Page 73 - IJPS-8-2

International Journal of
Population Studies Projecting sex ratio at birth in Pakistan

compared to those in earlier periods because the Appendix B. Bayesian model for provincial SRB
sample sizes are usually larger in recent years. By estimation and projection
merging backwards, we are able to preserve more
observations in recent years without merging them. B. 1. Notations
If we merge forward, it is likely that some of the The notations are listed in the table below.
observations in recent periods would be merged with
observations in early periods. B. 2. Scenario-based simulated projections for SRB
• Step 4 if CV < 0.1 or t − t > 1: When time t is at the inflation
n−1
n
most recent year t , the condition to stop merging the The province-specific SRB imbalance process δ α is
n
p
p,t
birth to t , the previous year where data is available is simulated using posterior samples from the model. The
n−1
(i) the CV < 0.1; or (ii) t is not the adjacent calendar simulated δ α is added to the projected bΦ for different
n−1
p
p,t
p,t
year of t , i.e. t − t n−1 > 1. starting years of the SRB inflation in each province.
n
n
• Step 9 if CV < 0.1 or t − t > 1 or t − t = 5: When For g ∈ {1,…,G}, the g simulated SRB inflation is denoted
th
n−1
n−1
n−2
n
time t is not at the most recent year t , the condition to ' g g
( ) ( ) g
n
th
stop merging the birth to t is (i) the CV < 0.1; or (ii) as α , pt δ p . p is the g posterior sample of parameter δ .
p
n−2
t is not the adjacent calendar year of t , that is, t ’ th
g
pt, is the g simulated SRB imbalance process, with the
n−2
n−2
n−1
− t > 1; or (iii) the merged period is already 5 years, ’
g
n−2
that is, t − t = 5. We set the upper limit of merging start year of inflation fixed at t ∈ {2021,…,2050}. pt, is
0
n
n−1
period to 5 years because we want to still generate simulated as below for g ∈ {1,…,G}:
(
a time series of observations even after merging. α ' ( ) g = ξ ( ) g / λ ( ) g )( −t t ), if t <<t t ( ) g
Without setting an upper limit, too few numbers of , pt p 1p 0 0 1p
observations may be generated just to satisfy the α ' ( ) g = ξ , if t ( ) g <<t t ( ) g
( ) g
CV condition. The 5-year upper bound follows the , pt p 1p 2p
(
conventional period length used in other population α ' ( ) g = ξ ( ) g −ξ ( ) g / λ ( ) g )( t −t ( ) g ) , if t ( ) g < <t t ( ) g
indicators as suggested by Pedersen & Liu, 2012. , pt p p 3p 2p 2p 3p
Symbol Description
Index
i Indicator of the i SRB observation across all province-years, i ∈ {1,…,531}
th
t Indicator of year, t ∈ {1980,…,2050}
p Indicator of provinces of Pakistan, p ∈ {1,7}
Unknown parameters
Θ Model fitting to the true SRB in Pakistan province p in year t
p,t
Φ p,t Province year-specific multiplier for capturing the natural fluctuation in SRBs around the national baseline b in Pakistan province p in year t
α p,t SRB imbalance in Pakistan province p in year t
t Start year of SRB inflation in Pakistan province p
0p
δ Indicator of the presence (δ = 1) or absence (δ = 0) of SRB inflation in Pakistan province p
p p p
ξ Maximum level of SRB inflation in Pakistan province p
p
λ Period length of the increase stage of the sex ratio transition in Pakistan province p
1p
λ 2p Period length of the stagnation stage of the sex ratio transition in Pakistan province p
λ Period length of the decrease stage of the sex ratio transition in Pakistan province p, which returns the SRB to the national SRB baseline
3p
ω Non-sampling error
r The i SRB observation
th
i
σ Sampling error for the ith SRB observation (computed in Appendix A.1)
i
b Baseline level of SRB over the whole of Pakistan (Chao et al., 2019a), where b = 1.063
ρ Autoregressive Indicator of Φ , where ρ = 0.9 (Chao et al., 2019a, b)
p,t
σ Standard deviation of distortion parameter for Φ , where σ = 0.004 (Chao et al., 2019a, b)
∈ p,t ∈
Volume 8 Issue 2 (2022) 67 https://doi.org/10.36922/ijps.v8i2.332