Learning Outcomes of School Children

Abstract

Borooah employs a unique set of data, encompassing India and its several social groups, to gauge the size of the educational gap between children, aged 8–11 years, belonging to the different social groups in India. These data, culled from the Indian Human Development Survey (Desai et al. 2015), tested approximately 12,000 children, aged 8–11, for their ability to read, write, and do arithmetic at different levels of competence. He analyses, on the basis of econometric investigation, why children have different levels of educational achievement. In particular, he investigates whether, after controlling for other factors – for example, parental education or household income – there might be a role for gender, caste, and religion in explaining differences between children in their learning outcomes.

Appendix Ordered Logit Methods

The idea behind the ordered logit model is that the ability of a child in reading, writing, or arithmetic may be represented by the value of the latent variable, ${H_i}$, with higher values of ${H_i}$ representing higher levels of ability. One may consider this latent variable to be a linear function of K “ability determining” factors whose values for the ith child are: ${X_{ik}},{\rm{ }}k = 1 \ldots K$. Consequently,

$${H_i} = \mathop \sum \limits_{k = 1}^K {X_{ik}}{\beta _k} + {\varepsilon _i} = {Z_i} + {\varepsilon _i}$$

((3.2))

where ${\beta _k}$ is the coefficient associated with the kth variable and ${Z_i} = \mathop \sum \limits_k {X_{ik}}{\beta _k}$. An increase in the value of the kth factor will cause a child’s ability to increase if ${\beta _k} < 0$ and to decrease if ${\beta _k} > 0$.

Since the values of ${H_i}$ are, in principle and in practice, unobservable, Eq. (1) orepresents a latent regression which, as it stands, cannot be estimated. However, what is observable is a person’s “reading status”¹⁸ and the categorisation of persons in the sample in terms of reading status is implicitly based on the values of the latent variable ${H_i}$ in conjunction with “threshold values”, ${\delta _1},{\delta _2}$ and ${\delta _3}$ ( ${\delta _1} < {\delta _2} < {\delta _3}$) such that:

$$\eqalign{ & {Y_i} = 1\,{\rm{if}}\,{H_i} \le {\delta _1} \cr & {Y_i} = 2\,{\rm{if}}\,{\delta _1} < {H_i} \le {\delta _2} \cr & {Y_i} = 3\,{\rm{if}}\,{\delta _2} < \,{H_i} \le {\delta _3} \cr & {Y_i} = 4s\,{\rm{if}}\,{H_i} > {\delta _3} \cr} $$

((3.3))

${\delta _1},{\rm{ }}{\delta _{\rm{2}}},{\delta _3}$ of Eq. (2) are unknown parameters to be estimated along with the ${\beta _k}$ of Eq. (1). A person’s classification in terms of his/her reading status depends upon whether the value of ${H_i}$ crosses a threshold and the probabilities of a person being in a particular reading status are:

$$\begin{array}{*{20}{l}} {\Pr ({Y_i} = 1) = \Pr ({\varepsilon _i} \le {\delta _1} - {Z_i})}\\ {\Pr ({Y_i} = 2) = \Pr ({\delta _1} - {Z_i} \le {\varepsilon _i} < {\delta _2} - {Z_i})}\\ {\Pr ({Y_i} = 3) = \Pr ({\delta _2} - {Z_i} \le {\varepsilon _i} < {\delta _3} - {Z_i})}\\ {\Pr ({Y_i} = 4) = \Pr ({\varepsilon _i} \ge {\delta _3} - {Z_i})} \end{array}$$

((3.4))

If it is assumed that the error term ${\varepsilon _i}$, in Eq. (1), follows a logistic distribution, then Eqs (3.2) and (3.3) collectively constitute an ordered logit model and the estimates from this model permit, through Eq. (3.4), the various probabilities to be computed for every child in the sample, conditional upon the values of the ability-determining factors for each child. ¹⁹