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1% ============================================================================2\section{Results}3\label{sec:results}4% ============================================================================56All estimates in this section are those of the original estimation run, whose console log is the only preserved numerical record of the study; the tables and figures are generated from that log by a versioned script (Appendix~\ref{app:audit}). Coefficients are in log points of the asking price.78\subsection{Conditional-mean benchmark and spatial autocorrelation}910Table~\ref{tab:moran} reports the fit of the OLS specification~\eqref{eq:hedonic} and Moran's $I$ on its residuals. The adjusted $R^2$ is 0.619 for houses, 0.669 for condominiums, 0.576 for multiplexes and 0.154 for vacant land; the last figure reflects a specification that contains only lot area, distance, region and six amenity indicators, when zoning, servicing and development potential are the main determinants of parcel prices \citep{colwell1997structure}. Moran's $I$ on the OLS residuals is 0.506 for houses, 0.551 for condominiums, 0.528 for multiplexes and 0.580 for land, with $z$-statistics between 22 and 67. These values are high by the standards of urban hedonic studies, which typically report residual Moran statistics of 0.2--0.4 on neighbourhood-level weights \citep{basu1998analysis, bourassa2007spatial}; two features of the design raise them. The samples span the whole province, so that the regional indicators and a single distance gradient leave considerable between-municipality variation in the residuals, and the ten-nearest-neighbour weights are very local, so that unmodelled street- and building-level price levels are strongly reflected in $\mathbf{W}\mathbf{e}$. Whatever its source, residual dependence of this magnitude means that the OLS and quantile standard errors computed under independence should be treated as lower bounds, and it motivates the spatial lag of Section~\ref{sec:sqr_results}.1112\input{tables/moran}1314\subsection{Quantile regressions: single-family houses}1516Table~\ref{tab:qr_houses} reports the amenity coefficients of equation~\eqref{eq:qr} at the five quantiles, with the OLS coefficient for comparison, and Figure~\ref{fig:coef} plots the profiles for all four segments; the structural, locational and regional coefficients for houses are in Appendix Table~\ref{tab:qr_houses_structural}. Four profiles stand out.1718The highway indicator has the steepest decline. Its coefficient is 0.205 at $\tau = 0.10$, 0.156 at 0.25, 0.113 at the median, 0.071 at 0.75 and 0.034, no longer significant, at 0.90; the OLS coefficient of 0.133 lies between the median and the first quartile. Transit proximity follows the same pattern from 0.178 to 0.053, with the OLS value of 0.113 again near the median. Park proximity declines from 0.122 at the 10th percentile to 0.038 at the median (significant at the 5 percent level) and to 0.014 at the top, where it is indistinguishable from zero. For these three indicators the conditional-mean coefficient is a poor summary of the schedule: it overstates the differential in the upper part of the conditional distribution by a factor of two to four and understates it in the lower part by a third to a half.1920The hospital indicator changes sign. It is $-$0.085 at $\tau = 0.10$, $-$0.061 at 0.25 and $-$0.027 at the median, all significant, and then $+$0.001 at 0.75 and $+$0.041 at 0.90, the last marginally significant at the 10 percent level. The OLS coefficient of $-$0.049 describes neither tail. The school indicator, by contrast, is uniformly negative, between $-$0.102 and $-$0.136, and the garage and golf-course indicators are positive and comparatively stable (garage 0.047--0.100; golf 0.015--0.063), with the smallest values at the 90th percentile.2122\input{tables/qr_houses}2324Table~\ref{tab:wald_houses} tests the equality of each amenity coefficient across three quantile pairs. For the $(0.10, 0.90)$ pair the statistic is 17.1 for highway, 12.1 for hospital, 11.6 for transit and 6.7 for park, each rejecting equality at the 1 percent level, against 0.05 for school, 0.6 for golf and 0.2 for garage. The $(0.25, 0.75)$ pair yields the same ranking with somewhat smaller statistics (13.8, 8.2, 11.7 and 3.7), and the $(0.10, 0.50)$ pair rejects equality for highway, transit and park at the 5 percent level but not for hospital. Four of the seven indicators therefore display significant heterogeneity between the tails, and the heterogeneity is concentrated in the lower half of the conditional distribution: the adjacent-quantile statistics in Figure~\ref{fig:wald} are largest between $\tau = 0.25$ and $\tau = 0.50$ for transit (6.8) and between 0.25 and 0.75 for highway, while the upper pairs are small.2526\input{tables/wald_houses}2728The pseudo-$R^2$ rises from 0.355 at $\tau = 0.10$ to 0.444 at $\tau = 0.90$ (Table~\ref{tab:pseudo_r2}, Figure~\ref{fig:pseudor2}). The covariates thus account for a larger share of the check-function loss in the upper part of the conditional distribution. One reading is that low-priced listings carry more idiosyncratic information---condition, deferred maintenance, motivated sellers---that is not in the listing fields; the structural coefficients in Appendix Table~\ref{tab:qr_houses_structural} are consistent with this, the floor-area elasticity rising from 0.51 at $\tau = 0.10$ to 0.69 at $\tau = 0.90$ and the Montreal premium from 0.28 to 0.44. The same appendix table also shows that the log-distance coefficient is implausibly large (1.6 to 2.5 with bootstrap standard errors of 0.4 to 0.8) and positive once the regional indicators are included; the coefficients on this control are not interpretable as a distance gradient, and Appendix~\ref{app:audit} flags the scaling of the distance variable in the original run as a point to be verified when the pipeline is re-executed.2930\input{tables/pseudo_r2}3132\begin{figure}[H]33\centering34\includegraphics[width=\textwidth]{figures/fig_coefficients_by_quantile.png}35\caption{Amenity coefficients by quantile and segment}36\label{fig:coef}37\begin{minipage}{0.95\textwidth}38\footnotesize39\textit{Notes:} Quantile regression coefficients of equation~\eqref{eq:qr} with 95 percent bootstrap confidence bands ($\pm 1.96$ bootstrap standard errors, $B = 200$). The garage indicator, absent from the land specification, is omitted. Rebuilt from the preserved log by \texttt{scripts/06\_tables\_figures\_from\_log.py}.40\end{minipage}41\end{figure}4243\subsection{Quantile regressions: condominiums, multiplexes and vacant land}4445Tables~\ref{tab:qr_condos} to~\ref{tab:qr_land} report the corresponding estimates for the other segments, and Table~\ref{tab:wald} the tail-equality tests for all four. For condominiums ($N = 1{,}031$) no amenity coefficient is significant at any quantile except the golf-course indicator, which is negative ($-$0.048 to $-$0.092) and significant at four of the five quantiles, and the garage indicator, which is large and significant throughout (0.105 to 0.164). The tail-equality statistics are all below 1.4. The bootstrap standard errors are two to three times those of the house sample, so the absence of heterogeneity is as much a statement about power as about the market; it is also consistent with a product whose urban location makes most amenity mentions nearly universal (Table~\ref{tab:descriptive}) and whose price is dominated by interior area, bathrooms and location, the OLS adjusted $R^2$ being the highest of the four segments.4647\input{tables/qr_condos}4849For multiplexes ($N = 3{,}755$) two indicators display significant tail heterogeneity. The highway coefficient is positive at the first quartile and the median (0.051 and 0.049) and negative at the 90th percentile ($-$0.084), with $W = 11.6$; the hospital coefficient is 0.054 at $\tau = 0.10$ and $-$0.042 at $\tau = 0.90$, with $W = 7.1$. Transit proximity is the strongest and most stable amenity in this segment, at 0.10 to 0.14 at every quantile, and the school discount is again negative everywhere but concentrated in the middle of the distribution ($-$0.131 at the median against $-$0.038 at the top). The garage indicator declines from 0.104 to 0.004 between $\tau = 0.10$ and $\tau = 0.75$ ($W = 4.4$ for the tails, 10.4 for the interquartile pair), which is natural for a product bought partly for rental income, where a garage adds less to the revenue of an expensive building than to the value of an owner-occupied one. The pseudo-$R^2$ declines with the quantile for this segment (0.417 to 0.341), the opposite of the house pattern.5051\input{tables/qr_plex}5253For vacant land ($N = 9{,}239$) the highway and transit indicators dominate, with coefficients of 0.43 to 0.57 and 0.36 to 0.57 respectively, that is, differentials of 50 to 80 percent in the asking price of a parcel; both are stable across quantiles. The tail tests reject equality for school ($W = 9.0$) and golf ($W = 13.1$): the school indicator is close to zero at the 10th percentile and reaches $-$0.249 at the 90th, and the golf indicator is positive at the lower quantiles (0.132 and 0.212) and negative at the top ($-$0.173). The pseudo-$R^2$ of the land regressions never exceeds 0.12.5455\input{tables/qr_land}56\input{tables/wald}5758Across the four segments, significant tail heterogeneity is found for four of six amenities for houses, none for condominiums, two for multiplexes and two for land. The pattern that is common to the three built segments is a transit differential that is positive at every quantile, and a school differential that is negative at most quantiles; the pattern that is specific to houses is the monotone decline of the highway, transit and park differentials with the quantile and the sign change of the hospital differential.5960\begin{figure}[H]61\centering62\includegraphics[width=\textwidth]{figures/fig_wald_adjacent_quantiles.png}63\caption{Wald statistics for equality of amenity coefficients at adjacent quantiles}64\label{fig:wald}65\begin{minipage}{0.95\textwidth}66\footnotesize67\textit{Notes:} Each cell is the statistic of equation~\eqref{eq:wald} for the pair of adjacent quantiles in the column, computed from the coefficients and bootstrap standard errors recorded in the log. $^{***}$, $^{**}$, $^{*}$: rejection at the 1, 5 and 10 percent levels. Rebuilt from the preserved log.68\end{minipage}69\end{figure}7071\subsection{Two-stage spatial quantile regression}72\label{sec:sqr_results}7374Table~\ref{tab:sqr} reports, for each segment, the fit of the first stage~\eqref{eq:firststage}, the spatial-lag coefficient of the median second stage~\eqref{eq:2sqr}, and the median amenity coefficients with and without the lag; Figure~\ref{fig:olsqrsqr} displays the three sets of coefficients side by side. The first-stage $R^2$ is 0.672 for houses, 0.719 for condominiums, 0.650 for multiplexes and 0.204 for land, so that the lagged covariates explain a large share of the neighbouring price level in the built segments and a small share for parcels; the overall first-stage $F$ implied by these values is in the hundreds, but the partial $F$ on the excluded instruments, which is the relevant weak-instrument statistic, was not recorded (Section~\ref{sec:limitations}). The median spatial-lag coefficient is 0.291 for houses, 0.203 for condominiums, 0.131 for multiplexes and 0.322 for land. The ordering is sensible: the undifferentiated product, land, is the one whose asking price co-moves most with neighbouring asking prices, and the rental building, priced partly on income, the one that co-moves least. Without standard errors for $\hat\rho$---the second stage was not bootstrapped---we do not claim significance, and we note that the lag coefficients at the other four quantiles were not recorded, so that nothing can be said here about the variation of $\rho$ across the conditional distribution.7576\input{tables/sqr}7778Introducing the lag attenuates the median amenity coefficients for houses: highway falls from 0.113 to 0.057, transit from 0.094 to 0.071, school from $-$0.110 to $-$0.072 and hospital from $-$0.027 to $-$0.006, while park (0.038 to 0.042) and garage (0.096 to 0.091) are essentially unchanged. The attenuation is largest for the indicators that are themselves spatially clustered---highways, transit lines and schools serve whole neighbourhoods, whereas a garage is a property attribute---which is what a spatial lag that absorbs neighbourhood price levels should produce \citep{kim2003measuring, small2012spatial}. The same qualitative pattern appears for multiplexes (garage 0.039 to 0.030, school unchanged) and for land, where the highway coefficient falls from 0.453 to 0.324 and transit from 0.485 to 0.349. Table~\ref{tab:effects} converts the median 2SQR coefficients into the scalar total effects of equation~\eqref{eq:effects}: with $\hat\rho = 0.291$ the multiplier for houses is 1.41, so that the total highway differential is 0.081 log points, or about \$48,800 at the median asking price, the transit differential 0.100 (\$60,900) and the school differential $-$0.101 ($-$\$55,600). Because the indirect share equals $\hat\rho$ by construction, these figures restate the second-stage coefficients at a different scale rather than add to them.7980\begin{figure}[H]81\centering82\includegraphics[width=\textwidth]{figures/fig_ols_qr_sqr_median.png}83\caption{Amenity coefficients at the median: OLS, quantile regression and 2SQR}84\label{fig:olsqrsqr}85\begin{minipage}{0.95\textwidth}86\footnotesize87\textit{Notes:} OLS coefficients of equation~\eqref{eq:hedonic}, median quantile regression coefficients of equation~\eqref{eq:qr}, and median second-stage coefficients of the 2SQR, equation~\eqref{eq:2sqr}. No standard errors are available for the 2SQR coefficients. Rebuilt from the preserved log.88\end{minipage}89\end{figure}9091\input{tables/effects}9293\subsection{Residual autocorrelation after the spatial lag}94\label{sec:moran_sqr}9596Table~\ref{tab:moran_resid} reports Moran's $I$ on the residuals of the quantile regressions and of the 2SQR second stages. For houses the QR residuals have $I$ between 0.515 and 0.524, slightly above the OLS value, and the second-stage 2SQR residuals between 0.518 and 0.545, slightly above the QR values; in the other three segments the two sets of statistics are of the same magnitude (0.53--0.61) and the 2SQR values are not systematically lower. The spatial lag therefore does not reduce measured residual autocorrelation. This is the expected outcome of the diagnostic as it was computed, not a failure of the model: the residuals of the second stage are formed with the fitted lag, which is a linear combination of the covariates and their spatial lags, so that the component of $\ln P_i$ that co-moves with the \emph{observed} neighbouring prices is still in the residual. Residual autocorrelation after a spatial lag must be assessed on the structural residuals, which use $\hat\rho(\tau)[\mathbf{W}\ln\mathbf{P}]_i$ rather than its projection; those were not computed in the original run. Until they are, the evidence on whether a first-order spatial lag is sufficient, or whether a spatial error component or finer spatial fixed effects are also needed, is simply unavailable, and we refrain from the statement that the 2SQR ``reduces but does not eliminate'' spatial dependence.9798\input{tables/moran_resid}99100\subsection{Dollar-equivalent implicit prices}101102Table~\ref{tab:implicit_prices} scales the house coefficients by the unconditional price quantiles (\$275,000, \$400,000, \$579,000, \$894,000 and \$1,485,000), and Figure~\ref{fig:implicit} plots the corresponding profiles for all segments. Two mechanisms interact in these figures. A declining coefficient applied to a rising price base can yield a flat or hump-shaped dollar profile: the highway differential is \$62,700 at the 10th percentile, peaks at \$69,200 at the median and is \$51,700 at the 90th, and the park differential falls from \$35,600 to \$21,300. A stable coefficient applied to a rising base yields a steeply increasing dollar profile: the school discount grows from \$26,600 to \$156,200 and the garage differential from \$17,300 to \$72,000 between the tails. The hospital differential turns from $-$\$22,400 to $+$\$62,800. These are differentials in asking prices between otherwise similar listings with and without the broker's mention of the amenity; they are reported to give a sense of scale and should not be read as values of the amenity to a household.103104\input{tables/implicit_prices}105106\begin{figure}[H]107\centering108\includegraphics[width=\textwidth]{figures/fig_implicit_prices_by_quantile.png}109\caption{Dollar-equivalent implicit prices by quantile and segment}110\label{fig:implicit}111\begin{minipage}{0.95\textwidth}112\footnotesize113\textit{Notes:} $\pi_k(\tau)$ of equation~\eqref{eq:implicit}, in thousands of Canadian dollars, using the unconditional asking-price quantile of each segment. Rebuilt from the preserved log; the figure replaces the figure of the May 2026 version, whose values could not be traced to the log.114\end{minipage}115\end{figure}116117\begin{figure}[H]118\centering119\includegraphics[width=0.75\textwidth]{figures/fig_pseudo_r2.png}120\caption{Pseudo-$R^2$ of the quantile regressions by segment}121\label{fig:pseudor2}122\begin{minipage}{0.95\textwidth}123\footnotesize124\textit{Notes:} $R^1(\tau)$ of equation~\eqref{eq:pseudor2}. Rebuilt from the preserved log.125\end{minipage}126\end{figure}127