A Bayesian Age-Period-Cohort approach for modeling fertility in Puerto Rico

This study presents the first Bayesian Age-Period-Cohort analysis of Puerto Rican fertility from 1948 to 2022, revealing that cohort effects, particularly among those born from 1963 onward, are the primary driver of the island's record-low fertility rates rather than period effects or birth postponement.

The Gist

Imagine you are trying to solve a mystery: Why are so few babies being born in Puerto Rico?

For decades, the island's birth rate has plummeted. In the late 1940s, the average woman had about 5 children. By 2023, that number dropped to less than 1. This isn't just a small dip; it's a cliff. The authors of this paper, a team of statisticians and demographers from the University of Puerto Rico, wanted to figure out why this happened.

They used a special detective tool called an Age-Period-Cohort (APC) model. To understand their findings, let's break down the three suspects in this mystery using a simple analogy.

The Three Suspects: Age, Period, and Cohort

Imagine fertility is a car driving down a highway.

1. Age Effects (The Driver's Fatigue): This is about biology. Just like a runner might sprint best in their 20s and slow down in their 40s, women have a biological "sweet spot" for having babies. This is the Age effect. It happens to everyone, regardless of when they were born.
2. Period Effects (The Weather): This is about what is happening right now that affects everyone at the same time. If there is a war, a pandemic, a massive economic crash, or a new law, it hits everyone on the road simultaneously. This is the Period effect.
3. Cohort Effects (The Car's Model Year): This is about the specific group of people born in the same years. Think of them as a specific "model" of car. A car built in 1960 has different features and habits than a car built in 1980. Similarly, women born in the 1960s grew up with different cultural values, education opportunities, and life goals than women born in the 1940s. This is the Cohort effect.

The Big Mystery: Who is to Blame?

In many countries (like South Korea or the US), when birth rates drop, it's usually because of Period Effects (bad weather). Maybe the economy crashed, or maybe women are just delaying having kids until they are older (postponement), hoping to have them later.

But Puerto Rico is different.

The researchers used a sophisticated "Bayesian" approach (think of this as a super-smart way of updating their guesses based on evidence) to analyze data from 1948 to 2022. Here is what they found:

• The Old Days (1948–1997): The drop in babies was mostly due to Period Effects. Big changes in society and the economy were slowing things down.
• The New Era (1998–Present): The plot twist! The main culprit shifted to Cohort Effects.

The "No Postponement" Surprise

Here is the most surprising part of the story. In many low-fertility countries, women are having babies later. They finish school, get a job, buy a house, and then have kids. This is called postponement. Usually, they eventually have those kids, just later in life.

In Puerto Rico, this isn't happening.

The researchers found that women born after the mid-1960s aren't just waiting to have babies; they are simply having fewer babies overall. It's not that they are driving the car slower; it's that they decided to take a different route entirely.

• The Analogy: Imagine a group of friends (a cohort) who grew up together. In the past, everyone planned to have 3 kids. But this new group of friends (born 1963–1967 and later) grew up with a different mindset. They decided, "We want 0 or 1 kid." They didn't just delay the decision; they changed the plan permanently.

Why Does This Matter?

If you think the problem is just "bad weather" (Period effects), you might try to fix it by fixing the economy or offering tax breaks for now.

But if the problem is the "Car Model" (Cohort effects), it means the issue is deeply rooted in culture, values, and identity. The women born in the 1960s and 70s have internalized a new way of living that doesn't prioritize large families.

The authors argue that because Puerto Rico isn't just "delaying" births, but actually reducing the total number of children per woman, public policies need to change. You can't just fix the economy; you have to understand and address the deep-seated cultural shifts that make women choose smaller families.

The "Secret Sauce" of the Study

The paper also highlights a technical breakthrough. Usually, trying to separate these three effects (Age, Period, Cohort) is like trying to untangle three knots that are tied together perfectly—it's mathematically impossible without making some guesses.

The authors used a new mathematical "knot-untangler" (using something called Scaled Beta2 priors and specific constraints). Think of it like using a new type of GPS that doesn't just guess the route but calculates the probability of every possible path, giving them a much clearer picture of the truth than older methods.

The Bottom Line

Puerto Rico is facing a unique demographic storm. It's not just that the economy is bad or that women are busy; it's that an entire generation has fundamentally changed its view on family size.

• The Problem: Birth rates are at historic lows (0.9 children per woman).
• The Cause: It's mostly about the generation (Cohort), not just the current times (Period).
• The Lesson: To solve this, Puerto Rico needs policies that respect and understand these deep cultural changes, rather than just trying to "speed up" the timeline of having children.

The island is now "super-aged" (more old people than young ones), and without a shift in these deep-seated cohort values, the population will continue to shrink.

Technical Summary

1. Problem Statement and Context

Puerto Rico faces a unique and severe demographic crisis characterized by one of the lowest Total Fertility Rates (TFR) globally (dropping from 5.2 in 1948–1952 to 0.9 in 2023), negative net migration, and a rapidly aging population (classified as "super-aged"). While global demographic trends often attribute fertility decline to period effects (e.g., economic crises, policy changes, postponement of births), Puerto Rico's situation differs. Unlike countries like Italy or South Korea, Puerto Rico shows no evidence of "birth postponement" (delaying childbearing to later ages).

The core problem addressed is determining whether period effects (external events affecting all ages simultaneously) or cohort effects (shared experiences of specific birth generations) are the primary drivers of Puerto Rico's fertility decline. Standard Age-Period-Cohort (APC) models face the identification problem due to the linear dependency $Cohort = Period - Age$, which creates a singular design matrix and infinite solutions. The authors aim to resolve this using a novel Bayesian framework to provide robust policy insights.

2. Methodology

The study employs a Bayesian Age-Period-Cohort (APC) model using data from the Puerto Rico Demographic Registry and US Census Bureau (1948–2022).

• Data Structure:

  • Age ($A$): 7 groups (15–19 to 45–49).
  • Period ($T$): 15 five-year intervals (1948–2022).
  • Cohort ($C$): 21 cohorts derived from the diagonals of the data matrix.
  • Response: Number of births ($y_{a,t}$) modeled via a Poisson likelihood with an offset for person-years ($p_{a,t}$).

• Model Specification:

  • Likelihood: $y_{a,t} \sim \text{Pois}(\lambda_{a,t})$ where $\log(\lambda_{a,t}) = \theta_a + \phi_t + \alpha_c - \log(p_{a,t})$.
  • Priors for Effects: Age ($\theta$), Period ($\phi$), and Cohort ($\alpha$) parameters follow Second-Order Random Walk (RW(2)) autoregressive priors, assuming smooth trends over time.
  • Precision Priors: The precision parameters ($\tau$) for the RW(2) priors are assigned Scaled Beta2 (SB2) distributions. This is a key innovation, replacing the commonly used (but often criticized) Inverse Gamma or Gamma priors to allow for heavier tails and greater robustness against extreme values.
  • Identification Constraints: To solve the identification problem, the authors impose specific constraints:
    • $\phi_1 = 0$ (First period fixed).
    • $\theta_1, \alpha_{10}, \alpha_{11} \sim N(0, \text{small variance})$.
    • Innovation: Instead of fixing parameters to exactly zero (which creates unreasonably narrow credible intervals), the authors impose constraints with uncertainty (small variance normal distributions) on specific age and cohort groups where complete fertility data is available.

• Inference & Computation:

  • Implemented using Stan (via RStan) with Hamiltonian Monte Carlo (HMC).
  • 15,000 iterations with centering to ensure convergence ($\hat{R} \approx 1$).

• Model Comparison:

  • Criteria: Leave-One-Out Information Criterion (LOOIC), Widely Applicable Information Criterion (WAIC), and Residual Sum of Squares (RSS).
  • Validation: The model was tested by fitting data from 1948–2017 and predicting the TFR for 2018–2022.

3. Key Results

• Dominance of Cohort Effects: The analysis reveals that cohort effects explain approximately 46% of the variation in the APC model, whereas period effects explain only 19%. This contrasts with studies in the US and South Korea where period effects are dominant.
• Fertility Patterns:
  • Phase 1 (1948–1962): Stable fertility in younger age groups.
  • Phase 2 (1963–1992): Drastic decline driven by period effects.
  • Phase 3 (1993–2022): Low, stable rates driven by cohort effects.
• No Postponement: Unlike "lowest-low" fertility countries where women delay childbirth (leading to recovery later), Puerto Rican women born in cohorts after 1963–1967 exhibit permanently low fertility without a "recovery" phase.
• Model Performance:
  • Priors: Models using Scaled Beta2 priors outperformed those using Gamma priors. The SB2 priors produced TFR estimates (median ~1.06–1.07) much closer to the actual 2018–2022 TFR (0.9) with reasonable credible intervals. Gamma priors produced excessively wide intervals (e.g., 0.15 to 9.07).
  • Constraints: The authors' "original" constraints (with uncertainty) yielded better LOOIC/WAIC scores and more realistic parameter estimates compared to traditional sum-to-zero constraints.
  • Predictive Accuracy: The APC model with SB2 priors successfully predicted the recent TFR decline, whereas simpler Age-Period (AP) or Age-Cohort (AC) models performed significantly worse (higher LOOIC/WAIC).

4. Key Contributions

1. Demographic Insight: The study challenges the global consensus that period effects drive fertility decline. It establishes that in Puerto Rico, cohort effects are the primary driver, suggesting that cultural and social values specific to generations born after 1963 are more influential than temporary economic or policy shocks.
2. Methodological Innovation (Identification): The authors propose a novel solution to the identification problem by imposing constraints with uncertainty (using small-variance normal priors) rather than hard equality constraints. This avoids the artifact of artificially narrow credible intervals.
3. Prior Distribution Advancement: The paper demonstrates the superiority of Scaled Beta2 priors over Gamma priors for precision parameters in APC models, offering better robustness and fit for demographic data.
4. Policy Implications: The findings suggest that public policies aimed at reversing fertility decline in Puerto Rico must address cohort-specific cultural and social values rather than relying solely on period-based interventions (like temporary economic incentives), as the decline is structural to specific generations.

5. Significance

This research provides a critical framework for understanding "lowest-low" fertility in unique demographic contexts where migration and aging compound the issue. By proving that cohort effects dominate in Puerto Rico, the study warns against applying standard period-based policy solutions that have worked elsewhere. Methodologically, it advances Bayesian APC modeling by offering a more robust way to handle the identification problem and precision priors, potentially applicable to other territories with similar demographic challenges. The authors also note the development of a future R package to make this framework accessible to the broader demographic community.