Dynamic Spatial Bayesian Machine Learning Model: Applications to Intergenerational Economic Mobility and Geographic Income Inequality in the United States

Introduction

• The introduction motivates a unified model for individual heterogeneity, regional spatial dependence, and temporal dynamics, and applies it to linked mobility and geographic income inequality outcomes.The outcomes covary across places in the Great Gatsby Curve.Source: Introduction · pp. 1–2

Model Framework

• DSP-BART-HS combines BART for individual-level nonlinearity, Horseshoe shrinkage for region-time covariates, CAR spatial effects, and state-space regional dynamics.The framework also adapts to aggregated regional cells by dropping individual identifiers and effects and weighting the likelihood by known precision.Source: Introduction · p. 2, Model Framework · p. 3

Introduction

• The simulation design evaluates six scenarios, including five adversarial departures from assumptions favorable to DSP-BART-HS, plus three extensions testing topology and information-structure choices.The adversarial scenarios vary effect density, spatial dependence and structure, the balance of individual- and region-time signal, and the presence of individual heterogeneity.Source: Introduction · p. 2

Identifiability between ϕj and the initial state of µjt

• The model fixes each region’s initial dynamic state at µ_j,1 = 0 so its CAR intercept captures the baseline and later states represent drift.This resolves non-identifiability between the time-invariant regional intercept and the initial random-walk value without changing fitted values.Source: Identifiability between ϕj and the… · p. 3

Sampling the dynamic component µjt

• Regional dynamic paths are sampled jointly with forward-filtering backward-sampling, using precision-weighted regional residuals as observations wherever training data exist.For a region with no training observations, this recursion alone yields an uninformative prior path; the zero-training-region diffusion procedure is described separately.Source: Sampling the dynamic component µjt · p. 3, Gibbs Sampler Full-Conditional… · p. 4

Gibbs Sampler Full-Conditional Distributions

• Under zero-training-region holdout, disabling diffusion raised test RMSE from 0.0306 to 0.0334 for mobility and from 0.0775 to 0.2813 for income.Diffusion had no effect under random, temporal, or individual-level simulation holdouts because every region retained training data.Source: Gibbs Sampler Full-Conditional… · p. 4
• The blocked Gibbs sampler updates BART trees, Horseshoe coefficients and scales, CAR effects, regional dynamic paths, individual effects, and variance components.The tree structures use Metropolis-within-Gibbs proposals, while the dynamic paths use forward-filtering backward-sampling.Source: Gibbs Sampler Full-Conditional… · p. 5

Beyond Known Heteroscedasticity

• The real-data applications use known-precision likelihood weighting, but estimating heteroscedasticity by region, time, or covariates is outside the implemented model.Such extensions would require a different model rather than a straightforward reparameterization.Source: Beyond Known Heteroscedasticity · p. 5

A Complementary Specification: BVARX-CAR

• BVARX-CAR is adapted to use exogenous covariates, with neighbor-restricted cross-region lags and a shared ridge prior replacing the published dense-lag, Horseshoe specification.A preliminary dense-B, full-Horseshoe implementation had effective sample size 10.8; the restriction resolved mixing and kept the sampler tractable, but this aggregate-level model does not separately represent individual heterogeneity.Source: A Complementary Specification: BVARX-CAR · p. 5

A Second Complementary Specification: MTS-CAR-X

• MTS-CAR-X is a lighter alternative with a per-region AR(1), CAR-correlated intercept and slope deviations, and no cross-region lag term.The study extends it with the shared exogenous-covariate block under its own Horseshoe prior; spatial dependence enters through the coefficient structure rather than a cross-region transition matrix.Source: A Second Complementary Specification:… · p. 5

Comparator Suite

• The comparator suite spans spatial-panel econometrics, small-area estimation, and Bayesian nonparametric methods; KKP serves as a floor reference because it cannot accommodate a high-dimensional covariate block.The included suite also contains SDPM, SLP, SAF, BCF, and Rao-Yu; the machine-learning baselines are limited to SAF and BCF because they retain explicit spatial smoothing or structural parameterizations.Source: Comparator Suite · p. 6

Simulation Design

• Scenarios A–F vary contextual-effect density, spatial strength and structure, variance allocation, and individual heterogeneity, but retain one 50-region rook lattice and fixed panel dimensions.Scenario E makes region-time structure dominant; Scenario F removes individual heterogeneity entirely. The fixed topology and dimensions leave network-size, density, and topology variation untested in these scenarios.Source: Simulation Design · pp. 6–7

Simulation Study Results

• DSP-BART-HS is best or statistically indistinguishable from the best estimator in every Scenario A–F, with large advantages in A–D and a tie with Rao-Yu in F.Against the region-time-aggregate cluster, its mean difference is ≈−2.69 to −2.70 in A–D (p < 10^-30); in E it is −0.055 (p < 10^-170), while in F the difference is +0.0002 (95% CI [−0.0002, 0.0006]; paired t p = 0.382, Wilcoxon p = 0.672).Source: Table 1 · p. 7, Simulation Study Results · p. 7
• Across Scenarios G–I, every model’s test RMSE stays within simulation noise of its Scenario A value, and DSP-BART-HS comparisons remain significant after Holm-Bonferroni correction.In Scenario G, cell-level BART, given the same aggregated information but no individual-level inputs, has test RMSE 4.1150 and does not close the gap to DSP-BART-HS, supporting individual-level access—not merely model flexibility—as a source of its advantage.Source: Table 2 · p. 9, Simulation Study Results · p. 9

MCMC Convergence Diagnostics

• DSP-BART-HS’s Gibbs sampler achieved R̂ ≤ 1.0075 for all ten checked parameters in Scenario A, with effective sample sizes well above the conventional baseline.The same validated sampler and production settings were used for Scenarios B–I, and no replicate in the 200-replicate runs showed evidence of non-convergence.Source: MCMC Convergence Diagnostics · p. 9

The Fitted Model for Real-Data Applications

• For both real-data applications, the fitted model drops the individual index and uses region-time-level covariates in its nonlinear component, while weighting each cell by its known sampling precision.The mobility and income panels contain county-cohort or county-year cell estimates rather than individual records; the likelihood uses w_jt = 1/se_jt^2 to account for cell-specific sampling standard errors.Source: The Fitted Model for Real-Data… · p. 10
• DSP-BART-HS’s likelihood uses known-precision weighting, but all reported RMSEs are unweighted and directly comparable across models.This is a known-heteroscedasticity specification, not an assumption-free extension to unknown heteroscedasticity.Source: The Fitted Model for Real-Data… · p. 10

Mobility

• The mobility panel covers 45,560 county-cohort rows across 3,175 counties, with a county-level split testing generalization to entirely unseen counties.The final covariate set has M = 12, including three PCA components explaining 86.4% of an eight-covariate cluster’s variance. Using 2023 covariates for earlier cohorts relies on an explicit, unverified assumption that relative county standing persists; their associations are not contemporaneous causal claims.Source: Mobility · pp. 10–11

County-level Split Comparison

• 0.0306 test RMSE makes DSP-BART-HS the best of six models under the genuine unseen-county holdout.SAF is closest at 0.0322, while Rao-Yu has the worst test RMSE (0.0416) and the largest generalization gap (0.0179). The primary criterion is test RMSE; the generalization gap is a secondary diagnostic.Source: County-level Split Comparison · p. 11, Table 3 · p. 12

Repeated-Split Uncertainty

• Across 20 paired county partitions, DSP-BART-HS beats SAF on every split, with a paired mean RMSE difference of −0.00219.The paired 95% CI is [−0.0024, −0.0020] and p < 10−15; DSP-BART-HS’s test RMSE across its repeats averages 0.0316 (SD 0.0010).Source: Repeated-Split Uncertainty · pp. 11–12

Row-Level Random and Temporal Holdout Comparisons

• 0.0307 test RMSE makes DSP-BART-HS best under the random row-level split, while MTS-CAR-X leads the temporal holdout at 0.0303.The random-split margin over Rao-Yu (0.0310) is modest; in the temporal holdout, MTS-CAR-X also has the lowest generalization gap (0.0008), ahead of DSP-BART-HS at 0.0310 test RMSE. These splits test interpolation or a future cohort for counties already observed, not unseen-county generalization.Source: Row-Level Random and Temporal Holdout… · p. 12

Cross-Model Consistency of Covariate Effects

• Higher Black population share and rent twobed2015 are associated with lower mobility, while higher White share and emp2000 are associated with higher mobility, holding other covariates fixed; these are regression-style associations, not causal estimates.Source: Cross-Model Consistency of Covariate… · p. 12

Validation Against Established Mobility Findings

• DSP-BART-HS’s mobility estimates align with established findings on racial differences, strong spatial persistence, and substantial within-state variation.The share black2000 effect is negative and share white2000 positive under both county and random holdouts. Estimated CAR spatial correlation is 0.992 for the county holdout and 0.993 for the random holdout; within-state variance shares are 41.7% overall and 38.3% among held-out counties, with 41.6% under the random holdout. The county-level decomposition is a proxy, not a literal replication of the prior literature’s statistic.Source: Validation Against Established… · pp. 13–14

Real Data Application: Geographic Income Inequality

• The companion geographic income inequality application models county-year log median household income, not an inequality index.Its common specification has M = 11 contextual covariates; income-denominated covariates are removed to avoid circularity, and the outcome’s log-scale RMSE describes relative error in income levels. Claims about inequality concern estimated patterns across counties and over time.Source: Real Data Application: Geographic… · p. 14
• Under the county holdout, DSP-BART-HS has the lowest test RMSE (0.0775) and train RMSE (0.0277) among six models; Rao-Yu’s test RMSE is 0.1020.SAF, BCF, and SLP form a weaker test-RMSE cluster of 0.1873–0.1912; DSP-BART-HS’s generalization gap is second-largest, at 0.0498.Source: Real Data Application: Geographic… · p. 15, Table 7 · p. 15
• On the random split, DSP-BART-HS’s test RMSE is 0.0557, near Rao-Yu’s 0.0469; temporally, MTS-CAR-X leads at 0.0696 and DSP-BART-HS is second at 0.1027.DSP-BART-HS’s temporal RMSE is 48% higher than MTS-CAR-X’s, while the other temporal results range from 0.1027 to 0.5803.Source: Real Data Application: Geographic… · p. 15, Table 7 · p. 15
• MTS-CAR-X remains best at all five rolling origins, with its test-RMSE margin over DSP-BART-HS ranging from 0.0046 to 0.0331.The five expanding-window cutoffs test years 16 through 20, indicating that the temporal advantage is not specific to the single final-year cutoff.Source: Real Data Application: Geographic… · p. 15, Table 8 · p. 16

Repeated-Split Uncertainty

• Across 20 county partitions, test RMSE averages 0.08194 (SD 0.00346; 95% CI [0.08032, 0.08356]), making the reported 0.0775 result favorable but within the observed range.It is about 1.3 SD below the repeated-split mean and below the confidence interval’s lower bound, so it should not be read as typical.Source: Repeated-Split Uncertainty · p. 16

Implied Covariate Effects (Partial)

• MTS-CAR-X reverses the sign of four covariate effects relative to every other model in both random and temporal splits, with each reversal statistically significant.The covariates are PCTPOVALL 2023, emp2000, rent twobed2015, and foreign share2010; the authors report no methodological explanation.Source: Implied Covariate Effects (Partial) · p. 16
• In the county split, five covariates have consistent, significant signs across DSP-BART-HS, Rao-Yu, and SDPM, while share black2000 lacks a stable pattern.SDPM’s share black2000 effect is significant and negative under the temporal split (−0.1040) but positive and insignificant under the county split (0.0245).Source: Implied Covariate Effects (Partial) · p. 16
• For Rao-Yu, rent twobed2015 is associated with lower mobility but higher income levels, while share hisp2000 is positively associated with mobility and negatively with income.Both share hisp2000 associations are significant in every split; the authors present these as descriptive patterns, not causal tests.Source: Implied Covariate Effects (Partial) · pp. 16–17

Concluding Remark

• The concluding analysis attributes DSP-BART-HS’s simulation advantage under individual-level heterogeneity to combining individual-level flexibility with spatial, temporal, and contextual structure.Its coefficient selection is a predictive association, not causal identification: the model does not address confounding with unobserved county characteristics.Source: Concluding Remark · pp. 17–18

Supplementary Material

• Full specifications and Gibbs sampler full-conditional derivations are provided for BVARX-CAR and MTS-CAR-X.These materials are in Section 1.Source: Supplementary Material for: Dynamic… · p. 20
• The supplement reports MCMC convergence diagnostics and supporting visualizations for Scenario A–F simulation results.These include Tables S1–S10 and Figures S1–S3.Source: Supplementary Material for: Dynamic… · p. 20
• A glossary defines the mathematical symbols, covariates, model abbreviations, and metric names used in the manuscript and supplement.The glossary is listed in Tables S13–S19.Source: Supplementary Material for: Dynamic… · p. 20
• DSP-BART-HS and SAF variable-importance outputs are reported for both applications.The supplement also provides application tables, a zero-training-region diffusion ablation, and real-data diagnostics.Source: Supplementary Material for: Dynamic… · p. 20

MTS-CAR-X

• Both specifications add the same exogenous contextual covariate block and a Horseshoe prior to their published base specifications.The contextual block is an addition; for BVARX-CAR, the study also replaces the published dense-B local-global prior with a shared ridge prior on restricted support.Source: MTS-CAR-X · p. 20, §1.1.2 · p. 21

1.1.1 Model

• BVARX-CAR uses the full cross-sectional lag vector, whereas MTS-CAR-X uses only each region’s own lag and has no cross-region spillover term in its likelihood.For MTS-CAR-X, spatial dependence instead enters through region-specific intercept and slope deviations.Source: §1.1.1 · p. 20, §1.2.1 · p. 22

where b′

• The study restricts each B row to a region and its spatial neighbors after a dense-B implementation showed severely degraded MCMC mixing.The neighbor-restricted version had effective sample size 364 under an otherwise identical setup and keeps the sampler tractable at county scale; no dense-B results are reported.Source: where b′ · p. 21

Region intercepts µ. With c the vector of region-level average residuals after removing b′

• Region-intercept updates combine residual information with spatial smoothing, while the neighbor-restricted B update retains a conjugate regression form on fixed support.The restricted update uses only the lag vector entries indexed by the region and its neighbors.Source: Region intercepts µ. With c the vector… · p. 21, Under this study’s… · p. 21

Residual variance σ2. With M1 = P

• The residual variance update follows a Jeffreys-prior form, and the spatial parameters ρ and ψ use a discretized grid sampler because their CAR determinants are nonlinear.The initial condition yj0 receives a conjugate Normal update.Source: Residual variance σ2. With M1 = P · p. 22, B instead has a fully conjugate update · p. 22

1.2.2 Priors

• The MTS-CAR-X specification links intercept and slope deviations through a sequential spatial prior rather than a symmetric joint bivariate CAR.Its diffusion coupling is detailed in the following specification section.Source: §1.2.2 · p. 23

Olayinka and Nandram [2026b] published specification is sequential,

• MTS-CAR-X couples intercept and slope deviations sequentially through a spatial diffusion matrix A, a structure a symmetric joint bivariate CAR would miss.The matrix is parameterized by two hyperparameters, m0 and m1.Source: Olayinka and Nandram [2026b] published… · p. 23

own conjugate Gaussian full conditional,

• The sampler analytically marginalizes the high-dimensional spatial-deviation block before drawing the global coefficients, then draws all deviations jointly.The collapsed Gaussian update removes strong posterior dependence between the blocks; a sparse Cholesky factorization is reused for both steps.Source: X = (1, yj,t−1), θ = (θ0, θ1)′, Z the… · p. 23, own conjugate Gaussian full conditional · p. 23

scales,

• The contextual coefficient and Horseshoe auxiliary-variable updates use the standard Inverse-Gamma mixture representation of the half-Cauchy.The contextual block is a study addition to the published specifications.Source: own conjugate Gaussian full conditional · p. 23, scales · p. 24
• The study adds contextual covariates governed by a Horseshoe construction to the published MTS-CAR-X and BVARX-CAR base models.This addition is not present in either cited published base model.Source: scales · p. 24

Spatial parameters ρ1, ρ2. Non-conjugate, updated on a discretized grid using the precomputed eigenvalues

• The spatial parameters ρ1 and ρ2 are updated on a discretized grid, while the diffusion parameters m0 and m1 receive a bivariate Normal full conditional.The respective updates use precomputed eigenvalues and a completed-square form.Source: Spatial parameters ρ1, ρ2.… · p. 24, bivariate Normal full conditional · p. 24

1.3 Comparison of the Two Specifications

• BVARX-CAR uses neighbors’ realized lagged outcomes in its likelihood, whereas MTS-CAR-X borrows spatially through priors on region parameters, not neighbors’ outcomes.This difference explains why county-level holdouts withhold information BVARX-CAR conditions on, while MTS-CAR-X parameters remain estimable from training counties.Source: §1.3 · p. 24

1.4 Empirical Test: MTS-CAR-X Under a County-Level Holdout (Income Application)

• Under the county-level holdout, MTS-CAR-X’s partial-run test RMSE was 0.3516, more than double its established random-split test RMSE of 0.1627, with a generalization gap of 0.2927.The authors interpret this as a genuine structural weakness under this specific holdout design, rather than an artifact of the abbreviated run.Source: §1.4 · p. 25

1.5 BVARX-CAR Under the Rolling-Origin Temporal Design (Income Application)

• BVARX-CAR’s rolling-origin test RMSE rose from 0.2335 at test year 16 to 0.3578 at test year 20, tracking its single-cutoff result closely.Across origins it stayed in the narrow band of BCF, SAF, and SLP, separated from DSP-BART-HS and MTS-CAR-X.Source: §1.5 · p. 25

2.1.1 Elements shared across all scenarios

• The shared simulation design uses 50 regions, 20 individuals per region, 10 periods, and 10,000 observations per replicate on a 5 × 10 rook-adjacency grid.Each scenario runs 200 replicates.Source: §2.1.1 · pp. 25–26, additive model used throughout this… · p. 26

2.1.2 Scenario A: Matched, Well-Specified Design

• Scenario A is the matched baseline; B makes all 50 contextual effects nonzero, C weakens spatial dependence, and D replaces the fitted CAR-generating structure with distance-decay dependence.Scenario D retains rook adjacency for model fitting despite using a smooth Euclidean distance-decay process to generate spatial effects.Source: §2.1.2 · p. 26, variance τ 2 · p. 26, §2.1.3 · p. 26

2.1.5 Scenario D: Misspecified Spatial Structure

• Scenario D’s distance-decay process gives nearest-neighbor correlation of about 0.72 and longer-range dependence than Scenario A’s CAR structure at the same distance.The scenario changes the true generating mechanism, not the rook-adjacency matrix used for fitting.Source: §2.1.5 · p. 27, kernel over the regions’ grid… · p. 27

2.1.6 Scenario E: Region-Time-Dominant Signal

• Scenario E shifts the signal toward region-time structure by increasing CAR variance, increasing dynamic-innovation SD from 0.2 to 0.6, and reducing nonlinear-function magnitude tenfold.The rescaling suppresses the observable nonlinear component’s variance, not the unobserved individual random effect.Source: §2.1.6 · p. 27

Identical to Scenario A in ρ, τ 2

• Scenario F removes individual heterogeneity while retaining unit-level covariates, testing whether individual-level models avoid fitting irrelevant signal and giving aggregate comparators their best-case comparison.The scenario sets f(X) to zero and removes the outcome variance aggregate methods cannot observe.Source: Identical to Scenario A in ρ, τ 2 · p. 27

2.1.8 Scenario G: Aggregated-Covariate Access

• Scenario G gives aggregate comparators region-period means of three individual covariates, while DSP-BART-HS retains its original aggregate inputs and individual-level covariate access.A cell-level BART comparator receives only the augmented aggregate information, without individual-specific covariates or spatial and dynamic random effects.Source: §2.1.8 · p. 27

2.1.9 Scenario H: Irregular Spatial Topology

• Scenario H replaces the regular rook lattice with a random geometric graph; its repaired realizations have mean degree 3.76–3.84 and all nodes are reachable.The graph is adjusted to match the lattice’s mean degree as closely as possible and ensure connectivity.Source: §2.1.9 · p. 28

2.1.10 Scenario I: Larger Panel

• Scenario I doubles the panel to 100 regions and 20 periods while retaining the regular-lattice topology, per-region individual count, and covariate dimension.This separates panel-size changes from the topology change in Scenario H.Source: §2.1.10 · p. 28

gives J × Nj × T = 40,000 observations per replicate, versus Scenario A’s 10,000.

• Scenarios G–I use 200 replicates and the same evaluation and paired-comparison methodology as Scenarios A–F, with results reported in Table 2.Scenario G adds cell-level BART, bringing its comparator count to nine; Scenarios H and I use the eight original comparators.Source: gives J × Nj × T = 40,000 observations… · p. 28

2.2 Full Simulation Study Results, Scenarios A–F

• For Scenarios A–F, the full supplementary results report mean train RMSE, mean test RMSE, and test-RMSE standard deviation across 200 replicates for every model.The main text instead gives a condensed table of mean test RMSE by model and scenario.Source: §2.2 · p. 28

2.3 Full Simulation Study Results, Scenarios G, H, and I

• For Scenarios G–I, the supplementary results report the same three RMSE summaries across 200 replicates for every model.The main text presents condensed mean test RMSE results for these robustness scenarios.Source: §2.3 · p. 30

2.4 MCMC Convergence Diagnostics

• Sampler convergence was assessed using four independent chains on a fixed representative Scenario A dataset, with Gelman–Rubin R-hat and effective sample size tracked across parameters.The tracked parameters include active and inactive coefficients, variance hyperparameters, and a representative spatial effect.Source: §2.4 · pp. 31–32

As presented in Table S10 all ten tracked parameters show ˆR ≤1.0075, comfortably below the conventional

• All ten tracked parameters have R-hat ≤1.0075, and mean ESS per chain ranges from 921.3 to 1986.1.Every parameter exceeds the ESS ≥800 baseline; the dynamic and idiosyncratic variance components have the lowest ESS, consistent with slower mixing in hierarchical state-space and CAR components.Source: As presented in Table S10 all ten… · p. 32, and idiosyncratic structure (σ2 · p. 32

2.5 Additional Simulation Visualizations

• The region-time-aggregate estimators’ test RMSE drops markedly in Scenario F, with BVARX-CAR and MTS-CAR-X showing the steepest decline, about 70%.Scenario F removes individual-level heterogeneity, while the four adversarial scenarios B–D retain it.Source: §2.5 · p. 32
• DSP-BART-HS ranks first in Scenarios A–E and second in Scenario F, behind Rao-Yu by 0.0002; KKP ranks last in every scenario.SDPM remains second-to-last even in Scenarios E and F, despite those designs favoring aggregate estimators.Source: §2.5 · p. 32
• Figure S1 traces test RMSE across Scenarios A–F, and Figure S2 reports within-scenario model ranks.Figure S3 combines the six scenarios into a model-by-scenario grid.Source: Figure S1 · p. 33, Figure S2 · p. 33, Figure S3 · p. 34
• Figure S3’s model-by-scenario grid shows DSP-BART-HS with the most uniform test-RMSE shading and KKP with the darkest, most variable row.The figure encodes log10(mean test RMSE), with exact values annotated in each cell.Source: §2.5 · pp. 32–33, Figure S3 · p. 34

2.6 Expanded Paired Comparisons Against All Comparators

• 47 of 48 paired comparisons across Scenarios A–F remain significant after Holm-Bonferroni correction, all favoring DSP-BART-HS; only its Scenario F comparison with Rao-Yu is indistinguishable.That comparison has mean difference 0.0002, 95% CI [−0.0002, 0.0006], and p = 0.38.Source: §2.6 · p. 34
• All 25 paired comparisons in Scenarios G–I remain significant after Holm-Bonferroni correction, each favoring DSP-BART-HS, including comparisons with BCF and cell-level BART.Against BCF, mean differences are −0.068, −0.069, and −0.072 in Scenarios G, H, and I, respectively.Source: §2.6 · pp. 34–35

3 Glossary of Notation, Covariates, and Abbreviations

• The glossary is intended to define the mathematical symbols, covariates, abbreviations, and metrics used in the manuscript and supplement.It allows readers to look up variable code names without returning to their first mention.Source: §3 · p. 35

3.1 Mathematical Notation

• The supplement includes separate notation tables for general DSP-BART-HS notation, BVARX-CAR, and MTS-CAR-X.It also lists a PCA-component table and tables for contextual and individual-level covariates.Source: Table S11 · p. 35, Table S12 · p. 36, Table S13 · p. 36

3.2 Covariates

• The PCA cluster contains eight variables for mobility and seven for income, with three components refit for each application and training split.Components are fit using training data and then projected onto test data.Source: §3.2 · p. 37

3.3 Metrics

• The supplement provides a table of real-data evaluation metrics and a table of model and method abbreviations.The supplied passages identify the tables but do not enumerate their entries.Source: Table S17 · p. 38, Table S18 · p. 39

standardized value and PCr = Pp

• Standardized covariates use their training-sample standard deviations, and the component effects used for back-projection differ by estimator type.Bayesian models use posterior means; Rao-Yu and SDPM use point estimates.Source: §3.5 · p. 39, standardized value and PCr = Pp · p. 39

3.5 Back-Projection Formula and Post-Selection Caveat (Metric C2)

• Equation 32 maps estimated effects on three retained principal components to implied effects on the original covariates.It uses the training-sample standard deviations and loading matrix as fixed quantities.Source: standardized value and PCr = Pp · p. 39, §3.5 · p. 39

standardized value and PCr = Pp

• For Bayesian models, applying the fixed linear map to each posterior draw yields covariate-effect distributions and 95% credible intervals.For frequentist models, the same map is applied using the delta method to the estimated sampling distribution.Source: standardized value and PCr = Pp · p. 39

Post-selection caveat

• The covariate-cluster choice and retention of three components were data-driven decisions made before coefficient estimation.The three components were retained using the Kaiser eigenvalue-≥1 criterion.Source: Post-selection caveat. Both the choice… · p. 39
• The reported intervals condition on the selected covariate cluster and retained-component count rather than accounting for uncertainty in those choices.Different training data could select different cluster membership or component counts.Source: Post-selection caveat. Both the choice… · p. 39
• The reported intervals propagate uncertainty in the estimated component effects while treating the loadings, standard deviations, cluster membership, and component count as fixed.The passage characterizes this as conditional-on-the-selected-model inference, not selection-adjusted inference.Source: Post-selection caveat. Both the choice… · p. 39
• The study does not use a selection-adjusted correction such as sample splitting or simultaneous post-selection confidence regions.The back-projected effects and intervals should therefore be read as conditional on this study’s specific clustering choice.Source: Post-selection caveat. Both the choice… · pp. 39–40

3.6 Comparator Tuning Settings: Real-Data Applications

• Table S19 reports comparator tuning settings for both the mobility and income applications.Settings are listed only for model types with a tunable quantity of the relevant kind.Source: §3.6 · p. 40, Table S19 · p. 40
• Rao-Yu and SDPM use single non-iterative optimizations, so MCMC iteration, burn-in, and thinning settings do not apply.The passage names optim() and splm::spml() as the respective fitting procedures.Source: §3.6 · p. 40
• SLP selects its LASSO penalty by cross-validation rather than fixing it a priori.KKP is used only in the simulation study and is omitted from the real-data tuning table.Source: §3.6 · p. 40

4.1 DSP-BART-HS Variable Importance, Mobility Application

• Singleparent share1990 accounts for 29–33% of mobility-application splits and ranks first under every split.Popdensity2000 ranks second, while the lower-ranked covariates’ ordering varies across splits.Source: §4.1 · p. 40, Table S20 · p. 41
• In the county split, the two lower-ranked covariates each register 14.76%, differing only in the fifth decimal place; their relative order varies across the three splits and is sensitive to held-out counties.Source: §4.1 · p. 40

4.2 DSP-BART-HS Variable Importance, Income-inequality Application

• These proportions describe split counts for the individual-level f(X) covariates, not coefficients or the separate Zβ contextual block.The income covariate ranking is a property of the variables supplied to the model, not a general claim about county characteristics that matter for income.Source: §4.2 · p. 41
• For income, the f(X) importance ranking is identical across all three splits: singleparent share1990 and frac coll plus2000 lead, traveltime15 2010 ranks third, and popdensity2000 last.Their reported shares are roughly 29–30%, 23–25%, and 16–18%, respectively, for the leading pair, third, and last covariates.Source: §4.2 · p. 41, Table S21 · p. 41

4.3 SAF Variable Importance, Income-Inequality Application

• SAF variable importance covers a disjoint contextual covariate block and is not directly comparable to DSP-BART-HS’s f(X) split counts or signed coefficient effects.The passage reports SAF’s importance for completeness.Source: §4.3 · p. 41

under the corrected M = 11 covariate specification described in the main text

• PC1 demog econ is SAF’s most important covariate under both splits, and PC2 demog econ ranks in the top three.The same PCA-derived demographic/economic composite also dominates the coefficient tables of DSP-BART-HS and the other Bayesian models, though those are different kinds of quantities.Source: Table S22 · p. 41, under the corrected M = 11 covariate… · p. 42
• Region idx ranks third under the random split and fifth under the temporal split, consistent with SAF partly recovering region-level structure from the county index.The cited passage notes that SAF has no explicit spatial term like the CAR-based comparators.Source: under the corrected M = 11 covariate… · p. 42

• The iterative diffusion step replaces the uninformative µ = 0 FFBS fallback for regions with zero training observations under a county-level holdout.Source: §5 · p. 42
• The refit targets zero-training regions in the county-level holdout.Source: §5 · p. 42
• The ablation refits both county-split applications with the diffusion step toggled off.Source: §5 · p. 42

• In the zero-training-region ablation, disabling diffusion increases test RMSE by 263% for income and 9% for mobility (0.0306 → 0.0334); the income effect is an order of magnitude larger.Source: Table S23 · p. 42, held at the µ = 0 fallback throughout · p. 42
• Test bias for income rises from 0.00235 to 0.21231 when diffusion is disabled, driving almost all of its RMSE increase.On the untransformed dollar scale, this corresponds to roughly 24% multiplicative bias.Source: held at the µ = 0 fallback throughout · p. 42
• The larger income impact lacks a confirmed explanation; more zero-training counties may increase the chance of adjacent zero-training chains.The proposed chain mechanism remains plausible rather than confirmed.Source: held at the µ = 0 fallback throughout · p. 42
• With diffusion disabled, random row-level and temporal-holdout results are identical to diffusion-on results to numerical precision.Every region retains training data in these splits, so diffusion is a no-op there.Source: held at the µ = 0 fallback throughout · p. 42

6 DSP-BART-HS Real-Data MCMC Diagnostics

• All eight tracked parameters have point-estimate R-hat < 1.03 and upper-CI R-hat < 1.05, below the conventional 1.1 convergence threshold.Source: Table S24 · p. 43, §6 · p. 43
• The maximum cross-chain range in variable-inclusion share is 0.0165 for popdensity2000, and chains agree closely on covariate ordering and approximate inclusion magnitudes.This is a descriptive proxy, not a formal ensemble-convergence guarantee, and cannot rule out finer-grained mixing differences.Source: §6 · p. 43, Table S25 · p. 43

the observed test-set mean (0.4608)

• The observed test-set mean falls within every chain’s 95% posterior predictive interval, with near-identical posterior means and intervals across chains.Source: Table S26 · p. 43, the observed test-set mean (0.4608) · p. 43