# Equal Access to University Education in Chile? An Application Using Spatial Heckman Probit Models

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Determinants of Access to Higher Education

#### 2.2. The Chilean Higher Education Admissions System

- The students had to pass the PSU, organized by the “Departamento de Evaluación, Medición y Registro Educacional” (DEMRE) of the Universidad de Santiago. To pass, they have to obtain a minimum score of 475 points out of 850.
- Once they pass the PSU, prospective students must decide whether to apply to the universities that belong to the Unified Admission System (SUA). Historically, only traditional universities used this system. In 2011, non-traditional universities were allowed to participate after evaluation by the Council of Chilean University Vice-Chancellors (CRUCH) to determine whether they met the necessary quality standards.
- After submitting their application, students received an admission decision, based on their PSU score.

## 3. Data and Variables

#### 3.1. Data Source and Descriptive Statistics

#### 3.1.1. Students’ Characteristics

#### 3.1.2. Location Factors

#### 3.1.3. Localized Social Capital

#### 3.2. Exploratory Data Analysis

#### 3.2.1. Higher Education System Design: Selection—Application—Admission

#### 3.2.2. Geography of Access to Higher Education: Distances and Neighborhoods

## 4. Estimation Strategy

#### 4.1. Heckman Probit Models

#### 4.1.1. Baseline Model 1

_{i}is observed, ${\mathsf{\Phi}}_{2}\left(\xb7\right)$ is the cumulative bivariate normal distribution function (with mean [0 0]′), $\mathsf{\Phi}\left(\xb7\right)$ is the standard cumulative normal, and ${w}_{i}$ is an optional weight for observation i.

#### 4.1.2. Baseline Model 2

#### 4.2. Heckman Probit Models with Spatial Effects

#### 4.2.1. Endogeneity Issues and Spatial Autocorrelation Test of the Residuals

**W**is the familiar n × n spatial weights matrix, which reflects the vicinity relations among the n spatial observations, where the main diagonal is equal to zero by convention; and ${\tilde{\sigma}}_{{Q}_{n}^{*}}$ is a normalizing factor [54]. The generalized residual values of the Heckit model are calculated as follows:

#### 4.2.2. A Spatial Heckit Model

**W**is the spatial weights matrix indicating “nearest neighbors”, $W{x}_{1i}$ are the spatial lagged variables representing local “spillovers”, and γ

_{1}is an additional vector of unknown parameters to capture interaction spatial effects [53].

## 5. Estimation Results

#### 5.1. Baseline Models

- Baseline Model 1
- Main equation:$$\mathrm{Pr}\left(\mathrm{APPLICATION}=1|\mathrm{PRE}-\mathrm{SELECTION}=1\right)\phantom{\rule{0ex}{0ex}}={\beta}_{0}+{\beta}_{1}\mathrm{FEMALE}+{\beta}_{2}\mathrm{Log}\left(\mathrm{LIT}\_\mathrm{SCORE}\right)\phantom{\rule{0ex}{0ex}}+{\beta}_{3}\mathrm{Log}\left(\mathrm{MATH}\_\mathrm{SCORE}\right)+{\beta}_{4}\mathrm{Log}\left(\mathrm{GRADE}\_\mathrm{PTS}\right)\phantom{\rule{0ex}{0ex}}+{\beta}_{5}\mathrm{Log}\left(\mathrm{DISTANCE}\right)+{\beta}_{6}{\left[\mathrm{Log}\left(\mathrm{DISTANCE}\right)\right]}^{2}\phantom{\rule{0ex}{0ex}}+{\beta}_{7}\mathrm{WORKING}+{u}_{1}$$
- Selection equation:$$\mathrm{Pr}\left(\mathrm{PRE}-\mathrm{SELECTION}=1\right)\phantom{\rule{0ex}{0ex}}={\gamma}_{0}+{\gamma}_{1}\mathrm{FEMALE}+{\gamma}_{2}\mathrm{Log}\left(\mathrm{GRADE}\_\mathrm{PTS}\right)\phantom{\rule{0ex}{0ex}}+{\gamma}_{3}\mathrm{RURAL}+{\gamma}_{4}\mathrm{Log}\left(\mathrm{DISTANCE}\right)+{\gamma}_{5}\mathrm{WORKING}\phantom{\rule{0ex}{0ex}}+{\gamma}_{6}SIBL\_UNIV+{\gamma}_{7}\mathrm{SOCIAL}\_\mathrm{CAP}+{u}_{2}$$

^{2}”.

- Baseline Model 2
- Main equation:$$\mathrm{Pr}\left(\mathrm{ADMISSION}=1|\mathrm{APPLICATION}=1\right)\phantom{\rule{0ex}{0ex}}={\beta}_{0}+{\beta}_{1}\mathrm{FEMALE}+{\beta}_{2}\mathrm{Log}\left(\mathrm{LIT}\_\mathrm{SCORE}\right)\phantom{\rule{0ex}{0ex}}+{\beta}_{3}\mathrm{Log}\left(\mathrm{MATH}\_\mathrm{SCORE}\right)+{\beta}_{4}\mathrm{Log}\left(\mathrm{GRADE}\_\mathrm{PTS}\right)+{u}_{1}$$
- Selection equation:$$\mathrm{Pr}\left(\mathrm{APPLICATION}=1\right)\phantom{\rule{0ex}{0ex}}={\gamma}_{0}+{\gamma}_{1}\mathrm{FEMALE}+{\gamma}_{2}\mathrm{Log}\left(\mathrm{LIT}\_\mathrm{SCORE}\right)\phantom{\rule{0ex}{0ex}}+{\gamma}_{3}\mathrm{Log}\left(\mathrm{MATH}\_\mathrm{SCORE}\right)+{\gamma}_{4}\mathrm{Log}\left(\mathrm{GRADE}\_\mathrm{PTS}\right)\phantom{\rule{0ex}{0ex}}+{\gamma}_{5}\mathrm{RURAL}+{\gamma}_{6}\mathrm{Log}\left(\mathrm{DISTANCE}\right)+{\gamma}_{7}{\left[\mathrm{Log}\left(\mathrm{DISTANCE}\right)\right]}^{2}\phantom{\rule{0ex}{0ex}}+{\gamma}_{8}\mathrm{WORKING}+{\gamma}_{9}\mathrm{SOCIAL}\_\mathrm{CAP}+{u}_{2}$$

^{2}”, “WORKING”, and “SOCIAL_CAP” as exclusion criteria (instruments) that correlate with selection (“APPLICATION”) but not with the binary outcome in the main equation (“ADMISSION”).

#### 5.2. Spatial Models

#### 5.2.1. Specification of the Spatial Weights Matrix

**W**. More specifically, based on the addresses of all the students, we performed Thiessen polygonization to define the spatial contiguity within the neighborhood. The most frequent number of neighbors was three, covering around 41% of the total number of students. Taking this into consideration, we specified three different

**W**matrices: (i) dispersed matrix (few neighbors), 3 neighbors; (ii) dense matrix (more neighbors), 100 neighbors; (iii) very dense matrix, 300 neighbors.

**W**matrices, and the dense matrix shows the most significant z-value.

#### 5.2.2. SLX Heckit Model Results

**W**

_{300}).

- Spatial Model 1
- Main equation:$$\mathrm{Pr}\left(\mathrm{APPLICATION}=1|\mathrm{PRE}-\mathrm{SELECTION}=1\right)\phantom{\rule{0ex}{0ex}}={\beta}_{0}+{\beta}_{1}\mathrm{FEMALE}+\cdots +{\beta}_{8}{W}_{300}\mathrm{Log}\left(\mathrm{GRADE}\_\mathrm{PTS}\right)\phantom{\rule{0ex}{0ex}}+{\beta}_{10}{W}_{300}\mathrm{WORKING}+{u}_{1}$$
- Selection equation:$$\mathrm{Pr}\left(\mathrm{PRE}-\mathrm{SELECTION}=1\right)\phantom{\rule{0ex}{0ex}}={\gamma}_{0}+{\gamma}_{1}\mathrm{FEMALE}+\cdots +{\gamma}_{8}{W}_{300}SIB{L}_{UNIV}\phantom{\rule{0ex}{0ex}}+{\gamma}_{9}{W}_{300}\mathrm{SOCIAL}\_\mathrm{CAP}+{u}_{2}$$

- Spatial Model 2
- Main equation:$$\mathrm{Pr}\left(\mathrm{ADMISSION}=1|\mathrm{APPLICATION}=1\right)\phantom{\rule{0ex}{0ex}}=\mathrm{No}\text{}\mathrm{spatially}\text{}\mathrm{lagged}\text{}\mathrm{variables}\text{}\mathrm{included}$$
- Main equation:$$\mathrm{Pr}\left(\mathrm{APPLICATION}=1\right)\phantom{\rule{0ex}{0ex}}={\gamma}_{0}+{\gamma}_{1}\mathrm{FEMALE}+\cdots +{\gamma}_{10}{W}_{300}\mathrm{WORKING}\phantom{\rule{0ex}{0ex}}+{\gamma}_{11}{W}_{300}\mathrm{SOCIAL}\_\mathrm{CAP}+{u}_{2}$$

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variable | Description | Coding | Year |
---|---|---|---|

Mother’s and father’s education level | No information | 0 | 2016 |

Illiterate | 1 | ||

Incomplete primary education | 2 | ||

Complete primary education | 3 | ||

Incomplete secondary education | 4 | ||

Complete secondary education | 5 | ||

Other studies | 6 | ||

Incomplete technical school education | 7 | ||

Complete technical school education | 8 | ||

Incomplete high school education | 9 | ||

Complete high school education | 10 | ||

Incomplete university education | 11 | ||

Complete university education | 12 | ||

Household monthly income (USD) | 0–213 | 1 | 2016 |

213–425 | 2 | ||

425–638 | 3 | ||

638–851 | 4 | ||

851–1064 | 5 | ||

1064–1276 | 6 | ||

1276–1489 | 7 | ||

1489–1702 | 8 | ||

1702–1914 | 9 | ||

1914–2127 | 10 | ||

2127–2340 | 11 | ||

2340+ | 12 | ||

Secondary school type | Public school | 1 | 2016 |

Subsidized school | 2 | ||

Private school | 3 |

Item | Obs. | Sign | Item–Test Correlation | Item–Rest Correlation | Average Inter–Item Correlation | Cronbach’s Alpha |
---|---|---|---|---|---|---|

Father’s education | 260,775 | + | 0.787 | 0.598 | 0.420 | 0.685 |

Mother’s education | 260,775 | + | 0.768 | 0.567 | 0.439 | 0.701 |

Family income | 260,775 | + | 0.799 | 0.616 | 0.408 | 0.674 |

School types | 260,775 | + | 0.700 | 0.463 | 0.509 | 0.756 |

Test scale | 0.444 | 0.762 |

Social Capital Model | ||||
---|---|---|---|---|

Coefficient | Z | OIM S.E. | ||

Father’s education | Factor score (CS) | 1 | - | - |

Constant | 5.032 *** | 677.7 | 0.007 | |

Mother’s education | Factor score (CS) | 0.850 *** | 307.9 | 0.003 |

Constant | 5.536 *** | 828.0 | 0.007 | |

Family income | Factor score (CS) | 0.774 *** | 258.9 | 0.003 |

Constant | 4.349 *** | 702.1 | 0.006 | |

High school type | Factor score (CS) | 0.116 *** | 204.4 | 0.001 |

Constant | 1.770 *** | 1427.6 | 0.001 | |

Var (Father’s Education) | Constant | 6.338 *** | 214.4 | 0.030 |

Var (Mother’s Education) | Constant | 5.858 *** | 247.2 | 0.024 |

Var (Family Income) | Constant | 5.190 *** | 239.4 | 0.022 |

Var (School Type) | Constant | 0.292 *** | 309.8 | 0.001 |

Var (CS) | Constant | 8.037 *** | 187.2 | 0.043 |

## References

- Espinoza, O. Creating (in) Equalities in Access to Higher Education in the Context of Structural Adjustment and Post-Adjustment Policies: The Case of Chile. High. Educ.
**2008**, 55, 269–284. [Google Scholar] [CrossRef] - Koljatic, M.; Silva, M.; Cofré, R. Achievement versus Aptitude in College Admissions: A Cautionary Note Based on Evidence from Chile. Int. J. Educ. Dev.
**2013**, 33, 106–115. [Google Scholar] [CrossRef] - OECD. World Bank Access and Equity. In Reviews of National Policies for Education: Tertiary Education in Chile 2009; OECD Publishing: Paris, France, 2009; pp. 73–121. [Google Scholar] [CrossRef]
- Speirs, N.M. Solidarity and Subsidiarity—How to Widen Access to Higher Education? In Social Responsibility and Sustainability: How Businesses and Organizations Can Operate in a Sustainable and Socially Responsible Way; Filho, W.L., Ed.; World Sustainability Series; Springer International Publishing: Cham, Germany; Hamburg, Germany, 2019; pp. 123–137. ISBN 978-3-030-03562-4. [Google Scholar]
- Millett, C. Designing Sustainable Funding for College Promise Initiatives. ETS Res. Rep. Ser.
**2017**, 2017, 1–55. [Google Scholar] [CrossRef] [Green Version] - Mitchell, M.; Leachman, M.; Masterson, K.; Waxman, S. Unkept Promises: State Cuts to Higher Education Threaten Access and Equity; Center for Budget and Policy Priorities, Virginia Tech: Washington, DC, USA, 2018; p. 23. [Google Scholar]
- Zarifa, D.; Hango, D.; Pizarro Milian, R. Proximity, Prosperity, and Participation: Examining Access to Postsecondary Education among Youth in Canada’s Provincial North. Rural Sociol.
**2018**, 83, 270–314. [Google Scholar] [CrossRef] - DesJardins, S.L.; Dundar, H.; Hendel, D.D. Modeling the College Application Decision Process in a Land-Grant University. Econ. Educ. Rev.
**1999**, 18, 117–132. [Google Scholar] [CrossRef] - Aguirre, J.; Matta, J. Walking in Your Footsteps: Sibling Spillovers in Higher Education Choices. Econ. Educ. Rev.
**2021**, 80, 102062. [Google Scholar] [CrossRef] - Kingdon, G.G. The Private Schooling Phenomenon in India: A Review. J. Dev. Studies
**2020**, 56, 1795–1817. [Google Scholar] [CrossRef] [Green Version] - McDonough, I.K.; Roychowdhury, P.; Dhamija, G. Measuring the Dynamics of the Achievement Gap Between Public and Private School Students During Early Life in India. J. Labor Res.
**2021**, 42, 78–122. [Google Scholar] [CrossRef] - Zuilkowski, S.S.; Piper, B.; Ong’ele, S.; Kiminza, O. Parents, Quality, and School Choice: Why Parents in Nairobi Choose Low-Cost Private Schools over Public Schools in Kenya’s Free Primary Education Era. Oxf. Rev. Educ.
**2018**, 44, 258–274. [Google Scholar] [CrossRef] - Ngware, M.W.; Mutisya, M. Demystifying Privatization of Education in Sub-Saharan Africa: Do Poor Households Utilize Private Schooling Because of Perceived Quality, Distance to School, or Low Fees? Comp. Educ. Rev.
**2021**, 65, 124–146. [Google Scholar] [CrossRef] - UNESCO, G.E.M.R. Education for All 2000-2015: Achievements and Challenges; EFA Global Monitoring Report. 2015. Available online: https://unesdoc.unesco.org/ark:/48223/pf0000232205 (accessed on 22 December 2021).
- Lira, J. Mind the Gap: Irrevocable Wage Differentials in Chile; Universidad de Chile: Santiago, Chile, 2008. [Google Scholar]
- Molina, A.; Lamb, S. School Segregation, Inequality and Trust in Institutions: Evidence from Santiago. Comp. Educ.
**2021**, 58, 72–90. [Google Scholar] [CrossRef] - Guzmán, P.; Cifuentes Gomez, G.; Santelices, M.V. Secondary Students’ Expectations on Transition to Higher Education. Educ. Res.
**2021**, 63, 164–179. [Google Scholar] [CrossRef] - Von Hippel, P.T.; Hofflinger, A. The Data Revolution Comes to Higher Education: Identifying Students at Risk of Dropout in Chile. J. High. Educ. Policy Manag.
**2021**, 43, 2–23. [Google Scholar] [CrossRef] - Ioannides, Y.M.; Topa, G. Neighborhood Effects: Accomplishments and Looking Beyond Them*. J. Reg. Sci.
**2010**, 50, 343–362. [Google Scholar] [CrossRef] - Ivemark, B.; Ambrose, A. Habitus Adaptation and First-Generation University Students’ Adjustment to Higher Education: A Life Course Perspective. Sociol. Educ.
**2021**, 94, 191–207. [Google Scholar] [CrossRef] - Dickerson, A.; McIntosh, S. The Impact of Distance to Nearest Education Institution on the Post-Compulsory Education Participation Decision. Urban Studies
**2013**, 50, 742–758. [Google Scholar] [CrossRef] [Green Version] - Butler, T.; Hamnett, C. The Geography of Education: Introduction. Urban Studies
**2007**, 44, 1161–1174. [Google Scholar] [CrossRef] - Zarifa, D.; Seward, B.; Milian, R.P. Location, Location, Location: Examining the Rural-Urban Skills Gap in Canada. J. Rural Studies
**2019**, 72, 252–263. [Google Scholar] [CrossRef] - Hango, D.; Zarifa, D.; Pizarro Milian, R.; Seward, B. Roots and STEMS? Examining Field of Study Choices among Northern and Rural Youth in Canada. Studies High. Educ.
**2021**, 46, 563–593. [Google Scholar] [CrossRef] - Weiler, W.C. Transition from Consideration of a College to the Decision to Apply. Res. High. Educ.
**1994**, 35, 631–646. [Google Scholar] [CrossRef] - Helland, H.; Heggen, K. Regional Differences in Higher Educational Choice? Scand. J. Educ. Res.
**2018**, 62, 884–899. [Google Scholar] [CrossRef] - Buckner, E.; Khoramshahi, C. Does the Private Sector Expand Access to Higher Education? A Cross-National Analysis, 1999–2017. Int. J. Educ. Dev.
**2021**, 84, 102410. [Google Scholar] [CrossRef] - Wilson, J.Z.; Harvey, A.; Mendes, P. Changing Lives: Improving Care Leaver Access to Higher Education. Oxf. Rev. Educ.
**2019**, 45, 573–586. [Google Scholar] [CrossRef] - Doolan, K.; Puzić, S.; Baranović, B. Social Inequalities in Access to Higher Education in Croatia: Five Decades of Resilient Findings. J. Furth. High. Educ.
**2018**, 42, 467–481. [Google Scholar] [CrossRef] - Amorim, J.P. Mature Students’ Access to Higher Education: A Critical Analysis of the Impact of the 23+ Policy in Portugal. Eur. J. Educ.
**2018**, 53, 393–413. [Google Scholar] [CrossRef] [Green Version] - Verdis, A.; Kalogeropoulos, K.; Chalkias, C. Regional Disparities in Access to Higher Education in Greece. Res. Comp. Int. Educ.
**2019**, 14, 318–335. [Google Scholar] [CrossRef] - Türk, U. Socio-Economic Determinants of Student Mobility and Inequality of Access to Higher Education in Italy. Netw. Spat. Econ.
**2019**, 19, 125–148. [Google Scholar] [CrossRef] [Green Version] - Prakhov, I.; Sergienko, D. Matching between Students and Universities: What Are the Sources of Inequalities of Access to Higher Education? Eur. J. Educ.
**2020**, 55, 261–274. [Google Scholar] [CrossRef] - Chea, P. Does Higher Education Expansion in Cambodia Make Access to Education More Equal? Int. J. Educ. Dev.
**2019**, 70, 102075. [Google Scholar] [CrossRef] - Fadhil, I.; Sabic-El-Rayess, A. Providing Equity of Access to Higher Education in Indonesia: A Policy Evaluation. Indones. J. Learn. Adv. Educ. IJOLAE
**2021**, 3, 57–75. [Google Scholar] [CrossRef] - Bulbul, T. Socio-Economic Status and School Types as the Determinants of Access to Higher Education. Educ. Sci.
**2021**, 46, 303–334. [Google Scholar] [CrossRef] - Carr-Hill, R. Inequalities in Access to Higher Education in Africa: How Large Are They? Do They Mirror the Situation in the Metropole 60 Years Ago? Int. J. Educ. Dev.
**2020**, 72, 102122. [Google Scholar] [CrossRef] - Benavides, M.; León, J.; Galindo, C.; Herring, C. Access to Higher Education of Afro-Peruvians: Disentangling the Influence of Skin Color and Social Origins in the Peruvian Stratification System. Sociol. Race Ethn.
**2019**, 5, 354–369. [Google Scholar] [CrossRef] - Molina, O.; Santa María, D.; Yamada, G. What Stops Poor Girls from Going to College? Skill Development and Access to Higher Education in a Developing Country; IZA Discussion Papers 12052; Institute of Labor Economics (IZA): Bonn, Germany, 2018; Available online: http://hdl.handle.net/10419/193346 (accessed on 30 December 2021).
- Gaentzsch, A.; Zapata-Román, G. Climbing the Ladder: Determinants of Access to and Returns from Higher Education in Chile and Peru; UNRISD Working Paper 2020-2; United Nations Research Institute for Social Development (UNRISD): Geneva, Switzerland, 2020; Available online: http://hdl.handle.net/10419/246234 (accessed on 30 December 2021).
- Looker, E.D.; Lowe, G.S. Post-Secondary Access and Student Financial Aid in Canada: Current Knowledge and Research Gaps (2001); Canada Millennium Scholarship Foundation: Otawa, ON, Canada, 2021; Available online: https://doi=10.1.1.482.7571 (accessed on 30 December 2021).
- Finnie, R.; Mueller, R.E.; Wismer, A. Access and Barriers to Postsecondary Education: Evidence from the Youth in Transition Survey. Can. J. High. Educ.
**2015**, 45, 229–262. [Google Scholar] [CrossRef] - Coleman, J.S. Social Capital in the Creation of Human Capital. Am. J. Sociol.
**1988**, 94, S95–S120. [Google Scholar] [CrossRef] - Furstenberg, F.F.; Hughes, M.E. Social Capital and Successful Development among At-Risk Youth. J. Marriage Fam.
**1995**, 57, 580–592. [Google Scholar] [CrossRef] - Madrid, S. Diversidad sin Diversidad: Los Colegios Particulares Pagados de Elite y la Formación de la Clase Dominante en una Sociedad de Mercado. In Mercado Escolar y Oportunidad. Educacional. Libertad, Diversidad y Desigualdad; Corvalán, J., Carrasco, A., García-Huidobro, J.E., Eds.; Ediciones Universidad Católica de Chile: Santiago, Chile, 2016; pp. 269–299. [Google Scholar]
- Van de Ven, W.P.M.M.; Van Praag, B.M.S. The Demand for Deductibles in Private Health Insurance: A Probit Model with Sample Selection. J. Econom.
**1981**, 17, 229–252. [Google Scholar] [CrossRef] - Wooldridge, J.M. Econometric Analysis of Cross Section and Panel Data, 2nd ed.; MIT Press: Cambridge, MA, USA, 2010; ISBN 978-0-262-23258-6. [Google Scholar]
- Pastore, F. To Study or to Work? East. Eur. Econ.
**2012**, 50, 49–78. [Google Scholar] [CrossRef] - Ahlin, L.; Andersson, M.; Thulin, P. Human Capital Sorting: The “When” and “Who” of the Sorting of Educated Workers to Urban Regions. J. Reg. Sci.
**2018**, 58, 581–610. [Google Scholar] [CrossRef] - Morrissey, K.; Kinderman, P.; Pontin, E.; Tai, S.; Schwannauer, M. Web Based Health Surveys: Using a Two Step Heckman Model to Examine Their Potential for Population Health Analysis. Soc. Sci. Med.
**2016**, 163, 45–53. [Google Scholar] [CrossRef] [Green Version] - Anselin, L. GeoDa on Github. Available online: https://geodacenter.github.io/ (accessed on 30 December 2021).
- LeSage, J.; Pace, R.K. Introduction to Spatial Econometrics; Chapman and Hall/CRC: New York, USA, 2008; ISBN 978-1-4200-6424-7. [Google Scholar] [CrossRef] [Green Version]
- Kelejian, H.; Prucha, I.R. On the Asymptotic Distribution of the Moran I Test Statistic with Applications. J. Econom.
**2001**, 104, 219–257. [Google Scholar] [CrossRef] [Green Version] - Vella, F. Estimating Models with Sample Selection Bias: A Survey. J. Hum. Resour.
**1998**, 33, 127–169. [Google Scholar] [CrossRef] [Green Version] - Halleck Vega, S.; Elhorst, J.P. The Slx Model. J. Reg. Sci.
**2015**, 55, 339–363. [Google Scholar] [CrossRef] - Fischer, M.M.; Scherngell, T.; Reismann, M. Knowledge Spillovers and Total Factor Productivity: Evidence Using a Spatial Panel Data Model. Geogr. Anal.
**2009**, 41, 204–220. [Google Scholar] [CrossRef] [Green Version] - Chasco, C.; Le Gallo, J. Hierarchy and Spatial Autocorrelation Effects in Hedonic Models. Econ. Bull.
**2012**, 32, 1474–1480. [Google Scholar] - Anselin, L. Spatial Externalities, Spatial Multipliers, And Spatial Econometrics. Int. Reg. Sci. Rev.
**2003**, 26, 153–166. [Google Scholar] [CrossRef] - Anselin, L. The Moran Scatterplot as an ESDA Tool to Assess Local Instability in Spatial Association. In Spatial Analytical Perspectives on GIS; Fischer, M., Scholten, H.J., Unwin, D., Eds.; Routledge: London, UK, 1996; ISBN 978-0-203-73905-1. [Google Scholar] [CrossRef]
- Quiroz, J.L.; Peeters, L.; Aroca, P. Access the University. A Spatial Heckman Probit Model [Data Set]. Available online: https://b2share.eudat.eu/records/b434ff0201074090a9bea92563c8c59f (accessed on 5 January 2022).
- LeSage, J.P. What Regional Scientists Need to Know about Spatial Econometrics. Rev. Reg. Studies
**2014**, 44, 13–32. [Google Scholar] [CrossRef] - Miller, P.M. Mapping Educational Opportunity Zones: A Geospatial Analysis of Neighborhood Block Groups. Urban Rev.
**2012**, 44, 189–218. [Google Scholar] [CrossRef] - Hillman, N.W. Geography of College Opportunity: The Case of Education Deserts. Am. Educ. Res. J.
**2016**, 53, 987–1021. [Google Scholar] [CrossRef]

**Figure 4.**Effect of distance from students’ home to Santiago city on the probability of applying to university.

**Figure 5.**Moran’s I test on the model residuals for ${W}_{300}$. (

**a**) Baseline model; (

**b**) Spatial Model.

Variables | Description | Obs. | Mean | Std. Dev. | Min. | Max. |
---|---|---|---|---|---|---|

A: Indicator (dummy) variables—model-dependent variables: | ||||||

$\mathrm{PRE}-\mathrm{SELECTION}$ | Successful pre-selection test | 260,775 | 0.603 | 0.489 | 0 | 1 |

$\mathrm{APPLICATION}$ | Application for university place | 260,775 | 0.539 | 0.498 | 0 | 1 |

$\mathrm{ADMISSION}$ | Admission accepted | 260,775 | 0.379 | 0.485 | 0 | 1 |

B: Indicator (dummy) variables—model-independent variables: | ||||||

$\mathrm{FEMALE}$ | Female | 260,775 | 0.530 | 0.499 | 0 | 1 |

$\mathrm{WORKING}$ | Working | 260,775 | 0.098 | 0.298 | 0 | 1 |

$\mathrm{RURAL}$ | Rural origin area | 260,775 | 0.020 | 0.140 | 0 | 1 |

$\mathrm{SIBL}\_\mathrm{UNIV}$ | Siblings in university | 260,775 | 0.300 | 0.458 | 0 | 1 |

C: Continuous variables—model-independent variables | ||||||

$\mathrm{DISTANCE}$ | Distance (km) | 260,775 | 312.8 | 426.1 | 0.040 | 3772 |

$\mathrm{LIT}\_\mathrm{SCORE}$ | Score literature test | 260,775 | 506.0 | 109.8 | 150 | 850 |

$\mathrm{MATH}\_\mathrm{SCORE}$ | Score mathematics test | 260,775 | 505.7 | 109.4 | 150 | 850 |

$\mathrm{GRADE}\_\mathrm{PTS}$ | High school grade points | 260,775 | 544.5 | 98.9 | 238 | 826 |

D: Continuous latent variable—model instrumental variable | ||||||

$\mathrm{SOCIAL}\_\mathrm{CAP}$ | Social capital | 260,775 | 0.000 | 2.505 | −4.237 | 6.467 |

Baseline Model 1 | Baseline Model 2 | |||
---|---|---|---|---|

A: Main equations | Pr (APPLICATION = 1| PRE-SELECTION = 1) | Pr (ADMISSION = 1| APPLICATION = 1) | ||

FEMALE | 0.176 *** | (22.1) | −0.248 *** | (−35.6) |

Log (LIT_SCORE) | 2.054 *** | (60.6) | 0.886 *** | (34.9) |

Log (MATH_SCORE) | 2.488 *** | (67.8) | 1.257 *** | (50.1) |

Log (GRADE_PTS) | 0.625 *** | (16.0) | 0.585 *** | (25.3) |

WORKING | −0.155 *** | (−12.1) | - | - |

Log (DISTANCE) | 0.056 *** | (27.5) | - | - |

[Log (DISTANCE)]^{2} | −0.002 *** | (−15.3) | - | - |

Constant | −31.924 *** | (−82.6) | −16.118 *** | (−82.2) |

B: Selection equations | Pr (PRE-SELECTION = 1) | Pr (APPLICATION = 1) | ||

FEMALE | −0.256 *** | (−44.8) | 0.144 *** | (24.4) |

Log (LIT_SCORE) | - | - | 2.700 *** | (139.9) |

Log (MATH_SCORE) | - | - | 2.069 *** | (111.5) |

Log (GRADE_PTS) | 3.470 *** | (196.9) | 1.210 *** | (62.2) |

Log (DISTANCE) | −0.019 *** | (−28.9) | 0.041 *** | (28.4) |

[Log (DISTANCE)]^{2} | - | - | −0.001 *** | (−12.5) |

RURAL | −0.573 *** | (−29.2) | −0.170 *** | (−9.4) |

WORKING | −0.056 *** | (−6.2) | −0.078 *** | (−9.4) |

SIBL_UNIV | 0.170 *** | (26.7) | - | - |

SOCIAL_CAP | 0.196 *** | (138.7) | 0.028 *** | (23.0) |

Constant | −21.252 *** | (−193.9) | −37.220 *** | (−259.9) |

Athrho | −0.267 *** | (−13.8) | −2.261 *** | (−61.8) |

Rho | −0.261 *** | −0.979 *** | ||

No. of observations | 260,775 | 260,775 | ||

No. of censored observations | 103,398 | 120,256 | ||

No. of uncensored observations | 157,377 | 140,519 | ||

Likelihood Ratio test | 206.4 *** | 7603.0 *** |

Baseline Model | Nearest Neighbors | Moran’s I | z-Value | Pseudo p-Value |
---|---|---|---|---|

Model 1 | 3 | 0.082 | 61.0 | 0.001 |

100 | 0.074 | 276.8 | 0.001 | |

300 | 0.066 | 453.9 | 0.001 | |

Model 2 | 3 | 0.014 | 10.2 | 0.001 |

100 | 0.015 | 57.0 | 0.001 | |

300 | 0.011 | 73.1 | 0.001 |

Spatial Model 1 | Spatial Model 2 | |||
---|---|---|---|---|

A: Main equations | Pr (APPLICATION = 1| PRE-SELECTION = 1) | Pr (ADMISSION = 1| APPLICATION = 1) | ||

FEMALE | 0.177 *** | (22.3) | −0.248 *** | (−35.6) |

Log (LIT_SCORE) | 2.047 *** | (60.2) | 0.886 *** | (34.9) |

Log (MATH_SCORE) | 2.455 *** | (66.6) | 1.258 *** | (50.1) |

Log (GRADE_PTS) | 0.602 *** | (15.4) | 0.587 *** | (25.3) |

WORKING | −0.145 *** | (−11.3) | - | - |

Log (DISTANCE) | 0.052 *** | (24.1) | - | - |

[Log (DISTANCE)]^{2} | −0.002 *** | (−13.1) | - | - |

Spatial lag W_{300} WORKING | −0.799 *** | (−6.2) | - | - |

Spatial lag W_{300} Log (GRADE_PTS) | 0.367 ** | (3.2) | - | - |

Constant | −33.747 *** | (−40.1) | −16.138 *** | (−82.3) |

B: Selection equations | Pr (PRE-SELECTION = 1) | Pr (APPLICATION = 1) | ||

FEMALE | −0.256 *** | (−44.8) | 0.144 *** | (24.4) |

Log (LIT_SCORE) | - | - | 2.698 *** | (139.6) |

Log (MATH_SCORE) | - | - | 2.061 *** | (110.8) |

Log (GRADE_PTS) | 3.491 *** | (197.1) | 1.207 *** | (61.8) |

Log (DISTANCE) | −0.020 *** | (−30.2) | 0.040 *** | (26.1) |

[Log (DISTANCE)]^{2} | - | - | −0.001 *** | (−11.6) |

RURAL | −0.537 *** | (−27.4) | −0.168 *** | (−9.3) |

WORKING | −0.065 *** | (−7.2) | −0.076 *** | (−9.1) |

SIBL_UNIV | 0.160 *** | (25.1) | - | - |

SOCIAL_CAP | 0.175 *** | (114.4) | 0.0236 *** | (17.4) |

Spatial lag W_{300} WORKING | - | - | −0.396 *** | (−5.1) |

Spatial lag W_{300} SOCIAL_CAP | 0.048 *** | (10.5) | 0.020 *** | (7.3) |

Spatial lag W_{300} SIBL_UNIV | 1.256 *** | (18.2) | - | - |

Constant | −21.746 *** | (−193.2) | −37.099 *** | (−257.7) |

Athrho | −0.269 *** | (−13.5) | −2.252 *** | (−62.1) |

Rho | −0.262 *** | −0.978 *** | ||

No. of observations | 260,775 | 260,775 | ||

No. of censored observations | 103,398 | 120,256 | ||

No. of uncensored observations | 157,377 | 140,519 | ||

Likelihood Ratio test | 195.7 *** | 7600.6 *** | ||

Likelihood Ratio test—spatial [d.f. = 4]/[d.f. = 2] | 1795.6 *** | 97.0 *** |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Quiroz, J.L.; Peeters, L.; Chasco, C.; Aroca, P.
Equal Access to University Education in Chile? An Application Using Spatial Heckman Probit Models. *Mathematics* **2022**, *10*, 280.
https://doi.org/10.3390/math10020280

**AMA Style**

Quiroz JL, Peeters L, Chasco C, Aroca P.
Equal Access to University Education in Chile? An Application Using Spatial Heckman Probit Models. *Mathematics*. 2022; 10(2):280.
https://doi.org/10.3390/math10020280

**Chicago/Turabian Style**

Quiroz, Juan Luis, Ludo Peeters, Coro Chasco, and Patricio Aroca.
2022. "Equal Access to University Education in Chile? An Application Using Spatial Heckman Probit Models" *Mathematics* 10, no. 2: 280.
https://doi.org/10.3390/math10020280