Project SENTRY: Architecture and Econometric Methodology

1. Sub-National Administrative Hierarchy

Project SENTRY formulates sub-national macroeconomic nowcasting as a hierarchical estimation problem across three nested administrative tiers in the Republic of the Philippines:

Level 0: National Macroeconomy

Philippines Real GDP / Quarterly National Accounts (QNA):

Level 1: Regional Administrative Tier

18 Administrative Regions (NCR, CAR, Regions I–XIII, BARMM, NIR)
Gross Regional Domestic Product (GRDP) Panel: $Y_{r, t}$

Level 2: Sub-Regional Provincial Tier

135 Provincial Product Accounts (PPA) Jurisdictions: Provinces & Highly Urbanized Cities (HUCs):

Hierarchy Invariants & Conservation Laws

1. Regional Growth Balance (Stone's Linear Balance):
   $$\sum_{r=1}^{18} \omega_{r, t-1} \cdot \tilde{g}_{r, t} = g_{\text{PH}, t}, \quad \text{where } \omega_{r, t-1} = \frac{Y_{r, t-1}}{Y_{\text{PH}, t-1}}$$
2. Provincial Additivity (Denton Proportional Conservation):
   $$\sum_{p \in R_r} \widehat{\text{PPA}}_{p, t} = \widehat{\text{GRDP}}_{r, t} \quad \text{with precision } \epsilon \le 10^{-6}$$
   3. Spatial Equilibrium Stability:
   $$\max \big| \text{eig}(\rho W) \big| < 1.0, \quad \sum_{j=1}^{18} W_{ij} = 1.0 \quad \forall i$$

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   2. Spatial Weight Matrix Formulations

   Pure NumPy Contiguity & Haversine 3-NN Maritime Augmentation

   Unlike continental economies, the Philippine archipelago comprises over 7,641 islands. Standard Queen contiguity leaves island regions (MIMAROPA, Central Visayas, Eastern Visayas, BARMM) completely isolated ($A_{i, \cdot} = 0$).

   To eliminate reliance on external geospatial C-libraries (gdal, geopandas, fiona, shapely), SENTRY implements pure NumPy analytical topology:

   1. Analytic Queen Contiguity ($A_{\text{Queen}}$): Direct land borders derived from official PSA/NAMRIA administrative geography.
   2. Haversine Great-Circle Distance:
      $$d(i, j) = 2 R \arcsin \left( \sqrt{ \sin^2\left(\frac{\Delta \phi}{2}\right) + \cos(\phi_i)\cos(\phi_j)\sin^2\left(\frac{\Delta \lambda}{2}\right) } \right)$$
      where $R = 6,371.0 \text{ km}$, $\phi$ is latitude, and $\lambda$ is longitude of regional administrative centroids. 3. Directed $k$-NN Augmentation ($k=3$): For any region with fewer than $k=3$ land neighbors, the $k - \deg(i)$ closest centroids are connected to model primary inter-island Ro-Ro (roll-on/roll-off) and maritime shipping channels.

      Economic Gravity Flow Matrix ($W_{\text{CFS}}$)

      Physical distance alone fails to capture commercial dominance. The Philippine Statistics Authority (PSA) Commodity Flow in the Philippines (CFS) measures domestic trade flows across air, water, and rail transport:

      1. Bilateral Flow Aggregation: $T_{ij} = \sum_m \text{Value}_{ij}^m$ (annual trade value in PHP millions from origin $i$ to destination $j$).
      2. Zero Diagonal: $T_{ii} = 0$ (self-flows excluded).
      3. Row Normalization:
         $$W_{\text{CFS}, ij} = \frac{T_{ij}}{\sum_{k=1}^{18} T_{ik}}$$
      4. Convex Hybrid Formulation:
         $$W(\alpha) = \alpha W_{\text{CFS}} + (1 - \alpha) W_{\text{geo}}, \quad \alpha \in [0, 1]$$
         Optimal $\alpha^* = 0.65$ was identified via grid search maximizing the 2SLS spatial autoregressive concentrated log-likelihood:
         $$\ln L_c(\alpha) = -\frac{n}{2} \big(1 + \ln(2\pi)\big) - \frac{n}{2} \ln \left(\frac{\text{SSR}(\alpha)}{n}\right) + \ln \big| I - \rho W(\alpha) \big|$$

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         3. Spatial Dynamic Factor Model (SDFM)

         Conventional nowcasting separates factor extraction from spatial econometric adjustment. SENTRY's unified SDFM incorporates spatial feedbacks directly into the Kalman state-space observation operator.

         State-Space Representation

         1. Latent Factor State Equation ($\text{VAR}(1)$):
            $$f_t = A f_{t-1} + u_t, \quad u_t \sim \mathcal{N}(0, Q)$$
            where $f_t \in \mathbb{R}^k$ ($k=3$ latent common factors), $A \in \mathbb{R}^{k \times k}$ satisfies stationary eigenvalues ($\max |\text{eig}(A)| < 0.95$), and $Q \in \mathbb{R}^{k \times k}$ is positive semi-definite.
         2. Spatial Observation Operator: The structural regional equation across all 18 regions is:
            $$(I - \rho W) y_t = \Lambda f_t + e_t, \quad e_t \sim \mathcal{N}(0, \Sigma_e)$$
            Inverting the spatial multiplier $S(\rho) = (I - \rho W)^{-1}$:
            $$y_t = Z(\rho) f_t + v_t, \quad v_t \sim \mathcal{N}(0, R(\rho))$$
            where:
            $$Z(\rho) = (I - \rho W)^{-1} \Lambda, \quad R(\rho) = (I - \rho W)^{-1} \Sigma_e (I - \rho W)^{-T}$$

         Information Transmission Mechanism

         When an indicator release for Region $j$ arrives at time $t$ (e.g., Cebu port traffic or NCR wholesale index), the Kalman measurement update:

         $$f_{t|t} = f_{t|t-1} + K_t \big(y_{t}^* - Z_t^* f_{t|t-1}\big)$$
         instantaneously propagates spatial spillovers to all other regions $i \ne j$ via $Z(\rho)$, ensuring continuous real-time cross-regional learning.

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         4. Finite-Sample Conformal Interval Recalibration

         Because $(I - \rho W)^{-1}$ concentrates observation errors, theoretical Kalman innovation covariance $F_t = Z P_{t|t-1} Z^T + R$ systematically underestimates empirical predictive uncertainty.

         To restore valid coverage without altering point nowcasts:

         1. Non-conformity scores are computed over out-of-sample walk-forward validation:
            $$\alpha_t = \frac{|y_{r, t} - \hat{y}_{r, t}|}{\hat{\sigma}_{r, t}}$$
         2. The calibrated conformal quantile $q_{1-\beta}$ for $\beta = 0.04$ ($96$ nominal) is derived empirically:
            $$q_{0.96} = 6.32$$
         3. Conformalized $95$ prediction intervals are constructed as:
            $$\text{PI}_{95}(y_{r, t}) = \left[ \hat{y}_{r, t} - q_{0.96} \cdot \hat{\sigma}_{r, t}, \; \hat{y}_{r, t} + q_{0.96} \cdot \hat{\sigma}_{r, t} \right]$$
            This restores empirical coverage from $59.72\%$ to **$95.83\%$** while preserving point nowcast optimality ($0.0251 \text{ pp MAE}$).

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         5. Domain-Partitioned Feature Gating

         Empirical analysis in Phase 9 revealed that pooling climate shock indicators (CHIRPS Standardized Precipitation Index and Cyclone Disturbance Index) globally across all 18 regions diluted tree split gains in manufacturing hubs (NCR, CALABARZON).

         SENTRY enforces Domain-Partitioned Feature Gating:

         • Metro / Industrial Hub Profile: Gating excludes raw precipitation and cyclone tracks, allocating capacity to foreign direct investment, US/China industrial production (INDPRO), remittance flows, and port transit volumes.
         • Agricultural / Island Profile: Retains high-dimensional climate proxies (CHIRPS precipitation anomalies, typhoon wind track intensity, Sentinel-2 NDVI vegetative vigor) to capture crop-cycle disruptions.

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         6. Quad-Model 3-Tier Simplex Meta-Ensemble

         SENTRY combines four mathematically orthogonal model architectures:

         1. LightGBM: Non-linear gradient boosted regression trees with asymmetric directional penalty ($\lambda = 2.5$).
         2. Regularized ElasticNet: Sparse linear model with expanding-window cross-validation ($\ell_1 = 0.5$).
         3. Unified Spatial DFM (SDFM): State-space dynamic factor model with endogenous spatial multiplier.
         4. Ridge U-MIDAS: Unrestricted mixed-frequency distributed lag regression bridging monthly indicators to annual output.

         3-Tier Simplex Stacking

         Weights $w = (w_{\text{LGBM}}, w_{\text{EN}}, w_{\text{SDFM}}, w_{\text{MIDAS}})^T$ are solved via constrained quadratic optimization on expanding-window out-of-sample errors:

         $$\min_{w} \sum_{t} \left( y_t - \sum_{m=1}^4 w_m \hat{y}_{m, t} \right)^2 \quad \text{s.t.} \quad \sum_{m=1}^4 w_m = 1, \quad w_m \ge 0$$
         Estimated independently across three economic archetypes: - **Tier 1 (National Capital Region):** High spatial and linear weights; low tree variance. - **Tier 2 (Industrial Regional Hubs):** Balanced spatial and gradient boosted tree weights. - **Tier 3 (Agricultural / Island Regions):** Balanced spatial, tree, and mixed-frequency autoregressive weights.

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         7. Constrained Lagrangian Indicator-Assisted Denton Downscaling

         To disaggregate annual regional GRDP nowcasts $\widehat{\text{GRDP}}_{r, t}$ into 135 provincial and HUC entities, SENTRY implements dynamic indicator-assisted Denton disaggregation.

         Let $p_{r, p, t}$ denote the unknown provincial output level, $p_{r, p, t-1}$ the historical PPA baseline level, and $z_{r, p, t} > 0$ a high-frequency composite proxy constructed from:

         $$z_{r, p, t} = 0.45 \cdot \text{NTL}_{p, t} + 0.35 \cdot \text{BankDeposits}_{p, t} + 0.20 \cdot \text{VehicleReg}_{p, t}$$

         The objective minimizes relative distortions from the dynamic indicator while satisfying regional conservation:

         $$\min_{p_{r, p, t}} \sum_{p=1}^{K_r} \left( \frac{p_{r, p, t}}{z_{r, p, t}} - \frac{p_{r, p, t-1}}{z_{r, p, t-1}} \right)^2 \quad \text{s.t.} \quad \sum_{p=1}^{K_r} p_{r, p, t} = \widehat{\text{GRDP}}_{r, t}$$

         Closed-Form Solution

         Using Lagrangian multipliers, the exact solution is evaluated in $\mathcal{O}(K_r)$ without iterative numerical solvers:

         $$p_{r, p, t}^* = p_{r, p, t-1} \left( \frac{z_{r, p, t}}{\bar{z}_r} \right) + \frac{z_{r, p, t}^2}{\sum_{k} z_{r, k, t}^2} \left( \widehat{\text{GRDP}}_{r, t} - \sum_{k} p_{r, k, t-1} \frac{z_{r, k, t}}{\bar{z}_r} \right)$$
         Non-negativity ($p_{r, p, t}^* \ge 0$) is strictly enforced with proportional residual rescaling, guaranteeing zero conservation error.

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         For the public-facing web workstation architecture, see Chapter 7: Web Frontend Architecture.