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1% ============================================================================2\section{Literature Review}3\label{sec:literature}4% ============================================================================56\subsection{The Hedonic Pricing Framework}78The theoretical foundation of hedonic pricing rests on \citet{rosen1974hedonic}, who showed that in a market with differentiated products, the equilibrium price function reveals marginal valuations of product attributes. In housing markets, the hedonic price function $P = f(S, L, N)$ relates property prices to structural characteristics $S$, locational attributes $L$, and neighborhood amenities $N$. The partial derivative $\partial P / \partial z_k$ yields the implicit (hedonic) price of attribute $z_k$, interpreted as the market's marginal willingness to pay.910The log-linear specification $\ln P = \alpha + \boldsymbol{\beta}' \mathbf{X} + \varepsilon$ has become the dominant functional form \citep{sirmans2005composition, malpezzi2003hedonic, halvorsen1981choice}. In this specification, coefficients on continuous variables represent semi-elasticities, while coefficients on dummy variables approximate percentage effects via $(\exp(\beta) - 1) \times 100$. \citet{cropper1988choice} compare alternative functional forms (linear, semi-log, double-log, Box-Cox) using Monte Carlo simulations and find that the log-linear form performs well under omitted variable bias, a pervasive concern in hedonic studies. \citet{cassel1985hedonic} further demonstrate that the semi-log form yields more interpretable and stable coefficient estimates than flexible alternatives when the true functional form is unknown.1112A recent methodological reflection by \citet{herath2025reflections} emphasizes persistent challenges: functional form misspecification, omitted variable bias, spatial dependence, and the assumption of a single market equilibrium. \citet{parmeter2010applied} provide a comprehensive review of applied hedonic methods, noting that the choice of functional form, variable selection, and spatial structure can substantially affect estimated implicit prices. These challenges motivate the multi-method approach adopted in this paper.1314The identification problem in hedonic models deserves particular attention. As \citet{ekeland2004identification} and \citet{bajari2005estimating} demonstrate, identifying the demand function for housing attributes requires either functional form restrictions or instruments. In a single cross-section, the hedonic price function reflects the equilibrium between supply and demand; the implicit prices we estimate are therefore market-clearing marginal values rather than structural demand parameters. We acknowledge this limitation while noting that for policy analysis (e.g., property taxation, cost-benefit analysis of amenity investments), the equilibrium implicit prices are the relevant quantities.1516\subsection{Spatial Hedonic Models}1718Housing prices exhibit strong spatial dependence: properties in the same neighborhood share unobserved characteristics (school quality, crime rates, aesthetic appeal) that generate correlated residuals. Ignoring spatial autocorrelation produces biased and inefficient OLS estimates \citep{lesage2009introduction, anselin1988spatial, can1992specification}.1920The spatial autoregressive (SAR) model augments the hedonic equation with a spatial lag of the dependent variable:21\begin{equation}22\ln \mathbf{P} = \rho \mathbf{W} \ln \mathbf{P} + \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}23\end{equation}24where $\mathbf{W}$ is a spatial weights matrix and $\rho$ captures the degree of spatial spillover. The spatial error model (SEM) places spatial structure in the disturbance term $\boldsymbol{\varepsilon} = \lambda \mathbf{W}\boldsymbol{\varepsilon} + \mathbf{u}$, while the spatial Durbin model (SDM) nests both by including spatially lagged explanatory variables $\mathbf{WX}$ \citep{lesage2009introduction, elhorst2010applied}. \citet{anselin2010thirty} provides a comprehensive review of thirty years of spatial econometrics, documenting the evolution from simple spatial lag and error models to more flexible specifications.2526The interpretation of spatial models requires attention to the distinction between direct and indirect effects. In the SAR model, the marginal effect of a change in $x_{ik}$ on $y_i$ includes not only the direct effect $\beta_k$ but also feedback effects through the spatial multiplier $(I - \rho W)^{-1}$. \citet{lesage2009introduction} formalize this decomposition, showing that the average total effect of a unit change in $x_k$ across all observations is $n^{-1} \text{tr}[(I - \rho W)^{-1}] \beta_k$, which can be approximated as $\beta_k / (1 - \rho)$ for global effects.2728The specification of the spatial weights matrix $\mathbf{W}$ is a critical modeling choice. Common approaches include contiguity-based weights, distance bands, and $K$-nearest neighbors (KNN) \citep{getis2004spatial}. \citet{stakhovych2009specification} demonstrate that misspecification of $\mathbf{W}$ can bias spatial parameter estimates, though the impact on non-spatial coefficients is often modest. KNN weights offer the practical advantage of avoiding ``island'' observations (properties with no neighbors under distance-band specifications) in datasets with irregular spatial density \citep{pace2004spatial}.2930Recent advances combine spatial structure with machine learning. \citet{gyger2025spatial} propose a hybrid spatial Gaussian process with tree boosting, tested on 1.5 million German rental listings. \citet{chen2025geoshapley} apply GeoShapley to decompose spatial effects in Beijing housing prices. For the Canadian context, \citet{bourassa2010spatial} document significant spatial dependence in Canadian housing markets, and \citet{dubois2018spatial} confirm this for the Quebec market specifically. \citet{dube2011spatial} estimate a spatial hedonic model of rail transit capitalization in Montreal, finding that the spatial lag absorbs approximately 20\% of the transit premium estimated by OLS.3132\subsection{Quantile Regression in Housing Markets}3334Quantile regression \citep{koenker1978regression, koenker2005quantile} estimates the conditional quantile function:35\begin{equation}36Q_\tau(\ln P | \mathbf{X}) = \mathbf{X}\boldsymbol{\beta}(\tau), \quad \tau \in (0,1)37\end{equation}38where $\boldsymbol{\beta}(\tau)$ varies with the quantile $\tau$. This reveals how the implicit price of each attribute differs across the price distribution---a dimension invisible to OLS, which estimates only $E[\ln P | \mathbf{X}]$.3940The econometric advantages of quantile regression over OLS are well-documented. First, quantile regression estimators are consistent under heteroskedasticity and do not require distributional assumptions on the error term \citep{koenker2005quantile}. Second, they are more robust to outliers than OLS, since the check function penalizes deviations asymmetrically rather than quadratically \citep{buchinsky1998recent}. Third, quantile regression reveals the full structure of the conditional distribution, enabling researchers to test whether the effect of a covariate is heterogeneous across the response distribution \citep{hao2007quantile}.4142\citet{zietz2008determinants} provide an influential early application to U.S.\ housing data, finding significant heterogeneity in housing attribute coefficients across quantiles. They document that structural variables (bedrooms, bathrooms, floor area) tend to have larger effects at upper quantiles, while locational variables show more complex patterns. \citet{liao2012hedonic} extend this to a spatial quantile framework using Shanghai housing data, documenting that implicit prices of structural and neighborhood attributes vary substantially between cheap and expensive properties. \citet{mak2010quantile} confirm these patterns for the Hong Kong market, finding that floor area effects are three times larger at the 90th percentile than at the 10th percentile. \citet{mcmillen2008quantile} applies quantile regression to land values in Chicago, finding that the distance gradient from the CBD varies substantially across quantiles. \citet{farmer2011exaggerated} use quantile regression to show that the premium for school quality is exaggerated at higher price quantiles in the U.K.\ housing market.4344In the Canadian context, quantile regression applications to housing markets remain rare. \citet{des2014quantile} apply quantile regression to assess the impact of environmental contamination on housing values in Montreal, finding that contamination effects are amplified for lower-priced properties. \citet{wen2017quantile} estimate quantile regression models for Toronto condominium prices, documenting heterogeneous effects of building age and floor level across the price distribution. These Canadian studies do not, however, combine quantile regression with spatial modeling.4546The combination of spatial structure and quantile estimation remains methodologically challenging. \citet{kim2004quantile} propose a two-stage instrumental variables approach: the spatial lag $\mathbf{W}\ln\mathbf{P}$ is instrumented with spatially lagged exogenous variables $\mathbf{WX}$ in a first stage, then the predicted spatial lag is included in a second-stage quantile regression. This two-stage spatial quantile regression (2SQR) avoids the computational intractability of jointly estimating spatial and quantile parameters. \citet{su2012profile} propose a profile likelihood approach to spatial quantile regression, while \citet{kostov2009spatial} introduces a penalized quantile regression method with spatial smoothing. The 2SQR framework has been applied to airport noise effects \citep{nz2025airport}, hospital accessibility \citep{hospital2023heterogeneity}, and urban green space valuation \citep{chen2020quantile}, but not to multi-segment Canadian housing data.4748\citet{salvati2023semiparametric} propose a spatial semiparametric M-quantile regression framework for hedonic pricing, offering robustness to outliers and distributional misspecification. Their approach accounts for nonlinear environmental effects while maintaining spatial trend modeling. \citet{chernozhukov2006quantile} develop instrumental variable quantile regression methods with formal weak instrument diagnostics, which inform our testing strategy.4950\subsection{Housing Market Segmentation}5152The assumption that a single hedonic price function governs the entire market has been challenged by the housing submarket literature. \citet{bourassa2003submarkets} demonstrate that segmented hedonic models outperform pooled models for property valuation, using data from Auckland, New Zealand. \citet{goodman1998housing} provides theoretical foundations for market segmentation based on supply rigidities and demand heterogeneity. \citet{reyblanco2024submarkets} use machine learning to identify spatial market segments endogenously in Madrid, finding that data-driven boundaries outperform administrative ones.5354Cross-segment analysis---comparing hedonic estimates across property types such as single-family homes, condominiums, and multiplexes---has received less attention. Property types differ fundamentally in their attribute bundles: condominiums lack land, multiplexes generate rental income, and vacant land has no building characteristics. These structural differences imply that amenity capitalization should vary across segments, yet most hedonic studies analyze a single property type \citep{sirmans2005composition}. \citet{palm1978spatial} is among the earliest to argue that hedonic models should be estimated separately by housing submarket, while \citet{watkins2001definition} proposes formal tests for market segmentation based on Chow statistics. \citet{pryce2013property} emphasizes that cross-segment analysis is essential for understanding how different buyer populations value neighborhood amenities.5556\subsection{Amenity Capitalization}5758The capitalization of neighborhood amenities into property values has been extensively studied. Parks generate premiums of 3--10\% for proximate properties \citep{crompton2001parks, anderson2001park, troy2008property}, though effects diminish with distance and may vary with park type and quality. School quality premiums are well-documented \citep{black1999housing, gibbons2012valuing}, though proximity to schools can generate disamenity effects from noise and traffic congestion. Transit accessibility premiums vary widely depending on transit mode and neighborhood context \citep{debrezion2007transit, mohammad2013meta}, with heavy rail stations typically commanding larger premiums than bus stops. Highway proximity produces competing effects: accessibility benefits versus noise, pollution, and visual disamenity \citep{highway2010effects, nelson2004meta}. Hospital proximity effects are less studied but potentially complex, combining accessibility benefits with institutional disamenity effects \citep{hospital2023heterogeneity}.5960For the Quebec context, \citet{dube2025park} find additional price premiums for houses exposed to urban parks in Qu\'ebec City, beyond simple proximity effects. \citet{dube2023yimby} document asymmetric impacts of social housing on nearby property values, with NIMBY effects dominating in upper-income neighborhoods. \citet{dube2011spatial} estimate a spatial hedonic model showing that proximity to Montreal's commuter rail network generates a 4--6\% price premium. However, no published study has examined whether these amenity effects are heterogeneous across the price distribution in the Quebec market.6162The literature on golf course proximity effects on property values is relatively thin compared to parks and transit. \citet{do1994impact} find that golf course proximity generates a 7.6\% premium, while \citet{grudnitski2002golf} estimate premiums of 4--8\% depending on lot position relative to fairways. These effects may vary with the socioeconomic profile of the neighborhood and the quality of the golf facility.6364\subsection{Bootstrap Inference in Spatial Models}6566Standard asymptotic inference for quantile regression can be unreliable when the data exhibit spatial dependence, heteroskedasticity, or clustering \citep{buchinsky1995estimating, hahn1995bootstrapping}. The bootstrap provides a flexible alternative that avoids the need to estimate the density of the error term at zero---a requirement of the asymptotic sandwich variance estimator for quantile regression \citep{koenker2005quantile}.6768In spatial contexts, the standard nonparametric (paired) bootstrap resamples observations with replacement, which breaks the spatial structure of the data. \citet{lahiri2003resampling} discusses block bootstrap methods that preserve local spatial dependence by resampling contiguous blocks of observations. \citet{jin2001simple} proposes a residual-based spatial bootstrap, while \citet{freedman1981bootstrapping} establishes the theoretical foundations for bootstrapping regression models. For instrumental variable quantile regression, \citet{chernozhukov2006quantile} develop formal inference methods, but these are computationally prohibitive for large datasets with spatial structure.6970We employ the nonparametric paired bootstrap with $B = 200$ replications, acknowledging that this approach may not fully account for spatial dependence in the resampling scheme. Our bootstrap standard errors should therefore be interpreted as potentially optimistic; we discuss sensitivity to the number of replications and the implications of ignoring spatial structure in the bootstrap in Section~\ref{sec:robustness}.7172\subsection{Gaps in the Literature}7374Four gaps motivate this study:75\begin{enumerate}[nosep]76\item \textbf{No spatial quantile hedonic analysis for Quebec.} Despite extensive hedonic work by the Universit\'e Laval group \citep{dube2023yimby, dube2025park, dube2011spatial} and emerging studies in other Canadian markets \citep{des2014quantile, wen2017quantile}, distributional heterogeneity in amenity capitalization has not been investigated using spatial quantile methods in Quebec.77\item \textbf{No multi-segment comparison.} Cross-property-type analysis of amenity effects is rare in any market \citep{sirmans2005composition}, and absent for Quebec. Comparing houses, condos, multiplexes, and land provides unique insight into how buyer heterogeneity drives amenity capitalization.78\item \textbf{Under-exploitation of Centris data.} The Centris platform provides the most comprehensive Quebec listing database, yet published academic studies using it remain few. The breadth and detail of Centris listings enable analyses that are infeasible with smaller, survey-based datasets.79\item \textbf{Insufficient robustness analysis.} Many spatial quantile hedonic studies report results without systematic robustness checks for weak instruments, multicollinearity, or sensitivity to spatial weights specification. Our comprehensive diagnostic framework addresses this gap.80\end{enumerate}81