## Servicios Personalizados

## Revista

## Articulo

## Indicadores

## Links relacionados

## Compartir

## Cuadernos de economía

##
*versión On-line* ISSN 0717-6821

### Cuad. econ. v.43 n.127 Santiago mayo 2006

#### http://dx.doi.org/10.4067/S0717-68212006000100003

Cuadernos de Economía, Vol. 43 (Mayo), pp. 81-112, 2006
University of Wisconsin-Madison
Traditional convergence tests implicitly evaluate the unit root hypothesis in per capita output series. Although these statistics do not have asymptotic distributions or critical standard values under the null hypothesis, the great majority of works on the subject uses them, with the corresponding problems for inference. For this reason, new tests have been developed in the literature on panel data estimation to analyze the convergence hypothesis correctly. The goal of this paper is to determine if the existing relatively weak evidence that favor the hypothesis of convergence of GDP and income among the regions of Chile holds when such new tests are applied and a more recent database is used. The analysis is conducted using not only the traditional tests (cross-section and panel data), but also the most recent unit root tests developed for panel data, which allow making correct inference. We also analyze convergence in dispersion, constructing confidence intervals for the variance of regional production and income and evaluating the presence of asymmetries or the formation of regional "clubs" through a nonparametric multimodality test. The theoretical foundation for the empirical analysis is framed within neoclassic growth theory, which attempts to explain whether the different economies converge to a single distribution of per capita output or income levels. In our analysis, these economies are the thirteen regions of Chile, and the time frame is 1960-2000. The implication of this literature is that per capita GDP is trend stationary as a contraposition of the endogenous growth literature, which predicts that this series would be difference stationary. The latter means that temporary shocks will have a permanent effect on the level of GDP, while in the case of the neoclassical model, temporary shocks willl have a temporary effect on the level of this variable. Therefore, the stochastic process that underlies the per capita GDP series is a key element for discriminating beween these two theories. The earliest works analyzing the dynamics of regional GDP per capita in Chile find that there is, in fact, convergence in per capita GDP levels, but that the speed or rate of unconditional convergence is relatively slow: 1.3% for the 1960-90 period (Fuentes, 1997) and 1.2% for the 1960-92 period (Morandé, Soto and Pincheira, 1997). A later analysis (Anríquez and Fuentes, 2001) shows that the speed is higher for income (8.9%) than GDP (1.3%), based on data from the National Socioeconomic Characterization (CASEN) survey for the 1987-1994 period. More recent studies find that this convergence rate fluctuates between 0.5% (Soto and Torche, 2004) and 1.0% (Díaz and Meller, 2004). Relatively few works find divergence among the Chilean economies. Some of them only find convergence in certain groups or "clubs" of regions, reporting, for example, that all regions converge except the Metropolitan and II regions (Díaz and Herrera, 1999) or that convergence occurs in all but the I, II, and XII regions (Araya and Oyarzún, 2001). One of the valid critiques of the empirical focus used in the majority of the previous works is that by regressing the per capita GDP growth rate (expressed as the change in the log of the level) and its initial level, the analyst is using the typical unit root test (augmented Dickey-Fuller) to estimate coefficients of convergence speeds whose This paper differs from previous works in three ways. First, we analyze the growth of regional economic activity using a more updated database that includes -in contrast to the other cases- regional GDP for the 1960-2000 period, with figures recently published by the Central Bank, and data on household income for 1987-2000, from the CASEN survey. We are thus able to capture the recent period of strong turbulence and external crisis (1997-2000) and to observe their potential effects on regional convergence. Second, we use unit root tests recently developed for panel data to evaluate the null hypothesis of the absence of convergence and avoid the problems of inadequate inference mentioned above. To contrast the results, we include estimates and tests of the convergence hypothesis using traditional cross-section and panel data techniques. Third, we use a nonparametric multimodality test to evaluate the presence of regional asymmetries or "clubs" and thus verify whether the apparent bimodality of the empirical distributions is statistically significant. The remainder of the paper is organized in four sections. The next section (section 2) briefly and intuitively explains the ideas of beta and sigma convergence. Section 3 then describes the construction, periods, frequency and sources of the data used in the estimations. Section 4 outlines the empirical strategy followed and presents our main results. Finally, section 5 concludes.
The fundamental paradigm of economic convergence stems from the Solow (1956) and Swan (1956) model, which was later formalized in a dynamic optimization framework by Cass (1965) and Koopmans (1965). This model predicts that, given an initial stock of capital per worker, an economy converges to a long-run (steady-state) equilibrium in which the output per worker grows at a constant rate equal to the rate of technical change. In the transition, the economy grows above the long-run rate and closes the gap with that rate asymptotically. This generates one of the main implications of the model -namely, that in the steady-state equilibrium, the per capita income of different economies converges to the same level once the savings rate, depreciation rate, and population growth rate are taken into account. The idea of convergence is also related to the idea that the poorest economy closes the gap with the richest economy in the dynamic transition to the steady state. In other words, the poorest economy grows faster than the richest. In the economic growth model with optimization, an economy's growth rate is an increasing function of the difference between the marginal product of capital and the agents' rate of intertemporal impatience. That is, economies with a larger difference grow faster. Poor economies have less capital and, therefore, a higher marginal productivity than richer economies. Thus the growth rate of the poorest economy (or the economy that is positioned farthest from its steady state) is higher. As the economy accumulates capital, marginal productivity falls and, therefore, the growth rate also drops until it finally reaches the rate of technical progress. There are two concepts of convergence related to this model: beta (b) convergence and sigma (s) convergence. 2.1. Beta Convergence Beta (b) convergence is said to occur when the poorest economies -which are the furthest from their long-run output or income level- grow faster then the richest economies until they achieve the same level of output. An issue that is closely linked with testing this hypothesis, and one that we need to address for the empirical analysis below, is the relationship between the type of growth and the type of stochastic process that underlies the output or income series. Modern economic growth theory can be divided in two strands: exogenous growth and endogenous growth. Our analysis and, therefore, our convergence hypothesis are consistent with the exogenous growth approach. 2.2. Sigma Convergence Sigma convergence is said to exist when the dispersion of per capita income or output, measured as its variance, diminishes over time. Formally, s convergence is confirmed when the cross-sectional variance of the regional per capital incomes or outputs shows a (statistically) significant decrease between the initial and final periods of the sample. In a world without stochastic shocks, the implicit idea is as follows: in the initial period there is a high dispersion of per capita output -due to the difference between rich and poor regions- that is expected to be lower at the end of the process of convergence toward the steady state. It can be shown that the existence of b convergence is a necessary but not a sufficient condition for the presence of s convergence. In practice, it is thus possible to find that the richest economies or regions grow less than the poorest, but ultimately the dispersion remains unchanged over time. This would be the case if the regions that were initially relatively poor grew faster than the wealthy regions and overtook them, leaving the dispersion of per capita income the same as at the initial period. However, since b convergence is a necessary condition, it is not possible to observe s convergence without b convergence 2.3. Absolute and Conditional Convergence From an empirical perspective, another dimension of economic convergence analysis involves the distinction between absolute (or unconditional) and conditional convergence. Absolute convergence is said to pertain when all the economies converge regardless of economy-specific factors (economic policies, investment rate, composition of output, and so forth); that is, the poorer economies always grow faster than the richer ones. This implicitly requires that the economies have similar population growth rates, preference parameters, and technology. In this sense, there is also a condition for convergence. Conditional convergence, in turn, occurs when the relation between the growth rate of per capita output and its initial level is negative once we have controlled for factors that condition the steady state. In other words, the economies converge only when we take into account the factors that are specific to the steady state toward which they are moving. 3. The Data To test the convergence hypothesis on the GDP and income of the thirteen regions of Chile, we use all the data available at the close of this study. 3.1. Per Capita GDP In the case of regional output, the period of analysis is from 1960 to 2000. Figure 1 shows the trends in gross real per capita output levels for each region (expressed in natural logs) during the period. These series were constructed from two sources. First, in the numerator, we use the Central Bank of Chile's regional GDP series, in millions of 1986-pesos for the 1960-96 period, and the growth rates from the new series in millions of 1996-pesos for the rest of the span (that is, 1997-2000). 3.2. Per Capita Income In the case of per capita income, the period of analysis is significantly reduced due to data availability. We used the average household income series Figure 2 shows the per capita income trend for each region during the available period.
Following the outline presented above, we first present the empirical tests and their respective results for b convergence in both GDP and regional income, and we then assess the case of s convergence. 4.1. Beta Convergence in Regional GDP We first hypothesize whether Chile's regions display absolute convergence in output levels -that is, b convergence. In order to present robust estimates and obtain solid conclusions, we then undertake the typical convergence tests through cross-section and pooled panel data regressions, for both GDP and income series. Finally, we apply recent panel data unit root tests that allow us to make correct inference on the hypothesis under study.
The most commonly used regressions in growth studies are cross-sectional. where The first evidence in favor of this hypothesis is presented in Figure 3, which shows the negative relationship between the growth rate (from 1960 to 2000) and the initial level of regional GDP per capita (in 1960). The estimated coefficient in this regression is negative, as expected, and statistically significant. This approach is limited, however, for making sufficiently valid inference because of the scarcity of observations (only thirteen, the number of regions); this limitation is of particular concern considering that the number of regions in a country is always small and finite. An alternative approach proposed in the literature is to test the b convergence hypothesis via the panel data technique, estimating an equation similar to the previous one:
where, in this case, One of the advantages of this technique is that it lets us take advantage not only of the cross-sectional dimension, but also of the time dimension, thus providing greater degrees of freedom. In the case of absolute convergence, we chose a pooled panel instead of fixed or random effects mainly because assuming the presence of unobserved idiosyncratic effects (whether fixed or stochastic) would be equivalent to assuming that each region's output or income converges to a different stationary level, or that they converge conditional on controlling for exogenous components specific to each region. Consequently, this alternative is particularly valid when dealing with conditional convergence, an issue we address below. On the other hand, we do include temporal effects common to all the regions, basically to control for the potential effects of the base change in the regional GDP series in 1997 and the change in methodologies in the different CASEN surveys. The inclusion of the temporal effects does not invalidate the hypothesis of absolute convergence to the extent that it is consistent with the series being trend stationary, which in turn is coherent with the type of growth model we are employing, as was explained in section 2.1. Thus if we estimate the coefficients that relate the ten- or five-year growth rate with the respective initial output levels, we obtain values of -0.74 and -0.85 percent, respectively (see table 1a). Once again the values are negative and statistically significant, which implies a process of regional convergence in GDP. These estimated values for the GDP convergence coefficient are, at nearly -1%, lower than the levels found in earlier studies on both regional convergence in Chile and developed countries (around 2%).
A valid criticism of regressions between the per capita GDP growth rate and initial per capita GDP is that the test does not have a standard distribution under the null hypothesis (b = 0), so making a comparison using the traditional statistics and related critical values can lead to erroneous conclusion. One possibility, then, is to examine whether each regional GDP (or income) series independently presents a unit root, but such a procedure suffers from serious power problems. The range of panel data unit root tests has grown in recent years. We apply only the most recent to contrast the results with earlier studies and only where such tests are adaptable to the requirements of our hypothesis. We thus apply four tests: Levin, Lee, and Chu (2002); Breitung (2000); the Fisher-ADF and Fisher-Phillips-Perron tests proposed by Maddala and Wu (1999); and Choi (2001). Prior to the application of the tests, we removed the temporal effects from the series, for two basic reasons. First, as mentioned earlier, we want to control for effects in the series stemming from the change in the base year, a factor that cannot be adequately addressed when undertaking unit root tests. Second, panel data unit root tests are generally constructed under the assumption of no cross-sectional correlation of errors. We therefore remove the (fixed) temporal (but not idiosyncratic) effects common to the regions to avoid drastic loss of power for the autocorrelation. Table 1b reports the modified 4.2. Beta Convergence in Regional Income Following the same empirical strategy as in the previous section, we analyze the data on regional per capital income.
While the relationship between the growth rate from 1987 to 2000 and the initial income level (in 1987) is negative, with a coefficient close to -0.015, it is not statistically significant at conventional levels (see Figure 4 and Table 3). The cross-section estimations thus do not provide clear evidence of convergence in income. Nevertheless, the estimations using panel data, which incorporate more information (greater number of observations) since they consider the time dimension, yield evidence in favor of the hypothesis under study. Both two- and four-year estimations confirm the hypothesis of convergence, with faster rates than those found with the GDP estimates: -0.078 and -0.038, respectively (see Table 3). Unit Root Tests
Analogously, we apply the same set of unit root tests to per capita income series and obtain similar results to the GDP tests. For the two-year data, only the Breitung test does not support the convergence hypothesis, although its probability ( 4.3. Conditional b Convergence To complete the sensitivity analysis, we performed conditional b convergence tests. We carried out tests similar to those described above for the 1960-2000 period, whenever feasible incorporating explanatory variables that allow us to approximate the potentially distinct steady states of each region. In this case we were able to carry out panel data tests. The results support the existence of conditional convergence of GDP (see Table 3). The importance of mining on the regional productive structure was the only statitiscally significant variable in the conditional convergence regression. In sum, the empirical evidence generally backs up the hypothesis of b convergence for both per capita income and GDP, showing higher convergence rates in the conditional case. The increase of the beta coefficients from absolute to conditional convergence tests is also observed in most of the regional evidence for developed countries. Barro and Sala-i-Martin (1995) found coefficients of 1.6% for Italy, 1.6% for Germany, 2.2% for the United States, and 3.1% for Japan. Although the coefficients are not strictly comparable, the average difference between the cross-section and panel coefficient values for theses studies is around 0.63 percentage points, closely similar to our findings (about 0.68 percentage points). This fact might reveal the omission of relevant variables -in our case, mining- in the absolute convergence regressions, since those variables may capture differences in the region's steady states. Statistically speaking, the sign of the bias due to relevant variable omission is governed by the correlation between the regressors (the lag of per capita GDP) and the omitted variable (mining). In our case, only a negative correlation between mining and the lag of GDP can explain the underestimation of the beta coefficient value. Empirically, regions with lower initial levels of per capita GDP present a higher share of mining on the regional productive structure in the transition to the steady-state equilibrium. 4.4. Sigma Convergence of Regional GDP As described earlier, another relevant type of convergence is s convergence. We have shown that, in general, the evidence clearly supports b convergence in GDP and income, which is a necessary but not sufficient requirement for the existence of s convergence. We must now analyze the behavior of the dispersion of the indicators of output and income across time. A first piece of evidence of s convergence in GDP can be seen in figure 1. A quick look suggests that in 2000 (the last available year), the regions presented a lower dispersion of the per capita GDP series than in 1960. Moreover, as can be seen in Table 6, the difference between minimum and maximum per capita output has lowered in recent years, in contrast with the 1970s. A way to verify s convergence commonly used in the literature consists of constructing the variance of the log of regional GDP and observing its evolution over time. As explained in section 2, if the variance decreases in the period of analysis, then there is evidence in favor of s-type convergence While the variance of regional GDP shows clear fluctuations, it generally follows a decreasing trajectory (see Figure 5). To test this claim, we construct a 90% confidence interval for the sample variance estimator, in contrast to earlier studies. In spite of a drastic decrease from 0.37 (1960) to 0.25 (2000), the confidence intervals do not allow us to affirm categorically that the reduction was stistically significant at 10%, since the final value of the variance in 2000 falls within the initial confidence interval constructed for 1960. Another way to analyze the dispersion of regional GDP is by estimating its empirical distributions in each year and observing its behavior over time. Figure 6 shows the distribution of regional GDP every five years since 1965 (end-of-period data). As shown in the figure, 1965 starts with a partially unimodal distribution that is highly volatile; this pattern is maintained until 1980 (see Table 6). From that year until 1990 -basically the period in which economic reforms were applied in Chile- we note the appearance of a bimodal distribution or "club", as it is known in the growth literature. To confirm this phenomenon, we carried out nonparametric multimodality tests (see Bianchi, 1997) on the per capita GDP series to determine whether the apparent asymmetries are statistically significant. Under the null hypothesis, the empirical distribution of the series has Table 7 presents the results of this test (values of the statistics and 4.5. Sigma Convergence in Regional Income Contrary to what we found in the case of GDP, s convergence does not appear to have changed greatly at the level of regional income in the 1987-2000 period. As shown in Figure 7, the variance of regional incomes has generally increased, although not significantly, despite having been reduced in the early 1990s. The evolution of other statistics, such as the difference between the maximum and minimum values, also points in this direction (see Table 8). When we look at the 90% confidence interval, however, the value of the variance toward the end of the period (in 2000) is not statistically different from the initial value (in 1987). This implies that while there is no s convergence in the period, neither is there a process of divergence that would contradict our results for regional GDP. When we calculate the empirical distributions through the kernel estimator, we also observe the appearance of supposed regional clubs in bimodal distributions and with high dispersion, especially in the years 1987, 1992, and 1998 (see Figure 8). However, they tend to disappear in 2000. The application of the multimodality test also reveals that the apparent bimodality of the regional income distribution is not, in fact, statistically significant (see Table 9). Under the null hypothesis of unimodality, the values of the statistics and probabilities are such that they allow us to reject the idea of multimodality in the estimated density of the series between 1990 and 2000, at standard levels of significance. In the case of the series expressed in relative terms, it is not possible to reject the unimodality of the series from 1987 on, which is consistent with the results of the same test applied to the GDP series.
The empirical evidence for the 1960-2000 period in Chile tends to support the hypothesis of convergence in regional GDP per capita, for both b-type convergence (convergence in levels) and s-type convergence (convergence in dispersion). The unit root tests employed mostly reject the presence of stochastic trend processes in favor of deterministic trend processes (trend stationary processes). That is, the evidence is strongly consistent with the neoclassical growth theory and the presence of convergence in both the regional per capita GDP and income series. However, this result -which we also found through traditional panel and cross-section convergence tests- is accompanied by slow rates of convergence (relative to the international evidence) of slightly under 1%. This implies that the period for closing half the gap between relatively poor and rich regions ranges from 81 to 96 years. Our analysis of conditional convergence finds that the speed increases to a range of 1.4% to 5.2% (with half the gap closing in a period of between 72 and 13 years, respectively) when we control for the share of the mining sector. This appears to be consistent with the idea that the regions are converging to their own steady state. Variables like average education of the work force do not appear to be good approximations for characterizing the steady states because they are not statistically significant. The analysis of s convergence or convergence of the variance of regional per capita GDP shows that the variance has undergone a (statistically nonsignificant) decrease for the full period. Nevertheless, this drop has not been constant and has included periods in which the estimator increased notably. The volatility of this indicator coincides with the period between the 1975 and 1982 crises, and the recent international turbulence starting with the Asian crisis. With regard to per capita income, the results on convergence are relatively favorable for b convergence, but we do not observe a statistically significant reduction in its dispersion. Given the short span of our income sample, this behavior is extremely similar to that shown by the variance of regional GDP for the same period, which indicates consistency in the findings. Moreover, the application of the multimodality test reveals that the apparent asymmetries or bimodalities of the regional income distribution are not, in fact, statistically significant.
^{36}
This test assumes that there is a common unit root process in the series. It considers the standard specification of an augmented Dickey-Fuller (ADF) test, but applied to panel data: where a = b - 1 is a common coefficient to the series, but different orders of lags ( y_{i}_{t} are allowed in the cross-section; and X is a vector of deterministic variables (for example, seasonal or trend dummies). The hypothesis to evaluate is H In general terms, the test follows four stages. (i) Estimate the ADF regressions (as in equation A1) for each region; find the optimal number of lags ( y_{i,t}_{-1} against y_{i,t}_{-L} (where L = 1,…, p) and the relevant deterministic variables (_{i}X). The idea here is to generate approximations of the variables Dy and _{i,t}y_{i,t}_{-1} that are free from autocorrelation, an assumption on which the test is built. (ii) Collect the residuals from the regressions and normalize them by dividing by the standard error of regression A1 (denote them as D (iii) Use the approximations to obtain estimators of a in the following specification: (iv) Finally, construct a modified
This test assumes the presence of a single nonstationary process in the series and constructs a modified
These tests were proposed by Maddala and Wu (1999) and Choi (2001). In contrast with the previous tests, these allow the presence of individual unit root processes. That is, regression A1 is run for each series, but now we evaluate In particular, the Fisher-ADF and Fisher-Phillips-Perron tests use the where F In all cases it is necessary to specify the number of lags used in the test. In the cases of Levin, Lin, and Chu (2002) and Fisher-Phillips-Perron, it is also necessary to specify the kernel method and the bandwidth selection for the zero-frequency spectral estimation.
We use the nonparametric multimodality test proposed by Bianchi (1997). The contrast is based on the estimation of the data density function using kernel methods and on testing the number of groups ("clubs" for the case of economic convergence) within a single distribution using the bootstrap technique. The key concept in the estimation of the density function ( where each p = 1, and _{j}g are the densities, with first and second moments m_{j} and s_{j}_{j}^{2}, respectively. For example, if we assume that the clubs are normally distributed, g will be defined as follows: A critical bandwidth h that generates a density with at least m modes, which implies that for h < h the estimated density function has at least _{m}m + 1 modes. This leads to the idea of using h as a statistic for evaluating: _{m}Thus, a high value for m modes, rejecting the null hypothesis. The value that is considered "high" is determined through bootstrapping (see Silverman, 1981, 1986; Efron and Tibshirani, 1993). (i) Starting from the data, generate where (ii) For each bootstrap sample ( (iii) Obtain an estimate of the achieved level of significance (ALS) of the test, defined as (iv) The null hypothesis of In short, the strategy pursued involves using this routine and applying the test beginning with the null hypothesis of
E-mails: duncantaraba@wisc.edu, rfuentes@bcentral.cl
Andrews, D. (1991). "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation." Anríquez, G., and R. Fuentes (2001). "Convergencia de producto e ingreso de las regiones en Chile: una interpretación." In Araya, I., and C. Oyarzún (2001). "Long-Run Dynamics of Regional Growth in Chile." Baltagi, B., and C. Kao (2000). "Nonstationary Panels, Cointegration in Panels and Dynamic Panels: A Survey." Working paper 16. Syracuse University, Center for Policy Research. [ Links ] Barro, R., and X. Sala-i-Martin (1991). "Convergence across States and Regions." _____ (1992). "Convergence." _____ (1995). Baumol, W. (1986). "Productivity Growth, Convergence, and Welfare: What the Long-Run Data Show." Bianchi, M. (1997). "Testing for Convergence: Evidence from Non-Parametric Multimodality Tests." Canning, D. (1999). "Infrastructure's Contribution to Aggregate Output." Working Paper 2246. Washington: World Bank. [ Links ] Cass, D. (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation." Cheung, Y., and A.G. Pascual (2004). "Testing for Output Convergence: A Re-examination." Choi, I. (2001). "Unit Root Tests for Panel Data." Chumacero, R. (2002). "Is There Enough Evidence against Absolute Convergence?" Working paper 176. Santiago: Central Bank of Chile. [ Links ] Díaz, L., and N. Herrera (1999). "Desigualdad de ingresos y bienestar 1990-1996: análisis comparativo desde un enfoque nacional/regional." Working Paper. Santiago: MIDEPLAN. [ Links ] Díaz, R., and P. Meller (2004). "Crecimiento económico regional en Chile: ¿Convergencia?". Working Paper 180. University of Chile, Department of Industrial Engineering. [ Links ] Efron, B., and R. Tibshirani (1993). "An Introduction to Bootstrap." Evans, P. (1998). "Using Panel Data to Evaluate Growth Theories." Fuentes, R. (1997). "¿Convergencia en las regiones en Chile? Una interpretación." In Fujita, M., and J-F. Thisse (2002). Hadri, K. (2000). "Testing for Stationarity in Heterogeneous Panel Data." Im, K. S., M. Pesaran, and Y. Shin (2003). "Testing for Unit Roots in Heterogeneous Panels." Inada, K-I. (1963). "On a Two-Sector Model of Economic Growth: Comments and a Generalization." Koopmans, T. (1965). "On the Concept of Optimal Economic Growth." In Levin, A., C. F. Lin, and C. Chu (2002). "Unit Root Tests in Panel Data: Asymptotic and Finite-Sample Properties." Maddala, G. S., and S. Wu (1999). "A Comparative Study of Unit Root Tests with Panel Data and A New Simple Test." Morandé, F., R. Soto, and P. Pincheira (1997). "Achilles, the Tortoise, and Regional Growth in Chile." In Phillips, P. C. B., and D. Sul (2003). "The Elusive Empirical Shadow of Growth Convergence." Discussion Paper 1398. Yale University, Cowles Foundation for Research in Economics. [ Links ]Quah, D. (1996a). "Empirics for Economic Growth and Convergence." _____ (1996b). "Ideas Determining Convergence Clubs." Working Paper (April). London School of Economics. [ Links ] _____ (1996c). "Twin Peaks: Growth and Convergence in Models of Distribution Dynamics." _____ (1997). "Empirics for Growth and Distribution: Stratification, Polarization, and Convergence Clubs." Silverman, B. (1981). "Using Kernel Density Estimates to Investigate Multimodality." _____ (1986). Solow, R. (1956). "A Contribution to the Theory of Economic Growth." Soto, R., and A. Torche (2004). "Spatial Inequality after Reforms in Chile: Where Do We Stand?" Swan, T. (1956). "Economic Growth and Capital Accumulation." White, H. (1980). "A Heteroskedasticity-Consistent Covariance Matrix and a Direct Test for Heteroskedasticity." Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons |