Journal of Probability and Statistics

Volume 2019, Article ID 3435626, 10 pages

https://doi.org/10.1155/2019/3435626

## Parameter Evaluation for a Statistical Mechanical Model for Binary Choice with Social Interaction

Mathematics and Statistics Department, University of Energy and Natural Resources, P. O. Box 214, Sunyani, Ghana

Correspondence should be addressed to Alex Akwasi Opoku; hg.ude.rneu@ukopo.xela

Received 1 October 2018; Accepted 17 February 2019; Published 4 March 2019

Academic Editor: Aera Thavaneswaran

Copyright © 2019 Alex Akwasi Opoku et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper we use a statistical mechanical model as a paradigm for educational choices when the reference population is partitioned according to the socioeconomic attributes of gender and residence. We study how educational attainment is influenced by socioeconomic attributes of gender and residence for five selected developing countries. The model has a social and a private incentive part with coefficients measuring the influence individuals have on each other and the external influence on individuals, respectively. The methods of partial least squares and the ordinary least squares are, respectively, used to estimate the parameters of the interacting and the noninteracting models. This work differs from the previous work that motivated this work in the following sense: (a) the reference population is divided into subgroups with unequal subgroup sizes, (b) the proportion of individuals in each of the subgroups may depend on the population size , and (c) the method of partial least squares is used for estimating the parameters of the model with social interaction as opposed to the least squares method used in the earlier work.

#### 1. Introduction

Education provides people with the knowledge and skills that can lead to better employment opportunities and a better quality of life. The educational level attained by an individual explicitly determines the occupational choice of that individual. All these attainments and choices of individuals are made under certain socioeconomic conditions such as peers, neighbours, family members, wealth quintile of the individual, gender, residence, etc. Those who reside in the rural areas with the poorest wealth quintile are more likely to have low education [1]. Here our focus is on how the collective behaviour of a reference group of individuals is determined by the intricate interactions among the individuals [2–5]. Collective behaviour such as self-organization has been observed in biological, ecological, and socioeconomical systems [6–8].

The collective behaviour of a large group of individuals may undergo sudden changes due to slight variations in the socioeconomic structure of the group. For instance a change in the pronunciation of a language due to a small immigrant population and a substantial decrease in crime rate is a result of actions taking by the authorities [9, 10]. This abrupt change in macroscopic behaviour caused by the changes in interactions among constituents is referred to as phase transition from the statistical mechanics literature. Phase transition has been shown to exist for some classes of spin models designed to explain the phenomenon of ferromagnetism [11, 12]. The simplest spin model within this class is the mean-field Ising model proposed in [11]. This model is tractable and has seen several applications in social sciences [13], finance [14], chemistry [15], and ecology [16]. An interesting family, which has naturally emerged in applications, is a multispecies version of the mean-field Ising model for studying magnetism in anisotropic materials [17]. This model has seen social science applications in recent works of Contucci, Gallo, and Barra [18–20].

The above and the works in [2–5] highlight the need for importing statistical mechanical models into the social science to offer insights into how social interactions determine social outcomes. Multipopulation Curie-Weiss model serves as a paradigm for certain binary discrete choice where individuals are to choose between two options, say to stay in school or drop out of school, use medicated mosquito net while sleeping at night or not etc., subject to the constrains of their socioeconomic environments. Here our key assumption is that individuals with the same socioeconomic attributes tend to behave the same way but people with different socioeconomic attributes may behave differently [19].

The authors of [18–20] considered a multigroup Curie-Weiss model where the fraction of individuals in each subgroup of the population is a constant that is independent of the total population size. In particular, the work in [19] provides an estimation procedure to estimate the parameters in a multigroup Curie-Weiss model for suicidal tendencies and mode of marriage in Italy. The authors used least squares estimation procedure to estimate the parameters of their model. The study considered residence as the only socioeconomic attribute. In this work we adapt partial least squares estimation procedure to estimate the parameters for a multigroup Curie-Weiss model for educational attainment for five developing countries. Here we use the socioeconomic attributes of residence and gender. That is, we study how the collective choice of educational attainment of a group of individuals is influenced by their gender and place of residence. Our focus is on educational attainment of individuals in developing countries due to our interest in comparing the estimates from the different countries. Because of this we wanted countries that share similar socioeconomic attribute from different parts of the globe. We chose Dominican Republic from North America, Kenya from East Africa, Egypt from North Africa, Ghana from West Africa, and Indonesia from Asia. The choice of these countries is based on the availability of data from the demographic and health survey program.

The rest of the paper is organized as follows: Section 2 addresses generalities on Curie-Weiss model, its multipopulation version, and the parameter estimation procedure for the model. A case study on how educational choices is influenced by gender and residence is presented in Section 3. Section 4 discusses the main findings of the case study and we conclude the work in Section 5.

#### 2. The Curie-Weiss Model

The Curie-Weiss model is made up of an energy function (Hamiltonian) that assigns interaction energies to spin configurations. This energy function takes the formwhere = . The energy function consists of two parts, namely, the interaction part modulated by the interaction strengths and the external field part also controlled by . The interaction between neighbouring spins tends to induce alignment of the neighbours if ; i.e., both neighbours will prefer to be either or both if . Otherwise the neighbours will prefer to assume different spin values. The parameter is the external magnetic field applied to site . If the magnetic field is positive it favours spin values. On the other hand if the field is negative it favours spin values. Hence, for each site , the external field contributes to the energy function by a term of [25]. This model is of a mean-field type as each spin interacts with the rest of the spins. The effect of all the other spins on any given spin is approximated by the average effect of the rest, and this makes computations easy. In this paper we will use the Curie-Wiess model as a benchmark model for discrete choice with social interaction. More precisely, we are interested in how educational attainment choices of individuals from some selected countries are influenced by the interaction among the individuals and the socioeconomic attributes of gender and residence. For related applications of statistical mechanical models to social science we refer the reader to [2–5, 26].

##### 2.1. Multipopulation Curie-Weiss Model

This work uses partial least squares estimation procedure developed in [27] for the multipopulation Curie-Weiss model to estimate parameters in an interaction based logit model for educational attainment in some selected developing countries. Here our key assumption is that individuals with the same socioeconomic attributes tend to behave the same way but people with different socioeconomic backgrounds behave differently. This assumption helps us to reparametrise the parameters in the Hamiltonian (1). Thus we will focus on finding a suitable parametrisation for the interaction coefficient and a systematic procedure that allows us to estimate the parameters characterizing the model from data. From our discrete choice model each individual is assigned socioeconomic attributes,

Therefore a population of size can be partitioned into groups that do not overlap. Each of the groups is identified by one of the elements of . Let be the set of individuals in partition for and =. Therefore . Let with , for . We suppose further that, for each ,Our assumption above implies that individuals in the same group or partition are characterised by the same socioeconomic attribute; i.e., individuals with the same socioeconomic attributes are characterised as a group or partition. Therefore, it follows from our assumption above that all individuals in a partition or a group have the same private incentive and for any pair of groups and , for every and . It follows from this assumption and (1) thatHereis the average decision for the individuals in group or partition . Our model has now been changed from individual choices to group choices. Note that returns the level of satisfaction for the entire population. measures the influence group has on group and also it is the social incentive of groups and to interact. When is positive, then it implies that the groups are satisfied if their empirical means have the same signs; otherwise the empirical means of the groups prefer to have different signs; i.e., it is not encouraging or rewarding for the groups to interact with one another. See Figure 1 for an artistic impression of the ’s. is the private incentive of the group , describing how the group is satisfied with itself.