Volume 46 Number 182,

July-September 2015

July-September 2015

Non-Linear Analysis of Regional Convergence

in Latin America, 1950-2010: A Panel Data tar Model

in Latin America, 1950-2010: A Panel Data tar Model

Date received: November 24, 2014. Date accepted: March 18, 2015

Abstract

This paper analyzes the hypothesis of regional convergence in Latin America through a non-linear growth model for the time period 1950-2010. The methodology combines three approaches: the threshold autoregressive model (tar), panel data unit root tests and calculating critical values with a bootstrapping simulation. The results of the tests applied to the per capita gross domestic product (gdp) of two groups of countries in Latin America (the wealthiest and then all nations in the region) suggest that the linear model is superior to the non-linear model and show no evidence of partial or absolute convergence. We did not identify a group of countries in the region with higher per capita income that would behave as a leading economy. Our results cast doubt on other studies conducted with linear tests that did find conditional convergence in some countries in the region.

Keywords: Economic growth, convergence/divergence, panel data tar model, linear and non-linear estimation methods, econometric models.

* Possibly in no other region of the Third World has the debate on development
engaged so many minds over the last twenty-five years. And in no other
region has it been so difficult to justify the highly precarious living
conditions of the great majority of the population, given the abundance
of natural resources and the proclaimed successes of development policies.*

Celso Furtado,

Trans. Suzette Macedo. 1976

[1970], p. 303.

** INTRODUCTION**

In analyzing economic convergence and divergence, it frequently emerges that the economies under study do not exhibit only one of these processes; long-term economic growth often coincides with the transition from convergence to divergence or vice versa, and moreover, these dynamics tend to be non-linear.

Between 1950 and 2010, Latin American (LA) countries underwent major social, cultural, and, principally, economic changes, which, in some time periods, have pushed the countries closer towards each other, while in others, under different circumstances, have pulled the countries further apart from one another. These actions of coming together and drifting apart are what we generally call economic convergence and divergence. The analytical and empirical challenge resides in understanding and explaining the interplay of these processes. In analytic terms, recent developments in regional economic theory, drawing on the neoclassical genre, the New Keynesian school of thought and the New Economic Geography, argue that the dichotomy of convergence versus divergence must be updated to account for the coexistence of the two processes at the same time. From an empirical perspective, the strategy proposed consists of combining the concepts of economic convergence and divergence with the econometric approach of the panel data unit root and/or stationarity hypothesis test to determine whether the Gross Domestic Product (GDP) per capita of LA countries follows a stationary (convergent) or random path process (divergent). But if the assumption is that LA economies have experienced economic changes that have prompted processes of convergence-divergence or divergence-convergence, then the econometric methodology to use would be the non-linear panel data unit root and/or stationarity hypothesis test.

The literature on long-term economic growth generally acknowledges that during the time period 1950-1980, LA was one of the most developed regions outside of the industrial world (Elson, 2005), with growth potential similar to that of the economies of Spain, Italy, and South Korea (Barboni and Treibich, 2010). However, this potential failed to solidify due to a combination of political, religious and human capital-related factors whose consequence was a structural change that led these countries to diverge from the other economies mentioned (Barboni and Treibich, 2010). In particular, it is known that the 1981-1982 crisis was a key turning point and that the “lost decade” was characterized by low growth in LA. Other milestones of the changing regime included the 1990s, when average growth was rather low. The next decade (2000-2010) witnessed higher economic growth rates alongside greater variability (Solimano and Soto, 2003).

Diverse conclusions have been drawn from the debate about convergence and divergence in LA over the past 60 years: Astorga (2010) contended, after analyzing the behavior of Argentina, Brazil, Chile, Colombia, Mexico, and Venezuela in the past century (1900-2000), that these countries tended towards economic and social convergence, principally because of similarities in their patterns of industrialization, urbanization and public provision, although this same process of convergence was not found for other countries in the region. Martín-Mayoral (2008), in turn, studied the disparities between countries in South America, Central America (not Belize), and Mexico during the period 1950-2008, and found, later on, a process of accelerated conditional convergence with different stationary states that can be primarily explained by the savings rate/public investment and spending.

Various studies have also analyzed the process of convergence using the unit root concept and method to test stochastic convergence and/or cointegration, comparing lead economies both within and outside of LA. For example, Holmes (2006) evaluated the hypothesis of convergence in eight Latin American countries in the time period 1900-2003, using the Markovian regime switching model to define the concepts of partial convergence (change from stationary to non-stationary regime) and varied convergence (degree of persistence). Holmes found that there was a change in process from a stationary process of convergence to another non-stationary or divergent process, which could also be identified as the existence of two different stationary regimes. Cermeño and Llamosa (2007), meanwhile, used the approach of Bernard and Durlauf (1995) to analyze potential convergence processes in six countries: Canada, the United States, Mexico, Argentina, Brazil, and Chile, for the period 1950-2000; in their cointegration analysis to compare LA countries and the United States, in the restricted and unrestricted versions, or absolute and conditional convergence, Cermeño and Llamosa did not find strong evidence, but rather weak evidence, of convergence for the pairs Argentina-United States, Chile-United States, and Brazil-Argentina. Another scholar, Escobario (2011) analyzed 19 countries in the time period 1945-2000; his unit root analyses compare pairs of countries with the same methodology employed by Bernard and Durlauf (1995) and found a process of convergence between the Dominican Republic and Paraguay; looking at groups of countries, he found that there was greater evidence of convergence between Central American and Caribbean countries than between South American economies. Finally, Rodríguez et al. (2012) studied the hypothesis of convergence between 17 LA countries and the United States economy in the period 1970-2010, using unit root and cointegration tests with panel data and finding no evidence for absolute convergence, but some evidence for conditional convergence.

What is interesting about the majority of these studies is that, despite the fact that Latin American countries and the structural changes that took place between the 1980s and the present are highly heterogeneous, the analytical approach is constructed by assuming that the phenomena of economic growth of GDP per capita are independent from one another. That is, they identify, on the one hand, historical moments in the process of convergence, or, on the other, divergence, sidestepping the need to explain the processes of convergence and divergence with a single non-linear model.

Unlike those studies, the specific contribution of this paper lies in the fact that it analyzes the regional convergence hypothesis in LA using a non-linear growth model for the period 1950-2010 (Bayeart and Camacho, 2008). The method chosen combines three approaches: the threshold model, the panel data unit root test, and the calculation of critical values using bootstrapping simulation. Besides this introduction, this article includes four sections: the first sketches out the hypothesis of conditional convergence with a linear panel data model; the second introduces the methodology to prove convergence with non-linear panel data; and the third provides the results of the non-linear panel data convergence tests, while the fourth offers some conclusions.

** CONDITIONAL CONVERGENCE IN LINEAR PANEL DATA**

Many studies have employed techniques with time series, some of which include the following: Linden (2000) analyzed the set of countries in the Organization for Economic Cooperation and Development (OECD), administering the pair-wise unit root tests ADF and KPSS, and found convergence only for Norway, Sweden, and the United Kingdom; Amable and Juillard (2000) applied the same tests to a sample of 53 countries with the result that the ADF test almost never confirmed convergence, with the exception of Denmark and Germany; Camarero, Flôres and Tamarit (2000) analyzed the Mercosur countries with pair-wise ADF tests and panel data models and found evidence of convergence in some countries; Easterly, Fiees and Lederman (2003) studied the hypothesis of convergence between Mexico and the United States with the Johansen test and found evidence of conditional convergence; Cheung and Pascual (2004) analyzed the G-7 countries using pair-wise ADF tests and found no evidence of convergence; finally, Cermeño and Llamosas (2007) used both the restricted and unrestricted version of model (2) to prove the hypothesis of convergence of GDP per capita in six emerging countries with respect to the United States, and administered cointegration tests under possible structural change pursuant to the approach of Gregory and Hansen (1996); their results suggest that in the majority of cases, there was no evidence of convergence in the presence of structural change, and that income gaps per capita in the countries studied * vis-à-vis* the United States are consistent with non-convergence.

The majority of these studies draw on basic economic growth models to prove convergence and transforms them in the framework of panel data integration and cointegration models. The concept of convergence typically employed is that of β-convergence. We say that there is β-convergence between countries or regions when there is a negative relationship between the per capita income growth rate and the initial value of income per capita, which means that the poorest countries grow faster than the wealthiest countries (* cf.* Barry and Sala-i-Martin, 1992; Mankiw * et al.*, 1992; Quah, 1993; Barro and Zavier Sala-i-Martin, 2004). In the 1990s, a series of studies analyzed the relationship between the income per capita growth rate and various living standards metrics in cross-sections of the population to investigate the growth process. These studies used a model in the following form:

(1) |

Where *g _{n}* is the growth rate at the country level,

The vast majority of studies that have employed equation (1) have disregarded the underlying patterns of heterogeneity in the data by using a regression model that is identical for all countries in the sample. Some have used dummy variables for LA or Sub-Saharan Africa to control for the differences in the growth processes of these groups of countries. However, this is insufficient to capture the statistics of the groups in the data set. In this regard, Bernard and Durlauf (1995) evaluated the possibility of convergence using the following model:

(2) |

Where * y _{nt}* is per capita income in the country in question, ӯ is average per capita income among the countries, and α

If absolute convergence is found, one simple and direct way to prove it would be to calculate the natural log of the difference between per capita income in the country in question and per capita income of the lead or reference country.

(3) |

Based on this series, the null hypothesis of non-convergence can be formulated as:

(4) |

This can be done using unit root tests. This version of the test is known as the restricted version. According to Cheung and García (2004), proving the null hypothesis established in equation (4) can bias the results towards the acceptance of the hypothesis of non-convergence due to the limited power of the unit root test, which is why they propose evaluating the convergence hypothesis in the following manner:

(5) |

If it is not possible to reject equations (4) and (5) at the same time, then the data cannot provide evidence to accept or reject the convergence hypothesis. The unrestricted version of the test does not * a priori *assume that model (2) will be used to estimate the parameters α_{n} and β. In this version of the test, the hypothesis of non-convergence is evaluated by applying the unit root test to the errors estimated in this model. From this approach, the null hypothesis states that there is no cointegration between per capita income of the country in question and the lead economy. Moreover, one advantage of this version of the test is that it is possible to determine whether the constant is significant or not, and therefore show evidence of conditional convergence, as well as verify whether the vector (1, -1) of the restricted model is fulfilled or not.

The test described in equation (2) to prove the hypothesis of convergence between two countries can be extended for a panel data model that encompasses a set of countries in the following fashion:

(6) |

Where *y _{nt}* is per capita income of country

** Methodology to Prove Convergence in Non-Linear Panel Data: Threshold Autoregressive (TAR) Models ^{1} **

Pursuant to economic growth models and linear panel data convergence tests, the methodology used here employs ideas set forth by Beyaert and Camacho (2008) and the tests conducted by Evans and Karras (1996), who used the following specification to prove the hypothesis of convergence with panel data:

(7) |

With * n* =1 and * t* = 1, … *T*. The subscripts _{n } and _{t } refer to units and time, respectively. The variable *g _{nt}* is defined as , where and

However, the process of convergence is not uniform. Some economies converge only when certain institutional, political, or economic circumstances are in place, and if they do not come about, there will be divergence. In other words, it may be that 0 < −ρ* _{n}* > 1 is met for all of the economies or regions considered in the sample under certain conditions, but that ρ

(8) |

Where *I _{{x}}* is an indicator function that takes the value of 1 when

According to Beyaert and Camacho (2008), in model (8), there is divergence if for all *n*; total convergence if for all *n* and *I = I, II*, and partial convergence if but for all *n* and . The variable *z* in equation (8) is known as the transition variable and can be either endogenous or exogenous: in the procedures described by Beyaert and Camacho (2008), it is estimated endogenously and so that is the criteria to which we adhere here. These same authors proposed estimating the transition variable using the following:

(9) |

For some *m*and some , where *m*and *d *are not fixed *a priori*, but are rather also determined endogenously. In this way, *z _{t}* can be stationary if the economies converge (all of them and for all the regimes) or not (for some regime or some regimes). Economically speaking, this means that the transition from one regime to another is related to the economic growth rate

Although it is possible to choose a *p *that is sufficiently high to make ε_{nt} distributed as normal with median zero and variance equal to one (white noise) for each *n*, it is not possible to exclude the possibility of contemporaneous correlation among the cross-section economies of the panel. This is crucial in economic terms, because even if the shocks are not serially correlated, it is likely that the converging economies are affected by these same shocks. Based on these assumptions the error matrix ε is not diagonal and will likely have the following structure:

(10) |

With and for all of *t*. Because the structure of matrix Ω is unknown, the model proposed in equation (10) is estimated using the feasible generalized least squares (FGLS) method. In this estimation process, the restriction is imposed, such that no regime exists in less than the fraction π_{1} of the total sample. Beyaert and Camacho set the value of π_{1} at around 0.10 and 0.15; if π_{1} falls below this limit, the linear model is preferred.

Now that the non-linear model of Beyaert and Camacho (equation (8)) has been estimated, we must prove that it is superior to the linear model from Evans and Karras proposed in equation (7). If the non-linear model is superior, the next step is to prove convergence in the coefficients ρ of equation (8); if there is evidence of convergence, we proceed to determine whether it is absolute or conditional using the coefficients δ of the same equation.

From this perspective of linearity, the null hypothesis to prove is that model (7) is more suitable than the alternative model (equation (8)). The problem here is that pursuant to the conventional statistics tests—the likelihood ratio, the Wald test, or LM tests—they do not follow the standard distribution under the null hypothesis, given that some of the parameters (known as δ, * m*, and* d*) are not identified under the null hypothesis, but they are under the alternative hypothesis. To overcome this problem, Beyaert and Camacho (2008) suggest a procedure similar to that proposed by Hansen (1999) and by Caner and Hansen (2001) in the single-equation TAR model that consists of obtaining critical values through bootstrapping simulations. The model used by Beyaert and Camacho (2008) consists precisely of extending the solution to the panel data TAR model. In this way, the idea is to prove the following hypothesis:

(11) |

For all *n* = 1,…,N and for all *i* = 1,…,p against the alternative that not all of the coefficients are equal in both regimes. In this way, model (7) is estimated by FGLS and model (8) by the FGLS grid method. Subsequently, for each model we calculate the value of the likelihood function at a point of estimation and we obtain L* _{12}* = -2 ln(L

In this way, the null hypothesis of linearity is rejected if L* _{12}* is relatively large. The critical values for L

If model (8) is more suitable, then the next step is to prove convergence over divergence, with the following null hypothesis:

(12) |

in equation (8). If it is not possible to reject the hypothesis proposed in equation (12), then the conclusion is that there is divergence in both regimes. The alternative hypotheses that are of economic interest are derived from equation (12):

(13a) | |

(13b) | |

(13c) |

These are interpreted as follows: alternatives (13b) and (13c) mean that convergence is present only under regime I or regime II, respectively. Beyaert and Camacho (2008) refer to the case in which the null hypothesis is rejected in favor of one of these two alternative hypotheses as "partial convergence." It should be noted that in meeting the hypothesis, either null or alternative, it is assumed that the coefficients ρ satisfy the same property for all of the economies in a specific time, which is consistent with the ideas that the panel data series *g _{nt}* are all stationary on the order of zero or one:

Aiming to discriminate among the three alternative hypotheses proposed in equation (13), Beyaert and Camacho (2008) suggest the use of various statistics, one of which is a Wald-type test to prove the alternative hypothesis *H _{ A, 2a'}* of total convergence. The statistic is:

(15) |

Where *t _{I}* and

Finally, to conclude the analysis of convergence, we must determine whether there is absolute or conditional convergence. Looking at model (8), under the hypothesis that , and , there is absolute convergence if , and . On the contrary, if the convergence process takes place in only one of the regimes, for example, in regime *I*, then there will be absolute convergence in this regime if , . Beyaert and Camacho (2008) mention the possibility that in the two-regime model there could be an interesting case of evidence of global convergence, that is, if for all *n* and * i*, but that in only one regime; in this case, we would say that there is absolute convergence in one regime and conditional convergence in the other. The statistics proposed by Beyaert and Camacho are based on the grid FGLS estimation model in equation (8). Analogously to the other cases, the statistics proposed to prove these hypotheses are extensions of the single-variable TAR model proposed by Evans and Karras (1996) for the linear case. The * t * statistics are given by with * i* = *I, II* and * n * = 1, …, *N*, associated with the values estimated for the constant terms, given by:

and |

Beyaert and Camacho (2008) argued that due to the endogenous nature of the transition variable, the p values of the bootstrapping method could be obtained by adjusting the linear model to the observed data. The decision-making rules are as follows:

1. If *H _{0, 2}* is rejected in favor of H

1.1 If is sufficiently large, then there is conditional convergence in both regimes.1.2 If is sufficiently large, but is not, then there is evidence of conditional convergence in regime I and absolute convergence in regime II.

1.3 If is sufficiently large but is not, then there is conditional convergence in regime II and absolute convergence in regime I.

2. Or, if *H _{0, 2}* is rejected in favor of

2.1 If is sufficiently large, then conditional convergence is present in regime I (II).

2.2 If is not sufficiently large, then there is absolute convergence in regime I (II).

** RESULTS**

This section presents the results of applying the Beyaert and Camacho (2008) method to the per capita GDP of a set of 17 LA countries in the time period 1950-2010. The procedure we followed is equivalent to that suggested by Beyaert and Camacho (2008), in which we considered * a priori *a subset of regions or countries with homogenous per capita income where it was thought that there might be convergence. We progressively added more countries and replicated the tests in turn; based on this assertion, we initially applied the test to a subset of countries.

Aiming to determine which countries or regions should be included in the first group subjected to the test, we initially included the countries in the region that could be classified as high per capita income by selecting countries above the average per capita GDP of the region in 2010. These countries were: Argentina, Brazil, Chile, Costa Rica, Mexico, Panama, Uruguay, and Venezuela. The data are annual and were incorporated into the tests as natural logs multiplied by 100. The per capita GDP data in natural logs for the period of study for the eight countries selected pursuant to this criterion are shown in Figure 1. The evolution of GDP per capita of the entire group of countries shows a slight trend towards convergence in the later years, with the exception of Brazil, which seemed to distance itself, principally in the final years of the sample.

The results of the tests conducted with this group, called the “higher” income per capita group, are shown in Table 1. Panel 1(* a*) displays the results of the linear model tests, that is, the results of the Evans and Karras (1996) test, modified with bootstrapping,^{2} while the second panel, 1(* b*) presents the results of the TAR model proposed in equation (8). Contrary to what was expected, the results of the linear model did not permit the rejection of the null hypothesis of divergence with a p-value of 0.1250, meaning that in this case, the absolute convergence versus conditional convergence test in the linear model was not applicable. Looking at the results of the TAR model, the linearity tests carried out did not permit the rejection of the null hypothesis that the linear model is the correct model either, by virtue of the fact that both tests, the restricted and unrestricted model, coincide with this result with p-values of 0.9999 in both cases.

Figure 1. Natural Log of Annual GDP Per Capita in the Eight Wealthiest Latin American Countries

Source: Created by the authors based on data from the Penn World Tables Version 7.1.

The time periods corresponding to each regime, as well as the position of the transition variables, are shown in Figure 2, which clearly shows that Regime 1 dominated.

In this way, the “higher” per capita GDP group in the region, consisting of Argentina, Brazil, Chile, Costa Rica, Mexico, Panama, Uruguay, and Venezuela, did not show a stable stationary trajectory or constant trajectory for some periods during the studied time.

In addition, as a matter of interest, we conducted a prior analysis of all LA countries. The evolution of per capita GDP in these countries is provided in Figure 2, and it is clear that there is a trend towards convergence among all of the countries in the region considered. The results of the tests conducted for all of the LA countries are displayed in Table 2. Pursuant to the results of the linear model applied to the countries in the region considered, it was not possible to reject the null hypothesis of divergence of the group in question at a significance level 5%, although it was possible at 10%. Even so, similar to the criteria used by Beyaert and Camacho (2008), we chose to conduct the analysis at the most conservative level of significance. Just as for the group of “higher income” countries, the linearity tests carried out with the TAR model revealed that the linear model was superior to the non-linear model, meaning that the rest of the tests under the linear model were not applicable to this group. As such, the results obtained for LA countries challenge the results of previous studies carried out using linear tests, which have maintained that there is evidence of some type of convergence in the region, whether absolute or conditional, because the tests carried out using the Evans and Karras (1996) model with bootstrapping point to a situation in which, far from signs of convergence, there is rather evidence of divergence among all the countries considered in the time period of the study.

Figure 2. Threshold Variable: Colombia (d=2) for the “Highest” Income Level Group

Source: Created by the authors based on data from the Penn World Tables Version 7.1.

Figure 3. Natural Log of Annual Per Capita GDP of the (17) Countries of Latin America: 1951-2010 (Multiplied by 100)

Source: Created by the authors based on data from the Penn World Tables Version 7.1.

Figure 4. Threshold Variable: Mexico (d=2) for All Latin American Countries

Source: Created by the authors based on data from the Penn World Tables Version 7.1.

** CONCLUSIONS**

This paper analyzed the hypothesis of per capita GDP convergence in a sample of LA countries in the period 1950-2010 using a variety of linear and non-linear panel data estimation methods, a methodology proposed by Beyaert and Camacho (2008).

Unlike the linear specifications of the models commonly used to prove convergence, the methodology employed here is a modification of the Evans and Karras (1996) method with bootstrapping simulations, which makes it more robust and reliable. The non-linear model used in this work belongs to the class of TAR models with two regimes and not only permits us to extend the TAR model to panel data models, but also adds to non-linearity the possibility of being non-stationary, which is attributable to the presence of a unit root in the series of the panel considered. This property is what makes the analytical approach described here to prove convergence or divergence in a group of regions or countries relevant, because, as is known, if the differences in per capita GDP in a group of countries with respect to the lead economy—which in this case is the cross-section average of the group—are stationary, then the economies considered converge; otherwise, they diverge, that is, if they possess unit roots. Upon applying this methodology to the study of a set of LA countries with “higher” per capita income levels, we found, in general terms, that the linear tests did not detect any type of convergence in the group. In this way, the test did not identify a lead group of countries with high per capita income that would have converged within the region. A similar result was found among all the countries in the region, when taken into account in our analysis.

Broadly speaking, in the two groups considered, we found evidence of divergence in the linear test model, which consisted of the Evans and Karras (1996) test enhanced with bootstrapping, because in both cases, the linearity tests suggest that the linear model is superior to the non-linear model. Because there was no evidence of convergence for the “higher” income group in the region, there was no possible group of countries whose average could act as the lead economy, and in this way, we were also unable to determine if the rest of the countries are followers of this average that would be considered the "lead economy" of the region.

These results are relevant because they cast doubt on the evidence reported in other studies conducted in the LA region that have found evidence of conditional convergence. The tests that we carried out using the Evans and Karras (1996) model with bootstrapping rather document a process of divergence among the group of "higher" income countries, as well as for all the countries in the region taken as a whole. In this way, we did not find evidence of absolute or conditional convergence in either LA country group, or in sub-periods of the time interval studied here. This also leads us to question the possibility that convergence clubs exist in LA, unlike the evidence found for other regions, such as in the Eurozone.

** APPENDIX: INDICATORS AND SOURCES USED**

The databases used in this study are the per capita GDP series at constant 2005 prices of the countries considered in LA for the time period 1950-2010, a series with the code “rgdpl” from Penn World Table Version 7.1: Alan Heston, Robert Summers and Bettina Aten, Penn World Table Version 7-1, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania, July 2012. Available at: https://pwt.sas.upenn.edu/php_site/pwt71/pwt71_form.php.

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^{*} School of Higher Education in Economics of the National Polytechnic Institute and the Post-Graduate Studies Division of the Faculty of Economics of the UNAM, Mexico. E-mail addresses: domr@economia.unam.mx, mendozag@unam.mx, iph@unam.mx, respectively.

^{1} The econometric method described in this section is based in large part on Beyaert and Camacho (2008). An application of the same methodology for Mexico can be found in Rodríguez, Mendoza and Venegas (2015).

^{2} In all cases, the bootstrapping was carried out with 1,000 repetitions.

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PROBLEMAS DEL DESARROLLO. REVISTA LATINOAMERICANA DE ECONOMÍA, Volume 49, Number 195 October-December 2018 is a quarterly publication by the Universidad
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