Sustainable thresholds, health outcomes, health expenditures and education nexus in selected African countries: quadratic and moderation modelling

Pre-estimation results

Table 1 contains the descriptive statistics. Furthermore, it displays the correlation coefficients between the target variables (lnMAT, lnCHD and lnLEX) and the regressor variables (lnHEXPC, lnEDUC, lnICT, lnHCON, lnHIV, lnTUB and lnTFP). The closer the correlation coefficient is to − 1 or 1, the stronger the association [14]. It is critical to indicate that while the correlation matrix may measure the direction and strength of association between dependent and independent variables, this does not always indicate causality. On the whole, all variables of interest are significant at 1% level as shown by the estimated correlation coefficients. The strength of the relationship in most cases is quite strong and have the expected signs with no indication of multicollinearity among the regressors. Also, the cross-sectional dependence test shows (last column of Table 1) that there is no cross-sectional independence as the coefficient are all significant.

Table 1 Summary statistics, pairwise correlations and cross-sectional dependence resultsComposite main results, PSCC-OLS and PSCC-FE

Table 2 is a composite result showing the quadratic results (Eq. 1) and moderation results (Eqs. 2 and 3) across the three dependent variables: maternal mortality (columns 1,4,7,10), child mortality (columns 2,5,8,11), and life expectancy at birth (columns 3,6,9,12) and two estimation techniques. Results are interpreted sequentially along modelling structure.

Table 2 Composite main results (HIV models)

For the quadratic models, we are only interested in the coefficients of HEXPC and HEXPCSQ to observe the shape of the curve. For maternal and child mortality, the relationship with health expenditures evidenced an inverted U-shape curve. That is, initial increase in health expenditures causes a rise in both maternal and child mortality. However further increase in health expenditures results in a decline in mortalities supporting the literature stance on the mortality-reducing effect of health expenditures [17]. In the same vein, life expectancy and health indicate a U-shape relationship. That is, initial increase in health expenditures causes a decline in life expectancy but additional increase in health expenditures results in improving life expectancy Again, this finding aligns with the empirical literature [22].

From the quadratic results, we obtain the sustainable thresholds or turning points beyond which health expenditures exert significant impact on health outcomes. From Eq. 1, the health expenditures turning point for maternal mortality is computed as \(\hat=0.5\ast ^\!\left/ \!_\right.\) = 3.16; for child mortality: \(0.5\ast ^\!\left/ \!_\right.\) = 2.73, and life expectancy: \(0.5\ast ^\!\left/ \!_\right.\) = 3.65. Since, these equations are computed using natural logarithm, it becomes important to take the exponents so as to confirm if these thresholds lie within the range of data. Therefore, for each health outcome, the corresponding sustainable health expenditures thresholds in real terms are as follows: maternal mortality: [exp(3.16)] = 23.57, child mortality: [exp(2.73)] = 15.33, and life expectancy: [exp(3.65)]= 38.47. Recall that from Table 1, the range of values for health expenditures per capita is 3.395 to 844 and since the shape for the parabola for the mortality models is an inverted U-shape suggesting that beyond 23.57 and 15.33, health expenditures per capita will contribute significantly to reducing both maternal and child mortality. In the same vein, following the U-shape relationship, it follows that beyond the threshold point of 38.47, health expenditures per capita will contribute significantly to improving life expectancy. We show this graphically in Fig. 2.

Fig. 2figure 2

Sustainable thresholds of health expenditures and educational quality. Source: Authors’ Computations

Having established the quadratic effects, this study proceeds with the results of the moderation models to compute the minimum sustainable thresholds at which educational quality could enhance the effect of health expenditures on each health outcome. These thresholds have policy implications because beyond the critical masses, the effect of health expenditures on each health outcome is dependent on the strength of educational quality. Given these clarifications, only the significant coefficients of HEXPC and HEXPC*EDUC are used in computing the threshold points. We observe an inverted U-shaped curve exists between health expenditures, educational quality and the mortality models while a U-shaped curve hold for life expectancy model. Following Eq. 4, the threshold points for lnEDU∗ across each health outcome are: Maternal Mortality: \(-\frac=0.42\); Child Mortality: \(-\frac=0.41\); and Life Expectancy: \(-\frac=0.61\). In these computations, 0.307, 0.126 and − 0.0604 represent the absolute value of the unconditional effect of health expenditures on each health outcome while − 0.737, − 0.310, and 0.0988 represent the moderation/conditional effect between health expenditures and educational quality on each health outcome. Hence, from these computed thresholds, it holds that threshold points beyond 0.42 and 0.41 induce a drop in maternal and child mortalities while a point beyond 0.61 exerts an improvement in life expectancy.

Similar to the quadratic thresholds, those computed from the moderation models are done using the natural logarithm. Therefore, to ascertain that these points lie within the range of educational quality we take the exponents to obtain the corresponding values as: maternal mortality: [exp(0.42)] = 1.52, child mortality: [exp(0.41)] = 1.51, and life expectancy: [exp(0.61)] = 1.84. Also, from Table 1, the range of values for educational quality is 1.069 to 2.939 and since the shape for the parabola for the mortality models is an inverted U-shape signifying that beyond 1.52 and 1.51, educational quality improves the mortality-reducing potentials of health expenditures per capita on maternal and child mortality. Similarly, beyond 1.84, educational quality improves the life-enhancing effect of health expenditures per capita on life expectancy per capita. Also, these outcomes are shown in Fig. 2.

The PSCC-OLS does not recognise the individual heterogeneities or fixed effects across the countries in the panel, hence, we re-estimated the models using the fixed effects routine (PSCC-FE) and the results are shown in the second half of Table 2. For the most part, the obtained results are consistent with those of PSCC-OLS. Though the quadratic models reveal that health expenditures have a U-shaped relationship with maternal and child mortality; and an inverted U-shape curve with life expectancy suggesting that more allocations of health expenditures beyond an identified threshold worsens health outcomes and the respective calculated thresholds are 3.46, 4.37 and 3.85, respectively. Analogous to the calculations done in Table 2, the real values for these threshold points lie within the range of health expenditures (3.395 to 844) are: 32.14, 79.04, and 46.99. The most plausible argument for these contradictions could be that individual differences across the countries is driving these anomalies.

Lastly, the moderation models reveal a U-shaped curve between health expenditures, educational quality and the mortality models while the relationship with life expectancy is not different from zero (that is, statistically not significant). Computing the critical mass for sustainable resulted in 0.58 and 0.98 for maternal and child mortality, respectively which that of life expectancy is inconclusive.Footnote 3 Converting to real terms, the threshold points which lie within the range of values for educational quality are 1.79 and 2.66 beyond which educational quality negatively influences the impact of health expenditures on maternal and child mortality.

Robustness results

To test the robustness of our analysis, we engage two sets of sensitivity checks. The first uses people living with tuberculosis (TUB) in place of those with HIV. The second uses 5-year averages of the sample data with HIV and TUB models across both empirical techniques. Starting with the results of the first set of robustness checks displayed in Table 3, the PSCC-OLS quadratic model reveals that health expenditure shows an inverted U-shaped relationship with maternal mortality and life expectancy. That is, health expenditures initially worsen both outcomes but later improved them as the coefficient of the squared health expenditure is negative. However, the relationship between health expenditure and child mortality is inconclusive. The thresholds of maternal mortality and life expectancy are 2.72 and 6.67 respectively. Accounting for the level of education as the moderation variable, health expenditure has an inverted U-shaped relation with maternal mortality but a U-shaped relationship with life expectancy. The respective conditional threshold points of health expenditures on maternal mortality and life expectancy are 0.25 and 0.37, respectively. From the PSCC-FE results, the quadratic models reveal that health expenditures exert a U-shaped relationship with maternal and child mortalities. The respective threshold points are 3.58 and 4.34. On the moderating relationships, we find that a consistent U-shaped nexus between health expenditures, maternal and child mortalities. Thus, the respective conditional threshold points are 0.62 and 0.92. For the most part, these results are consistent with those of Table 2.

Table 3 Robustness checks (tuberculosis models)

The results from using 5-year averages are displayed in Tables 4 and 5. Starting with Table 4 (HIV model) the results from the PSCC-OLS are similar to those displayed in Table 2. The inverted U-shaped relationship between health expenditures, maternal and child mortalities is sustained with a threshold point of 3.22 and 2.86 while the relationships with life expectancy is inconclusive though the coefficients have the expected signs. In addition, the moderating effects of education level on the nexus is sustained with inverted U-shaped relations with maternal and child mortalities and a U-shaped relation and life expectancy. The respective threshold points are 0.44, 0.46, and 0.60. From the PSCC-FE results, the quadratic effect reveals a U-shaped nexus with maternal and child mortalities and an inverted U-shaped relationship with life expectancy. The respective threshold points are 3.46, 4.27, and 4.04. Likewise, the moderation relationship depicts a U-shaped nexus with maternal mortality with the threshold point at 0.60 while the relationship between health expenditures with child mortality and life expectancy are inconclusive. These results bear semblance to those shown in Table 2.

Table 4 Robustness checks, 5-year averages (HIV models)Table 5 Robustness checks, 5-year averages (TUB models)

We further substituted HIV with those living with tuberculosis (TUB) and the results are displayed in Table 5. From the PSCC-OLS results, we find that the quadratic relationship depicts an inverted U-shaped relation with a threshold point at 2.86 while the relationships with child mortality and life expectancy are inconclusive. On the moderating relationships, we find an inverted U-shaped and a U-shaped nexus with maternal mortality and life expectancy while that of child mortality is inconclusive. The threshold points are 0.30 and 0.44, respectively. From the PSCC-FE results, a U-shaped relation with maternal and child mortalities and an inverted U-shaped nexus with life expectancy. The respective threshold points are 3.62, 4.16, and 4.29. From the moderation models, a U-shaped nexus exists between health expenditures and maternal and child mortalities with the respective threshold points at 0.66 and 0.84. Again, these results somewhat sustain those in Table 3. Overall, we submit that the relationship between health expenditures and the three health outcomes is nonlinear.

Applicability and results implication

This discusses the applicability of the results for SSA countries. Table 6 presents the mean of public health expenditure vis-à-vis its thresholds. We show that almost, all the countries in SSA have not met the computed threshold of 3.65 public health expenditure, except, Lesotho and Namibia. Although, the computed threshold of 3.65% of GDP (Gross Domestic Product) relates to life expectancy at birth, it is taken as the highest bench mark of threshold that must be achieved by each country since all the health outcomes are achieved simultaneously. This is because in the process of achieving life expectancy at birth, all other health outcomes like maternal mortality and infant mortality would have been achieved. Therefore, every country must aspire to allocate at least 3.65% of their GDP to health for an all-encompassing health performance. It is therefore evident that public health financing is underfunded among all the countries in SSA and there is a need for urgent campaign towards increase in public health expenditure in SSA.

Table 6 Mean of public health expenditure and thresholds

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