Use of the pseudo-panel approach

Using Deaton’s approach, the data set to be used in the analysis is compiled by grouping households sharing some common features into cohorts. Here, divisions/districts/cities are used as cohorts, after confirming that there was a sizable number of individuals in each cohort. The averages within these cohorts are then treated as observations in a pseudo-panel. Using this approach, the average value of all the observed variables in cohort c in period t, becomes the observation.

Consider the following simple model

(1) $y_{it}=x_{it}\beta +{\alpha }_i+{\epsilon }_{it}t=1\dots T$

where y_it is the dependent variable and $x_{it}$ is a (K×1) vector of explanatory variables, assumed to be exogenous, i is an index of individuals (children aged 10–14years in this case) and t refers to time periods. Following Reference DeatonDeaton (1985), cohorts C, based on divisions/districts/city are therefore identified, such that child i is a member of one and only one cohort for each t. Averaging over the cohorts gives

(2) ${\overline{y}}_{ct}={\overline{x}}_{ct}\beta +{\overline{\alpha }}_{ct}+{\epsilon }_{ct}c=1\dots C$

Let the size of the group c at time t be $n_{ct}$ such that the average value of all observed x_it s in cohort c at time t can be written as

(3) ${\overline{x}}_{ct}=n_{ct}^{-1}\sum _{i\in c}^{s=t}{x}_{is}$

The resulting data set is a pseudo-panel with repeated observations over T periods and C cohorts. The main problem in estimating equation (2) is that ${\overline{\alpha }}_{ct}$ depends on t and could be correlated with $x_{ct}$ . Considering ${\overline{\alpha }}_{ct}$ as a random error term would lead to inconsistent estimators, while considering them as fixed unknown parameters would result in an identification problem unless the variation over t can be ignored $\left({\overline{\alpha }}_{ct}={\overline{\alpha }}_c\right)$ . This assumption holds only if the number of observations used to create cohort averages is large. Reference Verbeek and NijmanVerbeek and Nijman (1993) have modified Deaton’s estimator to achieve consistency for a fixed t and a fixed i per cohort. If the number of children in each cohort is large, then the size of the cohort $n_c=N/C$ tends to infinity, and the measurement errors as well as their estimates tend to zero. This yields the cohort estimator β, which is asymptotically identical to the Reference DeatonDeaton (1985) estimator of

(4) $\stackrel{⌢}{\beta }={\left(\sum _{c=1}^C\sum _{t=1}^T{n}_{ct}{\left({\overline{x}}_{ct}-{\overline{x}}_c\right)}^2\right)}^{-1}\left(\sum _{c=1}^C\sum _{t=1}^T\left[n_{ct}\left({\overline{x}}_{ct}-{\overline{x}}_c\right)\left({\overline{y}}_{ct}-{\overline{y}}_c\right)\right]\right)$

where ${\overline{x}}_c=\left(\sum _{t=1}^T{\left(n_{ct}\right)}^{-1}\sum _{t=1}^T{\overline{x}}_{ct}{n}_{ct}\right)$ , the temporal average of the observed cohort means for cohort c. The properties of the β estimator depend upon the asymptotic properties employed. Reference DeatonDeaton (1985) considers the asymptotic properties when the number of cohorts C tends to infinity, that is, the number of individuals tends to infinity with constant cohort sizes. Furthermore, Reference MoffittMoffitt (1993) assumes that the number of cohorts should be fixed (114 in this case), while the cohort sizes should increase with the increase in the number of individuals. In the Moffitt approach, the problem of errors in variables disappears.

Furthermore, Reference Verbeek and NijmanVerbeek and Nijman (1993) has suggested that when the cohort size is at least 100 individuals, and the time variation in the cohort means is sufficiently large, the bias in the standard fixed effects estimator is small enough so that the measurement error can be ignored. Given the size of the LFSs and of the divisions/districts/cities chosen as cohorts, this size criterion is fulfilled. The construction of pseudo-panel data is done by computing cohort or cell means in each available cross section, where the cells are defined by district codes. This yields a balanced panel consisting of 114 groups (cohorts) for seven time periods, having a total of 798 observations.

Finally, in order to see how labour market factors influence child labour behaviour, a fixed effect approach, as suggested by Reference Verbeek and VellaVerbeek and Vella (2005), is applied. Reference Verbeek and VellaVerbeek and Vella (2005) suggest that the fixed effects estimator based on the pseudo-panel of cohort averages provides an attractive choice for estimation. When the sizes of the cohorts are large, the sample mean of the fixed effects provides a consistent estimator of the time-invariant population mean. Therefore, when $n_c$ is large, the errors-in-variables problem caused by the possible time variation in ${\overline{\alpha }}_{ct}$ is usually ignored and standard estimators like the fixed effect (within) estimators are usually used. Therefore, the relationship among the variables of interest is estimated by the standard fixed effects estimator.

Econometric specification

Based on the discussion in section ‘Theoretical foundations’, the study has formulated a simple regression model to understand how the labour market indicators, along with some other control variables, impact on child labour incidence. The significance of each variable with its endogenous/exogenous relation with child labour incidence is discussed in detail below. The model is specified thus

(5) $C{L}_{ct}=W{G}_{ct}{\beta }_1+U{N}_{ct}{\beta }_2+IF{S}_{ct}{\beta }_3+AG{R}_{ct}{\beta }_4+C{T}_{ct}{\beta }_5+{\epsilon }_{ct}$

The dependent variable child labour $CL$ consists of the average number of children aged 10–14years engaged in work, including both paid and unpaid employment and work in household enterprises (including farms), in a cohort. In the above model, WG (based on the Davidson and Mackinnon (1985) test, the log function of wages is included), UN, IFS, AGR and CT are the time-varying exogenous variables (averages over cohorts) representing adult wages, unemployment rate, size of the informal sector, size of the agriculture sector and some control variables such as the child’s own and household characteristics, respectively. Here, sizes are measured in terms of proportion of adult employment in the respective sectors. βs are the set of coefficients and ϵ is the unobservable error term.

As far as the independent variables explaining the effect of the labour market are concerned, first the effect of monthly adult wages is explored. Monthly adult wages of paid employees are used here because the LFS reports the wages of paid employees only. Use of this information results in a missing data problem. To overcome this problem, individual monthly wages of adults are regressed on personal and household characteristics. More specifically, the explanatory variables of the wage equation include age, age squared, gender, household size, gender of the head of the household and education level. The predicted wages from the wage equation are then aggregated at household level to come up with household wages (income).

The main hypothesis regarding wages is that the higher the household wages, the less will be the child labour supply. But, this is the supply side effect. There may also exist a demand side effect which will be generated when higher wages result in a higher cost of production, pushing employers to substitute adult labour with cheap child labour. This could result in an ambiguous net effect. Children either substitute for adults in the case of unskilled labour, or they may enter into employment as parents are priced out of the market and become unemployed. In order to check whether or not the demand side effect exists, median wages at division/district/city level were also included in the model. Here, real wages were used instead of nominal wages, using the CPI (Consumer Price Index) as a deflator.

As the effect of wages may vary with skill composition, in the preliminary analysis both median wages of skilled and unskilled labour were included in the model. Skilled wages are represented by median wages earned by labour having secondary education across divisions/districts/cities, while unskilled wages are represented by median wages of the illiterate (as defined by PBS: those persons 10years of age and above who could not read and write in any language with understanding labour force across divisions/districts/cities). As the effect remains insignificant in the main analysis, only adult wages at household and division/district/city level in real terms are included in the model. Furthermore, in the latter, household wages were also dropped from the main analysis as well, first, because household wages did not show any significant effect, and second, because the study acknowledges that household wages may include the contribution of unpaid child family work and hence inclusion of this variable may overestimate the effect of household wages on child labour supply.

The unemployment rate in equation (5) represents the ratio of the unemployed adult labour force to the total adult labour force. Adult unemployment could again exert two types of effects: supply side and demand side. First, the higher the adult unemployed proportion in an area, the harder it is for children to find employment, especially in the sectors where they are substituted by unemployed adults. Therefore, the higher the adult unemployed proportion in an area, the lower will be the demand for child labour. Second, and on the other hand, the higher adult unemployment rate also means a lower household income, forcing parents to send more and more children to work, in order to overcome the loss in income. In order to check how adult unemployment affects the child labour supply, the proportion of unemployed adults within the household (showing the supply side effect) is also included in the model.

Points discussed above also raise a question regarding the availability of work opportunities, that is, ‘where do children get jobs?’ The available variable that best reflects employment opportunity for children is the degree of informalisation in the economy. As explained earlier, this sector absorbs a high proportion of child labour. Children often start working in this sector to acquire skill, but after some time become regular workers (Reference BurkiBurki,1989). The LFS defines the informal sector as enterprises owned and operated by employers with less than 10 persons engaged. ^{Footnote 2} As there is a low skill requirement in this sector, children more easily gain employment and may sometimes substitute for adult unskilled labour. Therefore, the higher the proportion of the informal sector in an economy, the higher will be the probability of child labour. The relation could however be the other way around as well. For example, owing to the high incidence of child labour, the informal sector may experience growth. In order to overcome this problem, the size of the informal sector was measured in terms of adult employment (excluding child labour) in the informal sector, as a proportion of the total adult employed labour force across divisions/districts/cities.

Furthermore, in order to empirically analyse the effect of the indicators outlined above, one cannot ignore the importance of the agriculture sector, as almost 67% of the children in the sample were working in this sector. Such a high proportion suggests that larger the size of this sector, the higher will be the probability of child work. In addition then, some control variables are also included in the model, grouped to capture variations at the household and child level. To control for the effect of household characteristics, the following variables are included: average household size in a cohort, the average years of schooling of male and female heads of the house in a cohort, average years of schooling of the spouse (female only) of the heads in a cohort and average number of male headed household in a cohort. To control for the effects of the child’s own characteristics, age of children and average number of male children in a cohort were also included in the model.

This study also acknowledges the possible problem of endogeneity in the relationship between child labour (denoted by CL in equation (5)) and four of the explanatory variables in equation (5). Adult wages have an element of endogeneity as the higher the adult wages, the higher will be the demand for child labour as explained earlier. Recall the substitution effect discussed earlier, that is, higher adult wages lead to higher costs of production and the substitution of cheap child labour for more costly adult labour. Hence, higher wages are hypothesised to reduce the demand for adult labour, thus affecting wages. This simultaneity issue is also present with the adult unemployment variable. Another variable, household size, was also checked for presence of endogeneity because parents often prefer more children in order to earn more. Use of the Davidson and Mackinnon test of exogeneity (Appendix A-1, Supplementary Files (http://journals.sagepub.com/doi/suppl/10.1177/1035304617690332)) resulted in rejection of the presence of endogeneity among the variables; therefore, the main analysis was performed assuming exogeneity among the variables.