A Multi-State Generalization of the HMD Methodology, Applied to Fertility by Parity in the United States, 1917-2000 John R. Wilmoth Department of Demography University of California, Berkeley Presentation for the 1st HMD Symposium “Human Mortality over Age, Time, Sex, and Place” Rostock, Germany, 18-19 June 2004

Lexis Diagram Age Lifeline for a single individual Time

Three Shapes for 1x1 Lexis Region Age Age-cohort Age-period Period-cohort Time

Lexis Model – Definition A complete description of a population over age and time, including detailed estimates of population size (and perhaps characteristics), as well as the transition rates (and events) associated with changes in population size (and characteristics), as a function of both age and time.

Lexis Model – Mortality example Population surface: matrix of estimated population sizes, P(x, t), for 1-year age intervals (from age x to x+1) on January 1st of each year, t. Mortality surface: matrix of death rates, m(x, t), for 1-year intervals of both age and time (i.e., in 1x1 Lexis squares).

Lexis Model – Applications Lexis models can be used to describe: The actual life history of cohorts The hypothetical life history of cohorts (for example, mortality/longevity of a cohort in the absence of migration) The hypothetical life history of synthetic cohorts (for example, mortality/longevity during some time period)

Single- vs. Multi-State Models of Demographic Processes In a single-state model, individuals are either in or out of the population. Exit can occur through mortality or emigration, and entry can occur through birth or immigration. In a multi-state model, members of a population are classified according to certain states (i.e., characteristics or statuses), which may change over the life course.

Single-State Model of Mortality and Migration, with Transition Rates b Alive Birth b birth rate m mortality rate i immigration rate e emigration rate i m+e Death / Migration

Three Types of Multi-State Models Closed population (no mortality or migration) Open population with invariant rates of exit/entry by state Open population with variable rates of exit/entry by state

Multi-State Fertility Model for Closed Population fn + f1 f2 fn -2 fn -1 Parity 0 Parity 1 … Parity n - 2 Parity (n -1) +

Multi-State Fertility Model for Open Population with Invariant Rates of Exit/ Entry by Parity fn + f1 f2 fn -2 fn -1 Parity 0 Parity 1 … Parity n - 2 Parity (n -1) + m+e i m+e i m+e i b i m+e Death / Migration Birth

Multi-State Fertility Model for Open Population with Variable Rates of Exit/Entry by Parity fn + f1 f2 fn -2 fn -1 Parity 0 Parity 1 … Parity n - 2 Parity (n -1) + i(n-1)+ m0+e0 m1+e1 in-2 mn-2+en-2 i1 b m(n-1)++e(n-1)+ i0 Death / Migration Birth

Two Kinds of Fertility Rates Transition rates (Type I) Accrual rates (Type II)

Multi-State versus Multiple-Decrement Models Multi-state models describe variation in characteristics of a population Multiple-decrement (or increment) models describe variation in modes of transition (e.g., competing causes of death, causes of migration, causes of pregnancy)

Lifelines in Single-State Lexis Model Age a x + 2 d · · Out migration x + 1 o b x o In migration x x x Death e c x - 1 o t - 1 t t + 1 t + 2 Time

Lifelines in Multi-State Lexis Model Age a z3 x + 2 d · Out migration · x + 1 o b o In migration x z2 x z3 x x Death e z1 c z1 z2 z3 etc. State Transitions z1 x - 1 z2 o t - 1 t t + 1 t + 2 Time

Population Changes in Single-State Model (mortality only) Age x + 1 P2 DL x DU for k = 1, 2, … , n+ P1 x - 1 t t + 1 Time

Demographic Accounting in Single-State Model (mortality only)

Population Changes in Multi-State Model (both fertility and mortality) Age x + 1 DL P2(k) BL (k) x where k = 1, 2, … , n+ for births but k = 0, 1, … , (n-1)+ for women BU (k) P1(k) DU x - 1 t t + 1 Time

Demographic Accounting, Multi-State Model (both fertility and mortality) for k = 1, 2, …, n - 2

Demographic Accounting, Multi-State Model (cont.)

Necessary or Simplifying Assumptions (for purpose of estimating models) For both single- and multi-state models, it is necessary to choose between: Uniform distribution of events Constant transition rates over some age-time interval (e.g., Lexis triangle) For multi-state model, it is convenient to assume that rates of death and migration are equal across parity states

Demographic Accounting in Two Directions Time moving forward versus Time moving backward

Available Fertility Data for the U.S. Age 50 14 1917 1940 2001 Time

Elementary Formulas for Estimating Single- and Multi-State Lexis Models

Some Results Age patterns of fertility rates Time trends in fertility rates Summary measures of lifetime fertility (for both periods and cohorts) Distributions of women by parity

Age Patterns of Fertility Rates

Parity-Specific Fertility Rates over Age by Decade, U. S Parity-Specific Fertility Rates over Age by Decade, U.S., Ages 14-49 a) Birth Order k = 1

Parity-Specific Fertility Rates over Age by Decade, U. S Parity-Specific Fertility Rates over Age by Decade, U.S., Ages 14-49 b) Birth Order k = 2

Parity-Specific Fertility Rates over Age by Decade, U. S Parity-Specific Fertility Rates over Age by Decade, U.S., Ages 14-49 c) Birth Order k = 3

Parity-Specific Fertility Rates over Age by Decade, U. S Parity-Specific Fertility Rates over Age by Decade, U.S., Ages 14-49 d) Birth Order k = 4+

Time Trends in Fertility Rates

Parity-Specific Fertility Rates over Time by Age Group, U. S Parity-Specific Fertility Rates over Time by Age Group, U.S., 1917-2000 a) Birth Order k = 1

Parity-Specific Fertility Rates over Time by Age Group, U. S Parity-Specific Fertility Rates over Time by Age Group, U.S., 1917-2000 b) Birth Order k = 2

Parity-Specific Fertility Rates over Time by Age Group, U. S Parity-Specific Fertility Rates over Time by Age Group, U.S., 1917-2000 c) Birth Order k = 3

Parity-Specific Fertility Rates over Time by Age Group, U. S Parity-Specific Fertility Rates over Time by Age Group, U.S., 1917-2000 d) Birth Order k = 4+

Summary Measures of Lifetime Fertility

Total Fertility Rate (TFR), 1917-2000, and Completed Fertility Rate (CFR), 1903-1950, U.S.

Distributions of Women by Parity

Observed and Hypothetical Distributions of Women by Parity, U. S Observed and Hypothetical Distributions of Women by Parity, U.S., 1917-2001 a) Age 30

Observed and Hypothetical Distributions of Women by Parity, U. S Observed and Hypothetical Distributions of Women by Parity, U.S., 1917-2001 b) Age 50

Possible Applications for Multi-State Lexis Model Fertility status (parity 0, 1, 2, 3, etc.) Marital status (single, married, divorced, widowed, remarried, etc.) Health status (healthy, mildly disabled, severely disabled, etc.) Criminal status (unrestricted, incarcerated, parole, probation, etc.)

Possible Applications (cont.) Immigration status (citizen, permanent resident, temporary resident, illegal, etc.) Residency status (United States, Canada, Mexico, etc.) Employment status (employed full-time, employed part-time, unemployed, etc.) Educational status (full-time student, part-time student, graduate, dropout, etc.)

Simplifying Characteristics of Multi-State Fertility Model Data are reliable and readily available (at least for some countries) States are well defined No reverse transitions