Construction of appropriate model for Enrolment Projection of India

Article History: Received: 10 November 2020; Revised: 12 January 2021; Accepted: 27 January 2021; Published online: 05 April 2021 Abstract: There are different models used for enrolment projection in the education system. Of them ARIMA, double exponential, logistic curve method, reconstructive cohort, and regression model, method are commonly used. Every model has its suitability under certain conditions and with prior assumptions. In India, no attempt was made to estimate enrolment except in 1996. The logistic curve method was adopted to estimate enrolment between 1996 – 2001. In the present situation, an attempt has been made to estimate school enrolment for the next years. The present paper discusses fitting the best model for estimating enrolment for the next few years.


Introduction
Every country has a unique school education system. India also has its school education system. It has fourlayer systems. From grade 1 to grade 5 is called as a primary stage, grade 6 to grade 9 is called as an upper primary stage, grade 9 and 10 is called as a secondary stage and grade 11 and 12 is called as a higher (or senior) secondary stage. All states and Union Territories of the country have adopted the same pattern of school education system. The country has a defined entry age in the school system that is 6 years of age. Once a child completes 6 years of age then s/he will be eligible to get enrolled in the school system at grade 1. Accordingly, the stage-wise age group is also defined by the Government of India, children in the age group of 6 to 10+ years, are supposed to be in the primary stage, children in the age group of 11 to 13+ years, are supposed to be in the upper primary stage, children in the age group of 14 to 15+ years, are supposed to be in secondary stage and children in the age group of 16 to 17+ years, are in higher (or senior) secondary stage. However, there are some exceptions in entry age of a child in the system.
Once the child gets enrolled in the school system, the flow of s/he movement is depends on her/his academic performance. Though, after the implementation of the Right To Education Act 2009, the country has adopted the No Detention Policy in the country at the Elementary stage that is in primary and upper primary stages (from grade 1 to 8). But 100% transition of the children from grade i to grade i+1 (1 ≤ i ≤ 7) is not happened. Always, there will be a number of dropouts or repeaters remain in the system. The education flow model of children transitions from grade i to grade i+1 (1 ≤ i ≤ 9), repetition, and moving out from the systems is very well explained in [1] and [2]. The model in figure1 demonstrates the implications for enrolment at future dates with explicit assumptions about the continuation of past and current trends. The flow model can also be used to reconstruct the flows of a chart of education systems. The flow model is used to explain the behavioral relationship which determines in part the parameters playing such an important role in the model. Such as repetition rates, dropout, promotions, new entrants in the system at any class. Sometimes dropout occurs due to the influence of social factors forced to leave the systems against their wishes or individuals' perceptions of their short-term and long-term employment and earning prospects.
In some cases, dropouts for a school may be transferred to another school, district, or administrative area without informing the parent-school or without issuing school leaving certificate document or any other legal document. In such a situation, he/she may be treated as a new entrant for the new school. From the above-given base model of the flow of students in the system, let us elaborate in detail the fundamental concepts involved in the Transaction Model.

Literature Review
While fitting the appropriate model of enrolment projection, several researchers have used different models according to their suitability and availability of resources. A structural model for projecting public school enrolment by grade, progrmme, and district used in [3]. The advantage of specifying a structural model is that many of the standard approaches to forecasting enrolment emerge as special cases of its reduced form equations explained in [3]. The National Center for Educational Statistics (NCES) 2025 used double exponential smoothing and multiple linear regression method for projection of education statistics to 2025 [4]. NCERT used the logistic curve method to project the enrolment of 1997 -2001. They used prior data collected under the third, fourth, fifth, and sixth All India School Education Survey 1 conducted during 1973, 1978, 1986, and 1993 respectively. The achievement of the study was that the error between actual data and projected data was inbetween 2% to 5% only, which is statistically acceptable [5]. Cohort method for projection is explained by [6] and [7]. [7] studied the cohort-survival ratio method in the projection of school attendance and [7] explained reconstructive cohort methods with other aspects such as basic concepts, educational planning, methods of analyzing and projecting school enrolment, flow methods, RCM, projection of teachers' supply and demand, etc.
Several methods of enrolment projections have been explored. Four methods are identified as an appropriate method in the Indian context namely; auto regression integrated moving average method (ARIMA), double exponential method, logistic curve method, and reconstructive cohort method.

Construction of Projection Model
The projection of enrolment for the coming years is useful for planning and policy decisions for a country. Allocation of infrastructure, the demand of human resources including academics and secretarial staff, provisions for socially disadvantaged groups, etc. are being estimated on the basis for projected figures. Enrolment has three important components that play a vital role in the enrolment projection; namely, promotion rate, repetition rate, and dropout rate. Since enrolment is the function of the population so the population of children also useful in enrolment projection. Change in population proportion directly reflects in the enrolment. In India, a child who complete 6 years of age, is eligible to get enrolled at grade 1 in a school. It means that in total population, the population of 6-year children is supposed to enroll at grade 1 in a school, so the equation of grade enrolment can be defined as = _6 (1) 1 All India School Education Survey is a census survey of all schools in India. It covers all schools including government schools, local body schools, private aided schools and private unaided schools from grade1 to 12 situated in any part of the country. It includes urban are schools, rural schools, tribal area school, forest schools, oriental schools, etc. It collects all information from the schools such as type of school, enrolment, teachers, available infrastructure, incentive schemes, accessibility of schooling facility in rural areas, etc. First survey conducted in year 1957. Till then nine surveys has conducted in the country.
where, _6 is the population of Ag = 6 in time t and is the enrolment in grade i = 1 in time t. But it is not true in a real-life situation. Because a good number of children of under aged children (i < 6 ) and over-aged children (i > 6) are also get enrolled with children of aged (i = 6) in grade 1. So the equation (1) is redefined as where, _5 , _6 , _7 , _8 , _9 , _10+ : child population in aged i = 5, 6, 7, 8, 9, 10+ years respectively and : coefficient of the child population of aged i (i = 5, 6, 7, 8, 9, 10+)

years in grade 1 in time t and
where, _ = _ * ( _6 ) −1 , the intake rate of the child. If all new entrants in grade 1, start schooling at the same age then the proportion of this age-group population in time t with the enrolled students in grade 1 in that time can be taken into consideration to estimate the proportion of the population.
While fitting the appropriate model of enrolment projection, several researchers have used different models according to their suitability and availability of resources. As discussed in the literature review, the several methods of enrolment projections have been explored, out of them four methods are identified as an appropriate method in the Indian context namely; auto regression integrated moving average method (ARIMA), double exponential method, logistic curve method, and reconstructive cohort method. A tryout exercise has been conducted with an available enrolment for the year 2012-2015 and to estimate enrolment for the years 2017 and 2018. The finding of the suggestive models is as • ARIMA method smoothened the data sharply and gave an almost straight uniform line however in actual situation it is not possible • Double exponential methods produced an increasing linear trend while in decreasing child population rate, linearly increasing enrolment was incorrect.
• The basic assumption for fitting logistic curve was that data should be in an increasing order however there is much noise in the data of enrolment available through UDISE. Even growth in enrolment of two consecutive years was not showing any trend. So the logistic curve was not fitting in the available data. In the study, it was assumed that there is no dropout in the system, hence the dropout rate Due to these results, it was decided to adopt the reconstructive cohort method for the projection of enrolment and construct the model according to the requirement of the study. Hence, for Reconstructive Cohort Method, some basic requirements are (i) child population at aged 6 years (according to the grade 1 entry policy of India); (ii) apparent Intake Rate for time t; (iii) grade enrolment data for time t; (iv) repetition rate at grade i (i = 1, 2, … , 9) for time t; and (v) promotion rate of class i (i = 1, 2, … , 9) for time t.
Hence the model is constructed in two levels. At the first level, the model is constructed for grade i (i = 1) In the second level, the model is constructed for grade i where (2 ≤ i ≤ 10) is as In the above model, it is assumed that there is no dropout in the system. So, the dropout will be assumed to zero, that is, the dropout rate, −1 = . That's implies = ( −1 −1 * −1 ) + ( −1 * ) The model is given in (4) and (5) are similar. The only difference is that when there are dropouts in the system, model (4) will be applied and when no dropout in the system then model (5) will be used.