Figures and data in Stay on Target: Population Projections and Microsimulation Design

Abstract

Even though standard cohort component models are relatively easy to comprehend, designing simulations to match expected population estimates and projections in multiple countries and over long spans of time is surprisingly challenging. We identify microsimulation design options that replicate United Nations population estimates and projections in three countries (Norway, United States, and India) over a 150-year time span. The design adapts the United Nations cohort component model, which implies a certain ordering of demographic events, and uses either cohorts or age groups to assign risk. We design four simple microsimulations that use United Nations demographic statistics as exogenous variables. One model adjusts event ordering and assignment of risk by age, called the Split Fertility design, and one model does not. Each model operates in either one- or five-year steps. The Split Fertility design has less than one percent divergence in total births, deaths, and population in all three countries. The simpler design produces varying magnitudes of divergence, as large as 20% in India. The Split Fertility design is suitable for simulations that seek to maintain population dynamics in multiple countries, whether operating in one- or five-year steps. The Split Fertility design is ideal for comparative simulations that adapt demographic statistics from cohort component models. Development of the design highlights the flexibility of simulations and the importance of careful interpretation of exogenous statistics in simulation design.

This article contains 7 figures and 5 tables.

Figures

**Figure 1.** Results of Norway Models, scaled up to actual population size

**Figure 2.** Results of USA Models, scaled up to actual population size

**Figure 3.** Results of India Models, scaled up to actual population size

**Figure 4.** Population Dynamics Among Females of Child-bearing Ages (15-49), UN Data WPP2019

**Figure A1.** Results of Norway Models with Additional Stochasticity Options, scaled up to actual population size

**Figure A2.** Results of USA Models with Additional Stochasticity Options, scaled up to actual population size

**Figure A3.** Results of India Models with Additional Stochasticity Options, scaled up to actual population size

Tables

Table 1. Assignment of Demographic Event Risk by Age for 1-year Split Fertility Design

	t=0	Rnd. 2	t=1	Rnd. 2	t=2	Rnd. 2	t=3	Rnd. 2	t=4	Rnd. 2
Agent Age Characteristics
Current Age	43	44	44	45	45	46	46	47	47	48
Starting Age	43		43		43		43		43
Ending Age	48		48		48		48		48
Corresponding Age Labels for UN Statistics
Fertility Rate	40-44	40-44	40-44	45-49	45-49	45-49	45-49	45-49	45-49	45-49
Survival Ratio	40-44		40-44		40-44		40-44		40-44
Migration Count	45-49		45-49		45-49		45-49		45-49

Table 2. Years of Risk among Newborn Cohorts

Cohort	Year 0	Year 1	Year 2	Year 3	Year 4	Total Years of Risk	Percent of Total
A	1	1	1	1	1	5	33
B		1	1	1	1	4	27
C			1	1	1	3	20
D				1	1	2	13
E					1	1	7
Total						15	100

Table 3. Model Design Characteristics and Formulas

	Intuitive Models		Split Fertility Models
Time	5-year step and interval t =T= [1950,1955,…2100]	1-year step t= [0,1,2,3,4], 5-year interval T=[1950,1955,…2100]	5-year step and interval t =T= [1950,1955,…2100]	1-year step t= (0,1,2,3,4), 5-year interval T=[1950,1955,…2100]
Event ordering	0) Initialization, 1) fertility, 2) infant initialization, 3) mortality, 4) aging, 5) migration, 6) time step/interval advancement, T+5.	0) Initialization, 1) fertility, 2) infant initialization, 3) mortality, 4) aging, 5) migration, 6) time step advancement, 7) repeat events 1-6 for t<5 until interval advancement, T+5.	0) Initialization, 1) fertility (round one), 2) infant initialization (round one), 3) mortality, 4) aging, 5) migration 6) fertility round two, 7) infant initialization (round two), 8) second round infant mortality, 9) second round infant aging 10) time step/interval advancement, T+5.	0) Initialization, 1) fertility (round one), 2) infant initialization (round one), 3) mortality, 4) aging, 5) migration 6) fertility round two, 7) infant initialization (round two), 8) second round infant mortality, 9) second round infant aging 10) time step advancement, 11) repeat events 1-10 for t<5 until interval advancement, T+5.
Current age (A)	5-year age groups, A=[0,5…105]	1-year age groups A=[0,1,…105]	5-year age groups, A=[0,5…105]	1-year age groups A=[0,1,…105] For immigrants: A=eA-t
Initial age (iA)	Not Applicable	Not Applicable	Not Applicable	iA=A-t
End age (eA)	Not Applicable	Not Applicable	Not Applicable	eA=A+(5-t)
Model Age-Specific Fertility Rates (ASFR)	$\frac{(U N A S F R)}{1000} * 5$	$\frac{(U N A S F R)}{1000}$ ; Apply to current age (A)	$\frac{(U N A S F R)}{1000} * \frac{5}{2}$	$\frac{(U N A S F R)}{1000} * \frac{1}{2}$ ; Apply to current age (A)
Infant initialization	Assigned age -5, and sex based on UN sex ratio	Assigned age -1, and sex based on UN sex ratio	Assigned age -5, and sex based on UN sex ratio	Assigned age -1, and sex based on UN sex ratio.
Model Mortality Rate (MR)	1 - (UNSR)	$1 - \sqrt[5]{(U N S R)}$ ; Apply to current age (A)	1 - (UNSR)	$1 - \sqrt[5]{(U N S R)}$ ; Apply to initial age (iA). For newborn cohorts: $1 - \sqrt[3]{U N S R [0]}$ ; For agents age 110+: 1. For immigrants: no mortality
Model Emigration count (EC)	(UNEC) * (Sample%)	$\frac{\left (UNEC\right)*\left (Sample\text{\% }\right)}{25}$ ; Apply to current age (A)	(UNEC) * (Sample%)	$\frac{(U N E C) * (S a m p l e \%)}{25}$ ; Apply to end age (eA). For newborn cohorts: $\frac{(U N E C) * (S a m p l e \%)}{15}$
Model Immigration count (IC)	(UNIC) * (Sample%)	$\frac{\left (UNIC\right)*\left (Sample\text{\% }\right)}{25}$ ; Apply to current age (A)	(UNIC) * (Sample%)	$\frac{(U N I C) * (S a m p l e \%)}{25}$ ; Apply to end age (eA). For newborn cohorts: $\frac{(U N I C) * (S a m p l e \%)}{15}$ Note: All immigrants receive a tag to provide immunity to mortality and emigration.
Immigrant initialization	Assign current age (A)	Assign current age (A)	Assign current age (A)	Assign end age (eA), then calculate A and iA by subtracting remaining time steps and 5, respectively.
Period advance	NA	t + 1	NA	t + 1
Interval advance	T + 5, remove any tags that identify newborn cohorts or immigrant agents

Table 4. Average Percent Divergence from UN Target Values in 5-year Models

		Births		Deaths		Total Population
Country	Period	Intuitive	SF*	Intuitive	SF*	Intuitive	SF*
Norway	1950-2099	2.7	0.0	0.8	0.0	1.5	0.0
	1950-1999	2.6	0.0	0.1	0.0	0.5	0.0
	2000-2049	2.7	0.0	0.2	0.0	1.5	0.0
	2050-2099	2.7	0.0	1.5	0.0	2.0	0.0
USA	1950-2099	3.7	0.0	1.8	0.0	2.7	0.0
	1950-1999	3.6	0.0	0.2	0.1	1.3	0.0
	2000-2049	4.2	0.0	1.0	0.0	3.0	0.0
	2050-2099	3.4	0.0	3.1	0.0	3.2	0.0
India	1950-2099	9.3	0.0	7.0	0.0	8.9	0.0
	1950-1999	7.3	0.0	3.2	0.0	4.0	0.0
	2000-2049	11.2	0.0	5.0	0.0	9.3	0.0
	2050-2099	9.4	0.0	10.1	0.0	10.7	0.0

Split Fertility Design

Table 5. Average Percent Divergence from UN Target Values in 1-year Models

		Births		Deaths		Total Population
Country	Period	Intuitive	SF*	Intuitive	SF*	Intuitive	SF*
Norway	1950-2099	1.6	0.3	3.3	0.5	2.1	0.3
	1950-1999	1.0	0.3	3.3	0.9	1.2	0.1
	2000-2049	0.8	0.3	3.8	0.7	1.3	0.2
	2050-2099	2.9	0.1	3.0	0.2	3.3	0.4
USA	1950-2099	2.0	0.5	2.2	0.6	3.1	0.3
	1950-1999	0.9	0.6	3.6	0.5	1.4	0.2
	2000-2049	1.6	0.2	3.1	0.6	2.7	0.3
	2050-2099	3.5	0.5	0.8	0.6	4.5	0.2
India	1950-2099	19.5	0.3	20.8	0.6	20.7	0.3
	1950-1999	8.1	0.3	18.3	1.1	11.1	0.3
	2000-2049	23.8	0.4	14.6	0.5	21.2	0.3
	2050-2099	29.4	0.3	25.7	0.4	24.4	0.4

Split Fertility Design