The respective results also testified to the presence of convergence: coefficient accounted for circa 0.028. Assessments of coefficient for single subperiods that had a wrong sign proved to be positive after inclusion of structural variables. Barro and Sala-iMartin also conducted assessments of convergence basing on the data on 90 regions in 8 European countries over the period between 1950 through 1990 (11 in Germany and UK, 20 in Italy, 21 in France, 4 in Holland, 3 in Belgium, 3 in Denmark and 17 in Spain).
In contrast to the US and Japan, to account inter-country differences, the original convergence equation was supplemented by country–specific logical variables. The result was convergence Barro, Sala-i-Martin (2004).
for the sample of regions in question, whose coefficient made up a.
0.02. At the same time, the researchers were assessing a system of simultaneous equations to find coefficients for the top five largest countries: Germany, UK, Italy, France and Spain. In that case coefficient could vary across different countries, but not over time. The respective values ranged from 0.012 (France) to 0.(UK).
Under empirical testing of the concept of convergence the focus of attention is on the dynamics of the index of dispersion (the mean-square bias) of distribution of per capita GDP logarithms.
Accordingly, if for a give sample of countries (regions) the dispersion of distribution of per capita GDP logarithms diminishes from the start through the end of the period in question, the hypothesis of convergence is not rejected. Such a method of testing this particular hypothesis appears purely descriptive. Thus, particularly Lichtenberg (1994) has modified the Fischer’s test for a statistical testing if the diminishment of dispersion between the start through the end of the period was statistically significant.
While testing the hypothesis of convergence for the noted countries (the US, Japan and the European countries), the researchers examined if –convergence was present there. In particular, as already noted above, the researchers considered the dynamics of a standard bias of the per capita GDP (GRP) over the respective time periods. As long as the period between 1880–2000 is concerned, the US had been experiencing the decline in the value of the standard per capita GDP. The exception was the period between through 1930, when this index was growing. That was likely to become an effect of a decrease of relative prices for agrarian products and thus affected the US agrarian states whose incomes had been relatively lower than elsewhere nationwide.
As concerns the Japanese prefectures, between 1930 through 1940 the GDP dispersion index had been growing there, but it con sequently began decreasing up till the end of the period in question (1990). The analysis of dispersion of GDP of the European countries also evidenced the presence of a tendency to diminishment of the income level dispersion across the countries between through 1990. Thus, results of the analysis of the presence of convergence and –convergence for the noted groups of countries or regions completely correspond to each other.
In addition, to test the accuracy of the hypothesis of – convergence, one can also employ other indices that testify to a change in the inequality level between the countries or regions in terms of per capita income. Specifically, along with dispersion or standard bias indices, some papers employ the variation coefficient, which is square root of dispersion to the average value across the sample ratio. By contrast to dispersion or standard bias, once computed in such a way, the index will no longer depend on the unit of the examined revenue index. Furthermore, one may use a weighted coefficient of variation which is computed in a standard fashion, but considers weighted indices of average and the dispersion.
Another, rather popular, indicator of the inequality level is Gini coefficient computed on the basis of the Lorenz curve. To built the Lorenz curve, on the absciss axis one lays the accumulated proportion of the population of a group of countries or regions, while on the ordinate axis – the accumulated proportion of the countries’ incomes in the total income across a given sample. While building the curve, one ranks all the countries concerned in the increasing order of the per capita income level. Once the Lorenz curve is built, one can compute Gini coefficient, which is a doubled difference between the area under the curve and a straight laid at angle of 45°.
The analytical expression for the computation of Gini coefficient takes the following form:
N N G = pi p Yi - Y (2.4) j j 2Y i=1 j =where pi and p – proportions of the population of countries i and j j in the total number of population, Yk – index of income of country k. Accordingly Gini index can vary from 0 (complete equality) up to 1 (complete inequality).
To analyze the level of inequality between countries or regions in terms of per capita income, i.e. to test the concept of – convergence, one can also employ Tale index. It is borrowed from the theory of information and originally is associated with the concept of entropy. Tale index can be computed using the following formula:
N Yi Yi 1 log T (1) = (2.5) N Y Y i =One of the features of this particular index is the possibility for decomposition into two items, of which the first mirrors an inequality within each of the singled out groups of countries, while the other – an inequality within these groups. In addition, upon the division of the right part of the inequality into log(N)), the index value can range from zero to one.
It would also be worthwhile to focus on such a concept in the frame of the convergence theory as polarization. Despite it shows some similarity to the inequality concept, there is a considerable difference between them, nonetheless. According to Esteban and Ray19, the polarization concept arise because of “inequality measurement axioms do not allow an adequate distinguishing of the convergence to the global median value from a clusterization Esteban, Ray (1994).
around regular ones”. So, the polarization concept framework allows consideration of a possibility for emergence of clusters around local steady equilibrium states. The polarization concept can be illustrated using the following example: let us assume that there exists an originally noted even distribution of countries in terms of the level of GDP along the segment from 1 to 6. Let us then consider the following transformation of the original distribution: the countries with income level from 1 to 3 converge to the state with income 2, while those with income level between 4 to 6– to the state with income 5. Despite Tale index showing a reduction of the inequality, polarization grows. To compute the polarization index, the following formula is used:
N N PI = pi p Yi - Y (2.6) j j i=1 j =where – index from 1 to 1.6 that measures sensitivity of polarization. So, the smaller is the sensitivity index, the closer the polarization concept to the inequality concept is.
The consideration of convergence concepts also necessitates singling out a theory of convergence clubs. Convergence club is a group of countries (regions) for which the convergence conditions are met. Accordingly, there may exist a great number of convergence clubs with no convergence between them, which is correct in respect to any originally examined sample of countries or regions.
The emergence of convergence clubs is largely explained by the role of initial conditions. From the perspective of endogenous growth, to trigger a process of convergence between countries (regions), it is necessary for them to possess both an identical structure of their economies and similarity of their starting conditions.
Galor (1995) showed that under certain prerequisites heterogeneity in the dynamics of savings can likewise lead to emergence of convergence clubs in the frame of the neoclassical theory of growth.
Justification of a possible composition of such clubs pose a fundamental complex problem under empirical testing of this particular convergence concept. There are two most frequently used methods to do this. According to the first method, the composition of clubs is originally determined on the basis of criteria the researcher selects, while the other method implies “endogenization” of the selection of clubs, under which the researcher has to find any statistical method that would allow to single out convergence clubs in the given data series. Once the composition of clubs is identified, the researcher would just need to examine if any of the existing convergence criteria is met within each singled out group of countries (regions).
In addition to the aforementioned methods of testing convergence concepts, there exist other methods of their analysis. For instance, it was suggested to employ methods of the tome series theory to test the convergence hypothesis econometrically: specifically, Bernard (1991), Quah (1992), Bernard, Darlauf (1998) have introduced the concept of stochastic convergence. The hypothesis of stochastic convergence is met if for two given countries the difference between their per capita income levels is a stationary process with zero average. In such particular case it can be argued that the economies in question have reached their own state o steady equilibrium and shocks that affect them are short-term.
Evans (1996), in turn, considered statistical characteristics of a time series of the per capita GDP logarithms, which was built upon indices of dispersion at each given moment of time along the whole given time interval. If, as the neoclassical theory of grows forecasts this, long–term paths of GDP per capita in given countries (regions) are parallel, their dispersion series should appear stationary relative to a positive constant. If, as the theory of endogenous growth argues, GDP=s per capita grow at different paces, the series of dispersions should be integrated of the first order with an immanent square trend.
There exist, at least, three interrelated concepts of stochastic convergence. The concept of strict convergence (asymptotically perfect convergence) is true only in the event the difference between two tome series of any given countries from a given sample does not contain either unit root, or trend (no matter determinist or stochastic). In other words, it should meet the following condition:
limT E(Yi,T - Y 0) = 0, (2.7) j,T where 0 – all the existing information as of the time period T.
However, this concept of stochastic convergence is often criticized for its rather strict conditions: it suggest that a long-term envisaged value of the difference between per capita income levels in two countries is zero. That is why researchers consider an alternative concept of a weak stochastic convergence (asymptotically relative convergence), according to which the noted difference between time series of two countries converge not to zero, but to some constant. This can be expressed formally in the following way:
E(Yi,T - Y t ) < (Yi,0 - Y ), (2.8) j,T j,where 0 corresponds to the current moment of time, while T – to some moment of time in the future. According to noted definition, the difference between two time series should also be stationery, but allows the presence of a determinist time trend.
Finally, a less rigorous concept of stochastic convergence is correct in the event when, despite the existence of different trends for two time series, there exists their linear combination that satisfy the following condition:
limT E(Yi,T - Y 0 ) = 0, > 0. (2.9) j,T Hence, the approach based on the time series theory suggests that series of per capita GDP logarithms can comprise both stochastic and determined trends. If so, then the analysis of time series should aim at studying into a correlation between determined and stochastic trends that determine the dynamics of per capita GDP.
The hypothesis of stationarity means that time series have both determined and stochastic trends, i.e. they appear co-integrated (a cointegral correlation allows constant, but not a linear trend), and their dynamics is determined by identical factors.
In other words, convergence is understood as the preservation over time at the level close to zero of the existing gap between two countries in the per capita GDP levels. Evidently this condition is to a greatest extent applicable to groups of countries with close per capita GDP levels, albeit it fails to explain differences between “rich” and “poor” countries’ living standards. It should be noted that the concept of convergence that assumes the gap between levels of per capita GDP=s contradicts the condition of convergence.
Bernard, Durlauf (1998) explained this contradiction arguing that a convergence with a constant gap falls under the case of economies moving along steady long-term paths of growth, while convergence describes a period of transition from one path to another.
While the concept of stochastic convergence allows to lift a number of problems that arise in the course of consideration of - and –convergence, it suffers a number of drawbacks. Thus, particularly, the existing tests of stationarity (the extended DickyFuller test or the Phillips-Perron test) enjoy a low statistical capacity, as far as final samples are concerned. With such a capacity, the probability of a non-rejection of zero hypothesis of the existence of the unit root is high.
Another concept of convergence considers an evolution of a relative position of each country vis--vis the others. The author of the concept was Quah20 who believed that the concepts of -convergence and unconditional and conditional -convergences have nothing to do with the notion of convergence per se. That is why he suggested studying the process of convergence basing on an evaluation of the dynamics of the overall distribution of the multitude of per capita GDP values of a give sample of countries. Quah does not reject a hypothesis of convergence, if the distribution of per capita GDP for the given group of countries or regions tends in time to a unimodal one, while in the event of a bimodal distribution, the polarization concept becomes correct. Under the polarization concept the group of countries with an average income level disappears. As well, Quah stresses the necessity of assessing the scale of a change of a relative position a given country holds within the overall distribution.
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