spb/wp1_uqo

1% ============================================================================2\section{Discussion}3\label{sec:discussion}4% ============================================================================56\subsection{Heterogeneous capitalisation as a property of the price schedule}78The central finding is that, for single-family houses, the implicit prices of the accessibility-type amenities---highway, transit, park---are large at the lower conditional quantiles and small or nil at the upper ones, while the discount attached to school proximity is flat and the garage and golf premia nearly so. The conditional-mean estimate, which is what a standard hedonic regression would report, lies near the median profile and is therefore a poor description of either tail: it overstates the highway differential at the 90th percentile by a factor of four and understates it at the 10th by about a third. This is the pattern documented for structural attributes by \citet{zietz2008determinants} and \citet{mak2010quantile} and for location by \citet{liao2012hedonic}, transposed to broker-reported amenities in a province-wide listing sample. Within the hedonic framework it says that the equilibrium price schedule is not additively separable in the amenity indicators and the rest of the bundle: the slope of the schedule in the amenity direction depends on where, in the conditional price distribution, the property sits.910Several mechanisms are consistent with the pattern; the design does not separate them. The most natural is sorting on transit dependence and car ownership: households that buy at the lower end of the schedule are more likely to depend on transit and highway access for commuting \citep{glaeser2008poor}, so that the bid functions tangent to the schedule in that region are steeper in those attributes, whereas households at the upper end substitute private green space and private mobility for public parks and transit. A second is supply-side: in Quebec, the housing near highways, transit lines and urban parks is older and denser, so that the lower part of the conditional distribution contains most of the amenity-rich stock and the premium reflects, in part, the scarcity of such stock at that price level. A third is measurement: a broker marketing a lower-priced house may be more inclined to list accessibility amenities, and one marketing an expensive house to emphasise other qualities, in which case part of the heterogeneity is in the reporting rather than in the capitalisation. The attenuation of the highway and transit coefficients when the spatial lag is introduced points in the same direction as the second mechanism, since it shows that part of the differential is the price level of the surrounding, similarly situated listings.1112\subsection{The hospital indicator}1314The sign change of the hospital coefficient---a discount of 8.5 log points at the 10th percentile and a premium of about 4 log points, marginally significant, at the 90th---is the most distinctive result and the one most in need of corroboration. A plausible reading is that ``hospital nearby'' describes two different neighbourhood types at the two ends of the schedule: institutional corridors with traffic, parking and emergency-vehicle noise where lower-priced houses adjoin a regional hospital, and the established, well-served residential districts in which the large university hospitals of Montreal and Qu\'ebec City are embedded. The results of \citet{gu2023hospital}, who find that hospital accessibility capitalises positively in the upper price quantiles and weakly or negatively in the lower ones, are consistent with this reading. The 2SQR estimate at the median ($-$0.006 against $-$0.027 without the lag) suggests that the median discount is largely a neighbourhood price-level effect, but without the second-stage estimates at the other quantiles we cannot say whether the sign change survives the spatial adjustment. Distances to actual hospitals, by type, would be required to go further.1516\subsection{Cross-segment patterns}1718The four product types differ in the strength of the spatial lag, in the fit of the specification and in the presence of heterogeneity, and the differences are coherent with the nature of the products. Vacant land, the least differentiated product, has the strongest co-movement with neighbouring asking prices ($\hat\rho = 0.32$), the weakest structural model and the largest accessibility differentials: a highway or transit mention is associated with asking prices higher by half or more, at every quantile, which is what one expects when the indicator proxies for serviced, developable land near settlements rather than for an amenity in the residential sense \citep{colwell1997structure}. The sign change of the golf indicator for land, from a premium at the lower quantiles to a discount at the top, is plausibly a composition effect---recreational lots near golf courses in the lower part of the schedule, large rural or agricultural parcels at the top---rather than a preference reversal. Multiplexes, priced partly on rental income, show the weakest spatial co-movement ($\hat\rho = 0.13$), a transit premium that is stable across the schedule and a garage premium that disappears above the median; both are consistent with an investor's valuation, in which tenant accessibility matters at every price level and a garage adds little to the rent roll of an expensive building. Condominiums show no detectable heterogeneity, but the sample is small, selected on the reporting of interior area, and nearly homogeneous in amenity mentions, so the segment is uninformative on the question rather than evidence of homogeneity.1920\subsection{Implications}2122Two implications follow if the pattern is confirmed on transaction data. For mass appraisal and automated valuation, a conditional-mean adjustment for accessibility amenities will systematically over-adjust high-value houses and under-adjust low-value ones; the quantile profiles indicate the direction and rough size of the error, and quantile or segmented models offer a remedy at little cost. For the evaluation of amenity investments, the distribution of capitalisation across the price schedule is itself a policy-relevant object: if transit access capitalises mainly into lower-priced housing, the incidence of a transit investment falls on the owners of that housing and on the renters who may be displaced by the resulting appreciation \citep{renne2016cost}, whereas a park investment that capitalises in the lower part of the schedule but not at the top redistributes differently from one with a uniform premium \citep{wolch2014parks}. These are statements about the incidence of capitalisation, not about willingness to pay \citep{kuminoff2014capitalization}, and they are conditional on the limitations set out below.2324\subsection{Limitations}2526The limitations are of four kinds. The data are asking prices from a single snapshot, with binary broker-reported amenity indicators that carry no distance or quality information and that may be reported differently across market segments; the estimates are cross-sectional associations subject to sorting and to the usual omitted-neighbourhood-quality concern, which a single distance gradient and three regional indicators cannot fully address. The 2SQR estimator uses a least-squares first stage and the fitted lag in the second stage, so that its consistency rests on an independence assumption on the errors, no standard errors exist for the spatial-lag coefficient or for the spatially adjusted amenity coefficients, the second-stage coefficients are available at the median only, and the residual diagnostic was computed on second-stage rather than structural residuals. The inference for the quantile regressions rests on a paired bootstrap that ignores spatial dependence and on equality tests that ignore the covariance between quantiles. And the numerical record consists of a console log produced by an earlier version of the estimation script, which could not be re-run for this revision; the robustness analyses that would ordinarily accompany the results are, for that reason, listed rather than reported. The paper should be read as a documented first estimate whose main qualitative result---that amenity capitalisation in Quebec house prices is not constant across the conditional price distribution---is clear in the record, and whose quantitative details await the re-estimation described in Section~\ref{sec:limitations}.27