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1% ============================================================================2\section{Results}3\label{sec:results}4% ============================================================================56\subsection{Spatial Autocorrelation}78Table~\ref{tab:moran} reports Moran's $I$ statistics for OLS residuals across segments. Spatial autocorrelation is highly significant in all four segments ($p < 0.001$), with $I$ values ranging from 0.506 (houses) to 0.580 (land). These values indicate strong positive spatial dependence: proximate properties have correlated unobserved determinants of price. By comparison, \citet{dube2011spatial} report Moran's $I$ values of 0.30--0.45 for Montreal housing prices, and \citet{bourassa2010spatial} find values of 0.25--0.50 across Canadian metropolitan areas. Our higher values likely reflect the province-wide scope of our analysis, which includes both within-city and between-city spatial correlation.910This confirms that OLS hedonic estimates are inefficient and potentially biased, motivating the spatial quantile approach. The Moran's $I$ for land parcels (0.580) is the highest, consistent with the well-documented importance of comparable-sales evidence in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}.1112\input{tables/moran}1314\subsection{OLS Baseline}1516The OLS hedonic model explains 68.5\% of price variation for houses (adjusted $R^2 = 0.685$), 74.5\% for condos, 66.5\% for plex, and 24.1\% for land. The low explanatory power for land reflects the absence of structural characteristics and the high heterogeneity of vacant parcels (agricultural, residential, commercial zoning). All models use HC3 robust standard errors.1718For the house segment, the structural variables show expected signs and magnitudes: each additional bathroom increases price by approximately 30\% ($\beta = 0.26$, $p < 0.001$); the floor area elasticity is 0.49, indicating that a 10\% increase in floor area is associated with a 4.9\% price increase; garage presence commands a 12.6\% premium (\$53,000 at the median price). The Montreal regional dummy indicates a 40--45\% premium over Gatineau, consistent with the observed median price differential.1920\subsection{Multicollinearity Assessment}2122Variance inflation factors for the house segment range from 1.05 (golf proximity) to 3.82 (log floor area), all well below the conventional threshold of 10. The condition number of the centered design matrix is 18.4, below the \citet{belsley1980regression} threshold of 30. These diagnostics confirm that centering the continuous variables successfully resolves the multicollinearity concerns that arise in uncentered specifications, where VIFs for bedrooms and bathrooms can exceed 15 due to their correlation with floor area \citep{obrien2007caution}.2324\subsection{Quantile Regression Results}2526\subsubsection{Single-Family Houses}2728Table~\ref{tab:qr_houses} reports amenity coefficients across quantiles for single-family houses, the most attribute-rich segment. Several patterns emerge:2930\textbf{Hospital proximity} is the strongest source of inter-quantile heterogeneity ($\chi^2 = 12.3$, $p < 0.001$): the coefficient drops from $-$0.082 ($p < 0.001$) at $\tau = 0.10$ to $+$0.013 (insignificant) at $\tau = 0.90$. Lower-priced homes near hospitals appear to suffer from institutional neighborhood effects (traffic, ambulance noise), while the association reverses direction---though not significantly---at the upper tail. This pattern is consistent with \citet{hospital2023heterogeneity}, who document heterogeneous hospital proximity effects across price quantiles in European markets.3132\textbf{Park proximity} shows the clearest declining gradient ($\chi^2 = 14.1$, $p < 0.001$): the coefficient drops from $+$0.087 ($p < 0.001$) at $\tau = 0.10$ to $-$0.044 (insignificant) at $\tau = 0.90$. Parks serve as a primary outdoor amenity for lower-priced homes, where private green space (pools, large lots) is limited. Luxury properties may substitute private for public green space. \citet{anderson2001park} find a similar pattern in U.S.\ data, noting that park premiums are strongest in neighborhoods with limited private outdoor space.3334\textbf{Public transit} shows a declining positive association: 0.074 ($p < 0.01$) at $\tau = 0.10$ versus $-$0.008 (insignificant) at $\tau = 0.90$. Households purchasing affordable homes are more transit-dependent, hence transit proximity is associated with a larger premium. This aligns with \citet{mohammad2013meta}, whose meta-analysis shows that transit premiums are larger in lower-income neighborhoods, and with \citet{glaeser2008poor}, who documents that lower-income households are systematically more transit-dependent.3536\textbf{Highway proximity} shows a modest positive association at the median (0.043, $p < 0.001$) that becomes insignificant at both tails. The Wald test for $\tau = 0.10$ versus $\tau = 0.90$ is not significant ($\chi^2 = 2.5$), suggesting that the highway gradient is less pronounced once distance-to-Montreal is properly controlled.3738\textbf{School proximity} is consistently negative across all quantiles ($\approx -$0.07 to $-$0.09), likely reflecting the density, traffic congestion, and noise associated with school sites rather than school quality per se (the binary variable does not distinguish school quality). \citet{gibbons2012valuing} demonstrate that while school quality premiums are positive, physical proximity to school buildings often generates disamenity effects from traffic and noise.3940\textbf{Golf course proximity} and \textbf{garage} are positive and relatively stable across quantiles, suggesting homogeneous capitalization for these attributes.4142\input{tables/qr_houses}4344The increasing pseudo-$R^2$ across quantiles (0.355 at $\tau = 0.10$ to 0.444 at $\tau = 0.90$) indicates that structural and amenity variables explain a larger share of variance at the upper end of the price distribution. This is consistent with the interpretation that luxury home prices are more systematically determined by observable characteristics, while affordable home prices are more influenced by idiosyncratic factors (deferred maintenance, aesthetic condition) not captured in listing data.4546\subsubsection{Condominiums}4748The condo segment ($N = 1{,}031$) shows less inter-quantile variation, with no amenity variable exhibiting significant heterogeneity between $\tau = 0.10$ and $\tau = 0.90$ at the 5\% level. This may reflect the smaller sample size (lower statistical power) or the more homogeneous nature of condo amenity bundles (predominantly urban, with similar access to services). The condo market in Quebec is concentrated in Montreal (62.4\% of our sample), limiting geographic variation in amenity access. The OLS adjusted $R^2$ is 0.669, the highest among all segments, suggesting that floor area and location explain most of the condo price variation.4950\subsubsection{Multiplexes}5152The plex segment ($N = 3{,}755$) exhibits significant heterogeneity for highway ($p < 0.001$) and hospital ($p < 0.01$) proximity. Investment-oriented buyers of lower-priced multiplexes value highway accessibility (tenant convenience), while higher-priced plex buyers focus on neighborhood quality. The plex market combines owner-occupant and investor demand, creating a natural source of preference heterogeneity \citep{pryce2013property}. The OLS adjusted $R^2$ is 0.576.5354\subsubsection{Vacant Land}5556The land segment ($N = 9{,}239$) shows significant heterogeneity for school ($p < 0.01$) and golf course ($p < 0.001$) proximity. The OLS model explains only 15.4\% of variance, reflecting the dominance of unobserved zoning, development potential, and site-specific characteristics. The spatial lag parameter is the highest among all segments ($\rho = 0.322$), consistent with strong neighborhood effects in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}.5758\subsection{Spatial Quantile Regression (2SQR)}5960\subsubsection{First-Stage Diagnostics}6162Table~\ref{tab:first_stage} reports first-stage regression diagnostics for the 2SQR model across all segments. The first-stage $R^2$ ranges from 0.548 (plex) to 0.672 (houses), and the $F$-statistic exceeds 10 in all segments, well above the \citet{stock2002survey} threshold for weak instrument rejection. These diagnostics confirm that the spatially lagged exogenous variables $\mathbf{WX}$ are relevant instruments for the endogenous spatial lag.6364\input{tables/first_stage}6566\subsubsection{Spatial Lag Parameter}6768Table~\ref{tab:2sqr} reports the 2SQR spatial lag coefficients across quantiles and segments. The spatial lag coefficient $\rho = 0.227$ ($p < 0.001$) at the median for houses confirms significant price spillovers: a 1\% increase in average neighboring prices is associated with a 0.23\% increase in the focal property's price.6970The inclusion of the spatial lag reduces the magnitude of most amenity coefficients, as part of the amenity ``effect'' captured by QR reflects spatial spillovers from neighboring properties with similar amenity access. For example, the highway coefficient drops from 0.043 (QR) to 0.029 (2SQR) at $\tau = 0.50$. This attenuation effect is consistent with \citet{dube2011spatial}, who document similar coefficient reductions when spatial structure is introduced into Montreal hedonic models. Notably, with the corrected distance variable, the $\rho$ values are substantially lower than in preliminary analyses, because the log-distance gradient now properly captures the Montreal-centric price hierarchy, leaving less residual spatial structure for the spatial lag to absorb.7172\input{tables/sqr_rho}7374The spatial lag varies across both quantiles and segments (Figure~\ref{fig:rho}). For houses, $\rho$ is relatively stable (0.28--0.33), suggesting a consistent spatial pricing mechanism across the distribution. For condos, it declines sharply from 0.34 at $\tau = 0.10$ to near zero at $\tau = 0.90$, suggesting that luxury condos are priced more independently of their neighbors---possibly reflecting unique amenity bundles (penthouses, custom finishes) that differentiate them from adjacent units. For plex, $\rho$ increases with the quantile (0.089 to 0.224), possibly reflecting stronger comparable-sales effects for higher-valued investment properties, where sophisticated investors rely more heavily on market comparables.7576\subsubsection{Moran's $I$ on 2SQR Residuals}7778Table~\ref{tab:moran_sqr} reports Moran's $I$ on residuals across models and quantiles for the house segment. The 2SQR model reduces but does not eliminate residual spatial autocorrelation: Moran's $I$ drops from 0.506 (OLS) to approximately 0.35--0.40 (2SQR), depending on the quantile. The persistent spatial autocorrelation likely reflects spatial structure in unobserved variables (school quality, crime rates, aesthetic neighborhood characteristics) that are not captured by our specification.7980\input{tables/moran_sqr}8182\subsection{Inter-Quantile Heterogeneity Tests}8384Table~\ref{tab:wald} reports Wald test statistics for the null hypothesis $\beta(\tau = 0.10) = \beta(\tau = 0.90)$ for each amenity, by segment.8586\input{tables/wald}8788For single-family houses, four of six amenities show significant heterogeneity: highway ($\chi^2 = 31.6$), hospital (17.1), transit (13.0), and park (10.8). The omnibus test strongly rejects joint homogeneity ($\chi^2(17) = 167.14$, $p < 0.001$). For the plex segment, highway and hospital are significant at the 5\% level. For land, school and golf are significant. The condo segment shows no significant heterogeneity, possibly due to the limited sample size ($N = 1{,}031$) providing insufficient power to detect moderate effect heterogeneity.8990\subsection{Willingness to Pay}9192Table~\ref{tab:wtp} translates the quantile coefficients into dollar-valued implicit prices for single-family houses. WTP is computed as $\text{WTP}_k(\tau) = P(\tau) \times [\exp(\hat{\beta}_k(\tau)) - 1]$, where $P(\tau)$ is the sample price at quantile $\tau$.9394\input{tables/wtp}9596The school discount escalates from \$26,608 at the 10th percentile to \$156,225 at the 90th percentile, reflecting the higher absolute price levels of luxury homes even though the semi-elasticity is relatively stable across quantiles. In contrast, the highway premium peaks at the median (\$69,207) and declines at the top (\$51,655) despite the rising price base, because the coefficient decline (from 0.205 to 0.034) more than offsets the price increase. The hospital implicit price reverses from $-$\$22,403 to $+$\$62,771, a swing of \$85,174 between the 10th and 90th percentiles.9798\subsection{Direct and Indirect Effects}99100Table~\ref{tab:effects} decomposes the 2SQR coefficients into direct and indirect (spatial spillover) effects at the median for single-family houses.101102\input{tables/effects}103104Indirect effects represent approximately 29\% of total effects (proportional to $\rho$). A property benefits not only from its own amenity proximity but also from the amenity proximity of its neighbors, which raises neighboring prices and spills over. The total transit effect (\$60,914) exceeds the direct QR coefficient because it incorporates the spatial multiplier $1/(1-\rho) = 1.41$. The hospital effect is nearly zero at the median in the 2SQR model, suggesting that the OLS hospital discount is largely attributable to spatial correlation with other neighborhood characteristics rather than a direct hospital effect.105