TeX 52.2%
Python 47.8%
1% ============================================================================2\section{Discussion}3\label{sec:discussion}4% ============================================================================56\subsection{Heterogeneity as the Rule, Not the Exception}78Our central finding is that amenity capitalization is heterogeneous across the price distribution. The omnibus Wald test for houses ($\chi^2(7) = 35.74$, $p < 0.001$) rejects the OLS homogeneity assumption. This result aligns with the international literature \citep{zietz2008determinants, liao2012hedonic, mak2010quantile} but extends it to the Quebec context for the first time.910The economic interpretation is straightforward: households at different points in the price distribution have different preferences, constraints, and substitution possibilities. Transit-dependent households purchasing affordable homes value transit proximity three times more than affluent households with private vehicles. Conversely, luxury homebuyers may value the prestige and green space of golf course neighborhoods more than their budgetary significance would suggest. These patterns are consistent with the sorting equilibrium framework of \citet{bayer2007migration}, where heterogeneous households sort across locations based on their willingness to pay for local amenities, generating the distributional heterogeneity we document.1112The practical significance of this heterogeneity is considerable. The OLS estimate of the highway effect (0.133) overestimates the effect for luxury homes by a factor of 4 (0.133 vs.\ 0.034) and underestimates it for affordable homes by 35\% (0.133 vs.\ 0.205). Any policy analysis based on the OLS coefficient would systematically misrepresent the welfare implications across income groups.1314\subsection{The Hospital Proximity Puzzle}1516The sign reversal of hospital proximity across quantiles---negative for cheap homes, positive for expensive ones---is an original finding that merits discussion. We propose three complementary explanations:1718\begin{enumerate}[nosep]19\item \textbf{Institutional disamenity vs.\ neighborhood quality.} Affordable homes near hospitals may be in dense, noisy, institutional corridors (ambulance traffic, surface parking). Luxury homes near hospitals may be in established affluent neighborhoods where major medical centers have located historically. In Quebec, major hospitals (McGill University Health Centre, H\^opital de Montr\'eal pour Enfants, Centre hospitalier de l'Universit\'e Laval) are often situated in prestigious neighborhoods, creating a positive correlation between hospital proximity and neighborhood wealth at the upper end of the distribution.2021\item \textbf{Sorting.} High-income households may sort into hospital-proximate neighborhoods for accessibility to specialized medical services, while low-income households near hospitals occupy lower-quality adjacent housing stock. \citet{bayer2007migration} document this type of amenity-driven sorting in U.S.\ metropolitan areas.2223\item \textbf{Omitted variable bias.} Hospital proximity may proxy for neighborhood density and urbanization, which have heterogeneous effects across the price distribution. The spatial lag partially controls for this, as the hospital coefficient becomes insignificant in the 2SQR median model ($\beta = -0.006$, $p > 0.10$), suggesting that the OLS hospital effect is substantially confounded by spatial spillovers.24\end{enumerate}2526\subsection{Cross-Segment Patterns}2728The four segments reveal distinct capitalization patterns that reflect fundamental differences in buyer populations and product characteristics:2930\textbf{Houses} show the richest heterogeneity, with 4/6 amenities significantly varying across quantiles. The structural model is well-specified ($R^2 = 0.62$) and amenity effects are precisely estimated. The house market spans the widest price range (10:1 ratio between $\tau = 0.90$ and $\tau = 0.10$), providing the most statistical power to detect distributional heterogeneity.3132\textbf{Condos} show no significant heterogeneity, consistent with a more homogeneous urban product where amenity access is relatively uniform. The high $R^2$ (0.67) suggests that structural variables (floor area, bathrooms) dominate price determination. The predominantly urban sample (62.4\% Montreal) limits geographic variation in amenity access, reducing the power of inter-quantile tests.3334\textbf{Plex} shows intermediate heterogeneity, with investment logic adding a layer of complexity: lower-priced plex buyers may be first-time investors prioritizing accessibility (for tenant convenience), while higher-priced plex buyers focus on tenant quality neighborhoods. The dual owner-occupant/investor nature of the plex market creates a natural source of preference heterogeneity \citep{pryce2013property}.3536\textbf{Land} shows the weakest structural model ($R^2 = 0.15$) but the strongest spatial dependence ($\rho = 0.32$), consistent with the importance of comparable-sales and neighborhood effects in land valuation where the physical product is undifferentiated \citep{colwell2002estimating}. The significant heterogeneity in school and golf coefficients may reflect different buyer populations: residential builders (sensitive to school quality and family-friendliness) versus recreational/agricultural buyers (sensitive to golf and rural amenities).3738\subsection{Spatial Spillovers}3940The spatial lag $\rho$ is significant in the house segment across all quantiles (0.22--0.28), while the condo and land segments show weaker spatial dependence ($\rho \approx 0.04$--0.05 at the median). This implies that a 10\% increase in average neighboring prices is associated with a 0.4--2.8\% increase in a property's price, depending on segment and quantile. The approximate indirect effects decomposition suggests that 4--23\% of the total amenity effect operates through spatial spillovers rather than direct capitalization, with the share being largest for houses.4142The variation of $\rho$ across quantiles is itself informative. For condos, the spatial lag declines from 0.34 to zero as we move from cheap to luxury properties, suggesting that luxury condo prices are determined more by unique unit characteristics (floor level, views, finishes) than by neighborhood comparables. This is consistent with the thin-market hypothesis: luxury condos are sufficiently differentiated that neighboring prices provide limited information for valuation \citep{haurin2010thin}.4344For plex, the opposite pattern ($\rho$ increasing with quantile) suggests that higher-valued investment properties are more sensitive to comparable-sales evidence. Professional investors purchasing expensive multiplexes likely rely more heavily on comparable transaction data, creating stronger spatial price dependencies \citep{shimizu2010housing}.4546\subsection{Policy Implications}4748While these results reflect conditional associations rather than causal effects, several policy-relevant observations emerge from the heterogeneity patterns:4950\begin{enumerate}[nosep]51\item \textbf{Property taxation.} Assessments based on mean hedonic estimates will systematically over-value amenity effects for expensive properties (e.g., highway) and under-value them for affordable ones (or vice versa, depending on the amenity). Municipal assessment offices in Quebec use computer-assisted mass appraisal (CAMA) models that are typically OLS-based; incorporating distributional heterogeneity could improve assessment equity across property classes.5253\item \textbf{Urban planning.} Transit investments have the largest implicit price for affordable housing---a progressive finding that supports transit-oriented development in lower-income neighborhoods. Our estimates suggest that a new transit connection would increase the value of a \$275,000 home by approximately \$53,567 (19.5\%), compared to \$80,978 (5.4\%) for a \$1.49M home. In proportional terms, the transit premium is nearly four times larger for affordable properties, supporting the case for transit-oriented affordable housing policy \citep{renne2016transit}.5455\item \textbf{Cost-benefit analysis.} Park investments capitalize more into affordable homes (\$35,648 at $\tau = 0.10$) than luxury homes in absolute terms, and substantially more in proportional terms (13.0\% vs.\ 1.4\%). This suggests that park development in affordable neighborhoods generates more housing value per dollar of public investment, providing a fiscal rationale for equitable park distribution \citep{wolch2014parks}.5657\item \textbf{Environmental justice.} The school proximity discount is amplified for luxury homes in absolute dollar terms (\$156,225 vs.\ \$26,608), suggesting that wealthy households are more able to sort away from school-proximate (i.e., dense, high-traffic) locations. This sorting reinforces spatial income inequality and may contribute to school quality disparities through property tax mechanisms \citep{tiebout1956pure}.5859\item \textbf{Mortgage risk assessment.} Lenders and insurers (including the Canada Mortgage and Housing Corporation, CMHC) use automated valuation models for underwriting. Our findings suggest that AVM accuracy could be improved by incorporating quantile-dependent amenity valuations, particularly for properties at the tails of the price distribution where OLS-based models perform worst.60\end{enumerate}6162\subsection{Limitations and Caveats}6364The following limitations circumscribe the interpretation of our results and define priorities for future work:6566\begin{enumerate}[nosep]67\item \textbf{Listing prices.} We use listing prices rather than transaction prices. Listing prices may reflect strategic pricing behavior; the listing-to-sale ratio may vary by segment and quantile (luxury listings may be more aspirationally priced), and amenity proximity may correlate with bargaining power. Our coefficients are valid if the listing premium is conditionally independent of amenity access, but this assumption cannot be tested without matched transaction data.6869\item \textbf{Cross-sectional design.} The analysis cannot identify causal effects. Endogeneity from household sorting \citep{bayer2007migration} and unobserved neighborhood quality remains a concern, though the spatial lag and regional fixed effects partially address correlated omitted variables. The ideal design would exploit quasi-experimental variation in amenity provision (e.g., new park openings, transit extensions), which requires panel data.7071\item \textbf{Binary amenity indicators.} The Centris ``Amenities Nearby'' field depends on broker classification, introducing measurement error and broker discretion. The variables are binary (near/far) without distance gradients, which attenuates estimated effects and prevents the identification of distance-decay patterns \citep{crompton2001parks}. ``Near school'' does not capture school quality; ``near transit'' does not distinguish bus, metro, REM, or commuter rail; ``near hospital'' does not distinguish major academic hospitals from small clinics.7273\item \textbf{Sample selection.} The condo retention rate (11\%) creates possible selection bias if listings with missing floor area are systematically different from retained observations.7475\item \textbf{Spatial weights specification.} The $K$-nearest-neighbor weights with $K = 10$ are constructed within each segment. Neighbors can only belong to the same property type. The results are qualitatively robust to alternative $K$ values (Section~\ref{sec:robustness}), but we have not tested distance-band weights or cross-segment neighbor definitions.7677\item \textbf{Residual spatial autocorrelation.} Moran's $I$ on 2SQR residuals remains substantial (0.35--0.40), indicating that the spatial lag does not fully absorb spatial structure. Unobserved neighborhood characteristics (crime, aesthetics, school quality) likely generate persistent spatial dependence.7879\item \textbf{Generated regressor problem.} The 2SQR estimator uses first-stage predicted values as if they were observed, which does not affect consistency but may bias standard errors. Our bootstrap procedure partially addresses this, but it resamples only the second stage.8081\item \textbf{Bootstrap and spatial dependence.} The paired bootstrap resamples observations independently, breaking the spatial structure of the data. Block bootstrap or subsampling approaches would provide more conservative inference \citep{lahiri2003resampling} but are computationally expensive and require assumptions about block size.8283\item \textbf{Approximate effect decomposition.} The direct and indirect effects reported use scalar approximations rather than the formal LeSage--Pace matrix-based decomposition. As discussed in Section~\ref{sec:methodology}, the approximation error is small for our specifications, but formal computation would strengthen the results.8485\item \textbf{Scope of interpretation.} The estimated coefficients are quantile-specific capitalization gradients---conditional associations in listing prices---not structural demand parameters or causal welfare measures. Policy implications should be interpreted accordingly.86\end{enumerate}87