not necessarily result in the same number of distinct modes.4

If there are groups among countries and it is assumed that GDP ®gures are normallydistributed within each group, then gj�x; mj; s

2j � � 1/�sj

��2pp � � exp fÿ�x ÿ mj�2/�2s2j �g in equation

(1). The density can be ®tted to the data by estimating the parameters of the model usingmaximum likelihood principles and the number of groups, m, can formally be tested by likeli-hood ratio tests (see Izenman and Sommer, 1988, Feng and McCulloch, 1996). If assumingnormality is regarded as restrictive, the alternative is to estimate the density non-parametricallyby the method of kernels and to test for the number of groups by bootstrap multimodality tests.Given a sample fxign1 of independent and identically distributed observations, a kernel densityestimator of f(x) is constructed as (see Silverman, 1986; HaÈ rdle, 1990)

fh�x� � �nh�ÿ1Xni�1

Kx ÿ xi

h

� �� nh�ÿ1

Xni�1

K�u� �2�

where h > 0 is the bandwidth or window width and K�u� � 1/��2pp � exp �ÿ1=2u2� is the Gaussian

kernel. Bandwidth h governs the degree of smoothness of the density estimate, with larger valuesof h producing a smoother density estimate.

A key concept in density estimation is the concept of critical bandwidth introduced by Silver-man (1981, 1983, 1986). A critical bandwidth hm is de®ned as the smallest possible h producing adensity with, at most, m modes, which means that for all h < hm the estimated density fh has atleast m � 1 modes. This idea of critical smoothing is naturally related to hypothesis testing and,in particular, to multimodality tests. Indeed, if the true underlying density has two modes, a largevalue of h1 is expected, because a considerable amount of smoothing is required to obtain aunimodal density estimate from a bimodal density. This suggests that hm can be used as a statisticto test

H0: f �x� has m modes versus H1: f �x� has more than m modes �3�Here, a `large' value of hm indicates more thanmmodes, thus rejecting the null. How large is largein this context is assessed by the bootstrap, as discussed by Silverman, and, among others,Izenman and Sommer (1988) and Efron and Tibshirani (1993).

Following Efron and Tibshirani (1993), bootstrap tests require us to de®ne two basicingredients: a test statistic t(x) and an estimated null distribution F0 for the data under H0. Giventhese, the achieved signi®cance level (ASL, or p-value) of the test is given by

ASL � ProbF0ft�x��5 t�x�g �4�

where x� � �x�1; x�2; . . . ; x�n�0 is the bootstrap sample drawn from the null distribution F0. Anestimate of ASL is obtained by generating a large number of B samples from F0 and by countingthe proportion of samples for which t�x��5 t�x�, that is,

dASL � # ft�x�b�5 t�x�g=B; b � 1; . . . ;B �5�

4 For example, a mixture of two normal distributions with similar means and large variances is likely unimodal.

396 M. BIANCHI


In the context of bootstrap multimodality tests, it seems reasonable to choose t�x� � hm andF0 � f

hm. However, because bootstrap samples drawn from f

hmhave variance larger than the

sample variance of the data (Efron and Tibshirani, 1993, pp. 231 and 234), a small adjustment isrequired on f

hm; we set F0 � g

hm, where g

hmdenotes the rescaled density estimate. Bootstrap

samples x�1; x�2; . . . ; x�n drawn from the rescaled density g

hmare given by Efron and Tibshirani

(1993, p. 231):

x�i � �y� � �1 � h2m=s

2�ÿ1=2� y�i ÿ �y� � hmei� i � 1; . . . ; n �6�

where y� � � y�1; y�2; . . . ; y�n�0 are sampled with replacement from x1; x2; . . . ; xn; �y� � mean�y��, s2is the sample variance of x; ei are standard normal variables generated by the computer.5 Thesteps to test for multimodality can then be summarized as:

(1) Draw B bootstrap samples x� of size n from ghm

using equation (6).(2) For each bootstrap sample x� compute the critical bandwidth consistent with m-modality,

h�m. Denote the values of h�m by h�m�1�, h�m�2�; . . . ; h�m�B�.(3) Obtain an estimate of the achieved signi®cance level of the test as dASLm � #fh�m�b�5 hmg/B.6(4) Fail to reject the null hypothesis of m modes in the density whenever dASLm is larger than

standard levels of signi®cance.

2.2. Discriminant Analysis and Intra-distribution Dynamics

If by testing for the number of modes in the density probability distribution of the data wereject the null hypothesis of unimodality (m � 1), f (x) is multimodal with m > 1 modes. Indiscriminant analysis, each of these modes is regarded as a group in the data, and a two-stepprocedure is implemented for the discrimination. In the ®rst step, a non-parametric estimate fh�x�of f (x) is derived using a bandwidth h consistent with the detected number of modes m (that is,hm < h < hmÿ1). Subsequently, each unit is allocated to one of the m groups, by the followingsimple rule:

i 2 Gj if xi 2�cj; cj�1� for j � 0; 1; . . . ;m ÿ 1 �7�

where cj are the cut-points de®ned as the values of x at which the estimated density fh�x� has alocal minimum, except for c0 � ÿ1 and cm � 1. According to the rule, country i is allocated togroup Gj at time t if xi falls into the interval �cj , cj�1�, for j � 0; . . . ;m ÿ 1.

For the particular case of m � 2, the discriminant rule is illustrated by Figure 1, showing abimodal density that indicates a group of poor and a group of rich countries. The densityshows that there are more units in the group of poor countries than there are in the group ofrich countries (in a ratio of approximately 28 to 5, that is, with mixing proportions of

5 It can be shown here (see Efron and Tibshirani, 1993, p. 234) that if y� is sampled with replacement from x, e has astandard normal distribution and hm is ®xed, then: (1) r� � y� � hme is distributed according to f

hm; (2) r� has the same

mean as y� and variance s2 � h2m; (3) x� de®ned as in equation (6) has the same mean as r� but variance approximately

equal to s2.6 It has been proven by Silverman that the event h�m > hm is equivalent to the event that f �

hmhas more than m modes.

This result implies that it is not necessary to compute h�m for each bootstrap sample; one needs only to check theproportion of cases when f �

hmhas more than m modes.



p1 � 5=28 � 0�18 and p0 � 1 ÿ p1 � 0�82). It also shows that the àverage poor' and the àveragerich' countries have a (relative per capita) GDP of approximately 0.0033 and 0.021, corres-ponding to the values associated with global and local maxima (that is, the two modes) in thedensity. The vertical dotted line drawn at the local minimum of the density estimate,approximately at c1 � 0�016, `separates' the two modes for the identi®cation of the groups.Thus, a country is classi®ed to be poor if it has a (relative per capita) GDP smaller or equal than1.6% and to be rich otherwise. This technique is used to analyse intra-distribution mobilitypatterns by comparing the group each country is allocated in di�erent years.

3. EMPIRICAL RESULTS

In this section we consider the annual per capita GDP at constant US dollars for n � 119countries, measured in 1970, 1980, and 1989, using data from the Penn World Tables, June 1993.Prior to testing for multimodality using kernel density estimation and the bootstrap method-ology, we consider two possible transformations of the original data. These are the relative percapita income, xi � zi=Szi, already mentioned in Section 2.1, and the logarithmic transforma-tion, xi � ln�zi�.

Both transformations are of speci®c interest, but for di�erent reasons. The former has beenrecently suggested by Canova and Marcet (1995), who have found it useful to alleviate potentialproblems of income cross-correlation among countriesÐdue, for example, to the fact thatrecessions and expansions may a�ect the world economy as a whole. A second reason for thetransformation is that xi � zi=Szi has a natural economic interpretation as the fraction, orpercentage, of total world income contributed by the ith country, if all countries had the samepopulation (that is, if we abstract from the `size' of the di�erent countries). This allows us to focus

Figure 1. Discriminant analysis. The cut-point at c1 � 0�016 discriminates GDP ®gures of di�erentcountries between those falling in the interval � ÿ 1; c1�, being allocated to the group of `poor' countries,

G0, and those falling in the interval �c1; �1�, being allocated to the group of `rich' countries, G1

398 M. BIANCHI


on the distribution of pure numbers, thus facilitating the comparison of densities in di�erentpoints in time. The log transformation, instead, is motivated by the notion of b-convergence inthe context of Solow's growth model. This is concerned with the logs of per capita income ratherthan income itself.

For our own purposesÐ that is, for a comparison of income distributions in di�erent points intimeÐ it is important to stress that whereas the relative income transformation is purely a scaletransformation which does not a�ect the shape of the density distribution of the original data, thelog transformation is not. This can be easily seen from the boxplots shown in Figure 2. In 1970,1980, and 1989 the distributions of per capita and relative per capita income clearly indicate

Figure 2. Boxplots of the actual data and the two transformations



skewness about the median (the horizontal line within the box) toward large values, plus theexistence of several outliers (represented by the horizontal lines outside the box). The log trans-formation, however, by exerting an amount of pulling and pushing on the tails of the distri-bution, has the e�ect of producing symmetric distributions with no outliers.7

We can also ask how well the distributions of the data in the original scale and for the twoabove-mentioned transformations are approximated by the normal distribution. This is shown bythe normal quantile plots in Figure 3, comparing the quantiles of the data (circles) with those of anormal distribution (points on the solid line). For the actual data and the data rescaled by thetotal world income, the ®rst six charts in Figure 3 qualitatively indicate the same structure in thedistributions, which are, in all cases, hardly approximated by the normal. The log trans-formation, in contrast, simpli®es the structure of the data by removing skewness and outliers,thus rendering the normal approximation legitimate.

Keeping in mind the e�ects of the logarithmic transformation on the shape of incomedistributions, we turn now to density estimation and bootstrap multimodality tests. For the percapita, the relative per capita income and the log transformation, we ®nd the critical bandwidthsshown in Table I. Figure 4 also plots the critical bandwidths for the relative per capita income.The results of bootstrap multimodality tests are shown in Table I.

For the distribution of both per capita and relative per capita income, we have virtuallyidentical results in Table I. In 1970, the unimodality hypothesis is not rejected at a 5% level ofsigni®cance. In fact, out of B � 1000 bootstrap samples, as many as sixty times we observedh�1 > h1 � 0�0041 (for the relative income), giving an achieved signi®cance level of 0.06. However,in 1980 and in 1989, the above inequality condition was only satis®ed once, thus rejectingunimodality with p-values of 0.01 and 0.00 respectively.

For the log transformation, on the other hand, completely di�erent results are obtained, asunimodality is never rejected. Although this may sound surprising,8 a likely explanation is givenin Figure 5. This shows the kernel density estimates of the actual GDP data and its twotransformations in the three years, 1970, 1980 and 1989, obtained using a bandwidth consistentwith both unimodality and bimodality.

The results for 1970 are shown in the ®rst column of Figure 5 (similar conclusions are obtainedfor the years 1980 and 1989). From the top panel it is apparent that there is a substantial gap inthe annual mean per capita income of rich and poor countries. According to the bimodal density(represented by the solid line) this gap is approximately of 8000 ÿ 2000 � 6000 US dollars. Asimilar gap in the modes of the distribution is observed in the densities of the relative per capitaincome transformation in the middle panel. However, when we consider the logarithmic trans-formation, the picture is quite di�erent. According to the estimated bimodal density (see bottompanel), the two modes centred on 7 and 9 do not appear to be well separated in this case, makingthe detection of multimodality very di�cult, particularly to a fully non-parametric approachwhich only imposes minimal assumptions on the data. In other words, the change in the shape ofthe overall distribution due to the logarithmic transformation of the data makes it much more

7 The logarithmic transformation is, in fact, just a particular case within a set of (so-called) power transformations (see,for example, Cleveland, 1993). The power transformation with parameter t; which is de®ned to be yt if t 6� 0 and ln(y) ift � 0, is often used in statistics, especially in a regression framework, as a way to obtain a density distribution for theresponse (or dependent) variable which matches the assumptions of the classical linear regression model.8 Indeed, we would expect that even though the logarithmic transformation changes the shape of the distribution, as amonotonic transformation it should not change its bimodality structure. Moreover, given the well-known gap betweenmost- and less-developed countries, we would expect to detect at least two groups in the data.

400 M. BIANCHI


Figure 3. Normal-quantile plots of the actual data and the two transformations



di�cult to distinguish a bimodal density (like the one represented by the solid line) from aunimodal one (represented by the dotted line). Because the modes are not well separated,however, unimodality does not exclude the presence of two groups in the data.

The above results are valuable as a preliminary analysis of convergence. However, in order toobtain evidence in favour of polarization and the formation of convergence clubs (and againstthe convergence hypothesis), further information is required on intra-distribution dynamics. Forthis reason, the remaining analysis concentrates on relative per capita GDP. According to ourbootstrap multimodality tests at 10% level of signi®cance, we assume the presence of two groupsin the data (two modes in the density distributions) in the years 1970, 1980, and 1989.

In the three years, by selecting a value of the bandwidth consistent with bimodality,h � 0�0025, we obtain the density estimates shown in Figure 6. These densities have local maximaand minima represented by the vertical lines (dotted lines for the local and global maxima,dashed line for the local minima) at the values of relative per capita incomes reported in Table II.Each is skewed to the right, indicating the presence of a large mass of poor countries and a smallproportion of rich countries. The vertical dashed line in 1970 indicates that a country is classi®edas being rich (poor) if it has a relative per capita income larger (smaller or equal to) than thecritical value of 1.60% (this is approximately the value of x to which corresponds a localminimum in the density); in 1980 this critical value rises to 1.73%; in 1989 it rises to 1.83%.

Implementation of the discriminant rule of Section 2.1 with a critical value of 0.016 leads toinclude in 1970 the following 21 countries in the group of the rich: Australia, Austria, Belgium,

Table I. Bootstrap multimodality tests, with B � 1000 replications

Per capita income: xi � zi

hm dASLm

m � 1 m � 2 m � 3 m � 1 m � 2 m � 3

1970 1552 897 818 0.07 0.36 0.071980 2236 864 613 0.01 0.40 0.651989 2779 946 779 0.00 0.67 0.58

Relative per capita income: xi � zi=Szi

hm dASLm

m � 1 m � 2 m � 3 m � 1 m � 2 m � 3

1970 0.0041 0.0023 0.0021 0.06 0.32 0.051980 0.0045 0.0017 0.0012 0.01 0.41 0.631989 0.0049 0.0017 0.0014 0.00 0.67 0.59

Logarithm of per capita income: xi � ln�zi�hm dASLm

m � 1 m � 2 m � 3 m � 1 m � 2 m � 3

1970 0.34 0.25 0.18 0.33 0.33 0.461980 0.41 0.33 0.21 0.30 0.09 0.321989 0.41 0.36 0.19 0.37 0.10 0.60

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Figure 4. Critical bandwidths hm obtained by kernel density estimation 1970 (top panel), 1980 (middlepanel) and 1989 (bottom panel) are the values of h where jumps in the step function occur. Note that the

number of nodes in the estimated densities is a decreasing function of the window width



Figure 5. Kernel density estimates in three years for the actual data (top panels) and the two trans-formations (centre and bottom panels), using a bandwidth consistent with uni- and bimodality (dashed and

solid lines)

404 M. BIANCHI


Figure 6. Cut-points for the density estimates of relative income distributions in 1970, 1980 and 1989.Selected bandwidth, h � 0�0025, consistent with bimodality in 1970, 1980 and 1989



Canada, Denmark, Finland, France, West Germany, Iceland, Italy, Japan, Luxembourg,Netherlands, New Zealand, Norway, Sweden, Switzerland, Trinidad and Tobago, the UnitedKingdom, the United States and Venezuela; the remaining 98 countries are allocated to the groupof poor countries.9 In 1980 and 1989, all countries listed among the poor or the rich in 1970continued to belong to the same group, with the only exceptions of Hong Kong, Trinidad andTobago and Venezuela. These are the only countries in our sample showing mobility betweengroups. For these countries, we have the mobility patterns reported in Table III. For bothTrinidad and Tobago and Venezuela the mobility patterns are likely explained by the alternatefortunes of the oil price, whereas it is not surprising to ®nd Hong-Kong joining the group of mostindustrialized countries in the eighties.

For a more detailed comparison, we also plot the estimated densities together in the samegraph, as in Figure 7. The latter suggests the following considerations:

. In contrast to 1970, the 1980 and 1989 distributions show a pronounced mode in the longright-end tail of the distribution. Given the low mobility mentioned above, this suggests theformation of `clubs' or clusters in the data, in support of the so-called `polarizationhypothesis'.10

. From 1970 to 1980, the mode centred about 0.33% slightly shifts to the left to a value ofapproximately 0.31% and the mode centred about 2% shifts to the right to a value ofapproximately 2.26%. This e�ect is also clearly visible in Figure 6 and indicates a wideninggap between less and more developed countries. Moreover, in the same decade, we alsoobserve that the proportion of poor countries relative to the `middle class', represented in thelong tail of the distribution with income shares in between 1% and 1.6%, has become smaller.

Table II. Maxima and minima in the estimated density distributions shown in Figure 6

Global maximum Local maximum Local minimum

Year x-value f h�0�0025�x� x-value f h�0�0025�x� x-value f h�0�0025�x�1970 0.0033 27.41 0.0210 4.64 0.0160 3.491980 0.0031 25.47 0.0226 6.46 0.0173 3.291989 0.0027 26.96 0.0236 6.12 0.0183 3.07

Table III. Mobility between groups according to the discriminant rule of Section 2.1 (equation 7) applied tothe estimated densities shown in Figure 6

Country 1970 1980 1989

Trinidad and Tobago Rich (1�76% > 1�60%) Rich (2�24% > 1�73%) Poor (1�50% < 1�83%)Venezuela Rich (1�99% > 1�60%) Poor (1�44% < 1�73%) Poor (1�00% < 1�83%)Hong Kong Poor (1�17% < 1�60%) Rich (1�75% > 1�73%) Rich (2�45% > 1�83%)

9 Note that the ratio is 21=199 � 0�18, that is, exactly the same as the mixing proportion p1 in Section 2.1. We knowtherefore now that the density presented as an illustrative example in Figure 1 is the estimated density of relative percapita GDP data in 1970!10 Recent theoretical work in growth theory, such as Baumol (1986), Esteban and Ray (1994), and Quah (1996a,b,c), hasrationalized phenomena like the formation of convergence clubs, polarization, and poverty traps. According to thesemodels, convergence clubs endogenously form and the distribution of income across countries has a tendency to polarizetowards a bimodal distribution.

406 M. BIANCHI


Thus, in the top panel of Figure 7, the dashed line for the distribution in 1980 lays belowthe solid line for the distribution in 1970 in the range 0±1%, but it stands above it in the range1±1.6%.

. In the decade from 1980 to 1989, a further drift apart in the modes is observed. The world percapita income share of the average poor falls from 0.31% to 0.27%, while that of the average

Figure 7. Density estimates of relative income distributions in 1970, 1980 and 1989. Selected bandwidth,h � 0�0025, consistent with bimodality in 1970, 1980 and 1989



rich country rises from 2.26% to 2.36%. The increasing gap in relative terms between rich andpoor countries in this decade is associated with a process of vanishing of the middle class(further evidence of polarization/strati®cation). This is indicated by the larger proportion ofpoor countries (larger than in 1980) relative to middle-income countries. Thus, in the bottompanel of Figure 7, the dashed line for the distribution in 1989 lays now above the solid line forthe distribution in 1980 in the range 0±0.7%, and it stands below it in the range 0.7±1.5%.11

4. CONCLUSIONS

In this paper we have empirically examined the convergence hypothesis from the perspective ofincome distributions in a cross-section of countries. By means of purely statistical techniquessuch as non-parametric density estimation and bootstrap multimodality tests, we have tested forthe number of modes and estimated, consistently with the detected number of modes, the incomedistribution of a cross-section of 119 countries in 1970, 1980 and 1989. We have also performed adiscriminant analysis on the estimated densities to learn how units moved within the modesdistributions.

For actual GDP data, we have found strong evidence for bimodality (i.e. polarization andclubs formation) occurring in the 1970s, a widening gap between less and more developedcountries in the 1970s and the 1980s associated with a process of vanishing of the middle class inthe 1980s and a very low mobility of countries within groups. For the logarithms of GDP data,we could not reject the null hypothesis of unimodality, but it seems likely that we still have groupsin the data, although these groups do not re¯ect in a multimodal density due to the change in theshape of the distribution implied by the logarithmic transformation.

Overall, the empirical evidence suggested in our study appears to support the view of clusteringand strati®cation of growth patterns over time, in sharp contrast to the convergence hypothesis.

ACKNOWLEDGEMENTS

I would like to thank Danny Quah and Stephen Redding for suggestions, the Editor and threeanonymous referees for very helpful comments which led to substantial improvements over theoriginal draft of this paper. I am also deeply indebted to my thesis advisor, professor WolfgangHaÈ rdle, for introducing me to non-parametric methods and smoothing techniques in statistics.The statistical analysis for this paper was performed using GAUSS and S-PLUS. The programsare available upon request from the author. The views expressed are those of the author and donot necessarily re¯ect those of the Bank of England.

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408 M. BIANCHI


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Testing for convergence: evidence from non-parametric multimodality tests