TeX 52.2%
Python 47.8%
1% ============================================================================2\section{Literature}3\label{sec:literature}4% ============================================================================56\subsection{Hedonic prices and what a cross-section identifies}78In \citet{rosen1974hedonic}, a differentiated good is a bundle of attributes $\mathbf{z}$ and the market price function $P(\mathbf{z})$ is the locus of tangencies between the bid functions of heterogeneous buyers and the offer functions of heterogeneous sellers. Its gradient $\partial P/\partial z_k$ is the implicit (marginal) price of attribute $k$: at the equilibrium, it equals the marginal bid of the buyer who selects that bundle. Recovering marginal willingness to pay \emph{functions} from the gradient is the second stage of Rosen's procedure, and it requires exclusion restrictions, multiple markets or functional-form assumptions that a single cross-section does not supply \citep{epple1987hedonic, ekeland2004identification, bajari2005estimating}. The distinction matters for the interpretation of quantile estimates. The coefficient of an amenity indicator at the $\tau$-th conditional quantile is a feature of the equilibrium price schedule in the part of the market occupied by properties whose price, given their characteristics, lies at that quantile; it is not the preference parameter of a particular type of household. \citet{kuminoff2014capitalization} show, moreover, that even a well-identified capitalisation effect of a change in a public good equals marginal willingness to pay only under restrictive conditions on sorting and on the shape of the price function. We therefore use the term \emph{implicit price} throughout and avoid ``willingness to pay'' except when referring to the literature.910The log-linear specification is the dominant functional form in applied work \citep{halvorsen1981choice, cropper1988choice, malpezzi2003hedonic, sirmans2005composition}. \citet{cropper1988choice} find in simulations that simple forms, including the semi-log, are the most robust to omitted variables and mismeasured attributes, and \citet{cassel1985hedonic} argue that flexible forms buy little when the true specification is unknown. In a semi-log model the coefficient of a binary attribute is converted to a proportional price differential as $\exp(\beta) - 1$ \citep{halvorsen1980interpretation}. \citet{kuminoff2010quasi}, \citet{parmeter2013quasi} and \citet{bishop2020best} review the specification choices that most affect the credibility of hedonic estimates---spatial fixed effects, functional form, the treatment of unobserved neighbourhood quality---and recommend reporting the sensitivity of implicit prices to them.1112\subsection{Spatial dependence in hedonic models}1314Residuals of hedonic regressions are spatially autocorrelated because neighbouring properties share unmeasured locational attributes and because prices are formed with reference to comparable nearby sales \citep{can1992specification, basu1998analysis}. Two canonical models address this. The spatial autoregressive (SAR) model adds a spatial lag of the dependent variable, $\ln\mathbf{P} = \rho\mathbf{W}\ln\mathbf{P} + \mathbf{X}\boldsymbol\beta + \boldsymbol\varepsilon$, where $\mathbf{W}$ is a weights matrix; the spatial error model places the dependence in the disturbance; and the spatial Durbin model adds spatially lagged covariates $\mathbf{WX}$ \citep{anselin1988spatial, lesage2009introduction, elhorst2010applied, anselin2010thirty}. Because $\mathbf{W}\ln\mathbf{P}$ is endogenous, the SAR model is estimated by maximum likelihood or by spatial two-stage least squares with $\mathbf{WX}$ (and higher-order lags) as instruments \citep{kelejian1998generalized, lee2003best, kelejian2010specification}. In a SAR model the marginal effect of a covariate is not its coefficient but a matrix expression involving $(\mathbf{I} - \rho\mathbf{W})^{-1}$, which generates direct, indirect and total effects \citep{lesage2009introduction, lesage2014biggest}; \citet{small2012spatial} show that for an amenity which is itself spatially distributed, the willingness-to-pay interpretation of these effects depends on whether the spillover reflects preferences or the mechanics of price formation. \citet{gibbons2012mostly} and \citet{gibbons2015spatial} caution that the exclusion restriction on $\mathbf{WX}$ is rarely credible and that spatial lag models are better regarded as descriptive adjustments than as structural models; we adopt that reading.1516Applications to hedonic pricing include \citet{kim2003measuring} and \citet{anselin2008errors} for air quality, \citet{bourassa2007spatial, bourassa2010spatial} for prediction, and the choice of $\mathbf{W}$ has been studied by \citet{getis2009spatial} and \citet{stakhovych2009specification}, who find that misspecifying $\mathbf{W}$ affects $\rho$ more than the remaining coefficients. Spatially varying coefficients are an alternative to a spatial lag \citep{bitter2007incorporating}, and recent work combines spatial structure with machine-learning predictors \citep{gyger2025explainable, chen2025geoshapley}; the implicit price of location itself, in a multinodal metropolitan area, is the subject of \citet{waddell1993exogenous}. In the Canadian context, \citet{zhang2018spatial} document spatial dependence in provincial house price dynamics; the Qu\'ebec City and Montreal studies of \citet{desrosiers2000sorting, desrosiers2002landscaping}, \citet{theriault2003modelling}, \citet{dube2011spatial, dube2013commuter, dube2014did} and \citet{dube2013spatiotemporal} use spatial and spatio-temporal weights built from transaction microdata, and \citet{dube2014spatial} provide a textbook treatment of these methods on microdata.1718\subsection{Quantile regression in housing}1920Quantile regression \citep{koenker1978regression, koenker2005quantile, koenker2001quantile} estimates the conditional quantile function $Q_\tau(\ln P \mid \mathbf{X}) = \mathbf{X}'\boldsymbol\beta(\tau)$. Because $\boldsymbol\beta(\tau)$ is allowed to vary with $\tau$, the method characterises the whole conditional distribution instead of its mean, does not require distributional assumptions on the error term and is robust to outliers in the dependent variable \citep{buchinsky1998recent, hao2007quantile}. Inference can be based on the asymptotic sandwich, which requires an estimate of the conditional density of the error at zero, or on the bootstrap \citep{buchinsky1995estimating, hahn1995bootstrapping, horowitz1998bootstrap}. For dependent data, cluster and block versions of the bootstrap are available \citep{lahiri2003resampling, parente2016quantile, hagemann2017cluster}, and the joint behaviour of estimates at several quantiles is treated by \citet{koenker2002inference}.2122\citet{zietz2008determinants} is the reference application to house prices: using Utah transactions, they find that most structural coefficients increase with the quantile and that the implicit prices of several attributes are not constant. \citet{mcmillen2008quantile} uses quantile regression to decompose the change in the distribution of Chicago house prices into changes in characteristics and changes in coefficients, and \citet{mcmillen2013quantile} develops the spatial extensions. \citet{mak2010quantile} report, for Hong Kong, that the floor-area coefficient at the 90th percentile is several times that at the 10th; \citet{farmer2010quantile} use the quantile pattern of coefficients to identify submarket competition in a United States metropolitan area. Spatial quantile applications include \citet{kostov2009spatial} for agricultural land, \citet{liao2012hedonic} for Shanghai apartments, \citet{wen2017spatial} for the Hangzhou canal landscape, \citet{gu2023hospital} for hospital accessibility in a Chinese city and \citet{squires2025airport} for airport proximity in New Zealand. The semiparametric M-quantile approach of \citet{schirripa2024mquantile} and the geographically weighted quantile regression of \citet{chen2012geographically} relax linearity or spatial stationarity of the coefficients. We are not aware of a published quantile hedonic study of the Quebec market.2324\subsection{Estimating a spatial lag inside a quantile regression}2526Three families of estimators exist. The first is the fitted-value two-stage approach: the endogenous regressor is replaced by its projection on the instruments, obtained by least squares \citep{amemiya1982two, powell1983asymptotic} or by a quantile regression at the same $\tau$ \citep{chen1996twostage, kim2004quantile}, and the quantile regression of interest is run on the fitted value. \citet{kim2004quantile} show that the quantile-based first stage yields a consistent estimator of the structural quantile coefficients under weaker conditions than the least-squares first stage, which is consistent for the slopes only when the composite first-stage error is independent of the instruments. The second family is the inverse quantile regression of \citet{chernozhukov2005iv, chernozhukov2006quantile, chernozhukov2008instrumental}, which does not replace the endogenous regressor but searches over its coefficient for the value at which the instruments are irrelevant in the quantile regression of $y - \rho\,\mathbf{W}y$; \citet{su2011instrumental} adapt this estimator to spatial autoregressive models with dependent, non-identically distributed errors. The third is the conditionally parametric or locally weighted approach of \citet{mcmillen2013quantile}, which lets the coefficients vary smoothly over space instead of modelling the spatial lag. The estimator used in this paper belongs to the first family with a least-squares first stage; Section~\ref{sec:2sqr} defines it precisely and states what it does and does not deliver.2728\subsection{Submarkets and product types}2930That a single hedonic function should not be imposed on an entire metropolitan market is an old theme \citep{palm1978spatial, goodman1998housing, watkins2001definition}. Segmented models predict better \citep{bourassa2003submarkets, bourassa2007spatial}, submarkets form a lattice of imperfect substitutes \citep{pryce2013housing}, and data-driven segmentations are now common \citep{reyblanco2024submarkets}. Most of this literature segments by location or by price; segmentation by product type is less studied, although the attribute bundles of a single-family house, a condominium unit, a small rental building and a vacant parcel are so different that pooling them is rarely attempted. \citet{kestens2006heterogeneity} show, on Qu\'ebec City transactions matched to buyer surveys, that implicit prices vary with household profile---a direct motivation for allowing them to vary across the price distribution.3132\subsection{Amenity capitalisation and the Quebec evidence}3334Parks and open space generally command premiums that decline with distance and depend on the type of open space \citep{crompton2001parks, anderson2006open, troy2008property}; \citet{dube2025park} find, for Qu\'ebec City, a premium for houses with a direct view of a park beyond the proximity effect. School quality is capitalised \citep{black1999housing, gibbons2008valuing}, but physical proximity to a school building is a different attribute from school quality and can carry a discount through traffic and noise. Rail transit premiums are positive on average but heterogeneous \citep{debrezion2007transit, mohammad2013meta}; for Montreal, \citet{dube2011spatial} and \citet{dube2013commuter} find positive commuter-rail effects that depend on distance and on the period, and \citet{dube2014did} propose a spatial difference-in-differences estimator for such changes. Highways and traffic generate competing accessibility and noise effects \citep{theebe2004planes, nelson2004meta}; hospital accessibility has been found to capitalise heterogeneously across price quantiles \citep{gu2023hospital}; golf courses carry premiums for adjacent lots \citep{do1995golf}. \citet{glaeser2008poor} document that lower-income households are more dependent on public transit, which suggests a larger transit premium at the lower end of the price distribution, and the sorting of heterogeneous households over neighbourhoods that underlies any such pattern goes back to \citet{tiebout1956pure} and is formalised by \citet{bayer2007migration}. Social-housing externalities in Qu\'ebec City are studied by \citet{dube2023yimby}. None of these Quebec studies examines the distribution of the capitalisation effect across price quantiles, which is the gap addressed here.35