% Author: Simon-Pierre Boucher — contact@spboucher.ai
% ============================================================================
\section{Beyond the scoreboard: what the models imply}
\label{sec:ext}

Accuracy is not the only output economists take from a hedonic model. This
section asks whether the specification choices that move the scoreboard
also move the two objects hedonic models are most often built to deliver:
constant-quality price indices and implicit attribute prices.

\subsection{Price indices are form-robust}

Figure~\ref{fig:index} plots the constant-quality monthly index implied by
the month fixed effects of four linear forms --- semi-log, log-log,
quadratic and spline --- together with a gradient-boosting index obtained
by repricing a fixed 20{,}000-sale reference portfolio at each month
(an imputation \emph{\`a la} \citealp{hill2013hedonic}). All five track
the same cycle --- the 2021--22 boom, the 2022--23 correction, the
2024--26 recovery --- and the four linear forms agree to within a few
index points throughout. The choice that dominates the accuracy
scoreboard barely perturbs the index: reassuring news for statistical
agencies, and consistent with the RPPI handbook's pragmatism about form
\citep{oecd2013handbook, silver2018house}.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig_index.png}
\caption{Constant-quality monthly price indices implied by competing
specifications (January 2021 = 100). Linear forms: exponentiated month
fixed effects. Gradient boosting: mean repriced value of a fixed
20{,}000-sale reference portfolio.}
\label{fig:index}
\end{figure}

\subsection{Implicit prices: the machine agrees with the quadratic}

Figure~\ref{fig:profiles} compares the ln-price profiles in building age
and floor area implied by the quadratic OLS coefficients with the partial-%
dependence profiles of the boosting model \citep{friedman2001greedy}.
The two agree over the bulk of the data: steep initial depreciation that
flattens with age, and strongly concave returns to floor space. The
visible divergences are where the quadratic \emph{cannot} bend --- the
machine sees a milder gradient for very old stock (a vintage effect) and
a small new-construction premium spike --- but through the ages and areas
where 90\% of sales live, the curves sit within a tenth of a log point. The boosting model's advantage evidently comes from
higher-order interactions and spatial flexibility, \emph{not} from a
different view of the main attribute gradients --- so the interpretable
quadratic form remains a faithful summary of the price surface even where
it loses the prediction race \citep{mullainathan2017machine,
athey2019machine}.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig_profiles.png}
\caption{Implicit ln-price profiles: quadratic OLS coefficients versus
gradient-boosting partial dependence, each normalized to its leftmost
point. Panel A: building age. Panel B: floor area.}
\label{fig:profiles}
\end{figure}

\subsection{Learning curves: data beat flexibility only at scale}

Figure~\ref{fig:learning} and Table~\ref{tab:learning} trace accuracy
against training size from 10{,}000 to 400{,}000 sales. The OLS quadratic
is essentially flat from the very first step (18.4\% at $n=10{,}000$,
18.1\% at $n=400{,}000$) --- its bias floor binds almost immediately ---
while the boosting model improves monotonically through the full sample
(17.6\% $\to$ 14.5\%): the machine's advantage grows from under one
error point at $n = 10{,}000$ to 3.6 points at
$n = 400{,}000$. Flexible methods
are not a free lunch in small markets: below a few tens of thousands of
training sales, the machine's edge is modest, one reason rural and thin-%
market AVMs underperform \citep{bogin2020house}.

\begin{figure}[t]
\centering
\includegraphics[width=0.72\textwidth]{fig_learning.png}
\caption{Learning curves: median absolute error against training-set
size (random holdout fixed), OLS quadratic versus gradient boosting.}
\label{fig:learning}
\end{figure}

\begin{table}[t]
\centering
\begin{threeparttable}
\caption{Learning curves}
\label{tab:learning}
\small
\input{../results/tables/learning}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes:} MdAPE (\%) on the fixed random holdout, training on
seeded subsamples of the indicated size.
\end{tablenotes}
\end{threeparttable}
\end{table}

\subsection{Where the machine wins: segments}

Table~\ref{tab:segments} and Figure~\ref{fig:segments} decompose the
random-holdout MdAPE by property class and municipality size, and the
pattern identifies the machine's edge with striking precision: it is
spatial resolution. The gap is enormous for condominiums (9.1\% versus
20.4\%) and in the largest cities (11.9\% versus 17.8\%) --- segments
where structure is homogeneous but \emph{micro}-location varies hugely
within a municipality, exactly the variation that municipality dummies
cannot see and raw coordinates can. Once location is resolved, a condo
is the most predictable object in the market. At the other extreme,
cottages defeat both models (32.1\% versus 33.9\%): idiosyncratic,
waterfront-driven stock is hard for everyone, and the machine's spatial
surface cannot rescue attributes it does not observe. The complement of
the assessment-inequity anatomy in our companion paper (UQO WP10) is
instructive: the segments where assessors are most regressive
(heterogeneous, land-heavy stock) are also those where \emph{no}
standard technology, linear or boosted, prices accurately from roll
attributes alone.

\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{fig_segments.png}
\caption{Median absolute error by market segment (random holdout):
OLS quadratic versus gradient boosting.}
\label{fig:segments}
\end{figure}

\begin{table}[t]
\centering
\begin{threeparttable}
\caption{Accuracy by market segment}
\label{tab:segments}
\small
\input{../results/tables/segments}
\begin{tablenotes}[flushleft]\footnotesize
\item \textit{Notes:} MdAPE (\%) on the random holdout, by property class
and by the number of 2021--2026 sales in the municipality.
\end{tablenotes}
\end{threeparttable}
\end{table}
