Theoretical or empirical rationale for choosing the variables
The theoretical justification for conducting this study is based on the different well-known and well-recognized theories and models, for example the human capital theory by Becker , the health care model by Grossman , and the neoclassical model by Romer .
The rationale of choosing our studied variables is relies on the availability of data, from contemporary and previous literary works. We have considered the variables relating to life expectancy at birth, following Rahman et al. , Shahbaz et al. , Rodrigues and Plotkin , among others; mortality rate following Rahman et al. , Hurt et al. , Nicholas et al. , among others; access to electricity in line with Wang , Chen et al. , Bridge et al. , among others; female education rate following , Hurt et al. , Keats ; public health expenditure in line with Rahman et al. , Novignon and Lawanson , Nicholas et al. , among others; immunization rate following Akinkugbe and Mohanoe , Owais et al. , Rodrigues and Plotkin , among others; urbanization rate in line with Rahman and Alam , Wang , Amouzou et al. , among others; and economic growth corresponding to the research undertaken by Rahman and Alam , Rahman et al. , Wang et al. , among others.
Variables and data
For this study the female health outcomes are considered as the dependent variables; for this, female life expectancy (FLE) at birth and female adult mortality rate (FAM) per 1,000 female adults (ages 15–60 years) are used as proxy variables. Along with these, the access to electricity (AEL), female education rate, immunization rate (IMM), urbanization rate (URB), gross domestic product (GDP) and public health expenditures (PUH) are taken as the independent variables. Access to electricity is the percentage of population having access to electricity; female education rate is the school enrolment of females at secondary level as a percentage of gross; immunization rate is taken as measles vaccination taking rate as the percentage of children ages 12–23 months; GDP is used to see the reflection of economic growth; urbanization is defined as the urban population referring to people living in urban areas as a percentage of total population; and the public health expenditure is taken as the domestic general health expenditure per capita at current US$ funded by the government.
All the data are collected from the World Development Indicator  of the World Bank over the years 2002–2018. Some missing data of female school enrolment at secondary level are linearly interpolated through E-views-11. To accomplish the estimation, we have used two well-known statistical software packages, STATA-16 and E-views1.
The models used for the estimation of this study are presented below:
$$textFLE = text f left( textAEL,text FED,text PUH,text GDP,text IMM,text URB right)$$
$$textFAM = text f left( textAEL,text FED,text PUH,text GDP,text IMM,text URB right)$$
To get the direct elasticity of each variable from the coefficients, we have transformed all the variables of the Eqs. (1) and (2) into natural logarithmic form. Thus, the Eqs. (1) and (2) can be written as:
$$mathrmLNFLE_mathrmt=mathrmalpha +upbeta _1mathrmLNAEL_mathrmt+upbeta _2mathrmLNFED_mathrmt+upbeta _3mathrmLNPUH_mathrmt+upbeta _4mathrmLNGDP_mathrmt+upbeta _5mathrmLNIMM_mathrmt+upbeta _6mathrmLNURB_mathrmt+upvarepsilon _mathrmt$$
$$mathrmLNFAM_mathrmt=mathrmalpha +upbeta _1mathrmLNAEL_mathrmt+upbeta _2mathrmLNFED_mathrmt+upbeta _3mathrmLNPUH_mathrmt+upbeta _4mathrmLNGDP_mathrmt+upbeta _5mathrmLNIMM_mathrmt+upbeta _6mathrmLNURB_mathrmt+upvarepsilon _mathrmt$$
where, α is the intercept, and β1, β2, β3, β4, β5, β6 are coefficients and εt is the error term.
For empirical estimation we employed a number of renowned econometric approaches. We conducted the mentioned tests as: a cross-sectional dependence test to identify the shock effect; the Modified Wald test for group wise heteroskedasticity and Wooldridge test for aut-1ocorrelation in panel data to observe the heteroskedasticity and autocorrelation respectively. We employed the Panel corrected standard error (PCSE) model and the Feasible generalized least square (FGLS) model to obtain results that will show the robust relations between the variables; and the pair-wise Granger causality to determine the direction of causality.
Because of the resemblance of the geographic, economic, historical, ethnic and political shocks, the cross-sectional dependence of the variables may be observed. In this study we have employed four well-known cross-sectional dependency tests: Breusch and Pagan  BP LM, Pesaran  scaled LM, Pesaran  CD, and Baltagi et al.  biased-corrected scaled LM.
The Breusch and Pagan  model of examining the cross-sectional dependence among the panel data is:
$$CD_BP= sum _i=1^N-1sum _j=i+1^Nwidehatp_ij^2$$
Pesaran  developed the LM statistics to address the limitations of the cure from the above model as:
$$CD_LM= sqrtfrac1N(N-1) sum _i=1^N-1sum _j=i+1^N(widehatp_ij^2-1)$$
If the cross-sectional size is greater than the time dimension, Pesaran  recommends the below test statistic:
$$CD= sqrtfrac2TN(N-1) sum _i=1^N-1sum _j=i+1^Nwidehatp_ij^2$$
Baltagi et al.  developed the simple asymptotic bias correction model, which is:
$$CD_BC= sqrtfrac1N(N-1) sum _i=1^N-1sum _j=i+1^N(widehatp_ij^2-1)-fracN2(T-1)$$
where (widehatp_ij) specifies a correlation among the errors. In this test, the null hypothesis is H0: denotes no cross-sectional dependence and the alternative hypothesis is H1: prevalence of cross-sectional dependence.
To make an efficient and robust estimation of the fixed effect model, the model should be homoskedastic with no autocorrelation. If the model suffers from heteroskedasticity, the estimation may be consistent but inefficient . To detect the heteroskedasticity the modified Wald test for group wise heteroskedasticity is performed [50, 71]. Similarly, the presence of of autocorrelation is identified with the aid of Wooldridge  auto correlation test for panel data .
To overcome the complications of panel data estimation arose due to cross-sectional dependence, heteroskedasticity, and autocorrelation both the panel corrected standard error (PCSE) and the feasible generalized least square (FGLS) models are considered to be best and most efficient. So, the difficulty created due to the panel nature of data, the PCSE, is the path finder . Alternatively, the FGLS model is also able to overcome autocorrelation, heteroskedasticity, and cross-sectional dependence of the estimation . Both the PCSE and FGLS methods are efficient and effective in addressing the heteroskedasticty, autocorrelation and outlier estimates [55,56,57, 73,74,75,76].
To observe the causality between the studied variables, pair-wise Granger  causality of stacked test (common coefficients) is employed, where three outcomes are revealed as one-way causality, two-way causality, and no causality. The pair-wise Granger causality equations for the panel data can be written as [59, 60].
$$Y_i,t= A_0,i+A_1,iY_i,t-1+dots dots + A_k,iY_i,t-1+ B_1,iX_i,t-1+Omega _i,t$$
$$X_i,t= A_0,i+A_1,iX_i,t-1+dots dots + A_k,iX_i,t-1+ B_1,iY_i,t-1+Omega _i,t$$
where, t indicates the time period dimension of the panel, and i shows the cross-sectional dimension.
The stacked causality test considers all the coefficients are the same across all cross sections as common coefficients [59, 60]. This can be portrayed as:
$$A_0,i=A_0,j, A_1,i=A_1,j ,dots dots dots .,A_k,i=A_k,j,forall i,j$$
$$B_0,i=B_0,j, B_1,i=B_1,j ,dots dots dots .,B_k,i=A_k,j,forall i,j$$
Hence the decision rule is H0: Y does not Granger causes X, and H1: Y Granger causes X.