\section{Theoretical foundations of hedonic housing valuation}
\label{sec:theory}

The hedonic method rests on the observation that a house is a bundle of characteristics that are not traded separately, so that
the price of the bundle reveals, at the margin, the value buyers place on each characteristic. \citet{rosen1974hedonic} gave this
idea its modern form: in a competitive market with a continuum of differentiated products, the equilibrium price schedule
$P(\mathbf z)$ over the characteristics vector $\mathbf z$ is the envelope of buyers' bid functions and sellers' offer functions,
and its gradient $\partial P/\partial z_k$ is the implicit (marginal) price of characteristic $k$. Two consequences shape every
study in the literature synthesised here. First, the implicit price is a market-level object, determined by the joint distribution
of tastes, incomes and the stock of characteristics in a given market at a given time, so that there is no theoretical reason for
implicit prices to be equal across cities, countries or periods. Second, the functional form of $P(\cdot)$ is not pinned down by
theory: non-linearity is the rule, and the log-linear, log-log and Box--Cox specifications that dominate the empirical literature
are approximations whose coefficients have different interpretations.

The first consequence is the reason a synthesis cannot aim at a single ``true'' implicit price. What it can aim at is a
characterisation of the distribution of implicit prices across markets, a decomposition of that distribution into economic
heterogeneity and specification differences, and tests of whether systematic patterns, by geography, period or property type,
survive the noise. The second consequence is the reason the synthesis must be organised around effect-size families rather than
raw coefficients. In a log-linear model $\ln P=\alpha+\beta_k z_k+\dots$, a binary characteristic shifts the expected log price by
$\beta_k$ and the expected price, to first order, by $100[\exp(\beta_k)-1]$ percent, with the retransformation corrections discussed
in Section~\ref{sec:extraction}; a continuous characteristic in levels has a semi-elasticity, $100\beta_k$ percent per unit, whose
magnitude depends on the unit; in a log-log model $\beta_k$ is an elasticity; and in a level-level model $\beta_k$ is a monetary
marginal price in the currency and price level of the sample. These objects are not interchangeable, and pooling them without
conversion produces averages that have no interpretation.

Three refinements of the Rosen framework matter for the moderators used later. Spatial dependence, both in the error term and in
the dependent variable, arises because unobserved neighbourhood attributes are shared by nearby houses and because prices of
comparable sales enter appraisals and negotiations; spatial-lag and spatial-error models \citep{anselin1988spatial} and
geographically weighted regression \citep{brunsdon1996geographically} are the standard responses, and whether they change the
implicit prices of structural characteristics is an empirical question addressed in Section~\ref{sec:methodological}.
Identification of the implicit price of an environmental or locational attribute is threatened by omitted variables correlated
with it, and the quasi-experimental turn in the literature \citep{kuminoff2010which,parmeter2013hedonic} replaces cross-sectional
hedonic regressions with difference-in-differences, boundary and panel designs; these designs produce coefficients for the
treated attribute only, which the synthesis records but which cannot inform the structural characteristics. Finally, the second
stage of the Rosen programme, recovering demand parameters from implicit prices, is outside the scope of this paper: the synthesis
concerns first-stage implicit prices, which is also what the Sirmans programme summarised.