spb/wp2_uqo

1% Author: Simon-Pierre Boucher — contact@spboucher.ai2%3%======================================================================4\section{Robustness of the Baseline} \label{sec:robustness}5%======================================================================67This section reports the robustness checks of the first version of the paper, all re-executed by the replication pipeline: heteroskedasticity diagnostics, variance inflation factors, quantile regressions, trimming of extreme prices, bootstrap inference, regularised selection and a comparison with principal components of the embeddings. They address estimation and sampling issues within the baseline specification; the threats to interpretation raised by language, location and measurement are the subject of Section~\ref{sec:extended}.89\subsection{Heteroskedasticity}1011The Breusch--Pagan test \citep{breusch1979simple} applied to Model~D rejects homoskedasticity (LM $= 768.0$, $p < 0.001$), which justifies the HC3 standard errors used throughout. The pattern---larger residual variance at both tails of the fitted distribution (Figure~\ref{fig:diagnostics}, panel a)---is typical of housing price equations \citep{goodman1995age}.1213\subsection{Variance Inflation Factors}1415The correlations among the similarities translate into substantial multicollinearity: the variance inflation factors of the 20 similarity variables in Model~D average 17.2, 16 of them exceed 10, and the maxima are Renovated (41.6) and Entry-Level (40.7). This is a consequence of the design: all 20 features are projections of the same embedding and share its language and register. As \citet{obrien2007caution} emphasises, a high VIF signals imprecision rather than bias and is not by itself grounds for respecification. Three facts bound the damage. Despite the inflated standard errors, 16 of the 20 coefficients are individually significant at 5\%; the block-level evidence (joint tests, $R^2$ gain) is unaffected by collinearity within the block; and the coefficients are stable under trimming and bootstrap resampling. What collinearity does undermine is the attribution of effects to particular dimensions---a point that Section~\ref{sec:extended} makes concrete by showing that several coefficients depend on the reference phrasing and on the encoder.1617\subsection{Quantile Regressions}1819Table~\ref{tab:quantile} reports quantile regressions of Model~D \citep{koenker1978regression, koenker2005quantile} at the 25th, 50th and 75th percentiles for the six dimensions with the largest baseline coefficients.2021\input{tables/tab_quantile}2223Three patterns emerge. The Luxury coefficient increases monotonically with the conditional price level, from a 9.7\% premium per standard deviation at $\tau = 0.25$ to 17.3\% at $\tau = 0.75$, consistent with the finding of \citet{zietz2007determinants} that many attributes are valued more in upper segments. The Motivated Seller and Needs Renovation discounts attenuate at higher quantiles---the Motivated Seller coefficient falls from $-8.7\%$ at $\tau = 0.25$ to an insignificant $-3.8\%$ at $\tau = 0.75$. Land \& Nature is stable across quantiles and Modern/Contemporary is larger at both tails than at the median. The New Construction coefficient, not shown in the table, becomes \textit{more} negative at higher quantiles ($-0.068$, $-0.079$ and $-0.142$ at the three quantiles); the first version of this paper stated the opposite and used it to support a location-proxy reading of that coefficient. We withdraw that statement; the location-proxy reading is instead supported by the fixed-effects and multilingual results of Section~\ref{sec:extended}.2425\subsection{Trimmed Prices}2627Re-estimating Model~D after removing the 341 listings below the 1st or above the 99th percentile of price yields an adjusted $R^2$ of 0.486, lower than 0.511 because trimming removes price variation, with no sign reversal and all 16 significant similarities remaining significant. The largest coefficient changes are for Entry-Level ($-0.024$) and Motivated Seller (+0.023), both within the baseline confidence intervals. The results are not driven by the extreme prices noted in Section~\ref{sec:data}.2829\subsection{Bootstrap Inference}3031A nonparametric pairs bootstrap with 1,000 replications \citep{efron1993introduction}, resampling the 17,087 listings with replacement, gives standard errors within 9\% of the HC3 estimates for every semantic coefficient and within 5\% for 15 of the 20; the percentile 95\% intervals coincide closely with the HC3 intervals. Asymptotic inference is reliable under the observed heteroskedasticity. The bootstrap resamples listings independently and therefore does not address within-area dependence, which is handled by clustering in Section~\ref{sec:extended}.3233\subsection{Regularised Selection}3435As an alternative to significance-based selection, Lasso \citep{tibshirani1996regression} and elastic-net \citep{zou2005regularization} regressions with five-fold cross-validated penalties retain only four similarities (Needs Renovation, Premium Location, Income/Investment and Family-Friendly) and shrink the other sixteen to zero. This is the textbook behaviour of $\ell_1$ penalties under strong collinearity: one representative per correlated block is kept and its near-duplicates are discarded, so the retained variables stand for their blocks rather than for the ``true'' non-zero effects. The exercise confirms that the semantic block carries predictive signal that survives penalisation, and it reinforces the message that variable-by-variable attribution within the block is fragile.3637\subsection{Comparison with Principal Components}3839Table~\ref{tab:text_comparison} compares the reference projections with alternative representations of the same text on the same sample.4041\input{tables/tab_text_comparison}4243Content matters more than quantity: length alone adds 1.2 points of $R^2$, content-based features 4.4--7.8 points. There is also a measurable price of interpretability: 20 principal components of the raw embeddings, which capture 49.5\% of the embedding variance but have no economic names, fit better than the 20 reference similarities ($\Delta R^2$ of +0.070 against +0.044 without the length control). The 20 similarities computed with the multilingual encoder (Section~\ref{sec:extended}) close most of this gap (+0.069). The reference-based approach trades roughly two percentage points of in-sample fit for named features; Section~\ref{sec:extended} shows that the same ordering holds out of sample.4445\subsection{Summary}4647Table~\ref{tab:robustness_summary} summarises the checks of this section together with those of Section~\ref{sec:extended}. Within the baseline specification, the joint explanatory power of the semantic block and the signs of the main coefficients are stable to the estimation method, to trimming and to the quantile considered. The diagnostics also delineate the limits of the evidence: the dimensions are strongly collinear, individual coefficients are measured imprecisely relative to the block, and regularised selection does not single out the same variables as significance testing.4849\input{tables/tab_robustness_summary}50