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A dynamical model of the U.S. mathematics graduate degree pipeline

Chad M. Topaz, Oluwatosin Babasola, Ron Buckmire, Daozhou Gao, Maila Hallare, Olaniyi Iyiola, Deanna Needell, Andrés R. Vindas-Meléndez

TL;DR

The paper addresses modeling degree-production dynamics in the U.S. mathematics pipeline when only completion counts $b(t)$, $m(t)$, and $p(t)$ are observed, by introducing a latent-stock two-compartment framework with latent stocks $M(t)$ and $P(t)$ and time-varying routing fractions $\rho_{BM}(t)$, $\rho_{BP}(t)$, $\rho_{MP}(t)$ and completion hazards $\gamma_M(t)$, $\gamma_P(t)$. It reframes the problem as an inverse stock-flow problem with discrete-time dynamics and estimates parameters by minimizing the SSE on the log scale, comparing models with AIC/BIC; the best specification is quadratic in time variation for both routing fractions and hazards, with no explicit forcing. Empirically, the results reveal a strengthening BA-to-MA pathway, a peak in the MA-to-PhD channel in the late 1980s/early 1990s, and rising hazards indicating faster turnover, all captured as smooth trajectories rather than abrupt regime shifts. The framework provides a principled reduced-form lens to summarize long-run pipeline dynamics from exit data and offers a foundation for extensions to include lags, additional compartments, or domestic/international substructures.

Abstract

We present a latent-stock compartmental framework for modeling degree production systems when only completion flows, rather than enrollments, are observed. Applied to U.S.\ mathematics degrees from 1969 to 2017, the model treats master's and PhD populations as latent compartments -- unobserved state variables that are inferred indirectly because they generate the observed completion flows -- with time-varying routing fractions and completion hazards. Using information-criterion model comparison across a grid of specifications, we find strong support for smooth nonlinear time variation in routing fractions and hazards, while models with explicit international forcing are disfavored. The preferred model achieves a log-scale root mean squared error of approximately 0.036, corresponding to a typical multiplicative error of about 4\% in fitted degree counts, and highlights key structural shifts in the graduate pipeline: the master's pathway became increasingly central to PhD production through the late twentieth century before weakening, while direct bachelor's-to-PhD entry remained small but persistent. Estimated completion hazards for both degrees rise over time, indicating faster effective turnover in the graduate compartments. Methodologically, our main contribution is a latent stock dynamical approach that recasts linked degreecompletion time series as a coherent stock-flow system when intermediate enrollments are unobserved, making explicit both what features of pipeline dynamics are identifiable from completion data alone and what limitations such data impose.

A dynamical model of the U.S. mathematics graduate degree pipeline

TL;DR

The paper addresses modeling degree-production dynamics in the U.S. mathematics pipeline when only completion counts , , and are observed, by introducing a latent-stock two-compartment framework with latent stocks and and time-varying routing fractions , , and completion hazards , . It reframes the problem as an inverse stock-flow problem with discrete-time dynamics and estimates parameters by minimizing the SSE on the log scale, comparing models with AIC/BIC; the best specification is quadratic in time variation for both routing fractions and hazards, with no explicit forcing. Empirically, the results reveal a strengthening BA-to-MA pathway, a peak in the MA-to-PhD channel in the late 1980s/early 1990s, and rising hazards indicating faster turnover, all captured as smooth trajectories rather than abrupt regime shifts. The framework provides a principled reduced-form lens to summarize long-run pipeline dynamics from exit data and offers a foundation for extensions to include lags, additional compartments, or domestic/international substructures.

Abstract

We present a latent-stock compartmental framework for modeling degree production systems when only completion flows, rather than enrollments, are observed. Applied to U.S.\ mathematics degrees from 1969 to 2017, the model treats master's and PhD populations as latent compartments -- unobserved state variables that are inferred indirectly because they generate the observed completion flows -- with time-varying routing fractions and completion hazards. Using information-criterion model comparison across a grid of specifications, we find strong support for smooth nonlinear time variation in routing fractions and hazards, while models with explicit international forcing are disfavored. The preferred model achieves a log-scale root mean squared error of approximately 0.036, corresponding to a typical multiplicative error of about 4\% in fitted degree counts, and highlights key structural shifts in the graduate pipeline: the master's pathway became increasingly central to PhD production through the late twentieth century before weakening, while direct bachelor's-to-PhD entry remained small but persistent. Estimated completion hazards for both degrees rise over time, indicating faster effective turnover in the graduate compartments. Methodologically, our main contribution is a latent stock dynamical approach that recasts linked degreecompletion time series as a coherent stock-flow system when intermediate enrollments are unobserved, making explicit both what features of pipeline dynamics are identifiable from completion data alone and what limitations such data impose.
Paper Structure (9 sections, 21 equations, 3 figures, 3 tables)

This paper contains 9 sections, 21 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Modeling
  4. Model with Constant Parameters
  5. Time-Varying Parameters
  6. Estimation
  7. Results
  8. Discussion
  9. Conclusion

Figures (3)

  • Figure 1: Log-scale annual observations of bachelor's, master's, and PhD degrees in mathematics awarded in the United States for 49 observation years between 1969 and 2017. Earlier nonannual observations prior to 1969 are omitted from the figure and from the model estimation. Source: National Center for Education Statistics (NCES), U.S. Department of Education.
  • Figure 2: Fitted results from the preferred model (quadratic branching fractions with quadratic time-varying completion hazards, no explicit forcing). Panels (a) and (b) show observed degree counts (points) and model-implied fits (lines) for master's and PhD completions, respectively, over the 1969--2017 window. Panel (c) displays the estimated branching fractions over time, with shaded 95% Monte Carlo confidence bands constructed from parameter draws based on the Hessian: $\rho_{BM}(t)$ (bachelor's to master's), $\rho_{BP}(t)$ (bachelor's directly to PhD), and $\rho_{MP}(t)$ (master's to PhD). Panel (d) shows the estimated completion hazards $\gamma_M(t)$ and $\gamma_P(t)$ with analogous 95% bands.
  • Figure 3: Residual diagnostics for the preferred model. Left: log-residuals for master's and PhD completions over time. Right: histograms of log-residuals. Residuals are centered near zero and mostly lie within approximately $\pm 0.1$, consistent with typical relative errors of a few percent.