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IW1.I.T. LIBRARIES - DEWEY Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/convergenceinintOObern working paper department of economics Convergence in International Output massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 Convergence in International Output Andrew Steven No. 93-7 B. N. Bernard Durlauf May 1993 AUG 1 2 1993' Convergence Andrew in International B. Bernard Steven N. Durlauf Department of Economics Department MA of Economics Stanford University M.I.T. Cambridge, Output 1 02139 Stanford, CA 94305 May, 1993 *We thank Suzanne Cooper, Chad Jones, and Paul Romer for useful discussions. The authors thank participants at the NBER Summer Workshop on Common Elements in Trends and Fluc- also and LSE, and 3 anonymous referees for helpful comments. Economic Policy Research provided financial support. Bernard gratefully acknowledges dissertation support from the Alfred P. Sloan Foundation. All errors are ours. tuations, seminars at Cambridge, Oxford, The Center for Summary This paper proposes and tests new definitions of convergence and We capita output. define convergence for a group of countries to common mean has identical long-run trends, either stochastic or deterministic, while for proportionality of the stochastic elements. These reject trends allow definitions lead naturally to the use of convergence but find substantial evidence for European countries that each country common cointegration techniques in testing. Using century-long time series for 15 we trends for per common also reject convergence but are driven OECD economies, trends. Smaller samples of by a lower number of common stochastic trends. 1 Introduction One of the most striking features of the neoclassical growth model is its implication for cross-country convergence. In standard formulations of the infinite- horizon optimal growth problem, various turnpike theorems show that steady-state per capita output dent of output initial levels. is indepen- Further, differences in microeconomic parameters will gener- ate stationary differences in per capita output and will not imply different growth rates. when one observes Consequently, differences in per capita output growth across countries, one must either assume that these countries have dramatically different microeconomic characteristics, such as different production functions or discount rates, or regard these discrepancies as transitory. Launched primarily by the theoretical work of Romer (1986) and Lucas (1988), much attention has been focused on the predictions of dynamic equilibrium models for long-term behavior when various Arrow-Debreu assumptions are relaxed. shown that divergence scale associated with in Lucas and Romer have long-term growth can be generated by social increasing returns to both physical and human capital. An empirical literature exploring convergence has developed in parallel to the new growth theory. Prominent contributions is the work of Baumol (1986), DeLong among (1988), Barro (1991) and these Mankiw, Romer, and Weil (1992). This research has interpreted a finding of a negative cross-section correlation between initial income The use and growth rates as evidence in favor of convergence. of cross-section results to infer the long-run behavior of national output ignores valuable information in the time series themselves, which can lead to a spurious finding of convergence. First, it is possible for a set of countries which are diverging to exhibit the sort of negative correlation described by is diminishing. Baumol et al. so long as the marginal product of capital As shown by Bernard and Durlauf (1992), a diminishing marginal product 1 of capital means that short-run be mixed up will dynamics and long-run, steady-state behavior transitional in cross-section regressions. Second, the cross-section procedures work with the null hypothesis that no countries are converging and the alternative hypothesis that all countries are, which leaves out a host of intermediate cases. In this paper Our research we propose a new and definition set of tests of the from most previous empirical work differs framework. explicitly stochastic If same across countries. we test convergence in an long-run technological progress contains a stochastic trend, or unit root, then convergence implies that the the in that convergence hypothesis. The theory permanent components output are in of cointegration provides a natural setting for testing cross-country relationships in permanent output movements. Our examines annual log analysis, which real output per capita for 15 OECD economies from 1900 to 1987, leads to two basic conclusions about international output fluctuations. 1 First, we find very little evidence of convergence across the economies. deviations do not appear to disappear systematically over time. Second, is common strong evidence of countries. As a result, Per capita output we find that there stochastic elements in long-run economic fluctuations across economic growth cannot be explained exclusively by idiosyncratic, country-specific factors. A relatively small set of common long-run factors interacts with individual country characteristics to determine growth rates. Our work Quah (1990) is related to studies by who have Campbell and Mankiw (1989), Cogley (1990), and explored patterns of persistence in international output. quarterly post- 1957 data, Campbell and exhibit both persistence and divergence Mankiw demonstrate in output. Cogley, examining using a similar data set to the one here, concludes that persistence countries; yet at the same time he argues that common OECD 9 OECD that 7 is Using economies economies substantial for many factors generating persistence imply Quah finds a lack of convergence for a wide range of countries on the basis of post- 1950 data. Our that "long run dynamics prevent output levels from diverging by too much." analysis differs from this previous work relationship between cointegration, to distinguish between common in three respects. common factors, First, we directly formulate the and convergence, which permits one sources of growth and convergence. Second, we attempt to determine whether there are subgroups of converging countries and thereby move beyond the all or nothing approach of previous authors. Third, we employ different econometric techniques and data sets which seem especially appropriate for the analysis of long-term 'The countries are: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Japan, the Netherlands, Norway, Sweden, the United Kingdom, and the United States. Italy, growth behavior. The spirit of our analysis has much in common (1993) and Lee, Pesaran, and Pierse (1992), the US and UK respectively. with work by Pesaran, Pierse and Lee who study multisector output persistence for These papers derive methods to measure the long run of a shock originating in one sector on all effects sectors in the economy. While the focus of that research has been on measuring persistence rather than convergence, a useful extension of the current paper would be the application of the multivariate persistence measures to international data to both provide additional tests of convergence as well as to provide a framework for measuring the sources of growth. The plan common of the paper is as follows: Section 2 provides definitions of convergence trends using a cointegration framework. Section 3 outlines the test statistics Section 4 describes the data. Section 5 contains the empirical results. use. and we The evidence from the cross-country analysis argues against the notion of convergence for the whole common stochastic work come from employing stochastic definitions sample. Alternatively there do appear to be groups of countries with elements. Convergence 2 The organizing for in stochastic principles of our empirical environments both long-term economic fluctuations and convergence. These definitions rely on the notions of unit roots and cointegration in time series. We model the individual output series as satisfying a(L)Yi<t = where a(L) has one root on the unit circle m4 and Equation 2.1: (2.1) £i,i £ 1){ is a mean zero stationary process. This formulation allows for both linear deterministic and stochastic trends in output. The interactions of both types of trends across countries can be formalized into general definitions of convergence and common trends. Definition 2.1. Convergence in per capita output Log per capita outputs in countries 1. Yi 2. /x, 3. Yi t t, • = t t, . converge if -,YPt t are cointegrated with a cointegrating matrix, is a p—\ Definition 2.2. Long-run 1. .,p Vi,j, 0' where Ip -\ . -lYpj satisfy Equation 2.1, fij . 1,. log identity matrix Common = [/„_!, and e v -\ /3', such that -e p _i] wop-lxl (2-2) vector of ones. treads in per capita output per capita outputs in countries l,...,p are determined by the individual output series, Yi it , . . -lYpj, satisfy Equation 2.1, common trends if = 2. m 3. yj.t, The the V fij . . , . first i,j, Yp j are cointegrated. definition gives us a formal definition of convergence. If countries are to attain same long-run growth rates with output levels separated only by a stationary difference, then they must satisfy Definition 2.1. Each series must contain the same time trend and be cointegrated with every other series with the cointegrating vector (1,-1). However, if a group of output series does not satisfy Definition 2.1, but instead satisfies the weaker Definition 2.2, then output levels will be cointegrated but the stochastic trends for the group across countries. sibility not be equal. will This is It will remain true that permanent shocks the natural definition to employ that there are a small number of we if be related will are interested in the pos- stochastic trends affecting output which differ in magnitude across countries. The role of linear deterministic trends in our analysis is straightforward. outputs contain linear trends, then long-run levels and growth rates those trends are identical across group require that output for Our al. all countries have the countries in our sample definition of convergence who have between all countries. all is Thus both convergence and common same well mean countries' be equal only will linear trend. In practice we Our if any that modeled by a stochastic trend. 2 that there is Baumol et a negative cross-section correlation income and growth, thereby inferring long-run output behavior from section behavior. in will find substantially different from that employed by defined convergence to initial is If cross- analysis studies convergence by directly examining the time series properties of various output series, which places the convergence hypothesis in an explicitly dynamic and stochastic environment. One potential difficulty with the use of unit root tests to identify convergence presence of a transitional component in the aggregate output of various countries. series tests moments assume that the data are generated by an invariant measure, of the data are interpretable as population moments the Time the sample for the underlying stochastic process. If the countries in our sample start at different initial conditions to, i.e. is and are converging but are not yet at a steady state output distribution, then the available data may be generated by a transitional law of motion rather than by an invariant stochastic process. Consequently, unit root tests may erroneously accept a no convergence null. Simulations in Bernard and Durlauf (1992) suggest that the Results are available size distortions are unlikely to upon request fiom the authors. be significant. Output relationships across countries 3 Econometric methodology 3.1 In order to test for convergence and common trends, we employ multivariate techniques developed by Phillips and Ouliaris (1988) and Johansen(1988) Let j/i tt denote the log per capita output output in country i from output of output deviations, The Z?j/,,t, 1, i.e. AY as the first and ADY the first of the individual output levels, t t work starting point for the empirical - j/i jt y,- >t . Wold t is r cointegrating vectors then Phillips A t then natural to write a It is C(l) is (3.1) . t the p output series are cointegrated in levels with if of rank p—r and number of linearly-independent there is a vector ARMA representation. stochastic trends has been developed by and Ouliaris (1988) who analyze the spectral density matrix second test vector representation of output as As shown by Engle and Granger (1987), the , vector the finding that the individual elements t first test for Z?y, if the deviation of differences of the deviations. AY = n + C(L)e A and t of the per capita output vector are integrated of order one. multivariate i Y is defined as the n x 1 difference of Y DY as the (n-l)xl country in country level of at the zero frequency. due to Johansen (1988, 1989) who estimates the rank of the cointegrating is matrix. For a vector of output series, convergence and common trends impose different restric- AY ,f^y(0)- on the zero frequency of the spectral density matrix of tions requires that the persistent parts be equal; common t Convergence trends require that the persistent parts of individual output series be proportional. In a multivariate framework, proportionality and equality of the persistent parts corresponds to as a condition if not of full rank. the number of distinct If all Vt, or equivalently, the have common which is formalized on the rank of the zero-frequency spectral density matrix. From Engle and Granger (1987), is linear dependence, stochastic trends in n countries are converging rank of /ady(0) is 0. On Y t in per capita is less = output, then /^dk(0),,, the other hand, persistent parts, the output deviations from a than n, then /Ay(0) if several output series benchmark country must all have zero- valued persistent components. Spectral-based procedures devised by Phillips and Ouliaris permit a test for complete convergence as well as the determination of the number of common trends for the 15 output series. The tests make use of the fact that the spectral density matrix of at the zero frequency will be of rank q < n where q is the first differences number of linearly-independent stochastic trends in the data and n rank If the zero frequency matrix eigenvalues will also be q test that We the number of series in the sample. This reduction captured in the eigenvalues of the zero frequency of the spectral density matrix. in is is < The n. examines the smallest than less is m full rank, q particular Phillips-Ouliaris test = n—q eigenvalues to determine use two critical values for the bounds test, C\ m values assess the average of the the eigenvalues. critical value, The first critical the number we employ if of positive is a bounds they are close to zero. = 0.10^ and Ci = These 0.05. critical smallest eigenvalues in comparison to the average of value, Cj, 5% C?, corresponds to < n then is m all x 10% of the average root. The second of the total variance. For the Johansen tests we impose some additional structure on the output assume that a finite- vector autoregressive representation exists We series. and rewrite the output vector process as, £Y = t T(L)&Yt + nYt-i+ii + et (3.2) where = -(A t+1 + ...-A k ), T, (t =!,...,*-!), and ll II = -(I-A l -...-A k ). represents the long-run relationship of the individual output series, while T(L) traces out the short-run impact of shocks to the system. relationships, matrix, II, and thus all must be stationary different choice of = a in Equation a(3' p. 3.2. /? (3.3) is t II is < If the rank of r < II is not uniquely determined; a II II is equals p, then Yt still is related to the number of a stationary process. If the p, there are r cointegrating vectors for the individual series and hence the group of time rank of /? satisfying Equation 3.3 will produce a different cointegrating matrix. cointegrating vectors. the rank of the matrix of cointegrating vectors, However, Regardless of the normalization chosen, the rank of Y come from the which can be written as with q and 0, p x r matrices of rank r < in are interested only in the long-run the tests and estimates of cointegrating vectors II as /3'Yt-k We series is being driven by p—r common shocks. If the equals zero, there are p stochastic trends and the long-run output levels are not related across countries. In particular, from Definition 2.1, for the individual output series to converge there long-run trend. must be p— 1 cointegrating vectors of the form (1,-1) or one common Two test statistics proposed by Johansen to are derived from the eigenvalues of the MLE rank of the cointegrating matrix test the estimate of II. If n of is full rank, p, then will have no eigenvalues equal to zero. will have p—r zero eigenvalues. Looking at the smallest p-r eigenvalues the however, If, it is of less than p trace full rank, r < p, it then it statistics are p = T ]T A, ; « -2ln(Q;r,p) ^ = -T «=r+l /n(l - A,) (3.4) i=r+l and maximum The eigenvalue = TA r+1 ss — 2ln(Q;r, r+ 1) = — Tln{\ — A r +i) (3.5) trace statistic tests the null hypothesis that the rank of the cointegrating matrix against the alternative that the rank hypothesis that the rank is is p. The maximum eigenvalue r against the alternative that the rank is r is statistic tests the null r+1. Critical values for the asymptotic distributions of both statistics are tabulated in Osterwald-Lenum (1992). Data 4 The data used in the empirical exercise are ternational dollars. GDP The series annual log real GDP per capita in 1980 in- run from 1900-1987 for 15 industrialized countries with the data drawn from Maddison (1989) and the population data from Maddison (1982). Population for 1980-1987 comes from IFS yearbooks. The population data as published in current national borders, while the GDP Maddison (1982) are not adjusted to conform data are adjusted. Failure to account for border changes can lead to large one time income per capita movements as population lost. For example, loss of the GDP per capita in the UK jumps in is gained or 1920 without a correction for the population of Ireland in that year. To avoid these discrete jumps we adjust the population to reflect modern borders. 3 The to year-to-year The GDP data set also has a few minor problems. movements during the two world wars Austria are constructed from GDP for Belgium and during WWI for estimates of neighboring countries. Empirical results on convergence and cointegration 5 In testing for convergence and common trends, we use three separate groupings of countries: 15 countries together, the 11 European countries and finally a subset of 6 European all This type of gain or loss affects Belgium, Canada, Denmark, Fiance, Italy, Japan, and the UK at If territory, and thus population, are lost by country X in year Ti, we adjust earlier years by extrapolating backward from T\ using the year-to- year population changes of country X. 3 least once. countries which exhibit a large degree of pairwise cointegration. 4 Results from the PhillipsOuliaris procedures on convergence and common trends are Results using the Johansen methods are in Table We convergence initially test for Ouliaris bounds off the US output levels tests output. If on the first indicates the presence of of the number of roots. If the first common p < n is n to find is tests for the for the 95% more than one we can cannot between these extremes, lies As an If the lower the lower If reject the 95% than less we can bound is in of eigenvalues that account for sum bound on the no convergence 95% this of the the upper p same distinct roots. more of the We is is often not significant for any are presented for null hypotheses on The evidence on convergence is samples the convergence hypothesis number common For quite striking. fails. no-convergence null for both The less than p or more also look for the maximum number eigenvalue statistics on number of cointegrating of different then we cannot VAR lag length of maximum statistics give different estimates of the cointegrating vectors; the eigenvalue test Ad- total variance. multivariate results from the Johansen trace and The two null. two bound critical value largest we conclude reject the null hypothesis that there are greater than the or if convergence and cointegration are presented in Tables 3a and 3b for a vectors. Test results trends ranging from all test statistics 1 and to 15. in all three direct convergence test in Table critical levels as the largest eigenvalue *The 6 European countries are Austria, Belgium, Denmark, France, Italy 5 Cogley (1990) uses a similar measure. K, the size of the Daniell window was chosen to be T 6 or 27 for our sample. 7 Reducing the lag length increased the number of trends somewhat. ' , is 1 cannot statistically and the Netherlands. Bernard (1991) finds cointegration in 10 of 15 pairs. 6 n alternative measure of the total variance each group. First, reject the hypothesis that there are at least reject the idiosyncratic distinct roots for stochastic trend for the group. Table 2 presents number of common trends distinct roots. 7 If more of the sum, then we conclude or groups mentioned above. 6 acounts for the critical value for a given p, 2. distinct root in stochastic trends for the block. 5 common ditionally, if the largest root The 1 at the cumulative percentage of the largest roots contribute greater than the critical level that there to find presents the Phillips- Ouliaris bounds tests for convergence and the cumulative 1 sums of the eigenvalues root of significant roots we look respectively. having subtracted t , from a benchmark country. trends in international output. trends, that there are p important Table number common ADY we would expect we would expect trends dominate for every country, then and 2 1 3. difference of output deviations, in the deviations countries in the levels. If the Tables country group by performing the Phillips- in the 15 countries converge, then and no roots in for all three groupings. Additionally, from zero different maximum both the trace and the eigenvalue statistics reject convergence in every group in Table 3a. Having failed to find evidence for convergence, or the test for the number country sample and more distinct roots of common the alternative critical value of more or is distinct roots but a large common 96.7% of the C\ (= ^) critical value and cannot The trends. a single long-run trend, we turn to Phillips-Ouliaris statistics for the fifteen reject the null hypothesis that there are 7 or reject the null that there are at least 4 distinct roots. 5% of the now cannot sum total, coinciding with the results from the test the largest root accounts for barely 50% and the 5% level The Johansen and 13 or fewer six largest roots statistics. On common account for the other hand, the largest two roots for about variance, which argues against the existence of just a single for convergence. reject for 7 reject for at least 5. This leads us to posit that there component over the sample. The stochastic we again of the eigen values, Ci, 75% factor, as is of total required trace statistic rejects 12 or fewer cointegrating vectors at 10% at the level for the entire fifteen country sample. This implies that there are only two or three long-run shocking forces for the entire group. maximum eigenvalue statistic does not reject for any Taking all there are 6 or statistic 11 of the more trends and cannot statistic rejects 5 or for European countries and that there are more at least 3 trends with the trends, while the Turning to the results for the six distinct trends with rejects 2 or Using maximum more trends with the CI statistic. The Johansen C2 trace eigenvalue test again cannot reject results suggest that there are on the order European countries. both the C\ and Ci MLE trends. reject the null hypothesis that European countries, we null that there are at least 3, again with three largest eigenvalues. we The reject that there are at least 4 trends with the of 4-5 long-run processes driving output in the more number of as a group, any number of cointegrating vectors. These are 4 or With reject the null that there are critical values and cannot reject the both values. 97.8% of the sum comes from the statistics, the smaller six European country group trace statistic and 5 or more with the maximum eigenvalue test. 6 Conclusions This paper attempts to answer empirically the question of whether there output per capita across countries. We first based on the theory of integrated time countries can fail to converge only if is convergence in construct a stochastic definition of convergence series. Time series for per capita output of different the persistent parts of the time series are distinct. 10 Our analysis of the relationship little among long-term output movements evidence of convergence. Virtually hypothesis of no convergence. cointegration across OECD On all of our hypothesis tests cannot reject the null the other hand, we find evidence that there is some set of is substantial economies. The number of integrated processes driving the 15 countries' output series appears to be on the order of 3 to 6. that there across countries reveals common Our results therefore imply factors which jointly determines international output growth. 11 References 7 Barro, R. (1991). "Economic Growth in a Cross Section of Countries." Quarterly Journal of Economics, 106, 407-443. W. Baumol, (1986). "Productivity Growth, Convergence, and Welfare: Run Data Show." American Economic Review, What the Long- 76, 1072-1085. Bernard, A. (1991). " Empirical Implications of the Convergence Hypothesis." Working Paper, Center for Economic Policy Research, Stanford University. Bernard, A. and sis." Working Paper, Center Campbell, of J. (1992). "Interpreting Tests of the Convergence Hypothe- S. Durlauf. Economic Policy Research, Stanford University. for Y. and N. G. Mankiw. (1989). "International Evidence on the Persistence Economic Fluctuations." Journal of Monetary Economics, Cogley, T. (1990). "International Evidence on The 23, 319-333. Size of the Random Walk in Output." Journal of Political Economy, 98, 501-518. DeLong, J. B. (1988). "Productivity Growth, Convergence, and Welfare: American Economic Review, W. Engle, R. F. and C. Comment." 78, 1138-1154. Granger. J. (1987). "Cointegration and Error- Correction: Representation, Estimation, and Testing." Econometrica, 55, 251-276. Johansen, S. "Statistical Analysis of Cointegration Vectors." (1988). nomic Dynamics and Control, Journal of Eco- 12, 231-54. (1991). 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Bounds Tests Bounds Tests** 6 European Countries Trends Lower Upper Phillips-Ouliaris All Countries Trends <1 Upper 1.60+ 3.09 <1 0.68+ 1.32 Convergence 11 European Countries Trends <1 Lower Upper 0.29+ 0.46 bound is below the critical value for the largest root, reject null of no convergence. bound is above the critical value for the largest root, cannot reject null of no convergence. If the upper If the lower + Lower for Significant for critical value of 0.05. ** These statistics are calculated countries, the 11 European US is subtracted countries, France off. is on the vector of first differences of GDPi — GDPk. For all For the 6 European countries, France is subtracted off. For the subtracted off. Table lb. Cumulative Percentage from p Largest Eigenvalues 6 European Countries All Countries 11 European Countries Trends Cumulated % Trends Cumulated % Trends Cumulated % Largest Smallest 1 0.74 1 0.69 1 0.69 2 0.88 2 0.89 2 0.89 3 0.93 3 0.98 3 0.95 4 0.96 4 1.00 4 0.97 5 0.97 5 1.00 5 0.98 6 0.98 6 0.99 7 0.99 7 1.00 8 0.99 8 1.00 9 1.00 9 1.00 10 1.00 10 1.00 11 1.00 12 1.00 13 1.00 14 1.00 14 L Table Bounds Tests for Cointegration European Countries 11 European Countries Trends Lower Upper Trends Lower Upper Phillips- Ouliaris 2a. All Countries 6 Trends Lower Upper 15 0.00 0.00 14 0.00 13 0.00 6 0.00 0.00 11 0.00 5 0.00 4 0.01 0.01 0.02 0.03*+ 0.00 0.00 10 0.00 0.00 9 0.00 0.01 0.11 8 0.01 0.01 0.40 7 0.01 0.02 6 0.02 0.03" + 12 0.00 0.00 3 0.06*+ 11 0.00 0.01 2 0.23 10 0.01 0.01 9 0.01 0.01 5 0.03 0.06 8 0.01 0.02 4 0.06+ 0.10 7 0.03 0.04*+ 3 0.12* 0.19 6 0.04 0.07 2 0.30 0.46 5 0.07+ 0.10 4 0.11* 0.17 3 0.19 0.29 2 0.39 0.57 Cumulative Percentage from p Largest Eigenvalues All Countries 6 European Countries 11 European Countries Trends Cumulated % Trends Cumulated % Trends Cumulated % Table 2b. Largest the upper bound is 0.52 1 0.69 1 0.63 2 0.76 2 0.91 2 0.84 3 0.87 3 0.98 3 0.92 4 0.92 4 0.99 4 0.95 5 0.95 5 1.00 5 0.98 6 0.97 6 1.00 6 0.99 7 0.98 7 0.99 8 0.99 8 1.00 9 0.99 9 1.00 10 1.00 10 1.00 11 1.00 11 1.00 12 1.00 13 1.00 14 1.00 15 1.00 Smallest If 1 bound above the is below the critical value, • Significant at O.lOm/n, n + Significant at critical value, reject null 5% of the is P cannot reject null of at least the sum of number of countries, of the roots. 15 m is more or P the distinct roots. If the lower distinct roots. number of roots = 0. Table Multivariate Tests for Convergence and Cointegration 3. (VAR lag length Table 3a. > 14 >13 > 12 > 11 > 10 553.99 >9 >8 >7 >6 >5 >4 >3 >2 European 6 62.00* 53.84* 31.89* Eig Trends > 71.00 51.06 147.60 38.29 292.99 48.89 244.10 41.07 156.82 34.66 162.59 36.03 126.56 32.61 93.95 31.23 > 62.71 24.52 38.19* 17.22 9.08 1.38 Rejects at 5%. t Distributed x 2 (p — 1) where p is the 79 62.41 23.27 17.55 21.59 14.02 34.80J > 1 74.51 22.49 >0 0.57 52.02 18.07 33.95 15.95 18.00 9.16 1 8.84 7.55 >0 1.28 1.28 number of countries. Q Eig 39.14* 7.57 109.31* 16 7'5 40.54* 53.06 59.38 1.38 102.95 198.66 352.37 >0 Trace 251.72 >9 >8 >7 >6 >5 >4 >3 >2 Max Trends >5 >4 >3 >2 68.62 10.51 European 6 Eig 60.46 62.00 10.46 Max 312.18 414.37 20.97 Trace 10 482.99 * Cointegration European Max 1 > Convergence* European 6 Table 3b. Trace 2) All All Trends = 7.00 0.57 MIT LIBRARIES DUPL 3 TDflD OOflEEflDO M