spb/wp2_uqo

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2%3%======================================================================4\section{Baseline Results} \label{sec:results}5%======================================================================67This section reports the five nested specifications of Equations~(\ref{eq:modelA})--(\ref{eq:modelE}), which constitute the baseline of the paper and reproduce the estimates of its first version. The extended specifications that control for language and location are presented in Section~\ref{sec:extended}; readers should keep in mind, when reading the individual coefficients below, that several of them change materially once those controls are added.89\subsection{Model Comparison}1011Table~\ref{tab:model_comparison} reports goodness-of-fit statistics and Figure~\ref{fig:model_comparison} visualises them.1213\input{tables/tab_model_comparison}1415\begin{figure}[!htbp]16\centering17\includegraphics[width=0.8\textwidth]{fig1_model_comparison.pdf}18\caption{Goodness-of-fit comparison across hedonic model specifications.}19\label{fig:model_comparison}20\begin{minipage}{0.9\textwidth}21\footnotesize22Notes: $R^2$ and adjusted $R^2$ across the five nested specifications of Equations~(\ref{eq:modelA})--(\ref{eq:modelE}). The dashed line marks the structural baseline (Model A).23\end{minipage}24\end{figure}2526The structural model (A) explains 45.2\% of the variation in log asking prices. This is modest by the standards of hedonic studies that include location variables, and Section~\ref{sec:extended} shows that area fixed effects alone raise the $R^2$ to about 0.65; Model~A should be read as a deliberately sparse benchmark with no locational information. Adding description length (Model~B) raises the $R^2$ by 1.2 percentage points, so the quantity of text is itself price-informative. Adding the 20 similarities without the length control (Model~C) raises it by 4.4 points, and the full model (D) by 6.0 points, to 0.512; the semantic block adds 4.8 points over Model~B. The parsimonious model (E), which retains the 16 similarities individually significant at the 5\% level in Model~D, fits almost identically ($R^2 = 0.512$) and has the lowest BIC (24,425 against 24,448), so that the four dropped dimensions---Renovated, Income/Investment, Heritage/Character and Pool \& Landscaping---contribute nothing independently once the others are included. Model~E's coefficients are reported in Appendix~\ref{app:parsimonious}.2728Figure~\ref{fig:r2_decomp} decomposes the $R^2$ of Model~D in the order structural variables, length, semantics: 88.3\%, 2.4\% and 9.3\% of the explained variance. The decomposition is sequential and therefore order-dependent; the semantic share is the increment of the block entered last, which is the most conservative attribution.2930\begin{figure}[!htbp]31\centering32\includegraphics[width=0.55\textwidth]{fig8_r2_decomposition.pdf}33\caption{Sequential decomposition of the explained variance ($R^2$) of Model~D.}34\label{fig:r2_decomp}35\begin{minipage}{0.9\textwidth}36\footnotesize37Notes: Structural variables entered first account for 88.3\% of the $R^2$ of Model~D, description length entered second for 2.4\%, and the 20 semantic similarities entered last for 9.3\%. The shares depend on the order of entry.38\end{minipage}39\end{figure}4041\subsection{Joint Significance}4243The $F$-statistic of Equation~(\ref{eq:ftest}) for the 21 text variables (length and 20 similarities), comparing Model~D with Model~A, is $F(21,\,17{,}059) = 99.53$, $p < 0.001$. The hypothesis that listing text contains no price-relevant information beyond the structural characteristics is rejected decisively. Equation~(\ref{eq:ftest}) assumes homoskedastic errors, which the Breusch--Pagan test rejects (Section~\ref{sec:robustness}); the heteroskedasticity-robust Wald test of the 20 semantic coefficients alone gives $F = 54.3$, and the version clustered by FSA gives $F = 20.1$ (Table~\ref{tab:spec_summary}), both with $p < 0.001$.4445\subsection{Structural Variables}4647Figure~\ref{fig:structural_coef} presents the structural coefficients of Model~D with 95\% confidence intervals.4849\begin{figure}[!htbp]50\centering51\includegraphics[width=0.8\textwidth]{fig10_structural_coefficients.pdf}52\caption{Structural coefficients of the full model (D), with 95\% confidence intervals.}53\label{fig:structural_coef}54\begin{minipage}{0.9\textwidth}55\footnotesize56Notes: Each point is the percentage change in price associated with a one-standard-deviation increase in the variable (Equation~\ref{eq:impact}). Error bars are 95\% confidence intervals based on HC3 standard errors.57\end{minipage}58\end{figure}5960Bathrooms have the largest structural association (+34.5\% per standard deviation), followed by half-bathrooms (+17.1\%), description length (+10.7\%), parking (+9.6\%) and storeys (+7.7\%); bedrooms add a modest +3.2\%. The raw lot-size field is not significant ($t = 0.00$), which is the expected consequence of mixing square metres, square feet and acres in one variable (Section~\ref{sec:data}); the unit-corrected lot area used in Section~\ref{sec:extended} is strongly significant and, within FSAs, has the largest structural coefficient of all. The absence of a floor-area variable in the baseline is a notable omission relative to standard hedonic practice \citep{sirmans2005composition}, and the floor-area subsample of Section~\ref{sec:extended} addresses it.6162\subsection{Semantic Dimensions}6364Table~\ref{tab:full_results} reports the complete coefficient estimates of Model~D and Figure~\ref{fig:coefficient_plot} displays the semantic coefficients.6566\input{tables/tab_full_results}6768\begin{figure}[!htbp]69\centering70\includegraphics[width=0.8\textwidth]{fig2_coefficient_plot.pdf}71\caption{Price impact of the semantic similarity dimensions in Model~D. Green bars indicate positive and red bars negative associations; grey bars are not significant at the 5\% level. Error bars are 95\% confidence intervals.}72\label{fig:coefficient_plot}73\end{figure}7475Three dimensions carry large positive associations. \textbf{Modern/Contemporary} (+16.4\% per standard deviation, $p < 0.001$) is the largest: listings closer to the modern-design reference---contemporary architecture, clean lines, smart-home and energy-efficient construction---are the most expensive conditional on structure. \textbf{Luxury} (+14.2\%) captures premium finishes, gourmet kitchens and spa-like bathrooms, quality variation that bathroom counts do not record. \textbf{Land \& Nature} (+13.0\%) captures wooded lots, landscaping, privacy and natural surroundings. \textbf{Waterfront} (+4.7\%) and \textbf{Panoramic View} (+3.0\%) are smaller and positive.7677Several dimensions carry negative associations. \textbf{Family-Friendly} ($-11.6\%$) and \textbf{New Construction} ($-10.5\%$) are the largest; both describe segments---suburban family markets, peripheral new developments---whose price levels are lower, so that the language marks location rather than a disamenity. \textbf{Quiet \& Peaceful} ($-9.5\%$) likewise characterises rural and peripheral settings. \textbf{Motivated Seller} ($-8.5\%$) and \textbf{Needs Renovation} ($-7.9\%$) have the signs expected from the disclosure of urgency and of deferred maintenance. \textbf{Bright \& Spacious} ($-8.9\%$), \textbf{Energy Efficient} ($-7.8\%$) and \textbf{Premium Location} ($-6.1\%$) are negative against expectation, and \textbf{Entry-Level} (+10.4\%) is positive against expectation. The first version of this paper did not discuss these four anomalies. They are the signature of the strong collinearity documented below: when 20 projections of the same embedding are correlated at 0.66--0.97, a coefficient measures the association with the part of a dimension that is orthogonal to the other nineteen, and that residual part need not carry the dimension's intended meaning. Entry-Level, for instance, is correlated at 0.97 with Motivated Seller, and its positive coefficient is the mirror image of Motivated Seller's negative one. Section~\ref{sec:extended} shows that Entry-Level, New Construction, Energy Efficient and Bright \& Spacious change sign when the encoder is changed, whereas Luxury, Land \& Nature, Family-Friendly, Quiet \& Peaceful and Needs Renovation do not.7879\subsection{Semantic Profiles Across Price Quintiles}8081Figure~\ref{fig:quintile_heatmap} displays the mean similarity on each dimension by price quintile.8283\begin{figure}[!htbp]84\centering85\includegraphics[width=0.85\textwidth]{fig4_quintile_heatmap.pdf}86\caption{Mean cosine similarity by price quintile across the 20 reference dimensions.}87\label{fig:quintile_heatmap}88\begin{minipage}{0.9\textwidth}89\footnotesize90Notes: Warmer colours indicate higher similarity. Each cell is the mean cosine similarity between the listings of a price quintile and the reference description.91\end{minipage}92\end{figure}9394Lower-priced listings score higher on nearly every dimension. The first version attributed this gradient to longer descriptions; the language analysis of Section~\ref{sec:data_language} shows that it is mainly the rising share of English listings in the upper quintiles. The relative differences across dimensions remain informative: from the bottom to the top quintile the steepest declines are for Motivated Seller, Entry-Level and Premium Location, and the smallest for Land \& Nature, Modern/Contemporary and Panoramic View---the dimensions with positive coefficients in the regression.9596\subsection{Inter-Reference Correlations}9798Figure~\ref{fig:corr_matrix} presents the correlation matrix of the 20 similarities.99100\begin{figure}[!htbp]101\centering102\includegraphics[width=0.75\textwidth]{fig5_correlation_matrix.pdf}103\caption{Pearson correlation matrix of the semantic similarity dimensions (lower triangle).}104\label{fig:corr_matrix}105\begin{minipage}{0.9\textwidth}106\footnotesize107Notes: Correlations across all $n = 17{,}087$ listings. The uniformly strong positive correlations reflect a common component of the similarity scores---mainly the language of the listing, secondarily its length and register.108\end{minipage}109\end{figure}110111The correlations are uniformly positive and strong, from 0.66 to 0.97 with a mean of 0.84, and the first principal component of the 20 similarities accounts for 86\% of their variance. The common component is, to a large extent, the language of the listing (Table~\ref{tab:language}): the first principal component is correlated at $-0.82$ with the English indicator and at only 0.20 with description length. The strongest pairwise correlations are between Entry-Level and Motivated Seller ($r = 0.97$), both characteristic of value-oriented French listings, and between Luxury and Renovated ($r = 0.95$), which share upgrade vocabulary. Individual semantic coefficients are therefore partial associations within a correlated block, and the block-level evidence is the most robust object of inference; Section~\ref{sec:robustness} quantifies the consequences for standard errors and Section~\ref{sec:extended} for the stability of the coefficients.112113\subsection{Illustrative Bivariate Relationships and Residual Diagnostics}114115Figure~\ref{fig:scatter} shows bivariate scatter plots for four dimensions; these relationships do not control for structure, length or language and are shown for illustration only. Figure~\ref{fig:diagnostics} presents residual diagnostics for Model~D. The residual-versus-fitted plot reveals heteroskedasticity, with larger dispersion at both ends of the fitted distribution, which motivates the HC3 standard errors; the normal quantile plot shows heavier-than-normal tails, especially in the upper tail, consistent with the right skew of housing prices even after the log transformation. Least squares remains consistent under these conditions and the robust standard errors remain valid.116117\begin{figure}[!htbp]118\centering119\includegraphics[width=0.8\textwidth]{fig6_scatter_plots.pdf}120\caption{Bivariate relationships between selected semantic similarities and log price.}121\label{fig:scatter}122\begin{minipage}{0.9\textwidth}123\footnotesize124Notes: Lines are ordinary least squares fits. The relationships are bivariate and do not control for structural variables, description length or language.125\end{minipage}126\end{figure}127128\begin{figure}[!htbp]129\centering130\includegraphics[width=0.85\textwidth]{fig11_residual_diagnostics.pdf}131\caption{Residual diagnostics for Model~D: (a) residuals against fitted values; (b) normal quantile plot.}132\label{fig:diagnostics}133\begin{minipage}{0.9\textwidth}134\footnotesize135Notes: Panel (a) plots least-squares residuals against fitted values and shows heteroskedasticity at the tails. Panel (b) compares the residual distribution with the normal and shows heavier-than-normal tails.136\end{minipage}137\end{figure}138