Estimations obtained through the Fully Modified Ordinary Least Squares (FMOLS) and the Dynamic Ordinary Least Squares (DOLS) regressions produced a negative and highly significant (p ≤ 0.01) coefficient for the determinant ‘births attended by skilled healthcare staff’ (Supplementary Table S2). This indicates that assuming all other factors remain constant, an increase in ‘births attended by skilled healthcare staff’ can reduce the child mortality rate, which is consistent under both FMOLS and DOLS estimations. A similar interpretation applies to the relationship between infant mortality and ‘births attended by skilled healthcare staff’ (Supplementary Table S3). These empirical findings are consistent with some earlier studies conducted on other developing countries [17, 38,39,40,41]. Skilled healthcare staff members are professionals with sufficient training and skills to manage normal pregnancy childbirth; during the immediate postnatal period they can identify, manage, and refer women and newborns with complications [42]. Several studies further confirm that assistance from skilled healthcare staff during pregnancy and until birth is associated with reduced child and maternal mortalities [17, 38,39,40].

The variable ‘pregnant women receiving prenatal care’ is conducive to reducing both child mortality rate and infant mortality rate in Bangladesh (Supplementary Tables S2, S3). The FMOLS estimation produced a negative and highly significant (p ≤ 0.01) coefficient for the variable, consistent with the finding obtained previously by other studies [22, 43, 44]. Similarly, the variable ‘newborns protected against tetanus’ is highly significant (p ≤ 0.01) with a negative coefficient, implying that it is a major factor in reducing both child mortality and infant mortality in Bangladesh (Supplementary Tables S2, S3). This important finding is consistent in both DOLS and FMOLS estimation procedures and is well-supported by recent studies [23,24,25,26, 45]. Although the World Health Organization (WHO) and its partners have taken several initiatives to eliminate child mortality caused by not taking tetanus vaccine for NNT, the case fatality rate from this disease remains high and treatment is limited by scarcity of resources and effective drug treatments in developing countries. There has been much progress, however, in improving vaccination coverage, birth hygiene, and surveillance, with specific approaches in high-risk areas. This means that the incidence of the disease continues to decline. We see a steady fall in infant morbidity and mortality in Bangladesh since the establishment of the Expanded Program on Immunization (EPI) in 1979. EPI has been and continues to be effective against fatal diseases such as tuberculosis, diphtheria, tetanus, pertussis, poliomyelitis, and measles in children less than a year old. While the WHO projected national coverage for these vaccines to be over 90% in 2009, the mortality rates remained high at 37 deaths per 1000 live born infants in Bangladesh in 2014 [25].

In the attempt to determine whether ‘healthcare expenditure per capita’ is an important determinant of child mortality and infant mortality in Bangladesh, the empirical finding is highly promising in reducing both child mortality and infant mortality. A negative but a highly significant (p ≤ 0.01) coefficient for the variables emerged from both DOLS and FMOLS estimations (Supplementary Tables S2, S3). This very important outcome lends support to the findings of some large-scale and cross-country studies [27,28,29]. Generally, infant and child deaths can be prevented by providing 16 simple and low-cost interventions [46]. These low-cost strategies should be accessible and available if public health spending rises. Interestingly, when regressed against the numbers of both infant and child deaths as the dependent variables, ‘female labor force participation’ is found to be a positive and highly significant (p ≤ 0.01) influence on them in Bangladesh (Supplementary Tables S2, S3). While studies of the relationship between ‘female labor force participation’ and child mortality and infant mortality are quite rare, our empirical findings build on the earlier work of Narayan and Smyth [33]. In both DOLS and FMOLS estimations, ‘population growth’ positively influences both child mortality and infant mortality but is also highly significant (p ≤ 0.01) in FMOLS estimation (Supplementary Tables S2, S3). The relationship between ‘population growth’ and child mortality and infant mortality has, so far, been controversial due to varying characteristics of populations and varying levels of access they have to health and other basic amenities and resources across countries. Yet, our finding is consistent with the one found earlier by the World Health Organization [2] and Baker et al. [47].

Again, when regressed against both the dependent variables in DOLS and FMOLS estimations, the ‘GDP growth’ has significantly (p ≤ 0.05) reduced both child mortality and infant mortality in Bangladesh (Supplementary Tables S2, S3). As in most countries ‘GDP growth’—if transformed such that it improves the public healthcare system—can reduce both child mortality and infant mortality; this corroborates the findings of some recent studies [1, 27, 30, 31, 48, 49]. Conversely, periods of economic contraction bring additional economic setbacks for the developing economies, leading to worse living conditions for people, particularly children, one of the most vulnerable groups in society. Also, during economic contraction, household income is likely to fall, thus many families have to cut back on expenditure for food and health, and governments often reduce public healthcare spending to control outlays.

This study also investigates whether all the independent variables determine the infant deaths, under-five deaths, and neo-natal deaths in Bangladesh. Accordingly, we have applied both DOLS and FMOLS estimation techniques to determine the relationships among the three dependent variables and the seven independent variables. This strategy helps us to understand which segment of child mortality is best explained by which determinants. Independent variables such as ‘births attended by skilled healthcare staff’, ‘pregnant women receiving prenatal care’, ‘newborns protected against tetanus’, ‘healthcare expenditure per capita’ and ‘GDP growth rate’ all negatively but significantly influenced the infant deaths; both ‘female labor force participation’ and ‘population growth rate’ positively and significantly influence the same (Supplementary Table S4). These findings on Bangladesh are consistent with several studies [1, 27, 30, 31, 48, 49], which have been cited earlier in relation to confirming the significant macroeconomic determinants of infant deaths in other countries.

Similarly, the independent variables ‘births attended by skilled healthcare staff’, ‘pregnant women receiving prenatal care’, ‘newborns protected against tetanus’, ‘healthcare expenditure per capita’ and ‘GDP growth rate’ all negatively but significantly influence the deaths of under-five years old and neonatal children in Bangladesh (Supplementary Tables S5, S6). The other two independent variables, ‘female labor force participation’ and ‘population growth’, influence the under-five child mortality and neonatal death rates both positively and significantly (Supplementary Tables S5, S6). These findings on Bangladesh are consistent with several studies [1, 27, 30, 31, 48, 49], confirming them as the significant macroeconomic determinants of under-five years old child and neonatal child mortalities in other countries. Since 1950, developing countries have implemented many initiatives on lowering the mortalities of under-five years old and neonatal children alongside several initiatives such as promoting economic growth, increasing levels of education, encouraging female empowerment, and provision and acceptance of family planning services. But nevertheless, it is ironic that countries with high child mortality rates have the fastest growing populations in the world. Also, countries with infant mortality rates of less than 20 per 1000 births have an average total fertility rate of 1.7 children, while countries with infant mortality rate of over 100 per 1000 births have an average total fertility rate of 6.2 children [36]. These statistics imply that faster population growth and higher mortality rate in developing countries could be due to resource constraints, as well as a lack of awareness among populations concerning their health and socioeconomic wellbeing.

The strength of empirical findings, as presented and discussed earlier, lies in consistency of the determinants of neonatal, infant, and under-five mortalities identified by both DOLS and FMOLS estimations. In Supplementary Table S7, we show such consistency in terms of the sign of coefficients for independent variables and their level of significance. Except for a few differences in the levels of significance, all the independent variables significantly influence the dependent variables consistently in both estimations. Such consistency of the findings is considered robust, and hence extremely useful for devising and implementing the effective policies on reducing the neonatal, infant, and under-five child mortalities in countries having similar developing and macroeconomic characteristics as Bangladesh. Child mortality is an important indicator of overall health [50]. Appropriate policies centered around its significant macroeconomic determinants could potentially improve the overall health of a nation. Wagner [51] argues that infant mortality is not a health problem but a social problem with health consequences. Thus, any effective solutions for reducing child mortalities would require appropriate social policies aimed at reducing those health consequences among mothers and children.

Because findings from this this study reveal the determinants of neonatal, infant, and under-five mortalities in the context of a developing country, a cross-country panel study across other developing countries would provide a deeper understanding. Inclusion of more countries and use of longitudinal data sets would provide more robust results. Considering the views of health practitioners and policy makers in the analysis would further advance understanding of neonatal, infant, and under-five mortalities and their macroeconomic determinants from both a single country and cross-country perspectives.