spb/wp23_uqo

1\section{Proposed forecasting architectures}\label{sec:arch}23All architectures are implemented in the accompanying repository (PyTorch for the neural ones) with fixed hyper-parameters (Appendix~\ref{app:hyper}). Each produces the five quantiles of Task~C; the point forecast is the median unless stated.45\subsection{HSHN: hierarchical spatiotemporal hedonic network (Task A)}67The network maps the standardised hedonic and market features $x_i$, three learned embeddings $g_i = e_{\mathrm{reg}(i)} + e_{\mathrm{mrc}(i)} + e_{\mathrm{mun}(i)}$ (dimension 8) and a horizon embedding $e_h$ (dimension 4) to five quantiles of $\log P_i$: an encoder of two GELU layers (256, 128) on $[x_i, g_i, e_h]$, an interaction block that combines the encoded representation with $[g_i, e_h]$, and a linear head. Partial pooling is implemented by weight decay on the municipality embedding only (coefficient $5\times10^{-3}$ against $10^{-4}$ elsewhere), so that a municipality's embedding is a penalised deviation from its MRC and region embeddings and thin municipalities fall back on their parents; \emph{full pooling} removes the municipality embedding, \emph{no pooling} removes the penalty. Training minimises the mean pinball loss over the five quantiles with AdamW and a one-cycle schedule (8 epochs, batch 4{,}096). The design is a neural counterpart of mixed-effects hedonics with a horizon input; the test is whether the learned pooling beats trees and an unpooled MLP out of time, especially in thin municipalities.89\subsection{DSGFN: dynamic spatial graph forecasting network (Task B)}1011Nodes are the cells of one class at one origin; each node is encoded from its standardised features and its thirteen-month relative history by a two-layer network (dimension 32). A single graph-attention layer aggregates neighbours with logits $q_u^\top k_v/\sqrt{d} + \omega \log A_{uv}$, where $A$ is a prior adjacency that multiplies a Gaussian distance kernel between municipal centroids (bandwidth 60 km), a co-movement term $\tfrac12 + \tfrac12\,\mathrm{corr}$ of the two nodes' monthly real-time changes over the history window, and an indicator that the nodes share the level and class; self-attention is excluded so that the aggregated vector is a neighbourhood signal, and $\omega$ is learned. The head reads the node's own encoding and the neighbourhood vector. Edges are re-computed at every origin from information available then (dynamic graph). Ablations drop the distance kernel, the co-movement term or the whole neighbourhood vector.1213\subsection{RAREF: regime-adaptive mixture of experts (Task B)}1415At each origin a three-state Gaussian hidden Markov model is estimated by Baum--Welch on the sequence of provincial real-time monthly growth rates known at that origin (one value per past origin, each taken from its own vintage, so that no smoothed full-sample growth enters), and its filtered state probabilities at the origin gate three ridge experts estimated on the training rows weighted by their own origins' state probabilities. The forecast is the probability-weighted expert forecast; the predictive quantiles are the mixture's. Regimes are therefore inferred only from origin-dated information and can be mapped to calendar episodes.1617\subsection{BHPF: Bayesian hierarchical probabilistic forecaster (Task B)}1819The target is modelled as $y = \beta_0 + x^\top\beta + u_{\mathrm{region}} + u_{\mathrm{cell}} + \varepsilon$ with Gaussian priors $\beta \sim N(0, \tau_b^2 I)$, $u_{\mathrm{region}} \sim N(0, \tau_r^2)$, $u_{\mathrm{cell}} \sim N(0, \tau_c^2)$ and $\varepsilon \sim N(0, \sigma^2)$. The posterior is conjugate and exact; the variance components are updated by fifteen iterations of an EM-type scheme (posterior second moments of the random effects). The predictive distribution is Gaussian with the posterior-predictive variance, which separates parameter uncertainty from residual variance; cells unseen in training receive the region effect only. This is a small-area model in the Fay--Herriot tradition applied to index changes; it is the natural null for the neural hierarchical claims.2021\subsection{QANSS: quality-adjusted neural state-space forecaster (Task B)}2223WP22 already separates the quality component from the latent market component: its real-time index is the filtered latent state. QANSS forecasts the latent state by a learned transition: a two-layer network (32 tanh units) maps the cell's state at the origin (recent drift, volatility, volume ratio, provincial and regional drifts, last two monthly changes) to the mean monthly drift over the horizon and the log standard deviation of the $h$-month change, estimated by maximising the Gaussian likelihood of the training targets; the predictive variance adds the published filter variance of the origin level. The model is a deep state-space model in the sense of \citet{rangapuram2018dssm} whose observation equation is the WP22 hedonic state-space index; the question is whether the learned drift transition beats the local-linear-trend and autoregressive benchmarks on the same vintages.2425\subsection{QREFA: integrated architecture}2627The integrated architecture stacks the components by a non-negative least-squares combination of their in-sample forecasts, normalised to sum to one, estimated on the training origins; its quantiles are the weighted quantiles. Each component can be removed from the stack, which is the ablation design of the protocol. The null hypotheses are the best single component and the equal-weight combination.28