spb/wp4_uqo

1% ==========================================================================2\section{Empirical Results}3\label{sec:results}4% ==========================================================================56% ------------------------------------------------------------------7\subsection{Nested Hedonic Models}8\label{subsec:nested}9% ------------------------------------------------------------------1011Table~\ref{tab:nested} reports the five nested specifications. The12structural block alone (M1) explains 44\% of the variance of log prices13in the analysis sample, a figure in line with single-city Airbnb studies14\citep{WangNicolau2017, ChenXie2017, Gyodi2021} and far above the 5\%15obtained on the contaminated price field. Adding the presentation16variables (M2) and the host and reputation variables (M3) raises $R^2$17by about one percentage point each; the two location variables (M4) add18two points, and the city and cell fixed effects (M5) a further three, to190.509 (adjusted 0.495). The fixed effects alone explain2013\% of the variance, so the structural and listing-level21characteristics carry most of the explanatory power and location acts22mainly through the composition of listings across places rather than23through large city-level price differences. The root mean squared error24of M5 is 0.475 log points, which is a typical prediction error of about2540\% of the nightly price for an individual listing.2627\begin{table}[!htbp]28\centering29\caption{Nested Hedonic Specifications, Dependent Variable $\ln$(Displayed Nightly Price)}30\label{tab:nested}31\begin{threeparttable}32\footnotesize33\setlength{\tabcolsep}{4pt}34\input{../results/tables/rev_ols_nested.tex}35\begin{tablenotes}36\footnotesize37\item \textit{Notes:} OLS on the analysis sample ($N = 2{,}480$).38Columns (1)--(5) report HC3 standard errors in parentheses; column39(5$'$) repeats M5 with standard errors clustered by fixed-effect group40(51 groups) in brackets. The reference title category is41\emph{Rental unit}. Rating is centred at 4.8 and set to zero for unrated42listings, so the ``Rated'' indicator is the premium of a listing rated434.8 over an unrated one. City / cell fixed effects: one indicator per44municipality with at least 15 listings (26 cities) and 2545$k$-means cells for the remaining listings. ***, ** and * denote46significance at the 1, 5 and 10\% levels.47\end{tablenotes}48\end{threeparttable}49\end{table}5051The coefficients are stable across specifications, which indicates that52the omitted-variable bias from leaving out reputation or location is53small for the structural variables. Clustering the standard errors by54fixed-effect group raises them by 20 to 60\% for the capacity and55location variables and leaves the inference qualitatively unchanged,56except that the bedroom elasticity falls from the 1\% to the 5\% level57and the rating coefficient becomes more precise. Variance inflation58factors in M4 peak at 4.2 (for $\ln$ bedrooms), so the capacity59variables, although correlated, are separately identified60(Appendix Table~\ref{tab:app_vif}).6162% ------------------------------------------------------------------63\subsection{Capacity, Type and Presentation}64\label{subsec:structural}65% ------------------------------------------------------------------6667Capacity is priced through three channels. Holding bedrooms and68bathrooms constant, a 10\% increase in guest capacity is associated with69a 3.3\% higher price (elasticity 0.331, HC3 standard error 0.037);70holding capacity constant, a 10\% increase in one plus the bathroom71count is associated with a 4.4\% higher price (0.437, 0.070), and a 10\%72increase in bedrooms with a 1.6\% higher price (0.160, 0.042). The73bedroom elasticity is modest because capacity and bathrooms absorb most74of the size effect; in M1 without the other capacity variables it would75be far larger. A shared bathroom, which in these data identifies the76room-in-a-home segment that the missing room-type field would otherwise77have captured, carries a premium of78$g = 100[\exp(-0.385 - \tfrac{1}{2}0.053^2) - 1] = -32\%$ by79Equation~\eqref{eq:kennedy}, the largest single implicit price in the80model. Studios are priced 16\% above an otherwise equivalent81one-bedroom listing of the same capacity, which reflects their urban82location and the fact that a studio of given capacity is a complete83dwelling rather than a room.8485Title categories matter less than their raw price differences in86Table~\ref{tab:price_category} suggest. Without location controls87(M1--M3) no category differs significantly from rental units; once88distance and density are held fixed (M4) and especially within cities89(M5), chalets, cottages and cabins are priced 10\% above rental units of90the same capacity (Kennedy-corrected premium 10.3\%, $p < 0.01$) and91houses 8\% above ($p < 0.05$), while condominiums and lofts, hotel rooms92and unconventional listings are not significantly different. The93reversal between M3 and M4 arises because chalets are located where94listings are sparse and far from city centres, two attributes that are95themselves priced; the within-city chalet premium is the better estimate96of the type's implicit price.9798Among the presentation variables the number of photos is the most99robust: a 10\% increase in one plus the photo count is associated with a1000.9\% higher price in every specification. The amenity count has a101smaller and less stable coefficient (elasticity 0.09 with fixed effects,102$p < 0.05$), and allowing pets is associated with a price about 4\%103lower, significant at the 10\% level. The first version of this paper104reported large negative ``amenity index'' effects; those indices were105noise and the amenity count, which is the only amenity information in106the data, is positively priced.107108% ------------------------------------------------------------------109\subsection{Reputation Signals}110\label{subsec:reviews}111% ------------------------------------------------------------------112113The reputation block adds little to the fit, and its coefficients are114small relative to the capacity effects. The displayed rating is the one115reputation variable with a sizeable and precisely estimated coefficient:116among rated listings, each additional point of rating is associated with117a 34\% higher price (0.337, standard error 0.068 in M5), so that the118typical spread between a listing rated 4.6 and one rated 5.0 corresponds119to about 14\%. Because ratings are compressed near the top of the scale,120with a tenth percentile of 4.57 \citep[the pattern documented121by][]{Zervas2021}, this coefficient is identified from small differences122and should be read as the price distance between the lower tail and the123mass of near-perfect ratings, not as a linear effect across the whole124scale. Unrated listings, mostly new ones, are priced 4\% below rated125listings at 4.8, a difference that is not significant.126127Superhost status and the Guest Favourite badge carry no premium once128capacity, type and location are controlled: the Superhost coefficient is1290.023 (standard error 0.025) and the Guest Favourite coefficient130$-0.035$ (0.027). Both badges are strongly associated with price in the131raw data, but in the direction of the stay-price regime (Appendix132Table~\ref{tab:app_stay_logit}), not of the nightly price. This null is133consistent with the mixed evidence in the literature, where134\citet{Teubner2017} and \citet{Gibbs2018} find Superhost premia of a few135percent in some cities and none in others, and with the view that the136badge rewards responsiveness and reliability rather than quality that137guests pay for. The number of reviews is likewise unrelated to price on138average (0.015, 0.010); the quantile results below show that this139average conceals opposite signs at the two ends of the distribution.140Figure~\ref{fig:partials} displays the partial relationship between the141rating and log price and between distance to the nearest centre and log142price as binned partial residuals from M4.143144\begin{figure}[!htbp]145\centering146\includegraphics[width=0.95\textwidth]{rev_fig_partials.pdf}147\caption{Partial Relationships: Distance to Centre and Rating}148\label{fig:partials}149\begin{minipage}{0.9\textwidth}150\footnotesize151\textit{Notes:} Binned means (20 quantile bins) of the partial residual152$\hat u_i + \hat\beta_k x_{ik}$ from specification M4, with 95\%153confidence intervals. Left: distance to the nearest of 22 reference154centres (symmetric-log axis). Right: displayed rating, rated listings155only.156\end{minipage}157\end{figure}158159% ------------------------------------------------------------------160\subsection{Location}161\label{subsec:location}162% ------------------------------------------------------------------163164The two location variables behave as the urban-economics literature165predicts once the sample is restricted to nightly prices. Local density166is positively priced: a 10\% increase in one plus the number of listings167within 1~km is associated with a 0.6\% higher price (0.064, 0.013 with168fixed effects), the opposite of the ``competitive pressure'' reading in169the first version of this paper and consistent with density proxying170for centrality and amenity within cities. Distance to the nearest171reference centre has a small negative coefficient that is significant172only at the 10\% level with fixed effects ($-0.035$, 0.019); across the173province, distance mixes the within-city gradient with the fact that174resort listings are far from any city centre and expensive, which the175category and density variables partly absorb. The fixed effects176themselves are jointly significant (robust $F$ test,177$p < 0.001$), and the robustness section shows that finer spatial cells178leave the listing-level coefficients essentially unchanged.179180% ------------------------------------------------------------------181\subsection{Quantile Regression}182\label{subsec:quantile}183% ------------------------------------------------------------------184185Table~\ref{tab:quantile} and Figure~\ref{fig:quantile} report the186quantile estimates. The pseudo-$R^2$ rises from 0.278 at $\tau = 0.10$187to 0.405 at $\tau = 0.90$: observed characteristics explain more of the188variation among expensive listings than among cheap ones, the reverse of189what the first version reported on the contaminated field.190191\begin{table}[!htbp]192\centering193\caption{Quantile Regression Estimates, Specification M5}194\label{tab:quantile}195\begin{threeparttable}196\footnotesize197\setlength{\tabcolsep}{4pt}198\input{../results/tables/rev_quantile.tex}199\begin{tablenotes}200\footnotesize201\item \textit{Notes:} Analysis sample ($N = 2{,}480$). Column OLS202repeats M5 (HC3 standard errors); quantile columns report kernel-based203standard errors. All regressions include the city / cell fixed effects204and the studio and ``Other'' category indicators (not shown). Pseudo-$R^2$205is one minus the ratio of the minimised check-function loss to the loss206at the unconditional quantile. ***, ** and * denote significance at the2071, 5 and 10\% levels.208\end{tablenotes}209\end{threeparttable}210\end{table}211212\begin{figure}[!htbp]213\centering214\includegraphics[width=0.95\textwidth]{rev_fig_quantile.pdf}215\caption{Quantile Regression Coefficients Across the Conditional Price Distribution}216\label{fig:quantile}217\begin{minipage}{0.9\textwidth}218\footnotesize219\textit{Notes:} Point estimates and 95\% confidence bands at220$\tau \in \{0.10, 0.25, 0.50, 0.75, 0.90\}$; the dashed red line is the221OLS (M5) coefficient.222\end{minipage}223\end{figure}224225Three patterns emerge. First, the implicit prices of bedrooms and226bathrooms increase along the distribution: the bedroom elasticity is227indistinguishable from zero at $\tau = 0.10$ and 0.19 at228$\tau = 0.90$, and the bathroom elasticity doubles from 0.24 to 0.55,229while the capacity elasticity moves the other way, from 0.37 to 0.31.230Cheap listings are priced per guest; expensive listings are priced per231room. The paired bootstrap puts the $\tau = 0.90$ minus $\tau = 0.10$232difference in the bedroom elasticity at 0.17 (standard error 0.10,233$p = 0.08$) and finds no significant difference for capacity234($-0.07$, $p = 0.37$); the bathroom gradient, the steepest of the three,235is visible in the non-overlapping confidence bands of236Figure~\ref{fig:quantile}. The chalet premium shows the same237gradient, from 2\% at the bottom to 21\% at the top of the distribution.238239Second, the reputation variables matter at the bottom. The Superhost240badge is associated with 8\% and 5\% higher prices at $\tau = 0.10$ and2410.25 and with nothing above the median; the number of reviews is242positively associated with price in the lower quantiles (0.042 at243$\tau = 0.10$) and negatively in the upper quantiles ($-0.042$ at244$\tau = 0.90$), the inter-quantile difference of $-0.084$ (standard error 0.023,245$p < 0.001$) being the most significant in the table. At the lower end, reviews and the badge certify a cheap246listing; at the upper end, heavily reviewed listings are high-turnover247properties that are priced below the less reviewed, more exclusive248ones. The rating coefficient, by contrast, is stable across quantiles249(0.26 to 0.38) and its inter-quantile difference is not significant250($0.03$, $p = 0.86$).251252Third, the location gradient strengthens with price: distance to the253centre is unpriced at $\tau = 0.10$ and carries an elasticity of254$-0.07$ to $-0.08$ in the upper half, while local density is priced most255strongly at the bottom (0.10 at $\tau = 0.10$, 0.04 at256$\tau = 0.90$). Budget listings gain from being where listings are257dense, premium listings from being close to the centre.258259% ------------------------------------------------------------------260\subsection{Spatial Dependence}261\label{subsec:spatial}262% ------------------------------------------------------------------263264The residuals of the hedonic model are spatially autocorrelated.265Moran's $I$ on the M4 residuals with the five-nearest-neighbour matrix266is 0.126 ($z = 10.8$) ($p < 0.001$ under 999 permutations) and it remains2670.072 ($z = 6.2$, $p < 0.001$) after the city fixed effects, so dependence exists within268cities as well as between them. Table~\ref{tab:spatial} reports the269three spatial models estimated on the M4 regressor set. The spatial lag270coefficient is $\hat\rho = 0.200$ (standard error 0.022) in the SAR and2710.261 (0.028) in the SDM; the spatial error coefficient is272$\hat\lambda = 0.284$ (0.028). All three models improve substantially on273OLS in log-likelihood; the SDM has the highest likelihood and the274likelihood-ratio test rejects the SAR restriction275$\boldsymbol{\theta} = \mathbf{0}$ ($\chi^2(20) = 52.8$, $p < 0.001$), while the SEM has the276lowest AIC because it adds a single parameter.277278\begin{table}[!htbp]279\centering280\caption{Spatial Lag, Spatial Error and Spatial Durbin Models (M4 Regressors)}281\label{tab:spatial}282\begin{threeparttable}283\footnotesize284\setlength{\tabcolsep}{4pt}285\input{../results/tables/rev_spatial.tex}286\begin{tablenotes}287\footnotesize288\item \textit{Notes:} Maximum-likelihood estimates on the analysis sample289with a row-standardised 5-nearest-neighbour weights matrix built on290coordinates projected to kilometres. The last column reports the291coefficients $\boldsymbol{\theta}$ on the spatially lagged regressors of292the SDM. Studio and ``Other'' category indicators included but not293shown. Standard errors in parentheses (HC3 for OLS, asymptotic for the294spatial models). ***, ** and * denote significance at the 1, 5 and 10\%295levels.296\end{tablenotes}297\end{threeparttable}298\end{table}299300The hedonic coefficients are robust to the spatial specification: the301capacity, shared-bathroom, photo and rating coefficients move by less302than one standard error between OLS and any of the spatial models. The303largest changes are in the variables that carry spatial information,304the chalet premium (0.154 in OLS, 0.097 in the SAR) and local density305(0.067 to 0.044 in the SDM), part of whose OLS coefficient is absorbed306by neighbours' prices. The SDM's spatially lagged regressors are mostly307insignificant; two exceptions are that a listing surrounded by308high-capacity listings is priced lower ($\theta = -0.153$) and one309surrounded by highly rated listings is priced lower ($-0.294$),310conditional on its own characteristics, which is the pattern expected311when neighbours are substitutes.312313Table~\ref{tab:impacts} translates the SAR and SDM estimates into314average direct, indirect and total impacts. In the SAR the spatial315multiplier is $1/(1 - \hat\rho) = 1.25$: the total impact of each316characteristic is 25\% larger than its coefficient, and a fifth of the317total impact accrues to neighbouring listings through the price318feedback. The direct impacts are within a few percent of the319coefficients, so the OLS coefficients are good approximations of the320own-listing associations. In the SDM the indirect impacts are imprecisely321estimated, with two exceptions: the bathroom count has a positive322spillover significant at the 5\% level (total impact 0.70 against a323direct impact of 0.45), and the rating has a negative spillover,324significant at the 10\% level, that offsets its direct impact, so the325total association between the average rating of a neighbourhood and its326prices is close to zero. These spillovers describe327the equilibrium price surface, not a causal effect of neighbours' ratings328on a listing's price.329330\begin{table}[!htbp]331\centering332\caption{Average Direct, Indirect and Total Impacts, SAR and SDM}333\label{tab:impacts}334\begin{threeparttable}335\footnotesize336\setlength{\tabcolsep}{4pt}337\input{../results/tables/rev_impacts.tex}338\begin{tablenotes}339\footnotesize340\item \textit{Notes:} Impacts computed as in \citet{LeSagePace2009}341from the dense inverse $(\mathbf{I} - \hat\rho\mathbf{W})^{-1}$.342Standard errors (in parentheses) from 300 draws of the parameters from343their estimated asymptotic distribution. ***, ** and * denote344significance at the 1, 5 and 10\% levels.345\end{tablenotes}346\end{threeparttable}347\end{table}348349% ------------------------------------------------------------------350\subsection{Machine Learning Benchmarks}351\label{subsec:ml_comparison}352% ------------------------------------------------------------------353354Table~\ref{tab:ml} compares the out-of-sample accuracy of the M5355hedonic model and five learners under random and spatially blocked356cross-validation. Under random folds the M5 hedonic357model reaches an out-of-sample $R^2$ of 0.473 and a median absolute358percentage error of 28.5\%; the elastic net, which regularises the same359linear specification, does no better. The tree ensembles are clearly360more accurate: XGBoost attains $R^2 = 0.559$, RMSE 0.450 and MdAPE36126.4\%, with gradient boosting, LightGBM and the random forest within one362point of it. The gain of 0.086 in $R^2$ over the hedonic model is the363value of the interactions and non-linearities that the semi-log364specification rules out, for instance between type and capacity or365between density and city.366367Spatially blocked cross-validation is the sterner test, and it lowers368every model by about seven points of $R^2$: the hedonic model falls to3690.407, the tree ensembles to 0.48. The ranking is unchanged and the gap370between the best learner and the hedonic model narrows slightly, from3710.086 to 0.076. Two conclusions follow. The random-fold figures372overstate how well any of these models would price listings in an area373absent from the training data, because listings in the same area share374unobserved determinants of price; and the predictive advantage of the375learners is not an artefact of their fitting local spatial structure376more closely, since it survives when that structure cannot be exploited.377A median error of 26 to 30\% of the nightly price is, in either design,378the practical accuracy limit of an automated valuation of Quebec379short-term rentals from listing-card information alone.380381\begin{table}[!htbp]382\centering383\caption{Out-of-Sample Accuracy Under Random and Spatially Blocked Cross-Validation}384\label{tab:ml}385\begin{threeparttable}386\footnotesize387\setlength{\tabcolsep}{4pt}388\input{../results/tables/rev_ml_cv.tex}389\begin{tablenotes}390\footnotesize391\item \textit{Notes:} Five-fold cross-validation on the analysis sample392($N = 2{,}480$); all models use the M5 regressors including the fixed-effect393indicators. Random folds assign listings at random; blocked folds assign39420 $k$-means spatial clusters of listings to folds as whole blocks, so395that each listing is predicted by a model that has seen no listing from396its cluster. $R^2$, RMSE and MAE are on the log scale; MdAPE is the median397absolute percentage error on the price scale. Hyper-parameters: elastic398net with cross-validated penalty; random forest 500 trees, minimum leaf 3;399gradient boosting 500 trees, depth 3, learning rate 0.05; XGBoost 600400trees, depth 5, learning rate 0.03; LightGBM 600 trees, 15 leaves,401learning rate 0.03.402\end{tablenotes}403\end{threeparttable}404\end{table}405406% ------------------------------------------------------------------407\subsection{SHAP Decomposition of the Boosted Model}408\label{subsec:shap}409% ------------------------------------------------------------------410411Table~\ref{tab:shap} ranks the features of the XGBoost model by mean412absolute SHAP value and Figure~\ref{fig:shap} shows the distribution of413the contributions. Guest capacity is by far the most important feature414(21\% of the total attribution), followed by bedrooms (10\%), local415density, the aggregate of the city and cell indicators, bathrooms and the416review count (9\% each), the rating (7\%), photos (5\%), distance to the417centre (5\%), amenities (4\%) and the shared-bathroom indicator (3\%).418The ranking mirrors the hedonic model in two respects: the capacity block419dominates, and the badges, pets and the rated indicator are at the420bottom, with the Superhost badge absent from the top sixteen. It differs421in one: the review count, whose average hedonic coefficient is zero,422receives a large attribution. The beeswarm plot explains why. The SHAP423values of the review count are negative for listings with many reviews424and positive for listings with few, exactly the sign reversal between425the lower and upper quantiles in Table~\ref{tab:quantile}; a linear426coefficient averages the two to nothing, whereas the learner uses the427variable in both directions. Together with the quantile results, this is428the clearest instance in these data of an implicit price that is not a429constant. SHAP values quantify how the fitted model uses each feature;430they are attribution weights for associations, computed from an431estimator that has no identification strategy, and they carry no causal432content.433434\begin{table}[!htbp]435\centering436\caption{Mean Absolute SHAP Values, XGBoost}437\label{tab:shap}438\begin{threeparttable}439\input{../results/tables/rev_shap_importance.tex}440\begin{tablenotes}441\footnotesize442\item \textit{Notes:} Exact SHAP values from the TreeExplainer for the443XGBoost model fitted to the analysis sample. $\overline{|\phi_k|}$ is the444mean absolute contribution of feature $k$ to the predicted log price;445the share is relative to the sum over all features, including the446individual fixed-effect indicators, which are reported as one aggregate447row.448\end{tablenotes}449\end{threeparttable}450\end{table}451452\begin{figure}[!htbp]453\centering454\includegraphics[width=0.9\textwidth]{rev_fig_shap.pdf}455\caption{SHAP Summary Plot, XGBoost (Twelve Most Important Listing-Level Features)}456\label{fig:shap}457\begin{minipage}{0.9\textwidth}458\footnotesize459\textit{Notes:} Each point is one listing; the horizontal position is460the SHAP value (contribution to predicted log price), the colour the461feature value (blue low, red high). Fixed-effect indicators are462excluded from the plot.463\end{minipage}464\end{figure}465