\section{Related literature}\label{sec:lit}

\paragraph{Hedonic models, repeat sales and index construction.} Quality adjustment rests on hedonic regression \citep{rosen1974} and on repeat-sales methods \citep{bailey1963,caseshiller1989}; the index methods relevant here, including rolling time dummies \citep{hill2022rtd}, multilateral time-series smoothing \citep{ishaak2024hmts} and hierarchical structural time-series models for thin markets \citep{francke2010thin,francke2017hierarchical}, are reviewed in \citet{hill2013survey}, the Eurostat et al.\ handbook \citep{eurostat2013handbook} and WP22. \citet{sijpfrancke2026laplacian} model spatiotemporal random effects in hedonic and repeat-sales models through graph-Laplacian priors, the parametric counterpart of the spatial sharing that the neural architectures below attempt to learn. Selection into transaction is the subject of \citet{heckman1979selection}, \citet{gatzlaffhaurin1997,gatzlaffhaurin1998}, \citet{judseaks1994selection}, \citet{munnekeslade2000selection}, \citet{clapp1991alltransactions} and \citet{hwangquigley2004}.

\paragraph{Forecasting housing markets.} Predictability of house price changes is documented by \citet{caseshiller1989,caseshiller1990forecasting}; momentum and mean reversion by \citet{capozza2004anatomy}, \citet{berachaskiba2011momentum}, \citet{titman2014dynamics} and, theoretically, \citet{glaeser2014dynamics}, \citet{head2014search} and \citet{guren2018momentum}. Regional index forecasting with cross-sectional information includes \citet{rapachstrauss2009}, \citet{gupta2011large,gupta2011forecasting} (factor-augmented and large Bayesian VARs), \citet{borkmoller2015} (dynamic model averaging), \citet{kouwenbergzwinkels2014}, \citet{plakandaras2015forecasting}, \citet{milunovich2020forecasting} (time series against machine learning for Australia) and \citet{yeo2025deepdfvar} (dynamic-factor VAR with deep learning); \citet{ghysels2013forecasting} survey the field. Spatial and spatiotemporal dependence of prices is modelled by \citet{pace1998spatiotemporal,pace2000spatiotemporal}, \citet{hollypesaran2010,holly2011uk} and \citet{brady2011diffusion}; the spatial-econometric toolkit is \citet{anselin1988}, \citet{lesagepace2009}, \citet{kelejianprucha2010} and \citet{elhorst2014spatial}.

\paragraph{Machine learning and deep learning for property prices.} Tree ensembles \citep{breiman2001rf,friedman2001gbm,chen2016xgboost,ke2017lightgbm,prokhorenkova2018catboost} dominate valuation comparisons \citep{mayer2019updating,hong2020randomforest,baldominos2018opportunities,kok2017bigdata}; \citet{pacehayunga2020residuals} examine what trees extract from hedonic residuals; \citet{deppnercajias2024spatialcv} show that random cross-validation is biased by spatial dependence; interpretability is treated by \citet{lorenz2023interpretable} and \citet{lenaers2023xai}. Graph neural networks \citep{kipfwelling2017gcn,velickovic2018gat,wu2021gnnsurvey} have been applied to valuation with transaction, point-of-interest and road graphs \citep{viana2021attention,zhang2021mugrep,das2021gsne,guo2026heteroroadnet}; spatiotemporal graph forecasting models \citep{li2018dcrnn,yu2018stgcn,wu2019graphwavenet,bai2020agcrn} are surveyed by \citet{jin2024urbanstgnn,jin2024tsgnnsurvey}. Deep time-series forecasters include LSTM and GRU \citep{hochreiter1997lstm,cho2014gru}, temporal convolutions \citep{bai2018tcn}, N-BEATS \citep{oreshkin2020nbeats}, DeepAR \citep{salinas2020deepar}, the Temporal Fusion Transformer \citep{lim2021tft} and PatchTST \citep{nie2023patchtst}; \citet{zeng2023transformers} and the M4 and M5 competitions \citep{makridakis2020m4,makridakis2022m5accuracy,makridakis2022m5uncertainty} are the cautionary references.

\paragraph{Regimes, state space, hierarchy and probability.} Markov-switching models \citep{hamilton1989regime,kimnelson1999statespace}, mixtures of experts \citep{jacobs1991moe,jordanjacobs1994hme,shazeer2017moe}, deep state-space models \citep{rangapuram2018dssm,krishnan2017dkf,debezenac2020nkf}, hierarchical Bayesian modelling \citep{gelman2006prior,hoffmangelman2014nuts,blei2017vi,rue2009inla}, forecast reconciliation \citep{hyndman2011hierarchical,wickramasuriya2019mint,athanasopoulos2024reconciliation}, forecast combination \citep{batesgranger1969combination,timmermann2006combinations}, quantile and conformal methods \citep{koenkerbassett1978quantiles,taylor2000quantilenn,romano2019cqr,vovk2005conformal} and proper scoring rules \citep{gneitingraftery2007scoring,gneiting2007calibration,gneitingkatzfuss2014probabilistic,bracher2021wis} supply the components. Forecast evaluation follows \citet{dieboldmariano1995}, \citet{harvey1997equality}, \citet{west1996asymptotic}, \citet{clarkmccracken2001nested,clarkmccracken2013advances}, \citet{giacominiwhite2006}, \citet{hansen2011mcs}, \citet{politisromano1994stationary}, \citet{kunsch1989jackknife}, \citet{hyndmankoehler2006mase} and \citet{diebold2015twenty}; dynamic factor and Bayesian VAR benchmarks follow \citet{stockwatson2002diffusion,stockwatson2002principal}, \citet{litterman1986bvar} and \citet{banbura2010bvar}; interpretation follows \citet{lundberglee2017shap,lundberg2020treeexplainer}, \citet{apleyzhu2020ale} and \citet{molnar2025iml}.

\paragraph{The gap and the classification of contributions.} Table~\ref{tab:lit_class} classifies the six architectures against their closest predecessors (full detail in the repository's novelty scan). None is unprecedented. The hierarchical hedonic network is a neural parameterisation of mixed-effects and graph-Laplacian hedonics; the graph network is an adaptation of traffic-forecasting models to municipal nodes; the regime model is a gated mixture of experts with origin-only regime inference; the Bayesian forecaster is a conjugate hierarchical model in the small-area tradition; the neural state-space model is a deep state-space model whose observation equation is the WP22 quality-adjusted index; the integrated architecture is a constrained forecast combination. The gap this paper fills is evaluative: a controlled, leakage-audited comparison of these mechanisms on transaction-level data over a geographic hierarchy, with inference that respects overlapping horizons, and a simulation study that attributes performance to mechanisms.

\begin{table}[htbp]\centering\footnotesize
\caption{Classification of the proposed architectures against the verified literature}\label{tab:lit_class}
\begin{tabular}{p{2.3cm}p{5.2cm}p{3.2cm}p{3.6cm}}
\toprule
Architecture & Closest predecessors & Classification & What is tested here \\
\midrule
HSHN & Mixed-effects / hierarchical hedonics \citep{mayer2019updating,gelmanhill2007}; graph-Laplacian hedonics \citep{sijpfrancke2026laplacian}; MugRep \citep{zhang2021mugrep} & Adaptation (neural partial pooling with a horizon input) & Out-of-time gains over trees and MLP; partial vs full vs no pooling in thin municipalities \\
DSGFN & DCRNN, Graph WaveNet, AGCRN \citep{li2018dcrnn,wu2019graphwavenet,bai2020agcrn}; STAR / FAVAR \citep{pace2000spatiotemporal,gupta2011large}; Deep-DFVAR \citep{yeo2025deepdfvar} & Application of adaptive-graph ST models to municipal indices & Gains over spatial-lag and parent-momentum benchmarks; edge-type ablation \\
RAREF & Markov switching \citep{hamilton1989regime}; mixtures of experts \citep{jacobs1991moe}; DMA \citep{borkmoller2015} & Adaptation (origin-only HMM gating of ridge experts) & Gains around regime transitions; stability with 45 origins \\
BHPF & Hierarchical repeat sales \citep{francke2017hierarchical}; Fay--Herriot \citep{fayherriot1979} & Application (conjugate hierarchical Gaussian model) & Calibration and sharpness vs quantile networks and conformal methods \\
QANSS & Structural time-series indices \citep{francke2010thin}; deep state space \citep{rangapuram2018dssm} & New combination (WP22 quality-adjusted state + learned drift transition) & Gains over the local-linear-trend and AR benchmarks on the same vintages \\
QREFA & Forecast combination \citep{batesgranger1969combination,timmermann2006combinations}; TFT \citep{lim2021tft} & Constrained combination of the five & Whether integration beats the best component and an equal-weight combination \\
\bottomrule
\end{tabular}
\end{table}