Levels and trends estimate of sex ratio at birth for seven provinces of Pakistan from 1980 to 2020 with scenario-based probabilistic projections of missing female birth to 2050: A Bayesian modeling approach

Most evidence on son preference in Pakistan is reflected in the higher child mortality among females than males. The sex discrimination before birth is rarely reported in Pakistan. This is the first study to quantify prenatal sex discrimination in Pakistan on a subnational level. We provide annual estimates of the sex ratio at birth (SRB) from 1980 to 2020 and scenario-based projections of the number of missing female births up to 2050 by Pakistan province. The results are based on a comprehensive database consisting of 832,091 birth records from all available surveys and censuses. We adopted a Bayesian hierarchical time series model to synthesize different data sources. We identified Balochistan with an existing imbalanced SRB since 1980. For the rest provinces without past or ongoing SRB inflation, we projected the largest female birth deficit to occur in Punjab in 2033 under the scenario that the SRB transition process starts in 2021. We demonstrated important disparities in the occurrence and quantification of missing female births up to 2050.


Introduction
The ratio of the number of male live births to the number of female live birth, namely, the sex ratio at birth (SRB), is an essential element in estimating and projecting population size and its dynamics. Furthermore, an imbalanced SRB in a population reflects discrimination and disadvantages that females face in social, political, economic, and cultural contexts (Gupta et al., 2003;Guilmoto, 2009). SRBs are distorted from their natural levels in several countries, primarily clustered in South Asia, East Asia, and East Europe. The imbalanced values reach as high as 1.2 (i.e., 120 male births per 100 female births) in certain regions (Attané & Guilmoto, 2007;Chao et al., 2019a;Duthé et al., 2012;Goodkind, 2011;Guilmoto et al., 2009;Guilmoto, 2012;Guilmoto & Ren, 2011;Lin, 2009;Park & Cho, 1995). SRB is the The objectives of this study are (i) to provide annual estimates of SRBs among seven provinces of Pakistan from 1980 to 2020, (ii) to provide scenario-based projections to 2050 using a reproducible Bayesian statistical model, and (iii) to identify provinces with SRB imbalance. Our study has several contributions as a result of achieving the research objectives. First, to the best of our knowledge, this is the first study on Pakistan SRB that has produced provincial estimates and projections during 1980 -2050. Second, it is the 1 st time Balochistan is identified with the existence and transition process of the sex ratio imbalance using a Bayesian model. Third, based on the SRB imbalance results of the Bayesian hierarchical time series model, we compute the number of missing female births over time in provinces with imbalanced SRB and quantify the female birth deficits in each Pakistan province. Our study included seven provinces of Pakistan: Balochistan, Khyber Pakhtunkhwa, Punjab, Sindh, Gilgit Baltistan, Islamabad (ICT), and Azad Jammu and Kashmir. The results for Federally Administered Tribal Areas are omitted because of the unavailability of the longer time series data on SRB.
The remainder of this paper is organized as follows. Section 1.1 provides the theoretical background of the study. Section 2.1 summarizes the database compiled for statistical modeling and Section 2.2 summarizes the Bayesian statistical model used for provincial SRB estimation and the post-modeling process (identifying provinces with imbalanced SRB and calculating the number of missing female births). Section 3 presents the SRB results by province, the provincial SRB imbalances, the corresponding missing female births, and the scenariobased missing female birth projections. Sections 4 and 5 summarize the primary contributions and limitations and conclude the study.

Theoretical background
Distortion in the SRB has been primarily attributed to three interlinked factors (Guilmoto, 2009;: (1) Son preference, (2) technological advances in prenatal diagnosis, and (3) preferences for smaller family size and consequent fertility decline. In countries with a patrilineal culture and shrinking family size, when prenatal sex determination and abortion technology are available, couples practice sex-selective abortion to secure at least one son. The SRB in such populations is male biased. SRB imbalance has been reported in 12 countries/areas since 1970 (Chao et al., 2019a).
Pakistan is a country that has a strong preference for sons (Atif et al., 2016;Hussain et al., 2000;Khan & Sirageldin, 1977;Sathar et al., 2015). Preference for male births in Pakistan stems from lineage, economic and social conditions, caste, and identity. At least one son in a strongly patrilineal society is essential for living arrangements in old age. One study suggested that the ideal family size in Pakistan (four children) has remained constant since the 1970s; moreover, the ideal sex composition of the children is more than 1 son (Wazir & Shaheen, 2016). Son preference is evidenced by the excess mortality of female children over male children under five in Pakistan, indicating possible differential treatment between girls and boys in this age group (Alkema et al., 2014;Sathar et al., 2015). The education attainment gap between females and males is large in Pakistan. Between 2017 and 2018, 30% of young women (age 15 -24) completed middle or higher education compared to 50% of young men (National Institute of Population Studies (NIPS) [Pakistan] & ICF, 2019). Between 2018 and 2019, 36% of girls (ages 5 -16) were out of school versus 25% of boys (Pakistan Bureau of Statistics (PBS) 2019). However, little evidence of prenatal sex preference has been reported in Pakistan. The previous studies identified no imbalanced SRB at the national level (Zaidi & Morgan, 2016;Chao et al., 2019a). Other studies suggested that, among couples in Pakistan, the desire for a large family might dominate preferences for children of a particular type (De Tray, 1984). A high prevalence of sexselective abortion was identified in two rural districts in Balochistan province (Qayyum & Rehan, 2017). However, the results mentioned above are based on survey data with small sample sizes.
The national scale levels and trends in SRB can mask the disparity among subregions in a country. Even in countries such as China and India, with an overall strong preference for sons, the SRB is not imbalanced in every province or state (Chao & Yadav, 2019;Chao et al., 2020;Ge et al., 2020;Jiang & Zhang, 2021 To the best of our knowledge, no study has provided the annual estimates of the provincial SRB in Pakistan using all available data since 1980. To accurately determine whether the SRB is imbalanced in Pakistan and if so, where the imbalance occurs, it is essential to estimate the SRB on the subnational level.

International Journal of Population Studies
Estimating the SRB in Pakistan is challenging for two reasons. First, limited data are available on birth histories in the past. Without a fully developed vital registration system in Pakistan, administrative birth records are lacking, and vital events are mostly estimated based on household surveys. Only a few sample surveys provided birth histories over different periods since the 1990s. Second, the data quality of census counts is typically low because of age heaping (Feeney & Alam, 1998). In historical census data, the number of children ever born in Pakistan is either unavailable or is unreliably reported. For example, birth histories were not collected in the 1981 Pakistan census (Ali et al., 2001). Accordingly, the individual-level data of the three most recent censuses in Pakistan (conducted in 1981Pakistan (conducted in , 1998Pakistan (conducted in , and 2017 contain only the populations of boys and girls under 1 year old. The SRB data from sample surveys such as Pakistan DHS are suffering from large uncertainties because of the small sample sizes and misreporting of female births. When estimating and projecting the provincial SRB in Pakistan, it is crucial to assess the levels and trends in the SRB by a reproducible statistical model. Using a Bayesian modeling approach for estimation and projection, observations from different data sources with varying levels of uncertainties can be synthesized and pooled together in a systematic and reproducible fashion. The Bayesian method can take into account both provincial SRB observations and external information on the SRB imbalance process to assist in model estimation and projection. Table 1 summarizes our database of provincial SRBs in Pakistan, with 531 SRB observations available in eight provinces of Pakistan. The reference years of these observations range from 1965 to 2019. The database contains 832,091 birth records by summing up the number of birth records where available. The number of birth records is unknown in some data series. Hence, the actual number of birth records involved in the study is more than what we reported here. The SRB observations were generated from the individual birth records in data sources with full birth histories (appendix for details of the data processing steps). The database is available as Supplementary File 1 (https:// doi.org/10.6084/m9.figshare.21548082).

Data sources
The DHS (ICF International, 2022) and Multiple Indicator Cluster Survey (MICS) provide the birth histories (either the full birth histories or the birth histories during the past 24 months before the survey interview) of women interviewed in retrospective survey questionnaires. Birth records are excluded if they were born more than 20 years prior to the year in which surveys were conducted to minimize recall bias from older women. Furthermore, the Pakistan Social and Living Standards Measurement Survey (PSLM) is a provincial-level survey with high coverage of households in Pakistan (Pakistan Bureau of Statistics [PBS], 2019). The PSLM records births over the 12 months before the date of the survey interview. The census is conducted once per decade and collects births in the 12 months preceding the census (Minnesota Population Center, 2019).
Given Pakistan's lack of reliable administrative birth data, it is essential to include all available data from different surveys to produce more reliable estimates and projections. The practice of making use of data from multiple data sources in estimation and projection has been widely used by international agencies, including the UNICEF, UN Population Division, and the Global Burden of Disease, and researchers in global and public health to reduce systematic bias from a single data source, to increase the length of the period that is covered by data (Alkema et al., 2016;Bearak et al., 2018;Gerland et al., 2014;Liu & Raftery, 2020;Masquelier et al., 2018;Wang et al., 2020;You et al., 2015). The data sources, we used as listed in Table 1, are based on provincial representative samples. If any future in-depth survey-specific consistency checks provide concrete evidence of bias in the examples of certain data sources, that particular data source should not be included  1990 -1991, 2006 -2007, 2012 -2013, 2017 -2018, 2019 301 253,580 MICS Two-stage stratified sample design 2010, 2011, 2014, 2003 -2004, 2007 -2008, 2016 -2017, 2017 -2018 37 153,772 † PSLM Two-stage stratified sample design 1995 -2016, 2005 -2006, 2007 -2008, 2013 -2014, 2018   International Journal of Population Studies in the modeling. As of now, no evidence of biased sampling has been reported for any of these data sources. With their own objectives of the survey, these data sources provide a wealth of information, and there is no supporting evidence to choose one at the cost of the others. Hence, we use all these sources in our model estimation and projection.

Methods
The model performance and predictive power were assessed by an out-of-sample validation exercise (leaving out recent International Journal of Population Studies observations) and simulation exercises (appendix for details). The validation and simulation results suggest good calibration and predictive power of the model.
The remainder of this section overviews the SRB Bayesian model.

Bayesian model for provincial SRB estimation and projection
The model for the SRB in Pakistan province is based on the model described previously by Chao et al. (2021a, b). In this study, we made a few modifications to the model to better address the data quality and availability of provincial SRBs in Pakistan. Subnational SRB models have been applied to other culturally and demographically heterogeneous countries with son preference such as Nepal (Chao, et al., 2022) and Vietnam (Chao et al., 2021c).
The outcome of interest Θ p,t , the SRB in Pakistan province p in year t is modeled as follows: .056 is the SRB baseline level for the entire Pakistan. The Pakistan SRB baseline b is estimated based on national SRB observations in Pakistan before the reference year 1970 (Chao et al., 2019a, b). p ∈ {1,…,k} is the province index where k = 7. t ∈{0,…,h} is the time index where t = 0 refers to the year 1980 and t = h refers to the year 2050. Φ p,t follows an AR (1) time series model on the log scale, which captures the natural fluctuations of SRB in each province over time. The values of ρ and σ ∈ (ρ = 0.9) and σ ∈ = 0.004) were not estimated but were borrowed from a previous study (Chao et al., 2019a,b), which robustly estimated the parameters from an extensive national SRB database. We assume that log (Φ p,t ) is causal and weakly stationary AR (1) process with parameters ρ and σ ∈ , and hence, log (Φ p,t-s ) is uncorrelated with ∈ p,t for all s>0. The unconditional variance at t = 0 is expressed as V U  2 2 1 / for log (Φ p,t ). Φ p,t is modeled as follows: . a 0 2 V δ p is the binary identifier of the sex ratio transition at the provincial level with only two possible values 1 and 0. δ p = 1 indicates an SRB imbalance in province p, whereas δ p = 0 indicates no imbalance in province p. The provincial sex ratio transition identifier parameter δ p is meant to detect whether the transition process exists based on the levels and trends in the SRB observations. To ensure that the probability parameter π p lies in the interval [0, 1], we use the logit transformed π p follows a hierarchical normal distribution with a global mean and variance µ π and V S 2 , respectively. δ p follows a Bernoulli distribution: Vague priors are assigned to the parameters related to the indicator that detects sex ratio transitions: α p,t refers to the province-specific SRB imbalance process. The process is assumed to be non-negative and is modeled by a trapezoidal function representing the three consecutive stages (increase, stagnation, and decrease) of the sex ratio transition. The trapezoidal function specification of α p,t is motivated by the patterns of nationallevel SRB observations in countries with strong statistical son preference to capture the three-stage transition process (Chao et al., 2021a). The trapezoidal functional form for α p,t can capture the shape of the observed SRB transition process according to Chao et al. (2021c) for those countries. α p,t is modeled as: The start year of the SRB inflation t 0p is modeled by a continuous uniform prior distribution with a lower bound at the year 1980 and an upper bound at the year 2050, respectively. For p ∈ {1,…,k}, we have: The province-specific period lengths of the three stages of the SRB inflation (λ 1p , and λ 3p ) are assigned with informative priors. The means of prior distributions are taken from a systematic study (Chao et al., 2021a) International Journal of Population Studies which modeled the sex ratio transition of multiple countries, including Pakistan. The standard deviations of prior distribution are set such that the CV is 0.1. The informative priors assist the provincial-level modeling of the sex ratio transition in Pakistan by exploiting the corresponding information at the national level. For p ∈ {1,…,k}, we have:

Data quality model
where i indexes all SRB observations across the provinces over time. r i is assumed to follow a normal distribution on the log scale with mean of log , Where, n = 531 is the total number of observations. σ i 2 is the sampling error variance of log (r i ), which reflects the uncertainty in log-scaled SRB observations because of the survey sampling design. σ i 2 is calculated using a jackknife method (Appendix A.1). ω 2 is the non-sampling error variance representing the uncertainty contributed by nonresponses, recall errors, and data input errors. We assume that ω 2 is immeasurable and is estimated using the model by assigning a vague prior: Z a  0 0 5 , . .

Posterior distribution Likelihood
For the i th observation r i , let v i = log (r i ) and V p t The likelihood on log-scale up to proportion is: For ∈{1,…n}, the likelihood can be written as:

Posterior Density
The posterior density for V p,t , the true SRB on the log scale for province p at time t, up to proportion is: , , , ®°°½ ¾°¿°U .

Statistical computing and Bayesian Inference
We obtained posterior samples of all the model parameters and hyperparameters using a Markov chain Monte Carlo (MCMC) algorithm, implemented in the open source software R 4.2.1 (R Core Team, 2022) and JAGS 4.3.0 (Plummer, 2003), using R-packages R2jags (Su & Yajima, 2015) and rjags (Plummer, 2018). Results were obtained from 10 chains with a total of 5000 iterations in each chain, while the first 1000 iterations were discarded as burn-in. After discarding burn-in iterations and proper thinning, the final posterior sample size for each parameter by combining all chains is 25,000. The convergence of the MCMC algorithm and the sufficiency of the number of samples obtained were checked through visual inspection of trace plots and convergence diagnostics of Gelman & Rubin (1992), implemented in the coda R-package (Plummer et al., 2006). In provinces of Pakistan without past/ongoing SRB inflation (assessed in the model), we simulate the SRB imbalance process after 2020 for different starting years of SRB inflation.

Post-modeling process
The simulated province-specific SRB imbalance process δ p α p,t is based on posterior samples in the model. The simulated δ p α p,t is added to the projected Θ p,t for different starting years of SRB inflation in each province. The simulation process is detailed in the appendix. Figure 2 shows the simulated SRB imbalance process δ p α p,t in a Pakistan province, with a given start year of the inflation process t 0 . The SRB inflation process spans 38 years. After approximately one decade, the imbalance reaches its maximum level and remains around that level for approximately 7 years. The SRB imbalance then deflated toward the normal/reference level of SRB (i.e., the SRB inflation becomes zero) over the next 15 years.

Computing the number of missing female births
Let ψ p,t and < p t inflation free , denote the estimated and expected inflation-free numbers of female live births respectively, in province p in year t. The estimated and expected numbers of female birth are calculated as ψ p,t = B p,t /(1+Θ p,t ) and in the respective given province year. The number of missing female births is calculated using a method introduced in Dréze and Sen (1990), which was reviewed and validated in Guilmoto et al. (2020).
The annual number of missing female births (AMFBs) in province p in year t is defined as: , , . The cumulative number of missing female births (CMFB) from t 1 to t 2 in province p is obtained by adding the AMFB from year t 1 to year t 2 :  Figure 4 illustrates the disparity in SRB across geographic locations. The SRB is most inflated in the southwest and northeast regions, including the Balochistan and Gilgit Baltistan provinces. Table 2 lists the modeled SRB inflation probability in each Pakistan province. The probability of having a past or    ongoing SRB inflation is the highest in Balochistan at 100%. As the probability in Balochistan is the only probability above the cutoff value (95%), we identify Balochistan as the only province in Pakistan with SRB imbalance that happened before 2020. International Journal of Population Studies period, the lower bound of the 95% credible intervals of SRB exceeded the national baseline.

Scenario-based missing female births simulation after 2020
Although we identify Balochistan as the only province with past/ongoing SRB inflation, we do not rule out the possibility that the imbalanced SRB will emerge in other International Journal of Population Studies provinces. Hence, we present the results of scenario-based projections of missing female births in provinces without current SRB inflation. Figure 6 shows the AMFB results in four provinces: Islamabad (ICT), Khyber Pakhtunkhwa, Punjab, and Sindh, in which the projected annual total numbers of births are available and are not identified with past SRB inflation.
Each row in each heatmap shows the simulated AMFBs over the projection period 2021 -2050 under the scenario that the SRB inflation process starts in a certain year.
In general, the later the assumed start year of the SRB imbalance process, the AMFBs in more years are projected near zero. In Figure 6, the heatmap, these trends manifested as increasing extents of blue areas as the rows are traversed from bottom to top. In the heatmap, darker blue means a smaller number of missing female births while redder means a higher number of missing female births. This implies that while the start year of the SRB inflation process delays year by year (i.e., moving up on the y-axis), there are delayed effects on the occurrence of International Journal of Population Studies the number of missing female births. When the simulated SRB becomes imbalanced during the projection period, the three stages of the sex ratio transition (increase, stagnation, and return to normal) are visible in the resulting AMFBs. However, the AMFB is influenced not only by the SRB imbalance process but also by the levels and trends in the total number of births over time. Accounting for both the SRB inflation and the total number of births, we project the maximum AMFB for different combinations of start-year scenarios and the year in the projection period. In Islamabad (ICT), the maximum AMFB is projected to occur in 2050 when the SRB imbalance process starts in 2035, with the AMFB at 1.7 thousand. In Khyber Pakhtunkhwa, the maximum AMFB is projected at 29.2 thousand in 2049 when the SRB inflation starts in 2034. The largest AMFB in Punjab is projected to occur in 2033 when the SRB inflation stats in 2021 and its projected value is 76.2 thousand. In Sindh, when the SRB imbalance process starts in 2022, the maximum AMFB across all scenarios is projected to occur in 2035 at 37.8 thousand.

Discussion
Policy planners can prepare guidelines for preventing prenatal sex discrimination using scenario-based projections of the number of missing female births in the provinces without ongoing SRB inflation. A recent study provided missing female birth projections at the national level for all countries in the world (Chao et al., 2021b). According to that study, if sex-selective abortion were to happen in Pakistan national wide, the missing female births may contribute as high as 14% of the global numbers during the 2021 -2100 period.
Our study reveals the potential future missing female births in Pakistan in four provinces, namely, Islamabad (ICT), Khyber Pakhtunkhwa, Punjab, and Sindh. For different start years of the SRB imbalance in each province, we identify the years in which the number of missing female births will possibly deficit the greatest. The projections results reflect the fact that the number of missing female births is a combined effect of the SRB inflation process and fertility transition. Given the speed of the fertility transition in Pakistan, and the estimates from the 2017 -2018 DHS, we revealed that the decline in fertility rates in Pakistan has slowed at both the national and subnational levels since the 1990s, and the total fertility rate at 3.6 remains higher than in neighboring countries (National Institute of Population Studies (NIPS) [Pakistan] & ICF, 2019).
The findings of this study reinforce the persistence of strong son preferences in Pakistan and identified high-level distortion in Balochistan. The three conditions of Guilmoto (2009) hypothesis for the distortion of SRB in Pakistan are currently well aligned: Inherited gender discrimination, preference for large families, and technological advancement. Pakistan fares poorly on gender equality, ranking 135 of 191 countries on the Gender Inequality Index (United Nations Development Programme, 2022). The deep-rooted social and cultural norms and practices continue to be the underlying cause of gender inequalities.
Furthermore, inflated SRB influences the demand of a larger number of children, currently manifesting as higher fertility in Pakistan for the sake of more sons. The fertility stagnation in Pakistan since the onset of the 21 st century is evident primarily attributed to the higher ideal family size with at least two sons (Wazir, 2018). The imbalanced SRB leads to prolonged consequences in both demographic and social aspects. Imbalanced SRB is one of the main factors that lead to the phenomenon of "missing women" first endeavored by Amartya Sen, referring to the females that should have survived or been born in the absence of sex discrimination and excess mortality among females (Sen, 1990). A large number of "missing women" results in a marriage squeeze, increased levels of antisocial behavior and violence, and may eventually have a long-term impact on stability and social development.
Despite the strong son preference persists in Pakistan, one study implied that son preference has not resulted in nationwide sex-selective abortions but occurred in subpopulations such as in urban clinics (Sathar et al., 2015). Although abortion is illegal in Pakistan, the abortion rate significantly increased from 27 to 50/1000 for women aged 15 -49 over the period 2000 -2012. Meanwhile, Balochistan experienced the highest abortion rate of 60/1000 for women aged 15 -49 among the provinces (Sathar et al., 2014). In addition, our study also reinforces the distortion of SRB in Balochistan is highest among the provinces. These numbers are associated with well-documented demographic phenomena of "missing women" at the national and provincial levels in Pakistan. Although, the compelling evidence of sex selection and excess mortality are not prevailing in Pakistan on the national level, the absence of evidence is not the evidence of absence. Our subnational study pinpoints the disparity in SRB and sex-selective abortion on the provincial level that is masked by national-level results. There is an urgent need to generate high-quality data in Pakistan, particularly through census at the subnational level followed by in-depth research on the persistence of discriminatory practices of sex selection and excess mortality for females. This is the first study on estimating SRB in Pakistan from 1980 to 2020 and provides scenario-based projections of missing female births up to 2050 by province based on a Bayesian hierarchical time series model. Our results revealed important SRB disparity across geographic locations in Pakistan. Among the seven provinces included in the study,

International Journal of Population Studies
Balochistan presents a decisively imbalanced SRB. In the other provinces without the existing SRB inflation, we demonstrate important disparities in the occurrences and quantities of female birth deficits before 2050.

Conclusions
Our study provides model-based and data-driven SRB estimates and projections for provinces in Pakistan from 1980 to 2050. Our model results demonstrate important disparities in SRB levels and trends across provinces over time. Balochistan is identified as the only province in Pakistan with an existing SRB imbalance and, consequently missing female births. In future work, in-depth provincial studies and the collection of high-quality birth data are required to monitor subnational SRB disparities in Pakistan.
Effective program and policy solutions to curb sex discrimination remain elusive in Pakistan because the practices, leading to excess mortality among females and sex selection, are often poorly understood. Therefore, the institutional response is primarily focused on the improvement of the provision of health care. The last two decades have witnessed the adoption of several pro-women laws such as prevention of sexual violence and harassment, protection, domestic violence, and early marriages. However, the implementation remains challenging, primarily because of federal and provincial autonomy to deliver basic social services. Advancing gender equality and discriminatory practices require accountability mechanisms for policy implementation and enforcement of laws, adequate financing at the provincial level, and community engagement to address discriminatory gender and social norms.
Both Demographic and Health Survey (DHS) and Multiple Indicator Cluster Survey (MICS) provide individuallevel data with the full birth history of each woman of reproductive age interviewed during the survey fieldwork period. We calculated the sampling error in the logtransformed sex ratio at birth (SRB) obtained from the DHS and MICS data series using the jackknife method (Efron & Gong, 1983;Efron & Tibshirani, 1994;ICF International, 2012). For a certain DHS or MICS data series, let U denote the total number of clusters (based on the cluster/primary sampling unit numbers in the survey data (Verma & Le 1996)). The u-th partial prediction of SRB is determined by the following equation: Where, n indexes the live births in each state-surveyyear; N is the total number of live births; and x n , d n , and w n are the sex, cluster number, and sampling weight for the n-th live birth, respectively. The sampling weight of each birth w n is extractable from the survey data and reflects the survey sampling design (Verma & Le, 1996). We define I n (.) = 1 if the condition inside the brackets is true and I n (.) = 0 otherwise. The u-th pseudo-value estimate of the SRB on the log-scale is: The sampling variance is: In the DHS or MICS data, the annual log-transformed SRB observations are merged such that the coefficient of variation (CV) for log-transformed SRB is below 0.1 or the merged period reaches 5 years (Pedersen & Liu, 2012).
For a certain DHS/MICS data series, let {t n , t n−1 ,…,t 1 } be years with recorded births from recent to past. The merge starts from the most recent year t n and goes back year by year to t n−1 ,…,1. The process is performed by the following algorithm for each DHS and MICS data series: Step Set t=t n−2 , merge births from t n , t n−1 , t n−2 by summing them up 13 Repeat steps 8 -12 for t ∈ {tn−2,…, t1}

A. 1. Further explanations of the motivation and assumptions of merging DHS/MICS data
The above algorithm is to merging observations from single calendar years into observations over short time periods from DHS and MICS surveys where full birth histories are available. The merge refers to summing up the number of sex-specific births across multiple years before computing the SRB for that period. The purpose of the merging process is meant to reduce uncertainties associated with observations to a reasonable level. Without merging the births from each calendar year, the sampling errors in these population indicators become unacceptably large due to the small sample sizes (Pedersen & Liu, 2012). The underlying assumptions for the following expressions are: • Step 1 for t ∈ {t n , t n−1 ,…,t 1 } do: this means that we are merging the 1-year observations from the most recent year t n to the earliest year with data t 1 . The reason we merge the observations backward in time rather than merge forward is that: Usually in countries where DHS and MICS surveys are conducted, more births were sampled in recent years than in older years. This is largely due to the improvement of surveying technology, more mothers were still alive to recall recent births than births born decades ago, and less recall bias happened to births born in recent years than in earlier periods. Hence, the 1-year observations in recent years are usually less likely to be merged Table D.1 summarizes the results of the left-out SRB observations in the out-of-sample validation exercise and one-province simulation. The median errors are nearly zero in the left-out observations. Although the median absolute errors are slightly higher than the median errors, the average coefficient of variance of the absolute errors for left-out observations (calculated as absolute errors divided by the left-out observation values) is only 5.6%. The coverage of the 95% and 80% prediction intervals is more conservative than expected. The wider-than-expected prediction interval in the left-out observations can be primarily attributed to larger uncertainty in more recent observations. Table D.2 compares the model estimates obtained from the full dataset and the training set in the out-ofsample validation exercise. Here, we examined the model estimates of the true SRB Θ p,t and the inflation process with province-specific probability δ p α p,t . The median errors and the median absolute errors are close to zero.
In summary, the validation results indicate reasonably good calibrations and prediction power of the inflation model with conservative credible intervals. Note: Error is defined as the difference between a left-out SRB observation and the posterior median of its predictive distribution. SRB observations with data collection years since 2018 are left out. Numbers in the parentheses after the proportions indicate the average number of left-out observations that fall below or above their respective 95% and 80% prediction intervals.