Educational Attainment and Regional Economic ... - Springer Link

International Advances in Economic Research (2005)11:231–242 DOI: 10.1007/s11294-005-3018-5

* IAES 2005

Educational Attainment and Regional Economic Performance in Mexico ARMANDO ARELLANO* AND THOMAS M. FULLERTON, JR.**

Abstract A growing number of studies confirm the importance of educational attainment and human capital investment as a means for improving per capita income performance. In developing countries, attention to this linkage has primarily been carried out using national data aggregates. For relatively large countries such as Mexico, it is helpful to conduct similar analyses that document regional market income patterns. This paper utilizes 2000 census data for all 31 states and the Federal District in Mexico City to quantify regional income performance. Similar to other studies conducted using regional data in higher income economies, results confirm strong links between education and incomes across Mexico. (JEL R11)

Introduction Widespread international recognition exists for the positive correlation between education and incomes. It is generally agreed that education leads to improved human capital, and the latter translates into higher productivity [Becker, 1993]. As economies develop, education tends top play a more critical role in raising worker productivity. While policy analysts and government officials understand this relationship, they often do not have access to specific estimates of the income gains that can be associated with labor force improvement. In particular, regional estimates of the quantitative impacts of greater educational achievement are often lacking. Mexico is geographically large. It includes 31 states plus a special Federal District in which Mexico City, the national capital, is located. Formal years of schooling has increased for the country as a whole, going from 2.76 years per person in 1960 to 6.72 years per person in 1990 [Lo ´pez-Acevedo, 2001]. While the national trend is encouraging, regional income and education patterns vary widely across the country. This paper estimates how income performance is enhanced in Mexico as a consequence of greater educational attainment. Subsequent sections are as follows. A review of the literature is provided in the next section. The data and methodology are discussed in the third section. Empirical outcomes are then summarized in separate sections for parameter estimation and simulation * Universidad Auto ńoma de Ciudad Jua ´rez — Mexico and ** University of Texas at El Paso — U.S.A. Work on this article was completed while Fullerton was a Visiting Professor at Helsinki School of Economics in Finland. Partial financial support was provided by El Paso Electric Company, El Paso Metropolitan Planning Organization, Southwest Center for Environmental Research and Policy Grant W-04-03, and Wells Fargo Bank of El Paso. Helpful comments were provided by Ce´sar Olivas, Santiago Ibarreche, Cely Ronquillo, Martha Patricia Barraza, Eduardo Mendoza, Roberto Tinajero, and an anonymous referee.

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results. Concluding remarks and suggestions for future research are presented in the final section. Literature Review Early studies to emphasize the role of formal schooling in human capital development include Shultz [1961] and Becker [1993]. Those efforts underscore the importance of education as a means of productivity enhancement and incomes improvement. Subsequent efforts examine a large number issues with respect to education and economic performance. National, as well as international, evidence points to numerous benefits that are correlated with greater levels of educational achievement [Welch, 1970; Barro, 1991; Mankiw et al., 1991; Miller et al., 1995]. Although the existence of national labor markets insures that some level of income effect homogeneity is likely to be observed, regional earnings profiles still exhibit a large degree of variability in many countries. Accordingly, much of the recent research on this topic has been conducted for regional and metropolitan economies [Sloboda, 1999; Levernier et al., 2000; Fullerton, 2001; Gottlieb and Fogarty, 2003]. Studies in this category often focus their efforts on lower income regions and use model simulations to illustrate the potential gains from policies designed to encourage higher enrollment and graduation rates. Multilateral poverty eradication efforts have led to similar studies for Latin American economies such as Mexico in recent years [Lachler, 1998; Lo ´pez-Acevedo, 2001]. One recent effort examined the education and income link for a single metropolitan economy [Ghiara and Zepeda, 2001]. Two separate efforts examine regional income variation across the country [Messmacher-Linartas, 2000; Paga ń and TijerinaGuajardo, 2000]. In Messmacher-Linartas [2000], the return to education is found to have increased in recent years as a consequence of shifts in the demand for labor toward secondary and tertiary segments of the Mexican economy. Because of their respective dates of publication, none of the regional income studies for Mexico was able to utilize the data that became available following the publication of the 2000 census results [INEGI, 2003]. While all of the prior research on regional income variations in Mexico confirms a statistically significant relationship between education and earnings, all of them are based on average years of schooling. To date, there has not been any effort in Mexico to examine the relationship between incomes and completed levels of education. This study attempts to partially fill that gap in the literature by employing the more detailed state level census data collected in 2000. The model specification is based on those developed in earlier efforts published for regional economies in the United States [Sloboda, 1999; Fullerton, 2001]. In addition to parameter estimation, model simulations are utilized to estimate potential gains associated with improved educational attainment. Data and Methodology Educational attainment data for all 31 states and the Federal District in Mexico City are collected from the 2000 census [INEGI, 2003]. The information is taken from the census tables for the percentage distributions of the population in Mexico of 15 years of age or older. A fairly wide variety of information is provided regarding the formal schooling achievements of citizens. Data collected for the analysis include the percentage of persons who have not attended school, the percentage of persons with incomplete primary school education, the percentage of persons who have completed primary school,

REGIONAL ECONOMIC PERFORMANCE IN MEXICO

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and the percentage of persons with at least some secondary school or higher level of education. Other data collected include population, gross state product (GSP), and population per square kilometer for each of the 32 political jurisdictions that jointly comprise the sample. GSP is used in lieu of personal income due to data availability at the state level in Mexico. It is measured in 1993 real pesos [INEGI, 2003]. Per capita GSP is calculated by dividing total GSP by the population of each respective state. That variable is used as the dependent variable in the regression analysis summarized below. Readers should note that many, if not all, of the variables employed in the study can also be downloaded directly from the INEGI web site http://www.inegi.gob.mx). Population density enters the analysis as an explanatory variable in addition to the educational variables described above. It is widely recognized that agglomeration effects cause metropolitan economies exhibit higher than average levels of productivity within national and regional economies [Glaeser, 1992, 1998]. Population density has been successfully deployed as a proxy variable to capture this effect in a wide variety of national and regional earnings models [Ciccone and Hall, 1996; Fullerton, 2001; Wheeler, 2003]. Parameter estimation is accomplished by minimizing the sum of squared residuals using ordinary least squares. To avoid perfect multicollinearity, at least one or more of the educational attainment variables has to be eliminated from any specification that is employed. Sample population densities range from 6 persons/km2 in Baja California to 5,799 persons/km2 in Mexico City. Because of the wide variation in population densities across Mexico, heteroscedasticity is tested using the White [1980] procedure. Once parameter estimation is completed, simulation exercises are conducted to examine the potential gains associated with improving educational achievements in different regions of Mexico. Steps involved with each simulation are fairly easy to carry out. A percentage point increase in the variable of interest is multiplied by the regression coefficient that is estimated for that particular explanatory variable. Because the dependent variable is measured in real pesos per capita, each result is also multiplied by the population of its corresponding state in order to calculate aggregate regional GSP gains.

Estimation Results All data used in the empirical analysis are reported in Table 1. In general, the information conforms to the null hypothesis of a positive relationship between education and productivity. Real per capita GSP is lowest in Oaxaca, a state where more than 45 percent of the population age 15 or older has not completed schooling through the sixth grade. Conversely, the highest per capita GSP occurs in the Mexico City Federal District where more than 72 percent of the population age 15 or older has completed studies beyond primary school. Mexico City also has the highest population density among all 32 geopolitical regions in the sample. To avoid matrix singularity, the percentage of persons 15 years or older who have no formal schooling is excluded from the regression equation. The resulting equation specification is shown in Equation (1). PCGSPi ¼ b0 þ b1 PINi þ b2 PCMi þ b3 PLUSi þ b4 URBi þ ei ;

ð1Þ

where i is the subscript for each state in Mexico, PCGSP is real per capita gross state product in 1993 pesos, PIN is percent of population age 15 or older with incomplete

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TABLE 1 State Data State Aguascalientes Baja California Baja California Sur Campeche Coahuila Colima Chiapas Chihuahua Durango Guanajuato Guerrero Hidalgo Jalisco Mexico City, DF Me´xico, State Michoaca ń Morelos Nayarit Nuevo Leo ń Oaxaca Puebla Quere´taro Quintana Roo San Luis Potosı´ Sinaloa Sonora Tabasco Tamaulipas Tlaxcala Veracruz Yucata ń Zacatecas

PCGSP 17,959 19,361 18,644 23,056 20,006 15,193 6,394 21,622 12,426 10,734 7,842 9,400 14,972 38,903 12,107 8,762 13,331 8,971 26,522 6,339 9,967 18,088 22,350 11,092 11,855 18,249 9,145 16,269 8,304 8,795 11,945 8,358

URB 168 35 6 12 15 96 53 12 12 152 48 107 80 5,799 586 68 318 33 60 37 148 120 21 38 44 12 76 34 241 96 42 18

NOS 6.0 6.4 6.4 12.3 4.9 8.7 23.1 5.8 6.6 14.9 21.7 14.1 8.2 3.6 7.2 16.0 10.4 10.6 4.3 20.5 14.1 11.6 8.1 12.0 9.5 6.1 9.1 6.3 7.8 15.2 11.1 9.2

PIN 17.1 13.2 14.6 21.8 13.9 18.5 27.2 17.4 22.2 20.9 20.3 20.0 18.5 8.6 13.6 24.1 15.4 21.4 12.2 25.0 21.1 14.5 17.0 22.1 20.6 16.3 23.2 17.1 15.6 24.1 25.7 28.3

PCM 21.3 18.0 17.4 18.2 20.5 18.2 17.5 24.4 23.1 23.4 17.4 20.3 21.7 15.5 19.4 20.2 17.3 16.4 17.1 20.9 21.5 20.9 17.2 19.1 17.2 16.5 19.2 19.4 23.4 18.6 16.7 23.2

PLUS 55.6 62.4 61.6 47.7 60.7 54.6 32.2 52.4 48.1 40.8 40.6 45.6 51.6 72.3 59.8 39.7 56.9 51.6 66.4 33.6 43.3 53.0 57.7 46.8 52.7 61.1 48.5 57.2 53.2 42.1 46.5 39.3

Notes: Sample data from 2000 census. PCGSP, Real per capita gross state product in 1993 pesos; URB, state population density per square kilometer; NOS, percentage of population age 15 or older with no formal schooling; PIN, percentage of population age 15 or older with incomplete primary schooling; PCM, percentage of population age 15 or older that completed primary school; PLUS, Percentage of population age 15 or older with studies beyond primary school.

primary schooling, PCM is percent of population age 15 or older that completed primary school, PLUS is percent of population age 15 or older with studies beyond primary school, URB is state population density per square kilometer, and e is a random disturbance term. The coefficient estimated for the population density variable, URB, is hypothesized to be greater than zero. The sign of the constant is ambiguous and can be either positive or

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negative. Because the percentage of persons age 15 or older with no formal schooling is the variable excluded from the analysis, the slope coefficients for the educational attainment explanatory variables are hypothesized to be greater than zero. Furthermore, each successive attainment regressor is expected to be larger than its predecessor and cause per capita GSP to increase by progressively higher amounts. Parameter estimation results appear in Table 2. Overall goodness of fit is satisfactory, with an adjusted coefficient of determination of 0.684 and an F-statistic that is significant at the 1 percent level. The White test fails to reject the homoscedasticity null hypothesis and it is not, therefore, necessary to re-estimate the covariance matrix [Pindyck and Rubinfeld, 1998]. All of the slope coefficients exhibit the hypothesized positive arithmetic signs. The parameter estimates for education beyond the sixth grade and population density both satisfy the 5 percent significance criterion. Similar to other regional studies [Sloboda, 1999; Levernier et al., 2000; Fullerton, 2001], multicollinearity affects a subset of the slope coefficient t-statistics. Both of the t-statistics for the incomplete primary school and the completed elementary school regressors fail to meet the 5 percent significance criterion. Experimentation with different specifications confirms, however, that these variables do raise per capita productivity in a statistically reliable manner. In addition to exhibiting to hypothesized arithmetic signs, the parameter magnitudes for each of the regressors conform with prevailing economic logic and all can be reasonably included in the model [McCloskey and Ziliak, 1996].

TABLE 2 Real Per Capita GSP Parameter Estimation Results Variable Constant PIN PCM PLUS URB R2

Coefficient j14,269.78 29.064 97.017 504.322* 2.368* 0.725

Adjusted R2

0.684

Std. error regression Error sum squares

3933.802 4.18108

Log likelihood White heterosced.

j307.563 4.793998

Standard Error 20,104.80 377.250 334.565 193.159 0.794 Dependent variable mean Dep. var. std. deviation F-statistic F-statistic probability Observations White chi square prob.

t-Statistic j0.710 0.077 0.290 2.611 2.984

Probability 0.484 0.939 0.774 0.015 0.006 14,581.28 6,994.936 17.754** 0.000 32 0.779

Notes: Sample data from 2000 census. PCGSP, Real per capita gross state product in 1993 pesos, dependent variable; URB, state population density per square kilometer; PIN, percentage of population age 15 or older with incomplete primary schooling; PCM, percentage of population age 15 or older that completed primary school; PLUS, percentage of population age 15 or older with studies beyond primary school. *Statistical significance at 5% level. **Statistical significance at 1% level.

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The interpretation of the slope parameters is fairly straightforward. For each additional percentage point of the population who attends at least some primary school, real per capita GSP rises by slightly more than 29 pesos. For each additional percentage point increase in the number of persons who successfully complete elementary school, real per capita GSP rises by approximately 97 pesos. For each additional percentage point gain in the number of persons who study beyond the sixth grade, real per capita GSP in Mexico grows by more than 504 pesos. Similarly, each one-person increase in the number of persons per square kilometer is associated with a 2.37 real peso increment in per capita GSP. These estimates conform with earlier studies regarding the positive impacts of education on productivity. They do not, however, answer all questions regarding improved income performance in Mexico. The regressors utilized do not provide any information regarding quality of education across the 32 states included in the sample. The nature of the census data reported in 2000 force the model to assume uniform educational quality among all regions. The latter assumption is probably unrealistic in a country as large as Mexico that faces numerous tax base difficulties in rural areas. It should also be noted that while improved educational attainment would apparently help improve regional income performance in Mexico, the data collected do not permit estimating the impacts associated with different levels of post-primary educational achievement. Those data will eventually become available and additional testing along the lines of Sloboda [1999] and Fullerton [2001] will be useful. Similarly, the material at hand also fails to account for regional differences in capital investment, public infrastructure, and government policies. Additional analysis incorporating those factors would also be helpful in examining regional growth patterns in Mexico. The estimated model does, however, permit answering some questions regarding the potential gains from greater educational achievement via the simulations conducted in the next section [Polzin, 1990; Tomek, 1993].

Simulation Results To examine the potential gains from additional educational achievement, model simulations are conducted for the states whose 2000 schooling profiles lagged behind the national averages reported by INEGI [2003]. Also examined are the improvements in real GSP that would result from making at least partial school attendance universal in all regions of Mexico. Outcomes from the four simulation exercises are summarized in Tables 3 through 6. Table 3 calculates the implied real GSP gains that would result from raising to the national average the percentage of the state population that attends elementary school without completing the entire six-year curriculum. The estimates are conducted for all states whose percentages for this variable fall below the national average in 2000. The largest per capita impact occurs in the northern state of Baja California, more than 142 pesos per person. With just under 2.49 million people, the per capita gain for Baja California translates into a statewide gain of approximately 354 million pesos. For the state of Mexico, the most populous region in the country, the aggregate effect of raising to the national average the percentage of residents with at least some primary schooling exceeds 1.7 billion pesos. By bringing up to the national average those 12 states with low percentages of residents with at least some elementary school experience, the national economy would increase its total output of goods and services by slightly more than 3 billion pesos.


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TABLE 3 Implied Real GSP Gains from Increased Partial Primary School Attendance Rates State Aguascalientes Baja California Baja California Sur Campeche Coahuila Colima Chiapas Chihuahua Durango Guanajuato Guerrero Hidalgo Jalisco Mexico City, DF Me´xico, State Michoaca ń Morelos Nayarit Nuevo Leo ń Oaxaca Puebla Quere´taro Quintana Roo San Luis Potosı´ Sinaloa Sonora Tabasco Tamaulipas Tlaxcala Veracruz Yucata ń Zacatecas

Per Capita Peso Gain 29.1 142.4 101.7 NC 122.1 NC NC 20.3 NC NC NC NC NC NC 130.8 NC 78.5 NC NC NC NC 104.6 32.0 NC NC 52.3 NC 29.1 72.7 NC NC NC

Population 944,285 2,487,367 424,041 NC 2,298,070 NC NC 3,052,907 NC NC NC NC NC NC 13,096,686 NC 1,555,296 NC NC NC NC 1,404,306 874,963 NC NC 2,216,969 NC 2,753,222 962,646 NC NC NC

Total State Gain (Million Pesos) 27 354 43 NC 281 NC NC 62 NC NC NC NC NC NC 1,713 NC 122 NC NC NC NC 147 28 NC NC 116 NC 80 70 NC NC NC

Notes: Population data are from the 2000 census. All estimates are calculated in 1993 inflation-adjusted pesos.

Table 4 calculates the effects of raising to the national average, 18.1 percent, the percentage of residents who successfully complete elementary school. The biggest per capita gains are tallied in Nayarit and Sonora, more than 280 real pesos per person. A northern border state, Nuevo Leo ń, experiences the greatest total impact, more than 850 million 1993 pesos. For the 16 states as a whole, raising their collective primary school completion rates to the national average would increase gross domestic product (GDP) in Mexico by more than 5.7 billion real pesos. Implied gains from raising to the national average the percentage of students who study past the end of the sixth grade are summarized in Table 5. Remarkably large

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TABLE 4 Implied Real GSP Gains from Increased Primary School Completion Rates State Aguascalientes Baja California Baja California Sur Campeche Coahuila Colima Chiapas Chihuahua Durango Guanajuato Guerrero Hidalgo Jalisco Mexico City, DF Me´xico, State Michoaca ń Morelos Nayarit Nuevo Leo ń Oaxaca Puebla Quere´taro Quintana Roo San Luis Potosı´ Sinaloa Sonora Tabasco Tamaulipas Tlaxcala Veracruz Yucata ń Zacatecas

Per Capita Peso Gain NC 135.8 194.0 116.4 NC 116.4 184.3 NC NC NC 194.0 NC NC NC NC NC 203.7 291.1 223.1 NC NC NC 213.4 29.1 213.4 281.3 19.4 NC NC 77.6 261.9 NC

Population NC 2,487,367 424,041 690,689 NC 542,627 3,920,892 NC NC NC 3,079,649 NC NC NC NC NC 1,555,296 920,185 3,834,141 NC NC NC 874,963 2,299,360 2,536,844 2,216,969 1,891,829 NC NC 6,908,975 1,658,210 NC

Total State Gain (Million Pesos) NC 338 82 80 NC 63 723 NC NC NC 598 NC NC NC NC NC 317 268 856 NC NC NC 187 67 541 624 37 NC NC 536 434 NC


impacts are tallied for the two most impoverished states, Chiapas and Oaxaca. These two southern states each observe real per capita GSP increases of more than 9,300 pesos. They are joined by the eastern state of Veracruz as regions whose aggregate GSP growth would exceed 30 billion 1993 pesos. For the national economy as a whole, raising the percentage of persons with at least some post-primary educational studies to the national would collectively lead to real national GDP increments of more than 217 billion pesos. Although illiteracy is not the problem it once represented in Mexico, there are still relatively large numbers of people across the country that have never attended school. In


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TABLE 5 Implied Real GSP Gains from Increased Post-Primary School Attendance Rates State Aguascalientes Baja California Baja California Sur Campeche Coahuila Colima Chiapas Chihuahua Durango Guanajuato Guerrero Hidalgo Jalisco Mexico City, DF Me´xico, State Michoaca ń Morelos Nayarit Nuevo Leo ń Oaxaca Puebla Quere´taro Quintana Roo San Luis Potosı´ Sinaloa Sonora Tabasco Tamaulipas Tlaxcala Veracruz Yucata ń Zacatecas

Per Capita Peso Gain NC NC NC 2,269.4 NC NC 10,086.4 NC 2,067.7 5,749.3 5,850.1 3,328.5 302.6 NC NC 6,304.0 NC 302.6 NC 9,380.4 4,488.5 NC NC 2,723.3 NC NC 1,866.0 NC NC 5,093.7 2,874.6 6,505.8

Population NC NC NC 690,689 NC NC 3,920,892 NC 1,448,661 4,663,032 3,079,649 2,235,591 6,322,002 NC NC 3,985,667 NC 920,185 NC 3,438,765 5,076,686 NC NC 2,299,360 NC NC 1,891,829 NC NC 6,908,975 1,658,210 1,353,610

Total State Gain (Million Pesos) NC NC NC 1,567 NC NC 39,548 NC 2,995 26,809 18,016 7,441 1,913 NC NC 25,126 NC 278 NC 32,257 22,787 NC NC 6,262 NC NC 3,530 NC NC 35,192 4,767 8,806


fact, there are 15 states in which double-digit rates of non-attendance occur in the 2000 census. Table 6 examines the potential effects of making at least partial school participation universal in Mexico. In the three southern states of Chiapas, Guerrero, and Oaxaca, real per capita GSPs rise by more than 595 pesos. The largest aggregate gains are observed in the states of Chiapas, Mexico, and Veracruz, more than 2.6 billion 1993 pesos in each case. For national output as a whole, achieving universal primary school attendance translates into more than a 29.7 billion real peso GDP gain. In 2000, real GDP in Mexico totaled 338.1 billion pesos. According to the model simulation in

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TABLE 6 Implied Real GSP Gains from Universal Partial Primary School Attendance State Aguascalientes Baja California Baja California Sur Campeche Coahuila Colima Chiapas Chihuahua Durango Guanajuato Guerrero Hidalgo Jalisco Mexico City, DF Me´xico, State Michoaca ń Morelos Nayarit Nuevo Leo ń Oaxaca Puebla Quere´taro Quintana Roo San Luis Potosı´ Sinaloa Sonora Tabasco Tamaulipas Tlaxcala Veracruz Yucata ń Zacatecas

Increase (%) 6 6 6 12 5 9 23 6 7 15 22 14 8 4 7 16 10 11 4 21 14 12 8 12 10 6 9 6 8 15 11 9

Per Capita (Pesos) 174 186 186 357 142 252 670 168 191 432 629 409 238 104 209 464 302 307 125 595 409 336 235 348 276 177 264 183 226 441 322 267

Population 944,285 2,487,367 424,041 690,689 2,298,070 542,627 3,920,892 3,052,907 1,448,661 4,663,032 3,079,649 2,235,591 6,322,002 8,605,239 13,096,686 3,985,667 1,555,296 920,185 3,834,141 3,438,765 5,076,686 1,404,306 874,963 2,299,360 2,536,844 2,216,969 1,891,829 2,753,222 962,646 6,908,975 1,658,210 1,353,610

State Gain (Million Pesos) 164 462 79 246 327 137 2,627 513 277 2,015 1,938 914 1,503 898 2,735 1,849 469 283 478 2,044 2,076 472 206 800 699 392 499 503 218 3,045 534 361


Table 6, universal partial education attendance would have resulted in an aggregate output figure that is 8.8 percent higher than what actually occurred. Higher output performances, such as those illustrated in Tables 3 through 6 are difficult to achieve. Better quality human capital is one way to make them possible. The benefits shown in the tables almost assuredly exceed the costs associated with obtaining them. Given the limited tax bases observed across Mexico, public policies designed to increase primary and secondary school enrollments (and graduation rates) offer attractive gains that would go a long way toward improving economic conditions throughout the country.


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Conclusion Formal education has been identified as an effective means for increasing the stock of human capital in an economy. In turn, human capital improvements lead to higher productivity and improved income performance. Recent studies have quantified the effects of this progression for regional economies in different parts of the world. To date, however, state level data from the 2000 census in Mexico had not been utilized in this context. Estimation results reported above are broadly consistent with other studies in this branch of the literature. Simulation exercises conducted with the model detail the per capita and aggregate real GSP gains that can be achieved by increasing school attendance and graduation rates for the different state economies in Mexico. Those results indicate that regional economies across the nation are, from a productivity standpoint, operating well below capacity. Raising educational performance in Mexico will go a long ways toward strengthening municipal, state, and federal tax bases. Results obtained here suggest that policymakers in Mexico should strive to increase educational attainment throughout the nation. The data utilized do not permit simulating the effects of greater high school attendance and graduation rates, or of increased college attendance and graduation. Eventual examination of those types of data for Mexico would be useful. While this initial study is helpful, the 2005 census in Mexico will generate additional data that can be used to confirm the results obtained. If nothing else, the 2005 data will provide additional degrees of freedom by doubling the number of sample observations. Replication of these empirical findings for other regions of Latin America and other developing economies elsewhere would be useful. While similar outcomes are possible, there is no reason to anticipate they will be identical to those discussed in this paper.

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