Robust Tests for Convergence Clubs

Faculty of Economics Cambridge Working Papers in Economics Cambridge Working Papers in Economics: 1873 ROBUST TESTS FOR CONVERGENCE CLUBS Luisa Corrado Thanasis Stengos Melvyn Weeks M. Ege Yazgan 21 December 2018 In many applications common in testing for convergence the number of cross-sectional units is large and the number of time periods are few. In these situations asymptotic tests based on an omnibus null hypothesis are characterised by a number of problems. In this paper we propose a multiple pairwise comparisons method based on an a recursive bootstrap to test for convergence with no prior information on the composition of convergence clubs. Monte Carlo simulations suggest that our bootstrap-based test performs well to correctly identify convergence clubs when compared with other similar tests that rely on asymptotic arguments. Across a potentially large number of regions, using both cross-country and regional data for the European Union we find that the size distortion which afflicts standard tests and results in a bias towards finnding less convergence, is ameliorated when we utilise our bootstrap test. Robust Tests for Convergence Clubs∗ Luisa Corrado† University of Rome Tor Vergata and University of Cambridge Thanasis Stengos University of Guelph Melvyn Weeks Faculty of Economics and Clare College, University of Cambridge M. Ege Yazgan Istanbul Bilgi University December 20, 2018 Abstract In many applications common in testing for convergence the number of cross-sectional units is large and the number of time periods are few. In these situations asymptotic tests based on an omnibus null hypothesis are characterised by a number of problems. In this paper we propose a multiple pairwise comparisons method based on an a recursive bootstrap to test for convergence with no prior information on the composition of convergence clubs. Monte Carlo simulations suggest that our bootstrap-based test performs well to correctly identify convergence clubs when compared with other similar tests that rely on asymptotic arguments. Across a potentially large number of regions, using both cross-country and regional data for the European Union we find that the size distortion which afflicts standard tests and results in a bias towards finding less convergence, is ameliorated when we utilise our bootstrap test. Keywords: Multivariate stationarity, bootstrap tests, regional convergence. JEL Classifications: C51, R11, R15. ∗ We are grateful to Andrew Harvey, Hashem Pesaran, Jonathan Temple, Mike Wickens, Gernot Dopplehoffer, Vasco Carvalho for their comments on the work. † Luisa Corrado gratefully acknowledges the Marie-Curie Intra European fellowship 039326. [email protected] 1 1 INTRODUCTION 1 Introduction The extent to which countries and or regions are similar across one or more dimensions is a question that has long been of interest to economists and policymakers. Within the European Union the ECB targets a single Euro Area inflation rate, and in this respect the degree to which there exists convergence in regional per capita incomes and output is of critical relevance to European regional development policies (Boldrin and Canova (2001)). Moreover, one of the core components of the European cohesion policy has been to reduce the disparities between income levels of different regions and in particular the backwardness of the least favoured regions; this objective has, in general, been manifest as the promotion of convergence between eu regions.1 In this context it is evident that the correct detection of the extent of convergence within a regional economy is paramount given that policy usually tries to achieve regional convergence by reducing the gap between the richest and the poorest regions. In this respect a test of convergence which exhibits bias, for example being oversized in small samples, will mislead, and in this instance imply less convergence suggesting the need for more policy initiatives than may actually be required. Economists have conceptualised the notion of similarity using formal definitions of convergence based upon growth theory. Standard neoclassical growth models (Solow (1956) and Swann (1956)) founded upon the key tenets of diminishing returns to capital and labour and perfect diffusion of technological change, dictate that countries will converge to the same level of per capita income (output) in the long run, independent of initial conditions. The New Growth theory (see, for example, Romer (1986); Lucas (1988); Grossman and Helpman (1994); Barro and Sala-i-Martin (1997)) allows for increasing returns to accumulable factors such as human capital in order to determine the (endogenous) long-run growth rate.2 The emergence over the past decade of New Economic Geography3 models of industrial location and agglomeration, has resulted in the identification of other forces which generate increasing returns, two notable examples being the relationship between location and transportation costs (Louveaux et al., 1982) and the effect of regional externalities 1 See Article 158 of the Treaty establishing the European Community. Other variants of the New Growth theories predict the emergence of multiple locally-stable steady-state equilibria instead of the unique globally-stable equilibrium of the neoclassical growth model as a result of differences in human and physical capital per worker across countries (Basu and Weil (1998)), their state of financial development (Acemoglu and Zilibotti (1997)) or other externalities caused by complementarity in innovation (Ciccone and Matsuyama (1996)). 3 In the ‘new economic geography’ models the sources of increasing returns are associated with Marshallian-type external localisation economies (such as access to specialised local labour inputs, local market access and size effects, local knowledge spillovers, and the like). These models provide a rich set of possible long run regional growth patterns that depend, among other things, on the relative importance of transport costs and localisation economies (Fujita, et al. 1999; Fujita and Thisse 2002). 2 2 1 INTRODUCTION (Cheshire and Hay (1989)). To the extent that the process of growth is different across regions in the sense that there are different long-run steady-states, the standard neoclassical growth model is not valid. In this context traditional approaches to test for convergence are hard to justify, difficult to interpret, and often difficult to implement. For example, a rejection of the omnibus null of convergence across a groups of regions provides increasingly less information as the number of regions increases and where prior knowledge over both the number and composition of convergence clubs is minimal. Moreover, the justification of constructing such a large intersection null hypothesis is often questionable at the outset. Faced with the emergence of larger panels, with an attendant increase in cross-sectional heterogeneity, there has been a number of significant developments in testing. For example, the use of a heterogeneous alternative hypothesis partially alleviates the problem of testing over a large group of potentially heterogenous regions (see, for example, Im et al., 2003). In a further progression away from the testing of general omnibus hypotheses, Pesaran (2007) conducts pairwise tests for region pairs, with inference focussed on the proportion of output gaps that are stationary. One drawback of this approach is that limited inference can be made as to the significance of individual gaps, or indeed whether a group of output comparisons form a convergence club. An approach which allows for an endogenous determination of the number of clubs using a sequence of pairwise stationarity tests has been developed by Hobijn and Franses (2000). In extending this approach Corrado, Martin and Weeks (2005) developed a testing strategy that facilitates both the endogenous identification of the number and composition of regional clusters (or ‘clubs’), and the interpretation of the clubs by comparing observed clusters with a number of hypothesized regional groupings based on different theories of regional growth. However, given that the time series are relatively short, there are potential problems in basing inference on asymptotic results for stationarity tests. Reliance on large T asymptotics is likely to impart a size distortion, biasing the results towards finding less convergence than actually exists. To circumvent this problem we propose in this paper a recursive bootstrap test for stationarity which is designed to detect multiple convergence clubs without prespecification of group membership. Monte Carlo simulations suggest that the proposed bootstrap based method performs quite well in identifying club membership when compared with the Hobijn and Franses (2000) approach that is based on asymptotic arguments. We implement our bootstrap recursive test of convergence using the original cross-country dataset used by Hobijn and Franses (2000) and the European regional data used in Corrado, Martin and Weeks (2005). We then compare the asymptotic and bootstrap generated cluster outcomes. Our results show that by 3 2 TESTS FOR CONVERGENCE resolving the size distortion which afflicts the asymptotic test we find considerably more evidence of convergence in both the applications considered. The paper is structured as follows. Section two reviews existing tests for convergence clubs, and in section three we present the bootstrap version of the test. In section four we propose a Monte Carlo experiment to compare the properties of the asymptotic and bootstrap tests. Section five describes the data and applies the proposed tests to two real word datasets. In section six we discuss our findings and conclusions. 2 Tests for Convergence In this section we briefly discuss a number of significant developments in tests designed to detect convergence and identify clubs which are able to address a number of questions such as whether a particular pair of countries have converged, or whether a group of regions or countries form a convergence club. We briefly discuss the different approaches to detect convergence tracking a gradual progression away from multivariate time series and panel data tests based on an omnibus null, towards sequential tests and tests that are founded upon multiple pairwise comparisons.4 Our focus here is to identify endogenously clubs using multivariate tests for stationarity. However, given that the time series are relatively short, we show that there are potential problems in basing inference on asymptotic results for stationarity tests. To circumvent this problem we bootstrap the stationarity test and assess the effect of the size distortion on the cluster outcomes using two different applications based on country and regional level data. The use of multivariate time-series to test for convergence was initiated by the seminal papers of Bernard and Durlauf (1995, 1996). Given a set ̥ of N economies, a multi country definition of relative convergence asks whether the long-run forecast of all output differences with respect to a benchmark economy, (denoted with the subscript 1) tend to a country-specific constant as the forecasting horizon tends to infinity.5 We may then write lim E(y(i1),t+s | It ) = µ1i s→∞ ∀i 6= 1, (1) 4 Corrado and Weeks (2011) provide a more detailed overview. A necessary condition for regions i and j to converge, either absolutely or relatively, is that the two series must be cointegrated with cointegrating vector [1, −1]. However, if output difference are trend stationary, this implies that the two series are co-trended as well as cointegrated. Hence a stronger condition for convergence is that output differences cannot contain unit roots or time trends (Pesaran (2007)). 5 4 2 TESTS FOR CONVERGENCE where y(i1),t+s = yit+s − y1t+s and µ1i is a finite constant.6 There exist a number of problems with multivariate time series tests. First, the testing procedure is sensitive to the choice of the benchmark country. Second, in keeping with the problems of omnibus tests, in the event of rejecting the non-convergence null we have no information as to which series are I(0) and I(1), nor the composition of any convergence groups. Third, given the system properties of the test, a dimensionality constraint means that it can handle only a small number of economies simultaneously. Panel unit root procedures have also been adopted to test for convergence by considering the stationary properties of output deviations with respect to a benchmark economy (Fleissig and Strauss, 2001; Evans, 1998; Carlino and Mills, 1993). First, the so called ‘first-generation’ panel unit-root tests,7 maintain that errors are independent across cross-sectional units which imparts a size distortion. To overcome this problem a ‘second generation’ of panel unit root tests have been developed which allows for different forms of cross-sectional dependence.8 However, as pointed out by Breitung and Pesaran (2008), panel data unit root tests poses similar problems in that as N becomes large the likelihood of rejecting the omnibus null increases with no information on the exact form of the rejection. The problem of identifying the mix of I(0) and I(1) series whilst still utilising the attendant power from a panel by exploiting coefficient homogeneity under the null, has been addressed by the sequential test proposed by Kapetanios (2003).9 Specifically, Kapetanios employs a sequence of unit root tests of panels of decreasing size to separate stationary and nonstationary series,10 facilitating an endogenous identification of the number and identity of stationary series. Although a positive development there are a number of limitations. Critically the utility of this approach depends on the use of a panel framework to add power in a situation where most series are stationary but very persistent. In addition, the method only permits the classification of the N series into two groups whereas there may be many more groups. As a consequence it is not possible to address a number of questions that may be of interest: such as whether a particular pair of countries have converged, or whether a group of regions or countries form a convergence club. When applied to output deviations, an additional problem with the Kapetanois 6 We consider this as a more reasonable definition of convergence in the sense that it allows the process of convergence to stop within a neighborhood of zero mean stationarity (absolute convergence) and is consistent with the existence of increasing costs of convergence. 7 See, for example, Maddala and Wu, 1999; Im et al., 2003; Levin, Lin and Chu, 2002. 8 For example, Taylor and Sarno (1998) adopt a multivariate approach and estimate a system of N − 1 ADF equations using Feasible GLS to account for contemporaneous correlations among the disturbances. Other notable example of second generation of panel unit root tests with cross-sectional dependence include Pesaran (2007) and Moon and Perron (2007). 9 See also Flores et al. (1999) and Breuer et al. (1999). 10 This method is referred to as the Sequential Panel Selection Method (SPSM). 5 2.1 Sequential Pairwise Tests 2 TESTS FOR CONVERGENCE test is that the testing procedure is still sensitive to the choice of the benchmark country. One approach which avoids the pitfalls of the choice of a benchmark country and, more generally, the dimensionality problem that afflicts the application of omnibus tests, is to conduct separate tests of either stationarity and/or nonstationarity. By considering a particular multi-country definition of convergence, Pesaran (2007) adopts a pairwise approach to test for unit-roots and stationarity properties of all N (N − 1)/2 possible output pairs {yit − yjt }. The definition of convergence that is adopted is that the N countries converge if Pr(∩i=1,...,N,j=i+1,...,N |yit+s − yjt+s | < c|It ) > π, (2) for all horizons, s = 1, ..., ∞, and c a positive constant. π ≥ 0 denotes a tolerance probability which denotes the proportion that one would expect to converge by chance. In testing the significance of the proportion of output gaps that indicate convergence, the dimensionality constraint that affects the application of system-wide multivariate tests of stationarity is circumvented. However, although the pairwise tests of convergence proposed by Pesaran (2007) is less restrictive than the omnibus tests proposed by Bernard and Durlauf (1995), the subsequent inference is limited in that it does not allow inference on which pairs of regions have converged, or the number and composition of convergence clubs. Below we examine various testing strategies for club convergence and in particular the sequential testing procedure proposed by Hobijn and Franses (2000). 2.1 Sequential Pairwise Tests Despite the use of multivariate time series and panel data methodologies to test for convergence, there has been relatively few attempts to utilize this approach to systematically identify convergence clubs (see Durlauf et al. 2005). The existing early methods were generally focused on the convergence of various a-priori defined homogeneous country groups which were assumed to share the same initial conditions. Baumol (1986) for example grouped countries with respect to political regimes (OECD membership, command economies and middle income countries), Chatterji (1992) allowed for clustering based on initial income per capita levels and tested convergence cross-sectionally, while Durlauf and Johnson (1995) grouped countries using a regression tree method based on different variables such as initial income levels and literacy rates that determined the different ”nodes” of the regression tree. Similarly, Tan (2009), utilizes a regression tree approach which again utilises 6 2.1 Sequential Pairwise Tests 2 TESTS FOR CONVERGENCE exogenous information in the form of conditioning variables. An alternative approach to the cross-sectional notion of β− convergence was introduced by Bernard and Durlauf (1995, 1996) based on a time series framework that makes use of unit root and cointegration analysis (see Durlauf et al. (2005) for a comprehensive literature review for convergence hypothesis). There are many other studies where the identification of convergence clubs has been achieved exogenously through testing in conjunction with a pre-classification of clubs using parametric techniques.11 For example, Weeks and Yao (2003) adopt this approach when assessing the degree of convergence across coastal and interior provinces in China over the period 1953-1997. As Maasoumi and Wang (2008) note, the principal problem with pre-classification is that as the number of regions increases such a strategy is not robust to the existence of other convergence clubs within each sub-group. A different approach is advanced using the notion of σ−convergence by Phillips and Sul (2007) who developed an algorithm based on a log-t regression approach that clusters countries with a common unobserved factor in their variance. In general it is straightforward to test whether two regions form a single group. We could simply construct a single output (income) deviation and test for stationarity or a unit root. In the context of differentiating between the stationarity properties of multiple series (or output deviations), the contribution by Kapetanios (2003) has provided new techniques that utilise the power of an omnibus null in conjunction with a sequential test that allows greater inference under the alternative hypothesis. However, for a large set of regions, ̥, locating partitions that are consistent with a particular configuration of convergence clubs generates difficulties given that the number of combinations is large and related, that we have little prior information.12 Hobijn and Franses (2000) propose an empirical procedure that endogenously locates groups of similar countries (convergence clubs) utilising a sequence of stationarity tests. Cluster or club convergence in this context implies that regional per capita income differences between the members of a given cluster converge to zero (in the case of absolute convergence) or to some finite, cluster specific non-zero constant (in the case of relative convergence). Below we illustrate the method. The Hobijn and Franses (2000) test represents a multivariate extension of the Kwiatkowski et al.(1992) test (hereafter KPSS test). We introduce the test by first 11 See Quah (1997) for an example of non-parametric techniques to locate convergence clubs. Harvey and Bernstein (2003), utilize non-parametric panel-data methods focussing on the evolution of temporal level contrasts for pairs of economies, identifying the number and composition of clusters. Beylunioğlu et al (2018) also propose a method based on the maximum clique method from graph theory that relies on Augmented Dickey Fuller (adf) unit root testing. However, as it will be explained below, the maximum clique method is a “top down” method that leads to a different definition of clubs and is not directly comparable to the “bottom up” Hobijn and Franses’ approach. 12 7 2.1 Sequential Pairwise Tests 2 TESTS FOR CONVERGENCE denoting yt = {yit } as the N × 1 vector of log per capita income and write yt as yt = α + βt + D t X vs + εt , (3) s=1 where α = {αi } is a N ×1 vector of constants, β = {βi } is a N ×1 vector of coefficients for the deterministic trend t, and vs = {vl,s }, l = 1, ..., m represents a m × 1 vector of first differences of the m stochastic trends in yt , m ∈ (0, ..., N ). D = {Di,l } denotes a N × m matrix with element Di,l denoting the parameter for the lth stochastic trend. εt = {εi,t } is a N × 1 vector of stochastic components. In considering the difference in log per capita income for regions i and j we write y(ij),t = α(ij) + β(ij) t + m X D(ij),l X t vl,s s=1 l=1  + ε(ij),t , (4) where (4) admits two different convergence concepts: absolute and relative. The restrictions implied by the null of relative convergence are β(ij) = 0 ∀i 6= j ∈ ̥ and D(ij),l = Dil − Djl = 0 ∀l = 1, ..., m, with the latter restriction indicating that the stochastic trends in log per capita income are cointegrated with cointegrating vector [1 − 1]. The additional parameter restrictions for the null hypothesis of absolute convergence are that α(ij) = 0 ∀i 6= j ∈ ̥. Both asymptotic absolute and relative convergence imply that the cross sectional variance of log per capita income converges to a finite level. t P Denoting the partial sum process St = y(ij),s , the test statistic for zero mean s=1 stationarity is given by13 Denoting ht = y(ij),t − 1 T 13 T P τb0 = T −2 T X ′ St [b σ 2 ]−1 St . (6) t=1 y(ij),t and the partial sum process as S̄t = t=1 t P hs the test s=1 For the stationary null, Hobijn and Franses (2000) utilise the Kwiatkowski et al. (1992) test. The KPSS test is operationalised by regressing the pairwise difference in per capita income y(ij),t against an intercept and a time trend giving residuals εb(ij),t = y(ij),t − α b(ij) − βb(ij) t. and σ b2 = (5) T L T X 1 X 1X 2 εb(ij),t + 2 ω(k, L) εb(ij),t εb(ij),t−k , T t=1 T k=1 t=k+1 represents the consistent Newey-West estimator of the long-run variance. ω(k, L) = 1 − k/(1 + L), k = 1, ..., L is the Bartlett kernel, where L denotes the bandwidth. 8 2.1 Sequential Pairwise Tests 2 TESTS FOR CONVERGENCE statistic for level stationarity is given by τbµ = T −2 T X ′ S̄t [b σ 2 ]−1 S̄t . (7) t=1 Examining (4), we note that in the case of two regions, and focussing on a test of relative convergence with restrictions D(ij),l = 0 ∀l = 1, ..., m and β(ij) = 0, it is obviously straightforward to test whether two regions form part of a single group. However, for a large number of regions locating the partitions over ̥ that are consistent with a particular configuration of convergence clubs is infeasible both because the number of combinations is large and related, that we have little prior information on the form of D and the likely combination of zeros restrictions over the differences β(ij) and α(ij) . The alternative testing strategy proposed by Hobijn and Franses (2000) forms groups from the bottom up using a clustering methodology to determine, endogenously, the most likely combination of restrictions, and as a consequence, the most likely set of convergence clubs. The cluster algorithm is based on the hierarchical farthest neighbour method due to Murtagh (1985). We illustrate the sequential test using the set of regions ̥ = {1, 2, 3, 4}. (i) We first initialise singleton clusters K(i) for each region i = 1, ..., 4. The null hypothesis of level stationarity is tested for all N (N −1)/2 = 6 region pairs. We b s=1 = {p(ij) }, where p(ij) = Pr(b collect p-values in the vector p τ(ij),µ < c(ij) |It ), τb(ij),µ denotes the test statistic and c(ij) the critical value. s = 1 denotes the first iteration. b s=1 , indicating the Clusters are formed on the basis of the max p-value in p pair of regions which are most likely to converge. If, for example, p(1,2) = {max{b ps=1 } > pmin } then regions 1,2 are the first pair of regions to form a i,j∈̥ 14 ′ club. We denote the first cluster as K(1 ) = {1, 2} and discard the singleton ′ cluster 2, which is now part of the two-region cluster K(1 ). ′ ′ (ii) In the second iteration (s = 2) we define the set of regions as ̥ = (1 , 3, 4). We form pairwise output differences between the N − 2 remaining singleton clusters and the two-region cluster K(1′ ). Once again we collect the p-values b s=2 . Letting p(r,v) = {max{b in the vector p ps=2 } > pmin }, then if, for i,j∈̥ example, p(r,v) = p(1′ 3) , the singleton cluster K(3) joins cluster K(1′ ) forming a ′′ three-region cluster K(1 ) = {1, 2, 3}. 14 The choice of pmin has a direct effect on the cluster size. Since the stationarity test is known to be oversized in small samples, this bias will generate inference towards finding less convergence. 9 3 A BOOTSTRAP TEST ′′ (iv) In this example we find a three-region cluster K(1 ) and a singleton cluster K(4), so the procedure stops. The principle difference between this sequential testing strategy and the SPSM approach of Kapetanios is that the SPSM test is designed to endogenously classify stationary and nonstationary series. This is achieved by sequentially reducing the size of the omnibus null by removing series with the most evidence against the unit root null, classifying these series as stationary. The stopping point is when the unit root null does not reject, such that all the remaining regions are declared nonstationary. In contrast the Hobijn and Franses method seeks to endogenously allocate N series to J ≤ N convergence clubs. This is achieved by only classifying regions that provide, at each recursion and conditional on exceeding pmin , the most evidence for convergence. Although the sequential multivariate stationarity test is consistent in that for large T the tests will reveal the true underlying convergence clubs, the principle shortcoming is that the test statistic is known to be oversized in small samples (Caner and Kilian, 2001). When testing for convergence using yearly data T is likely to be small, and as a result inference is likely to be biased in the direction of finding less convergence. Similar size distortions also emerge when the series are stationary but highly persistent: in this case the partial sum of residuals which are used to derive the KPSS test resemble those under the alternative in the limit. Below we outline a bootstrap approach which circumvents the pitfalls of inference based upon asymptotic arguments since it is able to generate independent bootstrap resamples using a parametric model which is conditional on the sample size and the dependence structure of the dataset. In section 5 we utilise this test to investigate the extent of convergence in two different applications: (i) the cross-country dataset originally adopted by Hobijn and Franses (2000); (ii) the European regional dataset used by Corrado, Martin and Weeks (2005). 3 A Bootstrap Test To derive the parametric model with which to create independent bootstrap samples under the stationarity null, following Kuo and Tsong (2005) and Leybourne and McCabe (1994), we exploit the equivalence in second order moments between an unobserved component model and a parametric ARIMA model (Harvey (1989)) for the differenced data. In demonstrating this equivalence we note that (4) may be rewritten in structural form as a function of a deterministic component (α(ij) + β(ij) t), 10 3 A BOOTSTRAP TEST a random walk (rt ) and a stationary error (ε(ij),t ): y(ij),t = α(ij) + β(ij) t + m X D(ij),l rl,t + ε(ij),t (8) l=1 rl,t = rl,t−1 + vl,t , (9) P where rl,t = ts=1 vl,s represents the l-th stochastic trend for regions i and j with r0 , the fixed initial value, set to zero. We also assume that ε(ij),t is a stationary P∞ error process ε(ij),t = s=0 ψ(ij),s u(ij),t−s = Ψ(L)u(ij),t where ψ(ij),0 = 1, u(ij),t ∼  P s 15 Under these assumptions ε(ij),t has i.i.d(0, σu2(ij) ) and Ψ(L) = 1 + ∞ s=1 ψ(ij),s L . an infinite order autoregressive representation ε(ij),t = ∞ X λ(ij),s ε(ij),t−s + u(ij),t , (10) s=1  P s where Λ(L) = Ψ(L)−1 = 1 + ∞ s=1 λ(ij),s L . Given (8), since ε(ij),t is a stationary process, the necessary condition for convergence of regions i, j is that the variance of the random walk error (σv2 ) is zero. Focussing on a test for relative convergence, below we describe the nature of the recursive multivariate stationarity test using critical values generated from the empirical distribution of the test statistic constructed using bootstrap sampling. In generating a bootstrap test for relative convergence we focus on relative convergence where β(ij) = 0, which rules out the presence of a deterministic trend.16 The idea is to estimate the null finite sample distribution of the KPSS test statistics by exploiting the equivalence between the unobservable component model and the parametric ARIMA model. Harvey (1989) demonstrates that the components from the structural model (8) can be combined to give a reduced form ARIMA(0,1,1) model. In particular, assuming independence of ε(ij),t and vt , (8) becomes a local component model which, after time differencing, can be expressed as the MA model ∆y(ij),t = (1 − θL)η(ij),t where η(ij),t ∼ i.i.d(0, ση2(ij) ) and ση2(ij) = σε2(ij) /θ. The reduced form parameter θ is derived by equating the autocovariances of first differences at lag one in the structural and reduced forms. This gives the following relationship between the parameters of the components model (8) and the P∞ ε(ij),t is assumed to be invertible s=0 s ψ(ij),s < ∞. 16 For the test of absolute convergence the restrictions are β(ij) = α(ij) = 0. 15 11 3 A BOOTSTRAP TEST ARIMA(0,1,1) model: where q = σv2 σε2  1  σv2 +2− θ= 2  σε2(ij) σv2 σε2(ij) + σ2 4 2v σε(ij) !1/2    . (11) is the signal to noise ratio. Under the stationarity null, namely (ij) that regions i, j are converging, the variance of the random walk component (σv2 ) is zero, which in turn implies that θ = 1 in the ARIMA representation. Therefore by imposing a moving average unit root in the ARIMA representation one can use the parametric model for sampling instead of the ”unobservable” component model. Our bootstrap sampling scheme is based on the following procedure. First, for each region pair i, j and contemporaneous difference y(ij),t = yi,t − yj,t , we fit an ARMA(p, 1) model to the differenced series ∆y(ij),t = y(ij),t − y(ij),t−1 , namely ∆y(ij),t = p X φ(ij),k ∆y(ij),t−k + η(ij),t − θη(ij),t−1 , (12) k=1 The MA component in (12) follows from the reparametrisation of the structural component model to reproduce the stationarity properties of the data in the ARMA representation. The AR(p) component17 represents an approximation to the assumed infinite-order moving average errors to capture the dependence structure in the data. By imposing a moving average unit root in the sampling procedure we can then construct the bootstrap distribution of the test statistic for level stationarity defined in (7). The accuracy of the bootstrap test relative to the asymptotic approximations hinges on the bootstrap sample being drawn independently. Given the presence of a known dependence structure, in this case a stationary ARMA(p, 1) model, we utilise the Recursive Bootstrap.18 To achieve independent re-sampling from (12) we estimate r φ̂(ij),k and η̂(ij),t , and we draw a bootstrap sample {η̄(ij),t }Tt=1 from the distribution of PT 1 b(ij),t . centered19 residuals {η̄(ij),t }Tt=1 , where η̄(ij),t = ηb(ij),t − T −1 t=2 η r Given the bootstrapped residuals, {η̄(ij),t }Tt=1 , the rth bootstrap sample for the 17 p denotes optimal lag length, chosen using the AIC criterion. See Horowitz (2001) on the merits of the recursive bootstrap for linear models, and Maddala and Li (1997) and Efron and Tibshirani (1986) for specific examples. 19 Centering the residuals reduces the downward bias of autoregression coefficients in small samples (Horowitz, 2001). 18 12 4 A MONTE CARLO STUDY r data {∆y(ij),t }Tt=1 is generated based on the recursive relation20 r ∆y(ij),t = p X k=1 r r r φb(ij),k ∆y(ij),t−k + η̄(ij),t − η̄(ij),t−1 . (13) We then recover the level of the series (where the level denotes the contemporaneous regional difference) directly from (13) r y(ij),t = r y(ij),t−1 + p X k=1 r Defining hrt = y(ij),t − 1 T T P t=1 r r r φb(ij),k ∆y(ij),t−k + η(ij),t − η(ij),t−1 . (14) r y(ij),t , then for rth bootstrap sample, and the i, j region r pair, a test statistic for relative convergence, τb(ij),µ , is given by where S̄tr = t P s=1 r τb(ij),µ =T −2 T X r,′ S̄t [b σ r,2 ]−1 S̄tr , (15) t=1 hrs . For each region pair we draw R bootstrap samples and construct B the empirical distribution of the test statistic under the null, which we denote τ(ij),µ . B Bootstrap critical values C(ij),µ can then be recovered at the required significance levels and we can implement the algorithm described in section 3 utilising a vector of bootstrapped empirical p-values, p̂B . 4 A Monte Carlo Study In this section we compare the performance of the bootstrap version of the kpss test, cw henceforth, with the original hf test based on the asymptotic version of the multivariate kpss testing procedure. To the best of our knowledge there is no other comparable Monte Carlo study in the literature that evaluates clustering methods in the same context as we do here. hf is a method that relies on a “bottom up” algorithm that clusters groups one by one. To determine whether a set of countries is convergent, hf applies a multivariate stationarity test to panels comprised of consecutive pairwise difference series set elements and confirms convergence if the null hypothesis of stationarity 20 r r r r r r Initial values, ∆y(ij),t−1 = y(ij),t−1 − y(ij),t−2 = ... = ∆y(ij),t−p = y(ij),t−p − y(ij),t−p−1 are set to zero. 13 4.1 Monte Carlo Structure 4 A MONTE CARLO STUDY of the panel is not rejected using the kpss test. For example, if we want to test the convergence of countries 1,2,3 and 4, a panel consisting of y 12 , y 13 , y 23 , y 14 , y 24 and y 34 is subjected to the kpss test, where for example y 12 ,y 23 , and y 34 denote the difference in log per capita between countries 1 and 2, 2 and 3 and 3 and 4 respectively. If stationarity cannot be rejected the panel is then augmented with series other than 1, 2, 3 and 4, each added separately. If then for each of these additional panels the stationarity null is rejected, then these four countries are said to be convergent. 4.1 Monte Carlo Structure In this subsection, we will discuss the data generating processes that is used in our Monte Carlo study. We generated a number of datasets to conduct the evaluation of the clustering methods cw and hf that we compare. We will examine the performance of these methods to determine success rates in detecting club membership for various parameter configurations including the number of countries, club size, time span and number of clubs. The analysis is carried out for two separate cases. In the first case we analyze single club data, while in the second case we include multiple clubs. Below we present the data generating processes and evaluation procedures employed in this study. A similar design was used by Beylunioğlu et al (2018) in assessing the properties of the maximum clique method, an alternative clustering mechanism that relies on Augmented Dickey Fuller (adf) unit root testing. However, the maximum clique method is a “top down” method that leads to a different definition of clubs and is not considered in the present comparison.21 Data Generating Processes The simulation assumes that the log gdp series for region i is given by yit = αi + di rt + ǫit , (16) where ǫit ∼ I(0) is the error term and rt is the common factor which affects all countries the same way (such as technology). If we assume non-stationarity of the factor, a pair of countries can only be convergent if the country specific constants, di , 21 Another method developed by Phillips and Sul (2007) stands out by means of not requiring a priori classification of countries. However, we exclude this method for the reason that it is based on the notion of σ convergence. The method depends on the definition of convergence by means of reduction of variance over time and thus convergence of series to a steady state. Therefore, it is not appropriate to compare this method with HF and the CW method developed in this study as both of the latter deal with convergence of the mean (function) of the series. 14 4.1 Monte Carlo Structure 4 A MONTE CARLO STUDY which measure the impact of common factor are equal. In other words, for the pair i and j, if i = dj , rt is canceled out and yit − yjt becomes αi − αj + ǫit − ǫjt . In this case, since the error terms are assumed to be stationary, we have αi − αj + ǫit − ǫjt ∼ I(0) and the pair i and j would be convergent by definition. Likewise, for any subset of countries having equal di , all pairwise difference series in that subset would be stationary and hence these countries would constitute a convergence club. Finally, the constants, αi are country specific and are generated once for all data sets. The non-stationarity of rt is modeled using the arima process rt = rt−1 + vt , vt = ρv vt−1 + et , et ∼ iid N (0, 1 − ρ2v ), (17) where we allow ρv = {0.2, 0.6} as separate cases. In addition we also allow the error term of the log gdp series in equation (16) to have serial dependence, following the specification ǫit = ρi ǫi,t−1 + vit , vit ∼ iid N (0, σv2i (1 − ρ2i )), (18) where we assume that the error terms vit are i.i.d. distributed Normal random variables. The autoregressive coefficient ρi and σv2i are country specific and invariant among the single and multiple clubs datasets. We generated the coefficients to have the following property. 2 σvi ∼ iid U [0.5, 1.5], ρi ∼ iid U [0.2, 0.6] To generate a dataset containing a single club the coefficients of the m convergent countries are assumed to be di = dj = 1. For the remaining (N − m) countries, di is generated randomly as di ∼ iid Xm2 . Similarly, we also generate country specific constants as αi ∼ iid Xm2 . For multiple clubs, in order to assess successful detection in club membership we want to make sure that there are some non-convergent countries present in the data that do not belong to any club. In that case, the value m of club size, when the number of clubs (k) and the number of countries (N ) are given, is chosen in such a way as to allow for at least a pair of non-convergent countries to be present in order to evaluate successful converging behaviour. For a given k and N , the clubs sizes m’s are randomly drawn from a Poisson distribution with a rate of λ = N/k. For each N , random draws are repeated k times.22 The simulations are repeated 2000 times using different combinations of T = 22 Obviously we did not allow the sum of m to exceed N , if this happens we redraw the last club size. 15 4.1 Monte Carlo Structure 4 A MONTE CARLO STUDY {50, 100} time intervals, N = {10, 20, 30} count of countries, m = {3, 5, 7, 10} number of club members for the single club case. In the multiple club case we considered T = {50, 100}, N = 10, m = {3, 5} and k = {2, 3} number of clubs.23 4.1.1 Testing and Evaluating Procedures To evaluate convergence we utilise evaluation tools from the literature on forecasting. The first one is the Kupiers Score (ks), while the second one is the Pesaran and Timmermann (1992) (pt) test statistic commonly used in the forecasting times series literature for the evaluation of sign forecasts. It is worth noting that sign forecasts are used for predicting whether an underlying series would increase relative to a benchmark such as, for example, a zero return threshold. This test is cast in terms of a binary process where success is the increase relative to the chosen benchmark. In our case we take a “success” as the correct detection of a country’s membership in a club. In the context of forecasting, this is equivalent to success in forecasting the sign of a time series. Since granting membership into a club or denying it can occur randomly,24 ks takes the correct forecasts and false alarms into account separately. ks is defined as H − F where OI II , and F = . H= II + IO OI + OO I (O) are binary indicators indicating whether the country under investigation is a member (not a member) of a given club. In considering the pairs of letters, the first letter indicates whether the country is found to be a member in the Monte Carlo experiment, while the second letter denotes its actual membership state (i.e. whether the country is actually in the club or not). II then indicates that a country as a member of the club is correctly identified; OO denotes that a country is correctly identified as not a member of the club. Furthermore, IO indicates that a country is detected to be a member of the club, while actually it is not (false detection). OI refers to the reverse case where the country is misclassified as being outside, even though it is a member of the club (false alarm). The ratio H captures the rate of “correct hits” in detecting club membership, whereas F denotes the “false alarm” rate, that is the rate of false exclusions. As in the case of sign prediction in the forecasting literature, success can be the outcome of a pure chance probability event of 0.5. Hence, to test the statistical 23 The computational burden for larger values of m, k and N proved to be be quite high at this point. 24 This is similar to expecting an unbiased coin to come up heads with 50% probability. 16 4.2 Simulation Results 4 A MONTE CARLO STUDY significance of ks, we will employ the following pt statistic PT = c∗ Pb − P ∼ N (0, 1). c∗ )] 21 [Vb (Pb) − Vb (P Pb refers to the proportion of correct predictions (correct detections of countries as c∗ denotes being a member or non member) over all predictions (N countries), and P the proportion of correct detections under the hypothesis that the detections and actual occurrences are independent (where success is a random event of probability c∗ ) stand for the variances of Pb and P c∗ respectively. 0.5). Vb (Pb) and Vb (P In simulations involving multiple clubs, it is not possible to use either the ks or the pt statistic given that the success/failure classification is no longer binary - as in the case of the single club case. In the multiple club case there are more than two distinct cases for the actual membership state: the country can be either a member of the correct club, belong to the “wrong” club, or not be a member of any club. To confront this problem, in the case of multiple clubs we utilise a much stricter criterion by counting the successful cases in our simulations in which all countries are detected correctly. We do not evaluate success as a binary outcome, country by country as in the case of a single club in each replication, but we only count as success having all countries satisfying the convergence condition. This is a much stricter criterion given that success depends on the overall results in each replication in which all countries are detected correctly. 4.2 Simulation Results Below we discuss the findings of the simulations based on the data generating processes of club formation. The comparison involves the bootstrap version of the kpss test (henceforth cw) proposed in this paper, and the original hf test based on the asymptotic version of the multivariate kpss test. 4.2.1 Single Club Results The results are presented in Table 1 for 0.05 and 0.10 significance levels.25 The total number of countries N are set at (N = 10, 20, 30); there are two choices of time span (T = 50, 100) that mimic the real data time span availability; and two choices of the persistence parameter (ρv = {0.2, 0.6}). It is expected that as the number of 25 We also have the results for the 0.01 significance level but to conserve space we do not report them. They are available from the authors on request. 17 4.2 Simulation Results 4 A MONTE CARLO STUDY countries N and club size m increase, the likelihood of an incorrect classification will also increase, but the opposite will be the case for an increase of the time span T for given N and m. As seen in Table 1, cw outperforms hf in all categories. For example, with m = 3, N = 10, T = 50 and ρ = 0.2 (configuration A), and significance levels 0.05 and 0.10, the ks results (the “correct hit” ratio net of “false alarms”) for cw are, respectively, 0.87 and 0.89. The comparable numbers for hf are 0.63 and 0.66. Similarly, for the cases with m = 10, N = 30, T = 100 and ρ = 0.6 (configuration B), cw with 0.77 and 0.76 outperforms the hf method - 0.60 and 0.60. The results are in line with our prior expectations that larger N and m values would result in lower success rates. However, in all cases the cw test does better. The pt statistics26 for configuration A yield values 1.91 and 2.02 for hf and 2.51 and 2.49 for cw; for configuration B the hf values are 2.86 and 2.97; with 3.37 and 3.48 for cw. Note that the rejections of the null hypothesis of random success outcomes are higher with the pt test for cw in all cases. The results clearly demonstrate that the cw, bootstrap kpss test offers a significant improvement over the hf procedure. 4.2.2 Multiple Club Results The results for the multiple club case are presented in Table 2. The multiple clubs cases involve classifications with k = 2 and 3 and N = 10 for T = {50, 100} and ρv = {0.2, 0.6}. The club sizes associated with each club are listed in the second column of Table 2 for each k. For example, the entry 4, 4 for club size m refers to two clubs of equal size 4, for k = 2. In the case of k = 3, m enters as 3, 3, 2, that is two clubs of size 3 and one club of size 2. The cw test outperforms hf in the majority of cases. For example, with N = 10, k = 2, T = 100 and ρv = 0.6, cw detects 54%, 45% and 37% correct classifications at the 0.01, 0.05 and 0.10 significance levels; hf does that with frequency 44%, 38% and 34.80% respectively. For the case when k = 2 and T = 50, hf does slightly better than cw, but when k = 3 the performance of hf deteriorates rapidly. In that case when N = 10, k = 3, T = 100 and ρv = 0.2, cw detects 27%, 25% and 22% correct classifications, while the comparable results for hf, respectively, are 10.40%, 8.20% and 5.60%. Since we have adopted a much stricter criterion where success is defined as all countries detected correctly, we do expect lower rates of correct detection than was the case for single clubs. In all cases, we see an improvement for cw when the number of clubs increases even when T is relatively small, but not for hf. 26 The PT statistic follows an asymptotic standard normal variate. 18 5 APPLICATIONS Overall, the multiple club results suggest that in terms of accuracy the cw does better in detecting the presence of clubs or clusters of countries. This gives us confidence that applying the above method to real data can provide us with useful insights about how countries over time collect themselves into different club formations of similar characteristics as far as economic activity is concerned. 5 Applications As shown in the Monte Carlo study a problem with the asymptotic test is that it does not permit reliable inference with only 30 years of data. Using an asymptotic test of the null of stationarity (convergence) tends to distort club membership detection due to size distortions which are ameliorated when we implement the bootstrap. That is, the test is oversized resulting in a tendency to reject the null hypothesis of convergence. In this section we assess the extent to which a size distortion affects our inference on the degree of convergence using two real-world datasets. We first compare the results of the asymptotic and bootstrap tests using the cross-country dataset originally adopted by Hobijn and Franses (2000). We then utilise data gathered at a finer geographical and sectoral scale by making the same comparison based upon the European regional dataset used by Corrado, Martin and Weeks (2005). We expect to resolve the size distortion which afflicts the asymptotic test and to find more evidence of convergence than what originally acknowledged by Hobijn and Franses (2000). 5.1 Cross-Country Convergence In this section we compare the asymptotic and the bootstrap results using the Hobijn and Franses (2000) dataset for the period 1960-1989 which comprises 112 countries from the Penn World Table listed in Table 3. Focussing upon log per capita gdp, the results based upon the asymptotic test have a striking feature, namely a very large number of convergence clubs. In particular, Hobijn and Franses (2000) find 63 asymptotically perfect convergence clubs and 42 asymptotically relative27 convergence clubs.28 As Table 4 shows, in the case of perfect convergence the lack of convergence is manifest in 29 singletons and 22 two-country clusters. A similar result can be observed in the case of relative convergence where Hobijn and Franses (2000) find a large number of two and three-country clusters. The lack of convergence is also 27 Note that perfect convergence implies convergence to identical log real GDP per capita levels. Relative convergence implies convergence to constant relative real GDP per capita levels. 28 These are the results using pmin = 0.01 and L = 2 as presented in Tables BII and BIII of the Hobijn and Franses (2000) paper. 19 5.1 Cross-Country Convergence 5 APPLICATIONS evident in the fact there are no clusters of size six or more for asymptotic perfect convergence and only one club of size six for relative convergence. We therefore implement the bootstrap version of the test on the same dataset and find a significant increases in the extent of convergence. Specifically, in the perfect convergence case and relative to the asymptotic results, we observe a 57% reduction in the number of convergence clubs (from 63 to 27); for relative convergence the reduction is 38% (from 42 to 26). In other words, there is evidence towards finding more convergence. Looking at the change in the distribution of cluster sizes for perfect convergence, we observe a dramatic reduction (by 96%) in the number of singletons (from 29 to 1) and by 81% in the number of two-country clusters (from 22 to 4). Commensurate with this finding, we note that countries are now clustering at a larger scale with two clubs having up to seven countries and with a substantial increase in the number of clusters containing five and six countries. A similar increase in the degree of convergence can be observed in the case of relative convergence. Tables 5 and 6 report the asymptotic and bootstrap cluster composition for relative convergence. A number of noteworthy observations can be made. We confirm the findings of Hobijn and Franses that convergence is more widespread among low income economies, and in particular Sub-Saharan Africa. Similarly we find that in general low income countries do not converge to high income. The two exceptions to this found in the asymptotic results, namely Kenya and Ecuador forming clubs alongside Australia and Denmark and Canada, are not found in the bootstrap results. In contrast to Hobijn and Franses we do find a significantly higher degree of convergence, both amongst low income and high income countries. The results based on the asymptotic test indicate very little convergence for the richer economies with all groups of size two. The bootstrap test locates a greater degree of convergence, for example, cluster 11 (Germany, Denmark, France, Luxemburg and New Zealand) and cluster 17 (Belgium, Great Britain, Netherlands and Norway). In the next section we apply the asymptotic and bootstrap version of the test to the European regional dataset originally used by Corrado, Martin and Weeks (2005). A critical difference with respect to the analysis undertaken at the aggregate country level, is that we allow for the possibility that convergence is more prevalent at a sector-specific level, and in addition consider a smaller geographical unit. Much of the theory of convergence highlights the potential role of technology spillovers as one of the possible drivers of convergence. As a result, in what follows we move away from an aggregate analysis to considering how convergence differs across agricultural, manufacturing and service sectors. 20 5.2 European Regional Convergence 5.2 5 APPLICATIONS European Regional Convergence In the following sections we examine the extent of regional convergence within the eu. Regional convergence – or what the European Commission calls ‘regional cohesion’ – is a primary policy objective, and is seen as vital to the success of key policy objectives, such as the single market, monetary union, eu competitiveness, and enlargement (European Commission, 2004). As a result, the theory of and evidence on long-run trends in regional per capita incomes and output are of critical relevance to the eu regional convergence and regional policy debate (Boldrin and Canova, 2001). Indeed, according to Fujita et al. (1999), the implications of increasing economic integration for the eu regions has been one of the factors behind the development of the ‘new economic geography’ models of regional growth. To date, however, very few of these models have been tested empirically on eu evidence. In response to the policy and research questions outlined above our empirical analysis will be framed around the identification of regional convergence clubs in the eu. To identify regional convergence clusters we use the method introduced by Hobijn and Franses (2000) which allows for the endogenous identification of the number and membership of regional convergence clusters (or ‘clubs’) and compare the results of the bootstrap and asymptotic versions of the test to assess the differences in terms of number, size and composition of the resultant clusters. 5.2.1 Data The so-called Nomenclature of Statistical Territorial Units (nuts) subdivides the economic territory of the 15 countries of the European Union using three regional and two local levels. The three regional levels are: nuts3, consisting of 1031 regions; nuts2, consisting of 206 regions; and nuts1 consisting of 77 regions. nuts0 represents the delineation at the national level and comprises France, Italy, Spain, uk, Ireland, Austria, Netherlands, Belgium, Luxemburg, Sweden, Norway, Portugal, Greece, Finland, Denmark and West Germany. We are aware of the problems that surround the choice of which spatial units to use.29 For example, many of the regional units used by eurostat have net inflows of commuters and in addition, these regions also tend to be those with the highest per capita income. Boldrin and Canova (2001) criticize the European Commission for utilizing inappropriate regional units. Whereas nuts1, nuts2 and nuts3 regions are neither uniformly large or sufficiently heterogeneous such that a finding of income divergence across regions cannot unequivocally be taken as evidence for the existence of an endogenous 29 Chesire and Magrini (2000) provide a useful discussion of these issues, focussing on the importance of centering the analysis on regions that are self-contained in labour market terms. 21 5.2 European Regional Convergence 5 APPLICATIONS cumulative growth processes. In fact, the smaller the geographical scale, the more incomplete and fragmented is the statistical information we can get. Although we do not wish to detract from the importance of these matters, in this study our primary focus is a comparison of two different tests for regional convergence for which the unit of analysis is the same. In conducting our analysis we choose to focus on nuts1 regions, achieving a compromise between the availability of reliable data at a regional level which is sufficiently homogeneous, and the need to move beyond national borders. The complete list of nuts1 regions30 used in this study is given in Table 7. We use regional data on Gross Value Added31 per worker for the period 1975 to 1999 for the agriculture, manufacturing and services sectors. Although data are available for more recent years, we focus on this particular time frame to facilitate a comparison with the results of Corrado, Martin and Weeks (2005). The service sector has been further sub-divided into market and non-market services: market services comprise distribution, retail, banking, and consultancy; non-market services comprise education, health and social work, defence and other government services. 5.2.2 Results In this section we present the main results of our analysis. Given the large number of eu regions in Figures 1 and 2 we first present the results for the asymptotic and bootstrap test of convergence in mapped rather than tabular form. Table 8 summarises this information in terms of the number and size of the convergence clubs and group characteristics, such as average per-capita income. 5.2.3 Graphing Convergence Clusters In Figures 1 and 2 clusters which contain the largest number of member regions are indicated with a darker shade on each map. Regions which belong to two-region clusters or do not cluster with any other region have no shading. In the key to the maps, the first number indicates the cluster size and the second letter denotes the cluster identifier. In Figure 1 maps a) and b) ( c) and d)) present the asymptotic and 30 For Portugal, Luxemburg and Ireland, data are only available at the nuts0 level. For Norway we have no data at the nuts1 level. Time series data for the sample period considered are not available for East Germany, which is therefore excluded from the analysis. 31 gva has the comparative advantage with respect to gdp per capita of being the direct outcome of various factors that determine regional competitiveness. Regional data on gva per-capita at the nuts1 level for agriculture, manufacturing, market and non-market services, have been kindly supplied by Cambridge Econometrics, and are taken from their European Regional Database. All series have been converted to constant 1985 prices (ecu) using the purchasing power parity exchange rate. 22 5.2 European Regional Convergence 5 APPLICATIONS bootstrap generated outcomes for agriculture (manufacturing). The relative pattern of convergence corroborates with our prior expectations, namely that the bootstrap test is obviously rejecting the stationary null with a lower frequency and thereby locating more evidence for convergence. In Figure 2 we find a similar pattern for market and non-market services. In Table 8 we present the frequency distribution of the cluster size for both bootstrap and asymptotic tests and for each32 economic sector. Row totals provide an indication of the degree of convergence for each economic sector. Column totals provide information on the number of convergence clubs across sectors by cluster size. The asymptotic results are displayed in panel I and the bootstrap results are displayed in panel II. Overall, we observe a common pattern, namely a shift in the probability distribution towards a fewer number of clusters of larger size, and a commensurate increase in the extent of regional convergence. The total number of clusters for the asymptotic tests is 81, which falls by 32% to 55 clusters for the bootstrap test. This pattern is repeated for all sectors. Comparing column totals across the two tests is also informative since it gives the total number of clusters by cluster size, also shown in Figure 3. For the asymptotic test, more than 80% of the probability mass is distributed in clusters of size 4 or less, with approximately 10% of clusters of size 6 or more. In contrast, for the bootstrap test, approximately 50% of the clusters have a cluster size of 4 or less, with approximately 40% of clusters of size 6 or more. Examining the results for each sector, for agriculture the size of the largest cluster generated by bootstrap critical values increases from seven to ten regions, with a commensurate decrease in the number of clusters of size 5 or less. Similarly for the manufacturing sector we observe an increase in the size of the largest cluster from six to nine regions and a decrease in the number of clusters of size 4 or less. In the market-service sector there is a reduction in the size of the largest cluster from nine to eight and for non-market services there is no change in the size of the largest cluster, but a substantial increase in clustering at the medium scale. In both service sectors there is a decrease in the number of clusters of size 4 or less. Cluster Composition In establishing whether the composition of the clusters (i.e. the constituent regions) is changing between the two tests, we first collect the asymptotic (A) generated cluster outcomes in a N × N matrix MA = {mA ij }; element A mij equals to 1 if regions i and j belong to the same cluster and zero otherwise. MB = {mB ij } denotes the same for the bootstrap (B) generated cluster outcomes. The correlation parameter between the asymptotic, MA , and the bootstrap cluster 32 In order to directly compare the bootstrap and asymptotic results in Corrado et al. (2005) we set pmin to be equal to 0.01 and the bandwidth L = 2. The number of bootstrap samples is set at 200. 23 5.2 European Regional Convergence 5 APPLICATIONS pattern, MB , is then given by  1/2 N P N P A   mB ij × mij   i=1 j6=i   ζl =   ! ! 1/2 1/2   P N P N N P N P   B A mij mij i=1 j6=i , (19) i=1 j6=i where l indexes the set {Agriculture, Manufacturing, Market Services, Non-Market Services}. The results are reported in panel III of Table 8. With correlation coefficients ranging between 50% for manufacturing and 67% for agriculture, we note further evidence of a significant difference in the composition of the clusters generated by the asymptotic and bootstrap tests.33 Mean Income In order to assess the properties of each cluster we compute mean log per-capita income,34 ȳg for each test. The top panel of Figure 4 shows that the asymptotic test generates a distribution with a large number of small clubs while in the bootstrap test there are a fewer number of clusters of larger size. A visual impression of the oversized property of the asymptotic test of convergence is also evident in the distribution of the cluster mean of log per-capita income and in a relatively higher right kurtosis of this distribution, as presented in the lower panel of Figure 4. In this case an overrejection of the convergent null generates a distribution with a large number of small clubs characterised by a higher mean log per-capita income which results in a widening of the gap between the poorest and the richest clusters. In examining the comparable bootstrap distribution we observe a marked decrease in right kurtosis and a commensurate narrowing of the gap between the richest and the poorest cluster. Summary statistics are provided in the last three columns of panels I and II of Table 8. Note that for the bootstrap distribution the reduction in the gap between the richest and the poorest clusters is evident in a lower standard deviation of mean cluster per-capita income (from 15.2 to 5.4). The narrowing of the gap between the richest and poorest cluster translates into an increase in mean log per-capita income of the poorest cluster, ȳmin , by around 24% (from 9.4 to 11.7) and a decrease in mean log per-capita income of the richest cluster, ȳmax , by almost 50% (from 103 to 62.6). These results demonstrate the importance of the correct identification of convergence clubs. Given that many policy instruments are designed 33 The method used in this paper to locate convergence clubs bypasses the particular problem of exactly how to utilize conditioning information in the model specification. Corrado and Weeks (2011) provide further information on how to interpret the results by confronting the resulting cluster composition, for both the asymptotic and the bootstrap tests, with a set of hypothetical clusters based on different theories and models of regional growth and development. 34 Mean income is the cluster mean of log per-capita GVA. 24 6 CONCLUSIONS to reduce the gap between the richest and the poorest regions, basing inference and policy decisions on the results of the asymptotic test would indicate the need for a stronger action than is actually needed when looking at the bootstrap test outcomes. 6 Conclusions This study represents an extension of the multivariate test of stationarity which allows for endogenous identification of the number and composition of regional convergence clusters using sequential pairwise tests for stationarity. The main drawback of this approach is the short time-horizon which affects the size of the test. Our proposed bootstrap based extension to the sequential pairwise multivariate tests for stationarity performs well in Monte Carlo simulations in identifying and detecting correctly cluster membership when compared with the asymptotic version of the Hobijn and Franses (2000) approach. Based upon Monte Carlo evidence comparing the performance of cw with hf varying the number of countries, data span, club size and degree of persistence, indicate that detection rates of club membership (net of misclassifications) improve considerably when we implement the bootstrap. In operationalizing a bootstrap test of multivariate stationarity our results confirm the oversized property of the asymptotic test, and reveal a significantly greater degree of convergence. This evidence is gathered using both cross-country and regional data for the European Union for a number of industrial sectors. 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(2003). ‘Provincial Conditional Income Convergence in China, 1953–1997: A Panel Data Approach,’ Econometric Reviews, 22(1), pp. 59-77. 30 Table 1: Single Clubs Results DataType N m ρ T 0.2 3 10 0.6 0.2 5 0.6 0.2 3 0.6 0.2 5 20 0.6 0.2 7 0.6 0.2 10 0.6 0.2 3 0.6 0.2 5 30 0.6 0.2 7 0.6 0.2 10 0.6 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 50 100 KS 0.01 0.60 0.73 0.69 0.73 0.61 0.69 0.67 0.71 0.47 0.60 0.58 0.65 0.60 0.67 0.63 0.71 0.61 0.64 0.63 0.68 0.52 0.61 0.56 0.65 0.54 0.59 0.60 0.65 0.54 0.59 0.56 0.64 0.48 0.60 0.52 0.63 0.43 0.53 0.49 0.61 HF 0.05 0.63 0.78 0.71 0.81 0.61 0.76 0.67 0.76 0.47 0.58 0.60 0.69 0.58 0.71 0.63 0.70 0.60 0.66 0.61 0.69 0.48 0.64 0.50 0.62 0.49 0.62 0.56 0.67 0.52 0.61 0.54 0.67 0.46 0.57 0.53 0.64 0.41 0.52 0.42 0.60 0.10 0.66 0.83 0.74 0.82 0.64 0.73 0.67 0.77 0.57 0.63 0.64 0.73 0.58 0.71 0.66 0.71 0.58 0.66 0.62 0.68 0.47 0.64 0.52 0.62 0.56 0.65 0.61 0.72 0.50 0.63 0.57 0.66 0.44 0.57 0.54 0.62 0.41 0.53 0.43 0.60 0.01 0.85 0.76 0.87 0.77 0.87 0.79 0.89 0.80 0.66 0.54 0.72 0.62 0.87 0.80 0.92 0.81 0.90 0.88 0.91 0.88 0.73 0.81 0.76 0.82 0.76 0.72 0.80 0.73 0.81 0.76 0.85 0.79 0.77 0.82 0.84 0.85 0.67 0.74 0.72 0.78 PT CW 0.05 0.87 0.79 0.88 0.81 0.88 0.82 0.89 0.83 0.74 0.61 0.81 0.72 0.88 0.82 0.91 0.82 0.87 0.88 0.88 0.88 0.72 0.79 0.74 0.80 0.80 0.76 0.84 0.76 0.82 0.77 0.85 0.79 0.76 0.82 0.82 0.85 0.67 0.73 0.71 0.77 31 0.10 0.89 0.81 0.91 0.81 0.87 0.82 0.88 0.84 0.73 0.65 0.81 0.72 0.87 0.83 0.90 0.84 0.86 0.88 0.86 0.89 0.70 0.79 0.74 0.79 0.78 0.77 0.81 0.77 0.78 0.76 0.85 0.80 0.73 0.81 0.80 0.85 0.65 0.73 0.69 0.76 0.01 1.67 1.96 2.04 2.02 1.68 2.10 2.15 2.21 1.60 2.39 1.82 2.64 2.74 2.92 2.34 3.05 2.78 3.03 2.13 3.22 2.42 3.09 2.24 3.08 1.76 2.11 2.01 2.13 1.93 2.22 2.13 2.29 1.67 2.06 2.09 2.22 2.46 2.68 2.63 2.85 HF 0.05 1.91 2.16 2.17 2.42 2.00 2.18 2.25 2.40 2.18 2.46 1.98 2.83 2.88 2.99 2.56 3.39 2.80 2.90 2.67 3.08 2.48 3.06 2.18 3.09 1.85 2.33 2.13 2.43 1.93 2.40 2.13 2.40 1.71 2.09 2.21 2.49 2.44 2.94 2.67 2.86 0.10 2.02 2.34 2.40 2.52 1.95 2.31 2.21 2.40 2.43 2.71 2.35 2.91 2.94 3.04 2.55 3.17 2.74 3.09 2.68 3.29 2.37 3.01 2.28 3.02 1.97 2.56 2.23 2.55 2.03 2.31 2.14 2.43 2.16 2.29 2.41 2.75 2.49 3.00 2.89 2.97 0.01 2.49 2.26 2.51 2.29 2.68 2.62 2.75 2.62 2.87 2.09 3.14 2.13 3.84 3.40 3.91 3.48 3.98 4.03 4.06 3.91 3.28 3.83 3.36 3.77 2.56 2.21 2.62 2.25 2.77 2.55 2.83 2.57 2.62 1.94 2.94 2.24 3.76 3.24 3.98 3.31 CW 0.05 2.51 2.28 2.58 2.33 2.73 2.68 2.76 2.63 3.08 2.39 3.46 2.57 3.86 3.56 3.89 3.37 3.79 4.01 3.90 3.85 3.25 3.70 3.33 3.62 2.65 2.31 2.69 2.38 2.77 2.62 2.83 2.66 2.98 2.20 3.32 2.62 3.82 3.34 3.98 3.37 0.10 2.72 2.49 2.64 2.29 2.69 2.69 2.73 2.69 3.53 2.98 3.51 3.21 3.82 3.75 3.72 3.62 3.79 4.11 3.83 3.94 3.24 3.64 3.36 3.61 2.71 2.38 2.77 2.37 2.75 2.62 2.79 2.67 2.97 2.35 3.32 2.65 3.80 3.40 3.94 3.48 Table 2: Multiple Club Results N Data Type m k ρ 0.2 4,4 2 10 0.6 0.2 3,3,2 3 0.6 T 50 100 50 100 50 100 50 100 0.01 34.20% 51.80% 38.80% 44.00% 1.20% 10.40% 13.00% 3.20% HF 0.05 28.00% 39.00% 26.60% 38.00% 1.00% 8.20% 14.00% 2.60% 32 0.1 17.00% 33.00% 19.00% 34.80% 1.00% 5.60% 10.00% 1.00% 0.01 28.00% 61.20% 35.00% 54.00% 15.60% 27.00% 13.00% 20.00% CW 0.05 22.00% 45.00% 24.00% 45.00% 15.60% 25.00% 15.60% 17.80% 0.1 14.40% 40.00% 17.00% 37.00% 14.40% 22.00% 14.40% 15.00% Table 3: List of Countries (PWT) AGO ARG AUS AUT BDI BEL BEN BGD BOL BRA BRB BUR BWA CAF CAN CHE CHL CIV CMR COG COL CPV CRI CSK CYP DEU DNK DOM DZA ECU EGY ESP FIN FJI FRA GAB GBR GHA Country Angola Argentina Australia Austria Burundi Belgium Benin Bangladesh Bolivia Brazil Barbados Myanmar Botswana Central African Rep. Canada Switzerland Chile Ivory Coast Cameroon Congo Colombia Cape Verde Is. Costa Rica Czechoslovakia Cyprus West Germany Denmark Dominican Rep. Algeria Ecuador Egypt Spain Finland Fiji France Gabon United Kingdom Ghana GIN GMB GNB GRC GTM GUY HKG HND HTI HVO IDN IND IRL IRN ISL ISR ITA JAM JOR JPN KEN KOR LKA LSO LUX MAR MDG MEX MLI MLT MOZ MRT MUS MWI MYS NAM NER NGA Country Guinea Gambia Guinea Bissau Greece Guatemala Guyana Hong Kong Honduras Haiti Burkina Faso Indonesia India Ireland Iran Iceland Israel Italy Jamaica Jordan Japan Kenya Korea Sri Lanka Lesotho Luxembourg Morocco Madagascar Mexico Mali Malta Mozambique Mauritania Mauritius Malawi Malaysia Namibia Niger Nigeria 33 NLD NOR NZL PAK PAN PER PHL PNG PRI PRT PRY RWA SEN SGP SLV SOM SUR SWE SWZ SYC SYR TCD TGO THA TTO TUN TUR UGA URY USA VEN YUG ZAF ZAR ZMB ZWE Country Netherlands Norway New Zealand Pakistan Panama Peru Phillipines Papua N. Guinea Puerto Rico Portugal Paraguay Rwanda Senegal Singapore El Salvador Somalia Suriname Sweden Swaziland Seychelles Syria Tcad Togo Thailand Trinidad/Tobago Tunisia Turkey Uganda Uruguay United States Venezuela Former Yugoslavia South Africa Zaire Zambia Zimbabwe Table 4: Joint Frequency Distribution (PWT) I: Asymptotic Number of Clusters Cluster size Perfect Relative 1 2 3 4 5 6 7 29 2 22 21 9 12 3 4 0 2 0 1 0 0 Total Clusters 63 42 II: Bootstrap Number of Clusters Cluster size 1 2 3 4 5 6 7 Perfect Relative 1 1 4 3 4 4 8 7 3 3 5 6 2 2 34 Total Clusters 27 26 Table 5: Asymptotic: Relative Convergence (PWT) No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34 36 38 40 41 42 Countries AUS DNK KEN LUX MUS AUT ESP ISR ITA PRI CAN CSK ECU GRC IRL BDI HVO MLI MWI BGD BUR HND NZL CAF IND NER UGA GUY JOR SLV SYC AGO GHA HTI BEN GIN VEN BOL LKA PNG BRB IDN THA CIV COG MAR CMR CRI NGA CPV GNB RWA FIN ISL TTO FJI NAM PER IRN PRT YUG MRT PAK SOM MYS SWZ TUR Clusters with two countries ARG GMB 21 BEL NOR BRA SUR 23 BWA MLT CHE USA 25 CHL GAB COL JAM 27 CYP SGP DEU FRA 29 DOM SWE DZA GTM 31 EGY ZWE GBR NLD 33 HKG KOR LSO TGO 35 MDG ZMB MEX URY 37 MOZ SEN PAN SYR 39 PRY TUN TCD ZAR Two separate countries JPN PHL Listed according to size. ZAF pmin = 0.01 and l = 2. 35 Table 6: Bootstrap: Relative Convergence (PWT) No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Countries COL JAM MYS NAM PAN AGO BGD CMR GHA MRT CHE CHL CSK GRC MEX GUY JOR PNG SLV SYC CRI GAB IRN MUS SUR CIV COG LKA MAR PHL AUT FIN ISL ITA JPN BDI BUR GIN HVO MLI LSO RWA TCD TGO ZAR CPV GMB GNB NER UGA DEU DNK FRA LUX NZL CYP PRT SGP YUG DZA ECU GTM SWZ BOL DOM KOR PRY BRA FJI MLT PER CAF IND KEN NGA BEL GBR NLD NOR BEN MDG ZMB ZWE AUS CAN SWE BRB HKG IRL BWA MOZ SEN ESP ISR PRI Clusters with two countries HTI IDN EGY HND ARG URY One separate country VEN SYR PAK USA TUN ZAF THA TTO MWI Listed according to size. pmin = 0.01 and l = 2. The number of bootstrap samples is set at 200. 36 TUR SOM Table 7: NUTS1 code Code Country AT AT1 AT2 AT3 BE BE1 Austria BE2 BE3 DE DE1 DE2 DE3 DE5 DE6 DE7 DE9 DEA DEB DEC DEG DK ES ES3 ES4 ES5 ES6 ES7 F1 FR FR1 FR2 FR3 FR4 FR5 FR6 FR7 FR8 GR GR1 GR2 GR3 GR4 Ostosterreich Sudosterreich Westosterreich Belgium Region Bruxelles-Capital-Brussels Hoofdstedelijke Gewest Vlaams Gewest Region Wallonne Germany Baden-Wurttemberg Bayern Berlin Bremen Hamburg Hessen Niedersachsen Nordrhein-Westfalen Rheinland-Pfalz Saarland Thuringen Code Country IE Ireland IT IT1 IT2 IT3 IT4 IT5 IT6 IT7 IT8 IT9 ITA ITB LU Italy NL NL1 NL2 NL3 NL4 PT PT1 SE Denmark Spain Comunidad de Madrid Centro Este Sur Canarias UK UKC UKD UKE Finland UKF UKG UKH UK1 UKJ UKK UKL UKM France Ile de France Bassin-Parisien Nord Pas de Calais Est Ouest Sud-Ouest Centre-Est Mediterranee Greece Voreia Ellada Kentriki Ellada Attiki Nisia Aigaiou, Kriti 37 Nord Ovest Lombardia Nord Est Emilia-Romagna Centro Lazio Abruzzo-Molise Campania Sud Sicilia Sardegna Luxembourg Netherlands Noord-Nederland Oost-Nederland West-Nederland Zuid-Nederland Portugal Continente Sweden United Kingdom North East North West Yorkshire and Humber East Midland West Midlands East of England London South East South West Wales Scotland Table 8: Joint Frequency Distribution I: Asymptotic Number of Clusters Cluster size 1 2 3 4 Agriculture Manufacturing Market Service Non-market Service 0 0 1 1 3 7 9 6 7 9 3 7 2 4 4 1 6 0 2 1 Total Clusters 2 25 26 14 5 6 Summary Statistics 7 8 9 10 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 0 0 6 3 3 1 1 0 7 8 9 10 Total Clusters 18 22 21 20 81 σȳ 15.2 ȳ min 9.4 ȳ max 103 σȳ 5.4 ȳ min 11.7 ȳ max 62.6 II: Bootstrap Number of Clusters Cluster size 1 2 3 4 Agriculture Manufacturing Market Services Non-market Services 0 0 0 0 3 2 1 1 1 5 3 3 1 1 1 2 2 2 2 3 Total Clusters 0 7 12 6 5 8 6 0 1 3 1 1 3 0 1 1 0 4 1 1 0 0 3 0 2 0 0 10 3 8 2 2 12 15 14 14 55 III Correlation Between Asymptotic and Bootstrap Cluster Outcomes Agriculture Manufacturing Market Services Non-Market Services NB: σȳ denotes the standard deviation of cluster means. 0.672 0.509 0.557 0.591 of cluster means. ȳ min and ȳ max denote the Min and Max 38 (a) Relative Convergence Asymptotic Results in Agriculture: (b) Relative Convergence Bootstrap Results (c) Relative Convergence in Manufacturing: Asymptotic Results in Agriculture: (d) Relative Convergence in Manufacturing: Bootstrap Results Figure 1: Asymptotic and Bootstrap Results for Agriculture and Manufacturing 39 (a) Relative Convergence in Market Services: Asymptotic Results (b) Relative Convergence in Market Services: Bootstrap Results (c) Relative Convergence in Services: Asymptotic Results (d) Relative Convergence Services: Bootstrap Results Non-Market in Non-Market Figure 2: Asymptotic and Bootstrap Results for Non-Market and Market Services 40 Figure 3: The Distribution of Cluster Size. Figure 4: The distribution of average log per-capita GVA by cluster: All sectors. Skewness (Asymptotic ) 1.29 (Bootstrap) 0.27 Kurtosis (Asymptotic ) 6.82 (Bootstrap) 2.20 41