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Technical Report: Bayesian Fertility Rate Projections

Summary

This report presents a Bayesian statistical model for projecting cohort fertility rates using US Census Current Population Survey (CPS) data spanning 1976-2024. The model uses a hierarchical log-linear structure with Gaussian random walk priors to capture smooth temporal changes in age-specific fertility patterns across birth cohorts. The primary model (Version 4) extends the base model to explicitly capture cohort-specific timing shifts in childbearing, separating quantum (how many children) from tempo (when they are born).

Key findings:

1. Introduction

Fertility rates in the United States have been declining for several decades. Period fertility rates (Total Fertility Rate, or TFR) provide a snapshot of births in a given year, but they can be influenced by timing effects—women delaying childbearing can depress TFR even if lifetime fertility remains unchanged.

Cohort fertility rates provide a more stable measure of long-term reproductive patterns by following specific birth cohorts throughout their reproductive years. The key question is: How many children will recent birth cohorts (those born in the 1990s and 2000s) have over their lifetimes?

This question has profound implications for:

Projecting cohort fertility presents several challenges:

  1. Incomplete data: Younger cohorts are still in their reproductive years

  2. Survey weights: CPS uses complex sampling; results must reflect the population

  3. Uncertainty quantification: Point estimates are insufficient; we need credible intervals

  4. Model validation: How can we trust projections for cohorts that haven’t completed fertility?

We address these challenges using a Bayesian hierarchical model that:

2. Data

2.1 Data Sources

Current Population Survey (CPS)

The CPS is a monthly household survey conducted by the US Census Bureau. In June of most years, it includes a Fertility Supplement asking women about their childbearing history.

IPUMS CPS Data (Extract 13):

Note on data quality: This analysis uses cumulative parity (children ever born) rather than annual birth events. Parity is generally reported more accurately than recent birth measures, especially at older ages, and is less subject to undercounting concerns that may affect annual birth statistics.

Sampling:

2.2 Data Preprocessing

The preprocessing pipeline involves several steps:

  1. Load data: Read CPS data from IPUMS extract (.dta.gz format)

  2. Variable selection: Keep only relevant variables for modeling (female respondents, age ≤ 54)

  3. Parity computation: Convert FREVER to parity (total children ever born)

  4. Cohort construction: Compute birth year (cohort = year - age) and assign to 3-year bins (birth_group)

  5. Age binning: Group respondents into 3-year age bins (16, 19, 22, ..., 52) labeled by midpoint

  6. Survey weight handling: Use resampling with replacement to incorporate FRSUPPWT weights

  7. Aggregation: Compute total parity (sum_df) and number of respondents (count_df) for each cohort-age group

  8. Validation: Compare computed CFR with Census historical data

  9. Save: Store preprocessed data in HDF5 format for modeling

See notebooks/process_cps.ipynb for full preprocessing code.

2.3 Survey Weight Handling: Resampling Approach

Challenge: The Bayesian model uses a Poisson likelihood that expects integer counts. Survey-weighted data produces continuous (non-integer) values.

Solution: Resampling with replacement according to survey weights:

  1. Normalize weights within each survey year

  2. Resample respondents with probability proportional to their weight

  3. This produces a pseudo-sample where each respondent can appear 0, 1, or multiple times

  4. The resulting data represents the population rather than just the sample

  5. Maintains integer count structure for Poisson likelihood

Justification:

2.4 Data Structure

After preprocessing, the data is organized as:

Example structure:

Cohort 1976 (birth years 1975-1977):
  Age 16: 5 children total among 1,200 respondents
  Age 19: 45 children total among 1,180 respondents
  ...
  Age 40: 2,100 children total among 1,050 respondents

The data forms a sparse matrix where older cohorts have observations at older ages, and younger cohorts only have observations at younger ages (since they haven’t reached older ages yet).

Total parity heatmap

This heatmap shows the total parity (sum of children) for each birth cohort and age group combination. The diagonal structure reflects the survey design: we can only observe cohorts at ages they have reached by the survey years.

3. Model

3.1 Model Structure

The model uses a hierarchical log-linear structure to capture fertility patterns:

Age-Specific Birth Rates (ASBR):

λ[cohort, age] = exp(α[cohort] + β[age])

Where:

Cumulative Parity: The cumulative number of children for each cohort at each age is the sum of ASBRs:

μ[cohort, age] = Σ λ[cohort, age'] × n[cohort, age']
                  age'≤age

Where n[cohort, age] is the number of respondents.

Likelihood: Observed parity follows a Poisson distribution:

observed_parity[cohort, age] ~ Poisson(μ[cohort, age])

3.2 Prior Specifications

Cohort effects (α): Gaussian random walk

α[0] ~ Normal(0, 1)
α[i] ~ Normal(α[i-1], σ_α) for i > 0
σ_α ~ HalfNormal(0.1)

This prior enforces smooth changes across cohorts while allowing flexibility to capture trends.

Age effects (β): Gaussian random walk

β[0] ~ Normal(0, 1)
β[j] ~ Normal(β[j-1], σ_β) for j > 0
σ_β ~ HalfNormal(0.1)

This captures the characteristic age-fertility curve (low at young ages, peak in late 20s/early 30s, decline thereafter).

3.3 Model Versions

This report presents results from Version 4, which extends the base model with cohort-specific timing shift parameters:

Version 4 (primary - with timing shifts):

Version 3 (comparison - base model):

3.4 Inference

MCMC Sampling:

Convergence diagnostics:

The nutpie sampler produces excellent convergence with good effective sample sizes across all parameters, even with the extended model including timing shifts.

3.5 Predictions

Cohort Completed Fertility Rate (CFR): For each cohort and each posterior sample:

  1. Compute λ[cohort, age] for all ages up to age 44 (end of reproductive years)

  2. Sum across ages to get predicted CFR

  3. This produces a posterior distribution over CFR for each cohort

Uncertainty quantification:

4. Results

4.1 Model Parameters (Version 4)

Hyperparameters (estimated from data):

The small value of σ_γ indicates timing shifts change smoothly across cohorts. The larger σ_α (compared to v3) reflects that some cohort variation is now captured by timing shifts rather than baseline fertility. The σ_β remains similar, indicating age effects are still the most variable component.

4.2 Timing Shifts Across Cohorts

The timing shift parameter γ_i reveals systematic changes in the age at childbearing across cohorts:

Timing shifts by cohort

Timing shift (γ) summary:

Interpretation: The additive formulation (γ_i × age_centered_j in log-space) creates a linear tilt in the log-fertility curve. Positive γ values shift fertility from younger to older ages (delayed childbearing), while negative values shift fertility toward younger ages (earlier childbearing).

This pattern is consistent with well-documented demographic trends: the postponement of childbearing in recent decades. Women born in the 1980s-2000s are systematically having children at older ages compared to earlier cohorts. The magnitude of the effect is substantial—a γ value of 0.05 combined with age_centered ≈ 10 adds 0.5 to the log-fertility rate at age 10 years above the mean, roughly increasing the rate by 65% relative to what it would be without the timing shift.

4.3 Cohort Effects

Cohort effects over time

Cohort effects (α) summary:

After accounting for timing shifts (γ), the cohort effects show that baseline fertility peaked for the 1960s cohorts, remained relatively stable through the 1980s, and has been declining steadily since then. The decline accelerates for cohorts born after 1990.

Note: The peak cohort shifted from 1934 (in v3) to 1964 (in v4) because v4 explicitly models timing—the 1934 cohort had high total fertility but earlier childbearing (negative γ), while the 1964 cohort represents high baseline fertility with more typical timing.

The rate of decline for recent cohorts (1980-2000) remains comparable to the post-baby boom decline, both showing drops of approximately 1 child per woman over 20 years. The key difference is that the current decline starts from a much lower baseline.

4.4 Age Effects

Age effects by age group

Age effects (β) summary:

The age effects show the characteristic age-fertility curve with peak fertility in the mid-20s, representing the baseline age pattern before timing shifts are applied. Individual cohorts’ fertility curves can shift based on their γ values—positive γ tilts the curve toward older ages, while negative γ tilts it toward younger ages. The wide error bars at the oldest ages reflect limited data in those age groups.

4.5 Model Validation

The model’s retrodictions (predictions for cohort-age groups where we have data) fit the observed data well:

Model fit for 1970s-1980s cohortsModel fit for 1990s-2000s cohorts

The dotted lines show model predictions, while points show observed mean parity. The model captures both the overall fertility levels and the age-specific patterns well across cohorts.

Comparison with Census data:

CFR predictions vs actual Census data

The model’s predictions align well with actual Census CFR data for completed cohorts, providing confidence in the projections for incomplete cohorts.

4.6 Cohort Fertility Projections

Projected CFR by birth cohort

Key findings (CFR measured at age 42):

Convergence diagnostics: All 77 parameters show excellent convergence with max r̂ = 1.01. Effective sample sizes are adequate (min ESS_bulk = 358, min ESS_tail = 653), though 21 parameters have ESS_bulk < 400 (still acceptable for inference).

The shaded region shows 94% credible intervals. The projections suggest a dramatic decline in completed fertility for cohorts born after 1980, with fertility rates dropping to levels comparable to current South Korea (CFR ≈ 0.8) for cohorts born in the 2000s.

Uncertainty and Model Assumptions:

5. Discussion

5.1 Interpretation

The model reveals two distinct dimensions of fertility decline:

  1. Quantum decline (captured by cohort effects α): Overall reduction in total lifetime births

    • Cohorts born after 1980 show sharply declining baseline fertility

    • Reflects decisions about whether and how many children to have

  2. Tempo shifts (captured by timing parameters γ): Changes in the age at childbearing

    • Recent cohorts (1980s-2000s) show positive γ (delayed childbearing)

    • Earlier cohorts (1920s-1940s) show negative γ (earlier childbearing)

    • Middle cohorts (1950s-1970s) show near-zero γ (stable timing)

The postponement of childbearing (positive γ for recent cohorts) has important implications:

The declining fertility trend reflects multiple factors:

5.2 Implications

The projected decline is dramatic:

If current patterns persist:

Timing: The most dramatic declines are projected for cohorts born 1980-2000:

Historical context: This rate of decline is not unprecedented. It is comparable in magnitude and duration to the decline following the baby boom (cohorts born 1934-1954 saw CFR decline from 3.10 to approximately 2.0). However, the current decline starts from a much lower baseline, pushing fertility to historically unprecedented low levels.

5.3 Model Strengths

  1. Comprehensive data: Nearly 50 years of CPS data (1976-2024)

  2. Proper weight handling: Resampling approach maintains population representativeness

  3. Uncertainty quantification: Full posterior distributions for all parameters

  4. Validation: Excellent agreement with Census historical data (correlation = 0.9998)

  5. Interpretability: Parameters have clear demographic meanings

    • α captures overall fertility levels by cohort

    • β captures the age-fertility profile

    • γ captures timing shifts (postponement/acceleration of childbearing)

  6. Separates tempo from quantum: The extended model (v4) explicitly distinguishes between changes in timing (when) vs. total fertility (how many)

  7. Conservative projections: Gaussian random walks do not extrapolate trends—they assume recent patterns persist in the absence of new data, avoiding overconfident predictions about distant futures

5.4 Limitations

  1. Severely limited data for youngest cohorts: The most recent cohorts have been observed over only a small fraction of their reproductive years:

    CohortObserved Ages% of Reproductive SpanData Quality
    198815-3673%Good
    199415-3053%Moderate
    200015-2433%Limited
    200315-2123%Very Limited
    200615-1813%Extremely Limited
    200915 only3%Single age

    The 2009 cohort has been observed at only age 15, providing essentially no useful information about lifetime fertility patterns. Cohorts born after 2000 have been observed primarily during ages 15-24, when fertility is relatively low. Projections for these cohorts rely almost entirely on:

    • The random walk priors assuming similarity to previous cohorts

    • The model structure (age effects, timing patterns)

    • Very limited direct observations

    Implication: CFR projections for cohorts born after 2000 should be interpreted with extreme caution. These are model-based extrapolations, not data-driven estimates. Actual completed fertility for these cohorts will not be observable until the 2040s-2050s.

  2. Parameter identifiability challenges for youngest cohorts: The limited data for recent cohorts creates a fundamental identifiability problem between quantum (total fertility, α) and tempo (timing, γ) effects. Analysis of posterior correlations between α and γ reveals a U-shaped pattern across cohorts:

    Alpha-Gamma Correlation by Cohort
    Cohort GroupCorrelation (α, γ)Data CoverageIdentifiability
    Oldest (1922-1940)r ≈ 0.985-0.995Post-reproductive onlyPoor
    Middle (1945-1985)r ≈ 0.950-0.955Full reproductive spanGood
    Recent (1988-2006)r ≈ 0.955-0.980Early reproductive onlyPoor

    Minimum correlation occurs around cohort 1970 (r ≈ 0.950), which has the most complete reproductive data in our dataset.

    Why the U-shape?

    • Oldest cohorts: Observed only after reproduction ended (ages 45-54+). The model sees final parity but not the age pattern during reproductive years, making it impossible to distinguish whether high/low fertility was achieved early or late.

    • Middle cohorts: Observed throughout reproductive years (ages 20-45). The model can observe the full age-fertility profile, allowing it to separately estimate baseline fertility (α) and timing shifts (γ).

    • Recent cohorts: Observed only at young ages (15-27). For a given level of observed early fertility, the model admits a continuum of explanations:

      • Low quantum + early timing: α = -0.8, γ = -0.02 → low baseline fertility, but having children earlier partially compensates

      • High quantum + delayed timing: α = -0.6, γ = +0.02 → higher baseline fertility, but delay subtracts from young-age fertility

      • Many combinations in between

    Implication: For the youngest cohorts, we cannot confidently distinguish whether projected low CFR represents primarily quantum decline (permanent reduction in lifetime fertility) versus tempo delay (postponement that may eventually recover). The posterior appropriately captures this uncertainty through high correlation between α and γ, but interpreting projections for cohorts born after 2000 requires acknowledging this fundamental ambiguity.

  3. Model assumptions:

    • Random walks assume each cohort is similar to the previous one in the absence of data

    • This is conservative but may not capture accelerating changes or reversals

    • The linear timing shift (γ × age_centered) is a simplification—actual timing changes may be more complex

    • Does not fully separate “tempo effect” (recoverable delays) from permanent reductions in completed fertility

  4. External shocks: Cannot predict sudden changes (e.g., pandemic effects, policy changes, economic crises)

  5. Data limitations: CPS survey has limitations (coverage, response rates, recall bias)

  6. Model simplicity: The model is intentionally parsimonious and is validated on cohorts with completed fertility. More complex models improve in-sample fit but do not materially change the projections for completed cohort fertility. The model does not account for:

    • Socioeconomic heterogeneity (education, income, race/ethnicity, region)

    • Nonlinear timing patterns (e.g., bimodal fertility schedules)

    • Period effects that might affect all cohorts simultaneously

5.5 Future Directions

Potential extensions:

6. Conclusions

This analysis uses Bayesian hierarchical modeling to project cohort fertility rates using 48 years of CPS data (1976-2024). Key findings:

6.1 Main Results

  1. Dramatic fertility decline: Projected CFR drops from 3.10 (1934 cohort) to 0.78-0.89 (2000s cohorts), over 70% decline

  2. Two-dimensional decline: The extended model (v4) reveals fertility decline has both quantum and tempo components:

    • Quantum: Baseline fertility (α) declining sharply for cohorts born after 1980

    • Tempo: Systematic postponement of childbearing (positive γ) for recent cohorts

  3. Timing shifts: Clear postponement transition detected

    • Early cohorts (1930s): γ ≈ -0.065 (earlier childbearing)

    • Middle cohorts (1950s-1970s): γ ≈ 0 (stable timing)

    • Recent cohorts (1980s-2000s): γ up to +0.052 (delayed childbearing)

  4. Recent acceleration: The steepest decline occurs for cohorts born 1980-2000, falling from 1.86 to 0.89 in just 20 years

    • This rate of decline (≈1 child per woman in 20 years) is comparable to the post-baby boom decline

    • However, the current decline reaches historically unprecedented low levels

    • Coincides with the postponement transition (increasingly positive γ)

  5. Below replacement level: Cohorts born after 1995 are projected to have CFR well below the replacement level of 2.1

  6. Model validation: Excellent agreement with Census historical data (correlation = 0.9998) increases confidence in projections

6.2 Methodological Contributions

  1. Resampling approach: Successfully handles complex survey weights while maintaining integer counts for Poisson likelihood

  2. Extended log-linear model: Separates quantum from tempo by adding cohort-specific timing shift parameters (γ) in an additive log-space formulation

  3. Random walk priors: Gaussian random walks with estimated hyperparameters capture smooth temporal changes:

    • σ_α = 0.149 (cohort baseline fertility)

    • σ_β = 0.781 (age effects)

    • σ_γ = 0.013 (timing shifts)

    • Reveals age effects are most variable, timing shifts change smoothly

  4. Uncertainty quantification: Full posterior distributions provide credible intervals for all projections, including uncertainty in hyperparameters

  5. Computational efficiency: nutpie sampler enables fast, reliable MCMC inference with excellent convergence even for 77-parameter model

6.3 Implications

The projected fertility decline has profound implications:

6.4 Limitations and Caveats

  1. Conservative projections: Gaussian random walks do not extrapolate declining trends—they level off for the youngest cohorts. Actual fertility could decline further or reverse depending on future conditions.

  2. No heterogeneity: Does not account for differences by education, race/ethnicity, or region. Subgroup trends may differ substantially.

  3. Incomplete cohorts: Youngest cohorts have high uncertainty and limited data. Projections for cohorts born after 2000 are particularly uncertain.

  4. External factors: Cannot anticipate policy changes (e.g., parental leave, childcare subsidies), economic shocks, or cultural shifts that could substantially alter fertility behavior. Demographic trends exhibit inertia, and historically, large fertility reversals have been slow and rare even in the presence of policy interventions.

  5. Tempo vs quantum distinction: The linear timing shift (γ × age_centered) captures changes in the age pattern but doesn’t fully separate recoverable “tempo effects” (delayed births that will eventually occur) from permanent quantum decline (births that will never occur). However, the extended model (v4) that explicitly accounts for cohort-specific timing shifts fits the data better but produces similar completed fertility projections, suggesting that delayed childbearing alone is insufficient to explain the observed declines in cohort fertility.

  6. Model simplicity: Linear timing shift is a simplification—actual timing changes may be more complex (e.g., bimodal fertility schedules).

While substantial uncertainty remains for incomplete cohorts, the model provides a principled, data-driven framework for understanding long-term fertility trends and their potential implications for US demographics. The projections should be interpreted as “what we would expect if recent patterns persist,” not as forecasts of what will definitely occur.

References

[To be added]

Appendix A: Model Versions and Selection

A.1 Overview of Model Versions

This project developed multiple model versions with increasing complexity:

The main report presents results from Version 4, which provides the most complete picture of fertility dynamics by separating quantum (total fertility) from tempo (timing) effects. This appendix compares v3 and v4 to document the rationale for model selection.

Note: An earlier Version 2 with fixed σ_α was also developed but is not discussed here; it produced nearly identical results to v3 (see sections A.2-A.4 for details).

A.2 Differences Between Versions 2.0 and 3.0

Both model versions use the same basic structure (log-linear model with Gaussian random walk priors for cohort and age effects), but differ in how the random walk hyperparameters are handled.

Version 2.0

Hyperparameter specification:

Rationale: Initially set σ_α to a small fixed value based on exploratory analysis, while allowing σ_β to be estimated.

Version 3

Hyperparameter specification:

Rationale: Let the data determine both smoothing parameters, following standard Bayesian practice.

A.2 Comparison of Results

The two versions produce nearly identical predictions:

Metricv2v3Difference
Hyperparameters
σ_α0.050 (fixed)0.094 ± 0.015Data suggests higher
σ_β0.487 ± 0.0480.786 ± 0.128Data suggests higher
CFR Projections
Peak CFR (1934)3.0963.099+0.003
1980 cohort1.8451.8450.000
2000 cohort0.8740.8740.000
Lowest CFR (2003)0.7000.697-0.003
Convergence
Max r̂1.011.01Excellent both
Min ESS_bulk~400800Good both

Key observation: Despite different hyperparameter values, CFR predictions differ by at most 0.003 children per woman (< 0.5%), demonstrating robustness.

A.3 Decision Rationale: Why Version 3?

We selected Version 3 as the primary model for the following reasons:

1. Statistical Principles

2. Data-Driven Insights

3. Robustness

4. Computational Feasibility

5. Publishability

A.4 Sensitivity Analysis (v2 vs v3)

The minimal differences between versions demonstrates that results are robust to:

This robustness is important for interpretation: the projected fertility decline is not an artifact of modeling choices but a strong signal in the data.

A.5 Version 4: Extended Model with Timing Shifts

Version 4 extends the base model to explicitly capture cohort-specific changes in the timing of childbearing.

Model Structure

Base model (v3):

log(λij)=αi+βj\log(\lambda_{ij}) = \alpha_i + \beta_j

Extended model (v4):

log(λij)=αi+βj+γi×age_centeredj\log(\lambda_{ij}) = \alpha_i + \beta_j + \gamma_i \times \text{age\_centered}_j

Where:

Prior specification:

Interpretation

The γ parameter creates a linear tilt in the log-fertility curve:

The additive formulation (in log-space) ensures:

A.6 Comparison: Version 3 vs 4

|| Metric | v3 | v4 | Difference/Notes | ||--------|------|------|------------------| || Model Complexity | || Parameters | 46 | 77 | +31 (adds 30 γ + 1 σ_γ) | || Hyperparameters | || σ_α | 0.094 ± 0.015 | 0.149 ± 0.034 | Increased (timing absorbs some variation) | || σ_β | 0.786 ± 0.128 | 0.782 ± 0.128 | Nearly identical | || σ_γ | N/A | 0.012 ± 0.002 | New parameter (small, smooth) | || Timing Shifts | || γ range | N/A | -0.065 to +0.052 | Clear postponement pattern | || Early cohorts (1930s) | N/A | γ ≈ -0.065 | Earlier childbearing | || Recent cohorts (1990s) | N/A | γ ≈ +0.052 | Delayed childbearing | || CFR Projections | || Peak CFR (1934) | 3.099 | 3.105 | +0.006 | || 1980 cohort | 1.845 | 1.861 | +0.016 | || 2000 cohort | 0.874 | 0.894 | +0.020 | || 2009 cohort | 0.871 | 0.786 | -0.085 (notable) | || Lowest CFR | 0.697 (2003) | 0.644 (2006) | Lower minimum | || Convergence | || Max r̂ | 1.01 | 1.00 | Excellent both | || Min ESS_bulk | 800 | 745 | Excellent both (>400 threshold) | || Params with ESS<400 | 0 | 0 | All parameters well-sampled | || Goodness of Fit | || MAE | 172.8 | 127.0 | v4 lower error | || RMSE | 232.9 | 167.4 | v4 lower error | || MAPE | 12.23% | 11.12% | v4 better prediction | || WAIC (ELPD) | -2844.4 | -2406.4 | v4 better fit (higher ELPD) | || LOO (ELPD) | -2845.6 | -2415.7 | v4 better fit |

Key observations:

  1. Timing shifts detected: Clear systematic pattern in γ (postponement in recent cohorts)

  2. Improved model fit: v4 shows better goodness of fit metrics across the board:

    • MAPE decreases from 12.23% to 11.12%

    • WAIC and LOO strongly favor v4 (ΔELPD ≈ 430)

  3. Modest CFR changes: Most differences < 0.02, except youngest cohorts

  4. Higher complexity: 31 additional parameters, but convergence remains excellent:

    • All 77 parameters have ESS_bulk > 400

    • Excellent convergence (max r̂ = 1.00)

  5. Cohort interpretation: Timing separated from quantity reveals distinct patterns

A.7 Decision Rationale: Why Version 4?

We selected Version 4 as the primary model for the following reasons:

1. Demographic Insight

2. Substantive Findings

3. Model Fit

4. Theoretical Motivation

5. Robustness

Tradeoffs:

Limitations of v4:

Conclusion: Version 4 is the recommended primary model. It provides richer demographic insights by explicitly modeling the postponement of childbearing, achieves superior model fit (ΔWAIC ≈ 438), maintains excellent convergence with excellent ESS across all 77 parameters, and produces projections that are substantively similar to v3 for most cohorts while revealing important timing patterns that are obscured in the base model. The timing shifts (γ) are not only statistically significant but also demographically meaningful, capturing the well-documented postponement transition in recent cohorts.

Appendix B: Code and Reproducibility

All code for this project is available at: https://github.com/AllenDowney/BayesFertility

Key notebooks:

Requirements:

See requirements.txt for full dependencies.

Reproducibility: