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\begin{document}

\title{Underquoting and the Dual Role of Price Guides: Evidence from Housing Sales}
\author{}



\date{}
\maketitle

\begin{abstract}
This paper studies underquoting as a price-guidance problem in residential property markets. We argue that advertised guides perform different roles across sales mechanisms: in auctions, they can attract bidders and intensify competition, while in private-treaty sales they more directly signal the seller's asking position and shape bargaining. The model predicts that auction guides should be more downward distorted, less informative about final prices, and more likely to be associated with underquoting. Using Melbourne detached-house listings and transactions, we show that only around one-quarter of auction prices fall within the advertised guide, compared with more than half of private-treaty prices, and that around two-thirds of auction properties sell above the upper guide. We then construct guide-independent valuation benchmarks using global hedonic and geographically weighted regression models estimated on prior transactions and applied out of sample. The GWR benchmark produces locally adaptive valuation ranges that are more balanced around realised auction prices than agent-provided auction guides and provides a scalable benchmark for assessing advertised price guidance. Under a conservative GWR-based measure, around 44\% of auction transactions and 14\% of private-treaty transactions are classified as underquoted. The findings support the dual-role interpretation of price guides and provide a scalable framework for measuring underquoting and improving price-guidance governance.
\end{abstract}


\vspace{1ex}
\noindent \text{Keywords}: Underquoting; Housing market\\[1ex]
\noindent \text{JEL Classification}: R31, D44, L85, C21, C51


\newpage
\section{Introduction}

Housing is the largest component of household wealth and an important channel linking household balance sheets, credit conditions, and the broader macroeconomy \citep{anari2002house,d2021residential}. Fair and transparent price discovery in residential property markets is therefore important for both household welfare and market efficiency. Buyers rely on advertised price guides to assess affordability, allocate search effort, and decide whether to inspect a property or participate in its sale. When these guides systematically diverge from plausible market values, buyers may incur unnecessary search and participation costs, while the competitive conditions surrounding the sale may also be altered.

Underquoting refers broadly to the practice of advertising a property at a price below its plausible market value or expected transaction range. The practice has attracted particular attention in Australian housing markets, especially in Victoria and New South Wales, where regulators have introduced disclosure requirements governing the preparation and revision of price guides and the use of comparable sales.\footnote{See, for example, the Property, Stock and Business Agents Amendment (Underquoting Prohibition) Act 2016 (NSW) and the Estate Agents Amendment Act 2016 (Vic), discussed in \cite{castle2017underquoting}.} These reforms were intended to improve transparency and reduce misleading price guidance. Nevertheless, underquoting remains a persistent source of consumer complaints, enforcement activity, and public concern.

One main issue is that the economic role of a price guide may differ across sales mechanisms. In an auction campaign, the guide is announced before the competitive bidding process begins. It may therefore be used strategically to attract inspections, increase auction attendance, and expand the pool of active bidders. The seller can benefit from additional participation while relying on a reserve price to protect against an undesirably low outcome. In a private-treaty sale, by contrast, the guide is more closely connected to the seller's asking position and the subsequent bargaining process. Buyers formulate offers relative to the advertised range, and a guide that is substantially below the seller's expectations may attract buyers with insufficient budgets, generate unsuitable offers, and increase negotiation or delay costs.

This distinction suggests that price guides perform two related but different functions. They act as participation devices by influencing buyer attention and entry, but they also act as information signals by communicating a plausible transaction range. The relative importance of these functions depends on the sales mechanism. In auctions, the participation role may dominate, making guides more susceptible to strategic downward distortion. In private-treaty sales, the information and bargaining role may impose greater discipline on the guide and produce a closer relationship between advertised and realised prices.

This paper develops a theoretical and empirical framework for examining this dual role. The theoretical model allows a seller to lower the guide below a market-consistent valuation benchmark in order to increase buyer participation. The benefit of doing so is assumed to be greater in auctions, where additional bidders directly intensify competition. The cost of distorting the guide is assumed to be greater in private-treaty sales, where misleading guidance can generate buyer mismatch, low offers, and bargaining frictions. The model consequently predicts that auction guides will be more strongly distorted downwards, underquoting will be more prevalent in auction campaigns, and final auction prices will be less likely to fall within the advertised range.

Testing these predictions requires a benchmark for plausible property value that does not rely exclusively on the final sale price or on a small set of agent-selected comparable transactions. Existing measures of underquoting frequently use listing-to-sale price gaps, listing-to-sale ratios, or comparisons with nearby sales \citep{stevenson2010,osterling2016underpricing,pardasani2020three,gargano2021cooling}. These measures provide useful descriptive information, but they can conflate the original listing decision with subsequent auction competition, bargaining, and changes in market conditions. A large gap between the guide and the final price does not, by itself, establish that the property was deliberately underquoted when listed.

We address this measurement problem by constructing model-implied valuation benchmarks for detached houses in metropolitan Melbourne. Two complementary approaches are used. The first is a fixed-effects hedonic pricing model that relates transaction prices to structural, locational, and local-market characteristics. The second is a geographically weighted regression model that allows the marginal values of housing characteristics to vary across space. The models are estimated separately for auction and private-treaty transactions, reflecting the possibility that properties sold through the two mechanisms follow different pricing relationships.

The valuation models are estimated using 65,768 detached-house transactions from 2023--2024 and applied out of sample to 3,991 matched listings and transactions from 2025. The evaluation sample contains 1,336 auction sales and 2,655 private-treaty sales. Advertised price guides are not used to estimate the valuation models. Instead, the resulting model-implied values are compared with the advertised lower and upper guide bounds and with realised transaction prices.

The descriptive evidence reveals a sharp difference between the two sales mechanisms. Only 26.27\% of auction prices fall within the advertised range, while 65.72\% exceed the upper guide and only 8.01\% fall below the lower guide. In private-treaty sales, 56.05\% of prices fall within the advertised range, 19.62\% exceed the upper guide, and 24.33\% fall below the lower guide. Auction guides are therefore not merely less accurate; their errors are strongly asymmetric and concentrated below realised transaction prices. Private-treaty guides exhibit a considerably more balanced relationship with final prices.

To distinguish likely underquoting from ordinary prediction error or unexpected competition, we adopt a conservative classification rule. A transaction is classified as underquoted only when its realised sale price exceeds both the upper bound of the advertised range and the lower bound of the model-implied valuation interval. Under the preferred GWR benchmark, 44.09\% of auction transactions satisfy this criterion, compared with 14.05\% of private-treaty transactions. The results are consistent with the theoretical prediction that auction guides are more strongly used as participation devices, whereas private-treaty guides convey more information about plausible transaction values.

The spatial valuation model also materially improves typical out-of-sample prediction accuracy relative to the global hedonic benchmark. This improvement is important because Melbourne contains highly differentiated local housing markets in which the value of land, accessibility, dwelling attributes, and neighbourhood conditions can vary substantially across space. The resulting GWR benchmarks provide a locally adaptive basis for assessing whether advertised guidance is low relative to the values implied by observable property and neighbourhood characteristics.

The paper makes three contributions. First, it develops a unified framework for understanding the dual role of housing price guides across auction and private-treaty sales. While existing explanations of underquoting commonly emphasise bidder entry and auction competition, our framework incorporates this participation channel while showing why price guides may perform a stronger information and bargaining function in negotiated sales.

Second, the paper provides new comparative evidence on the informativeness of advertised price guides across sales mechanisms. The substantially lower within-guide rate and strongly asymmetric guide errors observed for auctions are consistent with auction guides being less closely related to expected transaction values than private-treaty guides.

Third, the paper develops a scalable model-based approach to measuring underquoting. By combining advertised guide ranges, realised sale prices, and out-of-sample valuation intervals estimated without using advertised guides, the measure reduces reliance on ex post price gaps or a small number of potentially unrepresentative comparable sales. The use of a spatially adaptive GWR benchmark further allows valuation relationships to vary across local housing submarkets and improves typical out-of-sample prediction accuracy relative to the global hedonic benchmark.

The analysis does not estimate the causal effect of the Victorian disclosure regime, nor does it observe bidder numbers, inspections, reserve prices, bargaining histories, or agents' subjective valuations. The theoretical mechanisms should therefore be interpreted as providing an economic explanation for the empirical patterns rather than as being structurally identified. Similarly, the empirical underquoting measure identifies transactions that satisfy a conservative valuation-based criterion; it does not establish the legal intent required for a regulatory finding in an individual case.

Melbourne provides a useful setting for the analysis. It is a large and spatially heterogeneous housing market with substantial activity under both auction and private-treaty mechanisms. It also operates under an established disclosure regime, making it possible to study the relationship between advertised guidance and transaction prices under current institutional conditions.

The remainder of the paper proceeds as follows. Section~\ref{Literature review} reviews the literature on strategic price guidance, bidder participation, asking-price signals, regulation, and housing valuation. Section~\ref{sec:theory} develops the dual-role theory of price guides. Section~\ref{SecData} describes the institutional setting and the data. Section~\ref{SecRes} presents the valuation models, constructs the underquoting measure, and reports the empirical results. Section~\ref{SecDiscussion} concludes.


\section{Literature review}\label{Literature review}


Price guidance plays an important role in housing markets because buyers must decide where to direct their search before the value of a property is fully revealed through bidding or negotiation. Advertised guides can convey information about plausible transaction values, but they can also be used strategically to influence buyer attention and participation. Underquoting arises when an advertised guide is set below a market-consistent valuation or expected transaction range. Related studies use terms such as underpricing, bait pricing, and strategic listing, but the present paper uses \emph{underquoting} to describe the listing-stage behaviour most closely associated with the Australian regulatory setting.

One strand of the literature examines price guidance as a participation device in housing auctions. A deliberately low guide can reduce perceived affordability barriers, attract inspections, and increase the number of buyers who attend or participate in the auction. Additional bidders may then raise the final price by increasing competitive pressure. Evidence from Ireland indicates that properties sold by auction can command a premium relative to comparable private-treaty sales, particularly in buoyant market conditions, which is consistent with the strategic use of low initial guidance to stimulate competition \citep{stevenson2010}. Evidence from Sweden also shows that lower guides can attract buyer attention, although increased attention does not necessarily translate into more bidders or higher prices in every market \citep{hungria2018}. Similarly, \cite{osterling2016underpricing} finds that requiring advertised guides to more closely reflect market values reduced buyer engagement at several stages of the sale process and increased the probability of an unsuccessful sale. These findings illustrate the central trade-off associated with auction guidance: more accurate guides may improve transparency, but they may also reduce participation and weaken competition.


The effect of low auction guides may be reinforced by behavioural responses during the sale process. Initial guides can provide salient reference points, while observable participation and bidding can generate informational herding, competitive arousal, and auction fever \citep{banerjee1992herding,bikhchandani1992theory,simonsohn2008rational,malmendier2011bidder}. In the housing-market context, \cite{han2014bidding} examine bidding wars and show how strategically low listing prices can stimulate competition among multiple buyers, increasing the likelihood that the transaction price exceeds the advertised price.\footnote{The prospect of a bidding war must be weighed against the uncertainty associated with the auction outcome. \cite{khezr2018why} shows that, when sellers are risk averse, they may prefer to accept a pre-auction offer rather than face the uncertainty associated with the auction. This may provide an additional rationale for using a low guide to attract a sufficiently large pool of potential bidders.} Buyers may also update their valuations in response to the actions of other participants or become increasingly committed after investing time and effort in inspections and bidding. These mechanisms can amplify the difference between the initial guide and the final transaction price. They also imply that the listing-to-sale price gap reflects both the seller's initial pricing strategy and the subsequent competitive process. A large gap is therefore consistent with underquoting, but does not by itself establish that the guide was below a plausible market value when it was announced.


A second strand of the literature studies the role of asking prices in negotiated housing markets. In private-treaty sales, the advertised price is not merely an instrument for attracting buyers; it also conveys information and establishes a starting point for negotiation. \cite{horowitz1992role} models the list price as a signal that affects buyers' beliefs about the property's value and the seller's willingness to accept an offer. \cite{yavas1995strategic} similarly show that the seller's listing-price decision balances competing objectives. A lower asking price can increase buyer arrival and shorten the search process, whereas a higher asking price may strengthen the seller's bargaining position and raise the transaction price conditional on sale. The optimal asking price therefore reflects both search and bargaining considerations. Announced prices can also influence seller commitment in auction settings. \cite{khezr2018auctions} characterise the asking price as a partial commitment device and show that, under certain conditions, it yields a higher expected payoff for the seller than a fully committed reserve price.



Empirical evidence also shows that asking-price strategies evolve during the marketing process. \cite{knight2002listing} documents the relationship between listing-price revisions, time on market, and eventual selling prices, indicating that the asking price responds to information revealed through buyer interest and unsuccessful search. Recent evidence from South Australia also links strategic underpricing to housing-market liquidity and time on market \citep{pilat2025strategic}. Using detailed records of offers and counteroffers, \cite{merlo2004bargaining} show that the timing and sequence of offers reveal information about valuations and bargaining positions. These studies imply that private-treaty guides should not necessarily be interpreted as truthful point forecasts. Nevertheless, because the guide frames subsequent offers and negotiations, setting it substantially below the seller's expected value can attract poorly matched buyers, generate unsuitable offers, and increase delay or bargaining costs.

The auction and private-treaty literature therefore assign different roles to the same observable object. In auctions, the guide can primarily operate as an entry device because the final price is determined through competitive bidding and the seller can use a reserve price to protect against an unacceptably low outcome. In private-treaty sales, the guide has a stronger signalling and bargaining function because offers are formulated and negotiated relative to the advertised range. Existing studies generally analyse these mechanisms separately. There is comparatively little work providing a unified explanation for why the informativeness and strategic use of price guides should differ across sales mechanisms. The theoretical framework developed in this paper addresses this gap by allowing the participation benefit and the information or bargaining cost of low guidance to vary between auctions and private-treaty sales.

A related literature examines the regulation of underquoting. Australian reforms have strengthened disclosure obligations, restricted the language agents may use in advertising, and imposed requirements concerning comparable sales and the revision of price estimates. Empirical studies of Sydney and Melbourne document a narrowing of listing-to-sale price gaps following stricter regulation \citep{pardasani2020three,gargano2021cooling}. These findings suggest that disclosure rules can constrain the most extreme forms of low guidance. At the same time, continued complaints and enforcement activity indicate that regulation has not removed the underlying incentives to attract buyers through advertised prices. The effect of regulation is therefore likely to depend on whether it transforms the guide into a binding constraint or merely increases the expected cost of setting it below a market-consistent value.

The empirical measurement of underquoting remains difficult. Most existing studies rely on listing-to-sale price differences, listing-to-sale ratios, or comparisons with a limited set of nearby transactions \citep{stevenson2010,osterling2016underpricing,hungria2018,pardasani2020three,gargano2021cooling}. These approaches provide informative descriptive measures, but they face two important limitations. First, realised prices are determined after the guide has been set and may reflect bidder competition, bargaining, market news, or changes in local demand. Second, comparable-sales approaches may depend on a small set of properties that differ in quality or location and may be selected strategically by the selling agent. Consequently, these measures can conflate deliberate low guidance with legitimate valuation uncertainty and subsequent price discovery.

This measurement problem connects the underquoting literature to the broader housing-valuation literature. Hedonic pricing models provide a conventional method for estimating the market value of heterogeneous properties by relating transaction prices to structural, locational, and neighbourhood characteristics \citep{rosen1974hedonic,blomquist1981hedonic,sheppard1999hedonic}. Their principal advantage is transparency: they provide an independently estimated benchmark that does not rely on the advertised guide itself. A standard hedonic specification, however, imposes common marginal effects across a metropolitan market, even though the value of land, accessibility, dwelling characteristics, and local amenities may vary substantially across space.

Geographically weighted regression addresses this limitation by allowing coefficients to vary with location \citep{brunsdon1996gwr,fotheringham2002gwr,fotheringham2009}. This flexibility is particularly relevant in a spatially heterogeneous housing market such as Melbourne, where otherwise similar property characteristics can be valued differently across local submarkets. Combining a global hedonic model with a spatially adaptive GWR benchmark provides a useful way to assess whether estimated underquoting is robust to alternative assumptions about the geographic structure of housing prices.

This paper contributes to these literatures in three ways. First, it develops a unified theory in which price guides serve as both participation devices and information signals, with the relative importance of these functions depending on the sales mechanism. The framework predicts that auction guides will be more strongly distorted downwards and less closely related to final prices than private-treaty guides. Second, the paper develops a conservative model-based measure of underquoting that combines advertised guide ranges, independently estimated valuation intervals, and realised transaction prices. This reduces reliance on listing-to-sale price gaps alone. Third, by estimating separate hedonic and GWR benchmarks for auction and private-treaty transactions, the paper provides new evidence on how the informativeness and strategic use of advertised price guides differ across housing sales mechanisms.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Dual Role of Price Guides in Housing Sales}
\label{sec:theory}

Price guides can serve two distinct economic functions in housing markets. First, they can act as participation devices by shaping buyer attention, inspection decisions, and entry into the sale process. Second, they can act as information signals by communicating the seller's expectations about plausible transaction values. The relative importance of these two functions depends on the sales mechanism. In auction campaigns, the participation role is likely to dominate because additional bidders directly increase competitive pressure and the seller can rely on a reserve price for downside protection. In private-treaty sales, the information role is likely to be more important because offers are negotiated relative to the guide, and misleadingly low guidance can attract mismatched buyers, generate low offers, and increase bargaining frictions.

This section develops a simple model that captures this dual role. The model shows that underquoting can occur under both auction and private-treaty sales, but it should be more severe in auctions. It also implies that final prices should be less likely to fall within the quoted price range in auctions than in private-treaty sales.

\subsection{Environment}

There is one seller and a set of potential buyers. The seller observes a signal \(s\) about the likely market value of the property. Let
\[
\mu(s)=\mathbb{E}[P^c\mid s]
\]
denote the seller's posterior expectation of the competitive transaction price under market-consistent price guidance, where \(P^c\) is the benchmark transaction price that would arise absent strategic distortion in the guide.

The seller publicly announces a price guide range
\[
[q_L,q_H], \qquad q_L\leq q_H.
\]
For the theory, we focus on the upper bound of the guide, \(q_H\), because it is the most conservative advertised benchmark for underquoting. To simplify notation, suppose $
q= q_{H}.$
The lower guide is captured by the width of the quoted range,
$
w= q_H-q_L,
$
so that the advertised range can be written as $[q-w,q]$.

Define the downward deviation of the upper guide from the seller's market-consistent benchmark as $d= \mu(s)-q.$
A larger value of \(d\) means that the upper guide is set further below the seller's posterior benchmark. The case \(d>0\) corresponds to underquoting.

\begin{definition}
A property is underquoted if the upper bound of the advertised price guide is below the seller's market-consistent benchmark:
$
q<\mu(s),
$
or equivalently,
$
d>0.
$
\end{definition}

The seller chooses the guide before the transaction price is determined. The sales mechanism is denoted by
\[
M\in\{A,PT\},
\]
where \(A\) denotes auction and \(PT\) denotes private-treaty sales.

\subsection{The two roles of the price guide}

The price guide affects the seller's expected payoff through two channels. The first is a participation channel. Let $m^M(d)$
denote the expected number of participating buyers under mechanism \(M\). In auctions, this is the expected number of bidders who attend and bid. In private-treaty sales, it is the expected number of serious buyers, inspections, or offers generated by the campaign. We assume that lower guides increase participation:
\[
\frac{d m^M(d)}{d d}>0.
\]
Since \(d=\mu(s)-q\), increasing \(d\) is equivalent to lowering the advertised guide.

The second channel is an information and bargaining channel. The guide communicates information about the seller's expected price and shapes buyer beliefs. This role is especially important in private-treaty sales, where offers are negotiated relative to the guide. If the guide is set too far below the seller's expected transaction value, the seller may attract buyers whose budgets are too low, receive offers far below the seller's acceptable price, or incur delay and bargaining costs.

We capture these two channels using a reduced-form payoff representation. Let $B^M(d;s)$ denote the gross benefit from setting the guide \(d\) below the market-consistent benchmark. This benefit reflects the participation effect of a lower guide. We assume
\[
B^M(0;s)=0,\qquad B_d^M(d;s)>0,\qquad B_{dd}^M(d;s)<0.
\]
Thus, a lower guide increases expected benefit through participation, but the marginal benefit is diminishing.

Let $K^M(d;s)$ denote the cost of setting the guide below the market-consistent benchmark. This cost includes regulatory exposure, reputational cost, mismatch costs from attracting poorly matched buyers, and bargaining or delay costs. We assume
\[
K^M(0;s)=0,\qquad K_d^M(d;s)\geq 0,\qquad K_{dd}^M(d;s)>0.
\]
Thus, the marginal cost of underquoting is non-negative and increasing in the degree of distortion.

The seller's payoff under mechanism \(M\) is
\[
\Pi^M(d;s)=\Pi_0^M(s)+B^M(d;s)-K^M(d;s),
\]
where \(\Pi_0^M(s)\) is the seller's expected payoff under market-consistent price guidance. The seller chooses
\[
d\in[0,\bar d],
\]
where \(\bar d\) is the largest feasible downward deviation of the guide.

For later use, define the marginal net gain from increasing the degree of underquoting as
\[
G^M(d;s)=B_d^M(d;s)-K_d^M(d;s).
\]
Given the assumptions above,
\[
G_d^M(d;s)=B_{dd}^M(d;s)-K_{dd}^M(d;s)<0.
\]
Thus, the marginal net gain from underquoting is strictly decreasing in the degree of underquoting.

\subsection{Auction and Private-treaty sales as different Mechanisms}

The key difference between auctions and private-treaty sales lies in the relative strength of the participation benefit and the information or bargaining cost.

In auction campaigns, additional bidders directly increase competitive pressure. A lower guide can therefore have a high marginal benefit. In addition, the seller can rely on a reserve price, which reduces the risk of being forced to accept a low price. This weakens the cost of quoting low.

In private-treaty sales, a lower guide can still attract attention and generate offers, but the final price is determined through negotiation rather than open ascending bidding. The seller cannot rely on the same public competitive process to convert extra attention into final-price pressure. Moreover, because offers are made relative to the advertised guide, a misleadingly low guide can generate low offers, attract unsuitable buyers, and create bargaining frictions.

We summarise this distinction with the following assumption.

\begin{assumption}
For all relevant \(d\) and \(s\),
\[
B_d^A(d;s)\geq B_d^{PT}(d;s)
\]
and
\[
K_d^A(d;s)\leq K_d^{PT}(d;s).
\]
At least one of these inequalities is strict on a non-empty interval.
\end{assumption}

This assumption formalises the dual-role interpretation. The participation benefit of lowering the guide is larger in auctions, while the information, mismatch, and bargaining costs of lowering the guide are larger in private-treaty sales.

Let
\[
d_M^*(s)\in\arg\max_{d\in[0,\bar d]}\Pi^M(d;s)
\]
denote the seller's optimal degree of underquoting under mechanism \(M\).

\begin{proposition}\label{pro1}
Underquoting is optimal under mechanism \(M\) if the marginal participation benefit of lowering the guide at the market-consistent benchmark exceeds the marginal cost.
\end{proposition}

\begin{proof}
At the market-consistent benchmark, \(d=0\) and the upper guide equals \(\mu(s)\). The derivative of the seller's payoff with respect to \(d\) is
\[
\frac{\partial \Pi^M(d;s)}{\partial d}
=
B_d^M(d;s)-K_d^M(d;s)
=
G^M(d;s).
\]
Suppose
\[
G^M(0;s)>0.
\]
Then, for a small increase \(\Delta>0\) in the degree of underquoting, a first-order Taylor expansion gives
\[
\Pi^M(\Delta;s)-\Pi^M(0;s)
=
\Delta G^M(0;s)+o(\Delta).
\]
Since \(G^M(0;s)>0\), this expression is positive for sufficiently small \(\Delta\). Therefore \(d=0\) cannot be optimal. The seller strictly prefers a positive degree of underquoting, so $d_M^*(s)>0.$
Equivalently, $
q_M^*(s)<\mu(s).$
Thus, underquoting is optimal whenever the marginal participation benefit at the market-consistent benchmark exceeds the marginal cost.
\end{proof}

This proposition shows why underquoting may occur even when price disclosure rules exist. Regulation and reputational concerns enter through \(K^M(d;s)\), but they do not eliminate underquoting if the participation value of a lower guide remains sufficiently large.

\begin{proposition}\label{pro2}
Auction guides are more strongly distorted downward than private-treaty guides.
\end{proposition}

\begin{proof}
By the mechanism-ordering assumption,
\[
B_d^A(d;s)\geq B_d^{PT}(d;s)
\]
and
\[
K_d^A(d;s)\leq K_d^{PT}(d;s)
\]
for all relevant \(d\) and \(s\). Therefore,
\[
G^A(d;s)
=
B_d^A(d;s)-K_d^A(d;s)
\geq
B_d^{PT}(d;s)-K_d^{PT}(d;s)
=
G^{PT}(d;s).
\]
Thus, at any given degree of underquoting, the marginal net gain from lowering the guide is weakly larger in auctions than in private-treaty sales.

Since \(G^M(d;s)\) is strictly decreasing in \(d\), an interior optimum satisfies
\[
G^M(d_M^*(s);s)=0.
\]
Consider first the case in which both optima are interior. Since
\[
G^{PT}(d_{PT}^*(s);s)=0,
\]
and since $G^A(d;s)\geq G^{PT}(d;s)$ for all \(d\), it follows that
$ G^A(d_{PT}^*(s);s)\geq 0$.
Because \(G^A(d;s)\) is strictly decreasing in \(d\), the point at which \(G^A(d;s)\) reaches zero must be weakly greater than \(d_{PT}^*(s)\). Hence
\[
d_A^*(s)\geq d_{PT}^*(s).
\]

The same ordering also holds when one or both optima are at the boundary. If \(d_{PT}^*(s)=0\), the result is immediate because \(d_A^*(s)\geq 0\). If \(d_A^*(s)=\bar d\), the result is immediate because \(d_{PT}^*(s)\leq \bar d\). If \(d_{PT}^*(s)=\bar d\), then \(G^{PT}(d;s)\geq 0\) throughout the feasible interval, and therefore \(G^A(d;s)\geq 0\) throughout the same interval. Hence \(d_A^*(s)=\bar d\) as well. Therefore, $d_A^*(s)\geq d_{PT}^*(s).$ Since \(q_M^*(s)=\mu(s)-d_M^*(s)\), this is equivalent to $q_A^*(s)\leq q_{PT}^*(s).$
Thus, auction guides are predicted to be lower relative to market-consistent value than private-treaty guides.
\end{proof}


This proposition captures the main theoretical distinction. In auctions, the guide is primarily an entry device. In private-treaty sales, the guide is more strongly disciplined by its information and bargaining role. The theory therefore predicts more severe underquoting in auctions, but it does not predict that private-treaty guides must be perfectly truthful.

\subsection{Price guides and final transaction prices}

The preceding result compares optimal guides across sales mechanisms. We now derive an implication for the probability that the final transaction price falls within the quoted range.

Let the final transaction price under mechanism \(M\) be written as
\[
P^M=q_M^*+z_M+\varepsilon^M,
\]
where \(q_M^*\) is the optimal upper guide, \(z_M\geq 0\) is the expected gap between the final price and the upper guide, and \(\varepsilon^M\) is a mean-zero pricing shock. The term \(z_M\) captures the extent to which the final price is expected to exceed the guide.

Assume
\[
z_M=a_M d_M^*(s), \qquad a_M>0.
\]
Thus, the expected final-price gap is increasing in the degree of downward guide distortion. The parameter \(a_M\) captures how strongly guide distortion translates into a final price above the guide. In auctions, this parameter is likely to be larger because low guides attract bidders but the final price is determined by competitive bidding. In private-treaty sales, it may be smaller because the guide also anchors negotiation.

The quoted range is \([q_M^*-w,q_M^*]\). Conditional on guide width \(w\), the final price lies within the quoted range if
\[
q_M^*-w\leq P^M\leq q_M^*.
\]
Using the expression for \(P^M\), this event is equivalent to
\[
-w\leq z_M+\varepsilon^M\leq 0.
\]

\begin{assumption}
The shock \(\varepsilon^M\) has a continuous distribution with density \(f\), symmetric around zero and weakly decreasing in \(|x|\) for \(x\geq 0\). Conditional on observed property characteristics and guide width, the distribution of the shock is comparable across mechanisms.
\end{assumption}

\begin{proposition}\label{pro4}
Final prices are less likely to fall within the quoted range in auctions than in private-treaty sales when auction guides are more strongly distorted downward.
\end{proposition}

\begin{proof}
For a given guide width \(w\), define
\[
\phi(z;w)=\Pr(-w\leq z+\varepsilon\leq 0).
\]
This is the probability that the final price falls within the quoted range when the expected final-price gap above the guide is \(z\). Using the distribution function \(F\) of \(\varepsilon\),
\[
\phi(z;w)=F(-z)-F(-w-z).
\]
Differentiating with respect to \(z\) gives
\[
\frac{\partial \phi(z;w)}{\partial z}
=
-f(-z)+f(-w-z).
\]
By symmetry of \(f\),
\[
f(-z)=f(z),
\qquad
f(-w-z)=f(w+z).
\]
Since \(w\geq 0\) and \(z\geq 0\), we have \(w+z\geq z\). Because \(f(x)\) is weakly decreasing in \(x\) for \(x\geq 0\),
\[
f(w+z)\leq f(z).
\]
Therefore,
\[
\frac{\partial \phi(z;w)}{\partial z}
=
-f(z)+f(w+z)
\leq 0.
\]
Thus, the probability that the final price falls within the quoted range is weakly decreasing in the expected gap between the final price and the upper guide.

From the previous proposition,
\[
d_A^*(s)\geq d_{PT}^*(s).
\]
If, in addition,
\[
a_A d_A^*(s)\geq a_{PT} d_{PT}^*(s),
\]
then $z_A\geq z_{PT}.$
Since \(\phi(z;w)\) is weakly decreasing in \(z\),
\[
\Pr(P^A\in[q_A^*-w,q_A^*])
\leq
\Pr(P^{PT}\in[q_{PT}^*-w,q_{PT}^*]).
\]
Therefore, conditional on comparable guide widths and observed property characteristics, final prices are less likely to fall within the quoted range in auction campaigns than in private-treaty campaigns.
\end{proof}

This proposition connects the theory directly to the empirical pattern that final prices are substantially less likely to fall within advertised guides in auctions than in private-treaty sales. In auctions, the guide is more strongly used to attract participation, so final prices often exceed the guide. In private-treaty sales, the guide is more informative and more closely tied to the negotiation process, so transaction prices are more likely to remain within the advertised range.

\subsection{Underquoting incidence across mechanisms}

The theory also delivers a prediction about underquoting incidence. Suppose properties differ in the strength of the participation incentive and in the cost of guide distortion. Let \(\omega\) denote a property-specific or local-market state, such as buyer demand, market tightness, or valuation uncertainty. The optimal degree of underquoting is then
\[
d_M^*(s,\omega).
\]
A property is empirically classified as underquoted when the guide is sufficiently below the market-consistent benchmark and the final transaction price confirms that the guide was low relative to realised market value.

Let \(\tau>0\) denote the minimum economically meaningful downward distortion required for a property to be classified as underquoted. The probability of underquoting under mechanism \(M\) can be written as
\[
\Pr(\text{Underquoted}\mid M)
=
\Pr(d_M^*(s,\omega)>\tau).
\]

\begin{proposition}\label{pro5}
Underquoting is more prevalent in auction campaigns than in private-treaty campaigns.
\end{proposition}

\begin{proof}
From the previous ordering result,
\[
d_A^*(s,\omega)\geq d_{PT}^*(s,\omega)
\]
for each state \((s,\omega)\). Therefore, for any threshold \(\tau>0\),
\[
\{d_{PT}^*(s,\omega)>\tau\}
\subseteq
\{d_A^*(s,\omega)>\tau\}.
\]
Taking probabilities over the distribution of \((s,\omega)\) gives
\[
\Pr(d_{PT}^*(s,\omega)>\tau)
\leq
\Pr(d_A^*(s,\omega)>\tau).
\]
Hence,
\[
\Pr(\text{Underquoted}\mid PT)
\leq
\Pr(\text{Underquoted}\mid A).
\]
Thus, underquoting is predicted to be more prevalent in auction campaigns than in private-treaty campaigns.
\end{proof}

This result does not imply that underquoting is absent in private-treaty sales. Lower guides can still attract buyer attention and generate more offers. The prediction is comparative: because the participation benefit is larger and the information cost is lower in auctions, underquoting should be more common and more severe in auction campaigns.

\subsection{Discussion and empirical implications}

The model highlights the dual role of price guides in housing markets. The guide is both a participation device and an information signal. The participation role is strongest in auctions, where additional bidders directly increase competitive pressure and the seller can rely on a reserve price for downside protection. The information role is strongest in private-treaty sales, where offers are negotiated relative to the advertised guide and misleadingly low guidance can create mismatch, delay, and bargaining costs.

The theoretical model is written in terms of the seller's market-consistent benchmark, $\mu(s)$, and the downward deviation of the upper guide, $d=\mu(s)-q$. These objects are not directly observed in the data. We therefore use model-implied valuation intervals as empirical counterparts to the unobserved market-consistent benchmark. The advertised upper guide corresponds to the observable guide chosen by the seller or agent, while the hedonic and GWR valuation intervals provide guide-independent benchmarks based on observed property characteristics and local market conditions.

The theory yields three empirical implications. First, Proposition \ref{pro2} predicts that auction guides should be positioned lower relative to market-consistent values than private-treaty guides. Empirically, this corresponds to testing whether model-implied valuation bounds exceed advertised guide bounds more frequently in auctions. Second, Proposition \ref{pro4} predicts that final transaction prices should be less likely to fall within advertised guide ranges in auctions than in private-treaty sales. Empirically, this corresponds to comparing the distribution of realised prices relative to advertised guide ranges across sale mechanisms. Third, Proposition \ref{pro5} predicts that underquoting should be more prevalent in auction campaigns. Empirically, this corresponds to comparing the conservative GWR-based underquoting indicator across auction and private-treaty sales, both descriptively and after controlling for observable property and locational characteristics.

These empirical tests should be interpreted as reduced-form tests of the model's observable implications rather than as a structural estimation of the seller's optimal guide choice. The data do not contain bidder numbers, inspections, reserve prices or bargaining histories, so the participation channel itself is not directly observed. Instead, the empirical analysis examines whether the observable patterns are consistent with that channel: whether auction guides are more often below guide-independent valuation benchmarks, whether auction prices are more likely to exceed the advertised range, and whether the resulting underquoting signal is more prevalent in auction campaigns.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Institutional Setting and Data}\label{SecData}
Melbourne provides a particularly suitable setting for studying underquoting because it has substantial activity under both auction and private-treaty sales, together with a strong auction tradition in which listing guides may shape bidder expectations and participation. This section describes Melbourne's housing market, its institutional setting, and the data used in the empirical analysis. High transaction volumes, spatially heterogeneous property values, and active disclosure regulation make Melbourne a useful setting for comparing advertised guides with model-implied valuations. We first outline the institutional context before describing the datasets used to construct property-level valuation benchmarks.
\subsection{Melbourne Housing Market}

The empirical analysis focuses on Melbourne, one of Australia’s largest and most active housing markets. Together with Sydney, Melbourne accounts for roughly 55\% of the nation’s total residential property value, making it a natural laboratory for studying price discovery, market frictions, and regulatory effectiveness in housing \citep{corelogic2018}. The combination of deep auction activity, high household leverage, and strong spatial price gradients means that even modest distortions in listing behaviour can have systemically relevant consequences for wealth distribution and market efficiency.

Underquoting is a particularly salient concern in Victoria’s auction‑driven institutional environment. Complaints to Consumer Affairs Victoria (CAV) about misleading price guides rose sharply in the mid‑2010s, prompting strengthened disclosure rules and expanded enforcement \citep{osterling2016underpricing}. Despite these interventions, underquoting remains a prominent feature of the regulatory docket. The dedicated Underquoting Taskforce received more than 2{,}800 complaints between 2023 and 2024, an increase of roughly 70\%, with a growing share lodged by industry professionals rather than buyers \citep{vicgov2024underquoting,consumeraffairsvictoria2024underquoting}. This pattern suggests that even compliant agents perceive strategic underquoting as a source of competitive pressure.

Experience in New South Wales provides a useful external benchmark. Following the 2016 reform, NSW Fair Trading continues to record high complaint volumes and to issue regular infringement notices \citep{abc2024,nswfairtrading2024}. In both Victoria and New South Wales, complaints have grown more rapidly than formal enforcement actions, indicating that underquoting remains difficult to detect and substantiate ex ante. From an economic perspective, this enforcement gap reflects a more fundamental measurement problem: regulators largely observe listing ranges, realised prices, and a small set of agent‑selected comparables, but lack a systematic benchmark for the market‑consistent value of individual properties.

This study frames underquoting in Melbourne as a problem of identifying systematic deviations of listing guides from model‑implied market values in a regulated housing market. By combining transaction-level data with econometric valuation benchmarks, we provide evidence on the prevalence and spatial distribution of patterns consistent with underquoting under the current Victorian disclosure regime. 

\subsection{Data}

This study uses two linked datasets for detached house transactions in metropolitan Melbourne. The raw 2023--2024 sales file contains a larger set of completed house transactions. After restricting the sample to detached houses and retaining observations with valid structural characteristics, land size, suburb-level rent, contract prices and method-of-sale information, the final training sample used in the pricing analysis contains 65,768 observations: 20,250 auction sales and 45,518 private-treaty sales. The second dataset is the 2025 evaluation sample, which contains properties for which advertised listing-price bounds and subsequent realised transaction prices are both observed.\footnote{The advertised guide variables refer to the most recent recorded lower and upper price bounds available before the recorded sale outcome. The data do not contain the full sequence of guide revisions during the marketing campaign.} After applying the same validity requirements, the 2025 evaluation sample contains 3{,}991 observations: 1{,}336 auction transactions and 2{,}655 private-treaty transactions.
This distinction is central because price guides serve different functions in auction and private-treaty sales. In private-treaty sales, the quoted range is more closely connected to the seller’s asking position and the subsequent negotiation between buyer and seller. In auctions, however, the guide is set before the competitive bidding process begins and can influence which buyers inspect the property, attend the auction and decide to bid. Combining the two sale mechanisms would therefore blur two quite different pricing environments. For this reason, the valuation models and underquoting measures are estimated separately for auction and private-treaty transactions.


The analysis is restricted to detached houses. This restriction reduces heterogeneity arising from apartments, townhouses and other dwelling types, whose pricing may reflect different ownership structures, land components and buyer segments. Each property record contains core structural characteristics, including the number of bedrooms, bathrooms and parking, land size, and realised contract price. The 2025 evaluation sample additionally contains advertised lower and upper listing-price bounds. The data also include locational information, including distances to the Melbourne CBD, public transport and education facilities, together with suburb-level median rent. Median rent is used as a local-market indicator because it captures neighbourhood demand conditions that are not fully summarised by physical housing attributes alone.

Geographic coordinates are used in the spatial modelling stage. For the 2023--2024 training sample, each property is assigned the centroid coordinates of its corresponding Mesh Block\footnote{Mesh Blocks are the smallest geographic areas defined by the Australian Bureau of Statistics (ABS) within the Australian Statistical Geography Standard (ASGS). Most residential Mesh Blocks contain approximately 30--60 dwellings. In this study, each property in the training sample is assigned the centroid coordinates of its corresponding Mesh Block, providing spatial precision while preserving confidentiality and consistency with the ABS geographic hierarchy; see the \href{https://www.abs.gov.au/census/guide-census-data/geography/census-geography-glossary}{ABS Census Geography Glossary}.}. For the 2025 evaluation sample, geocoded property-level coordinates derived from the property address are used. These coordinates allow the GWR model to estimate locally varying pricing relationships across Melbourne rather than imposing a common set of marginal effects across all locations.

\begin{table}[H]
\centering
\caption{\small Summary statistics for Melbourne house sales in the 2023--2024 training sample, reported separately by sale segment. Panel A reports auction transactions and Panel B reports private-treaty transactions. The table presents the minimum, median, mean, maximum and standard deviation for key property characteristics, contract prices and suburb-level median rent. Auction sales are properties recorded as sold at auction, while private-treaty sales are negotiated non-auction transactions. Statistics are calculated after applying the validity filters used in the pricing analysis.}
\label{tab:summary_house_training_by_segment}

\small
\renewcommand{\arraystretch}{1.15}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrrr}
\toprule
\textbf{Variable} & \textbf{Min} & \textbf{Median} & \textbf{Mean} & \textbf{Max} & \textbf{SD} \\
\midrule
\multicolumn{6}{l}{\textbf{Panel A: Auction} ($N = 20,250$)} \\
\midrule
\multicolumn{6}{l}{\textit{Property characteristics}} \\
\midrule
Bedrooms & 1 & 3 & 3.35 & 8 & 0.83 \\
Bathrooms & 1 & 2 & 1.77 & 6 & 0.67 \\
Parking & 0 & 2 & 1.69 & 6 & 0.93 \\
Land size (m$^2$) & 31 & 576 & 555.07 & 2,999 & 247.04 \\
\midrule
\multicolumn{6}{l}{\textit{Prices (AUD)}} \\
\midrule
Contract price (AUD) & 285,000 & 1,090,000 & 1,191,584 & 2,800,000 & 507,057 \\
\midrule
\multicolumn{6}{l}{\textit{Suburb-level indicator}} \\
\midrule
Median rent (AUD per week) & 251.14 & 530.00 & 539.51 & 1,375.00 & 85.58 \\
\midrule
\multicolumn{6}{l}{\textbf{Panel B: Private-treaty sales} ($N = 45,518$)} \\
\midrule
\multicolumn{6}{l}{\textit{Property characteristics}} \\
\midrule
Bedrooms & 1 & 3 & 3.40 & 8 & 0.79 \\
Bathrooms & 1 & 2 & 1.85 & 6 & 0.61 \\
Parking & 0 & 2 & 1.77 & 6 & 0.89 \\
Land size (m$^2$) & 33 & 553 & 588.84 & 2,996 & 338.04 \\
\midrule
\multicolumn{6}{l}{\textit{Prices (AUD)}} \\
\midrule
Contract price (AUD) & 200,000 & 762,000 & 902,680 & 2,800,000 & 421,027 \\
\midrule
\multicolumn{6}{l}{\textit{Suburb-level indicator}} \\
\midrule
Median rent (AUD per week) & 317.62 & 490.00 & 506.11 & 1,375.00 & 84.56 \\
\bottomrule
\end{tabular*}
\end{table}
Table~\ref{tab:summary_house_training_by_segment} reports summary statistics for the 2023--2024 training sample. Auction properties are more expensive than private-treaty properties in the training period. The median auction contract price is AUD1.09 million, compared with AUD762{,}000 for private-treaty sales, and the mean auction price is also substantially higher at AUD1.19 million. The physical characteristics are broadly similar across segments, but auction dwellings are slightly smaller on average in bathrooms and land size. Auction properties have a mean of 3.35 bedrooms, 1.77 bathrooms and 555 m$^2$ of land, while private-treaty properties have 3.40 bedrooms, 1.85 bathrooms and 589 m$^2$ of land. The auction sample is also located in somewhat higher-rent suburbs, with a median suburb-level rent of AUD530 per week compared with AUD490 for private-treaty sales. This suggests that auctions are more concentrated in higher-value local markets.\footnote{Note that the analysis focuses on completed transactions with matched advertised guides and realised sale prices. Consequently, withdrawn, unsold, or unmatched passed-in properties are outside the evaluation sample. The findings therefore relate to successfully completed sales.}\\
Table~\ref{tab:summary_house_2025_by_segment} summarises the 2025 evaluation sample. The auction segment again has higher prices than private-treaty sales. The median realised auction price is AUD1.012 million, compared with AUD820{,}000 for private-treaty sales. The advertised price ranges also differ sharply across segments. In auctions, the median lower and upper listing bounds are AUD900{,}000 and AUD990{,}000, while the median realised sale price is AUD1.012 million. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
    \centering
    \includegraphics[width=0.9\linewidth]{melb_sales_listings_points_side_by_side.png}
    \caption{Spatial distribution of the 2023--2024 training sample ($N=65{,}768$, left) and the 2025 evaluation sample ($N=3{,}991$, right) across Greater Melbourne. Training-sample locations are represented by Mesh Block centroids, while evaluation-sample locations are based on geocoded property addresses.}
    \label{fig:spatial_dist}
\end{figure}

\begin{table}[H]
\centering
\caption{\small
Summary statistics for Melbourne house listings and sales in the 2025 evaluation sample. Panel A reports auction transactions and Panel B reports private-treaty transactions. The table presents property characteristics, advertised price bounds, realised contract prices and suburb-level median rent. One private-treaty observation with reversed bounds was corrected before calculation.
}

\label{tab:summary_house_2025_by_segment}
\small
\renewcommand{\arraystretch}{1.15}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrrr}
\toprule
\textbf{Variable} & \textbf{Min} & \textbf{Median} & \textbf{Mean} & \textbf{Max} & \textbf{SD} \\
\midrule
\multicolumn{6}{l}{\textbf{Panel A: Auction} ($N = 1,336$)} \\
\midrule
\multicolumn{6}{l}{\textit{Property characteristics}} \\
\midrule
Bedrooms & 1 & 3 & 3.55 & 7 & 0.76 \\
Bathrooms & 1 & 2 & 1.76 & 4 & 0.62 \\
Parking & 1 & 2 & 2.26 & 6 & 0.96 \\
Land size (m$^2$) & 127 & 604 & 610.05 & 2,000 & 194.57 \\
\midrule
\multicolumn{6}{l}{\textit{Prices (AUD)}} \\
\midrule
Listing price (min quoted) & 320,000 & 900,000 & 998,103 & 2,980,000 & 401,369 \\
Listing price (max quoted) & 350,000 & 990,000 & 1,074,020 & 2,980,000 & 428,411 \\
Contract price (AUD) & 370,000 & 1,012,500 & 1,110,517 & 2,860,000 & 438,677 \\
\midrule
\multicolumn{6}{l}{\textit{Suburb-level indicator}} \\
\midrule
Median rent (AUD per week) & 390.00 & 540.00 & 552.73 & 980.00 & 71.67 \\
\midrule
\multicolumn{6}{l}{\textbf{Panel B: Private-treaty sales} ($N = 2,655$)} \\
\midrule
\multicolumn{6}{l}{\textit{Property characteristics}} \\
\midrule
Bedrooms & 1 & 4 & 3.59 & 8 & 0.72 \\
Bathrooms & 1 & 2 & 1.84 & 4 & 0.57 \\
Parking & 1 & 2 & 2.26 & 6 & 0.97 \\
Land size (m$^2$) & 38 & 608 & 653.38 & 2,590 & 313.89 \\
\midrule
\multicolumn{6}{l}{\textit{Prices (AUD)}} \\
\midrule
Listing price (min quoted) & 202,500 & 790,000 & 899,393 & 3,000,000 & 355,456 \\
Listing price (max quoted) & 250,000 & 850,000 & 959,393 & 3,200,000 & 378,603 \\
Contract price (AUD) & 235,000 & 820,000 & 925,960 & 2,800,000 & 357,715 \\
\midrule
\multicolumn{6}{l}{\textit{Suburb-level indicator}} \\
\midrule
Median rent (AUD per week) & 390.00 & 525.00 & 537.30 & 1,250.00 & 75.23 \\
\bottomrule
\end{tabular*}
\end{table}
This gap is consistent with the later evidence that auction properties frequently sell above the advertised maximum. In private-treaty sales, the median lower and upper listing bounds are AUD790{,}000 and AUD850{,}000, compared with a median realised price of AUD820{,}000, indicating a more balanced relationship between listed and realised prices. The 2025 auction and private-treaty samples are similar in bedrooms, bathrooms and parking, but private-treaty properties have larger median land size, at 608 m$^2$ compared with 604 m$^2$ for auctions.

Extreme observations are removed using the same trimming approach applied in the pricing analysis. Observations with extreme log contract prices or extreme log land sizes are excluded to reduce the influence of outliers on the hedonic and GWR estimates. In addition to this log-price and log-land-size trimming, the sample is restricted to transactions with contract prices and land sizes lying within the support used for model estimation. This removes extremely high-value transactions and unusually large land parcels that are unlikely to be comparable with the suburban detached-house market analysed in this paper.
\footnote{Observations are trimmed symmetrically at five standard deviations from the sample mean of log contract price and log land size. Trimming is conducted after log transformation to reduce the influence of scale and skewness. Additional support restrictions are applied to ensure that the estimation and evaluation samples remain within the suburban detached-house market considered in the analysis.}
For both datasets, suburb-level median rent sourced from CoreLogic is merged to proxy local housing market conditions. As shown in Figure~\ref{fig:spatial_dist}, the spatial concentration of observations varies across Melbourne.

\section{Empirical analysis}\label{SecRes}
This section takes the theoretical predictions to the data in three steps. First, we construct independent valuation benchmarks for each sale mechanism using completed 2023--2024 transactions and apply them out of sample to the 2025 evaluation sample. The purpose is to obtain model-implied valuation ranges that do not depend on the advertised guide. Second, we compare advertised guide bounds with model-implied valuation bounds and realised transaction prices. This allows us to examine whether auction guides are positioned lower relative to estimated market values and whether auction outcomes are less likely to remain within the quoted range. Third, we use a conservative GWR-based classification rule to identify likely underquoted properties and compare incidence across auction and private-treaty sales. The empirical analysis therefore links directly to the propositions: auction guides should be more downward distorted, auction prices should less often fall within the quoted range, and underquoting should be more prevalent in auction campaigns.

\subsection{Fixed-effects hedonic benchmark}
As a first valuation benchmark, we estimate a fixed-effects hedonic pricing model in the spirit of \cite{rosen1974hedonic}. Hedonic models are widely used in housing economics to relate transaction prices to observed structural and locational characteristics, such as dwelling size, land size and neighbourhood attributes (see, e.g., \cite{blomquist1981hedonic,sheppard1999hedonic,diao2010residential,diao2016railnoise}). In this study, the hedonic model provides a transparent benchmark against which the more flexible spatial model can be compared.

The model is estimated on completed transactions from the 2023--2024 training sample and then used to predict prices for the 2025 evaluation sample. Estimation is conducted separately by sale mechanism. The auction benchmark is estimated using only auction sales, while the private-treaty benchmark is estimated using only private-treaty sales. This avoids imposing a common pricing relationship on two different sale environments and means that method of sale is used to define the estimation sample rather than entering the regression as an explanatory variable.

Let the transaction price of property $i$ depend on observed structural attributes $\mathbf{A}_i$, locational and neighbourhood characteristics $\mathbf{N}_i$, and an idiosyncratic error term:
\begin{equation}
\label{eq}
\text{Price}_i = f(\mathbf{A}_i, \mathbf{N}_i) + v_i .
\end{equation}

We operationalise this relationship using a log-linear ordinary least squares specification. The dependent variable is the logarithm of the realised contract price. The main explanatory variables are the number of bedrooms, bathrooms and parking, log land size, distance to public transport and distance to the Melbourne CBD.  Suburb fixed effects absorb persistent neighbourhood-level differences, while seasonal indicators capture common within-year variation in market conditions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
The baseline hedonic benchmark is estimated in log-linear form as
\begin{align}
\label{eq:current_study_hedonic_price}
\ln(\text{SalePrice}_i)
=~& \beta_0
+ \beta_1 \text{Bedrooms}_i
+ \beta_2 \text{Bathrooms}_i
+ \beta_3 \text{Parking}_i \nonumber \\
&+ \beta_4 \ln(\text{LandSize}_i)
+ \beta_5 \ln(\text{MedianRent}_{s(i)}) \nonumber \\
&+ \beta_6 \text{DistTransport}_i
+ \beta_7 \text{DistCBD}_i
+ \lambda_{q(i)}
+ \delta_{\ell(i)}
+ v_i .
\end{align}

Here, $\lambda_{q(i)}$ denotes seasonal fixed effects and $\delta_{\ell(i)}$ denotes suburb fixed effects. The term $\text{MedianRent}_{s(i)}$ is measured at the suburb level and varies across both suburbs and years. The model is estimated separately within each sale segment: the auction model is estimated using only auction transactions, while the private-treaty model is estimated using only private-treaty transactions. The method-of-sale variable is therefore used to define the estimation sample rather than being included as an explanatory variable.

The revised specification uses the same core structural and locational covariates as the preferred GWR benchmark, while retaining suburb fixed effects in the global hedonic model. The structural controls are bedrooms, bathrooms, parking and land size, with land size entering in logarithmic form. Suburb-level median rent controls for local rental-market conditions and broader neighbourhood desirability. The locational controls include distance to public transport and distance to the Melbourne CBD. Seasonal indicators account for within-year variation in market conditions, while suburb fixed effects capture broad spatial differences.

The fitted values from Equation~\eqref{eq:current_study_hedonic_price} provide model-implied expected property values for the 2025 evaluation sample. For each 2025 property, the estimated log price is exponentiated to obtain the predicted price level. These predictions are then used to construct model-implied valuation intervals, which are compared with advertised listing ranges and realised transaction prices. Advertised guide prices are not used as explanatory variables in the hedonic specification; they enter only at the comparison stage when assessing whether public price guidance is low relative to model-implied market values.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the real-time adjustment exercise, geographic coordinates are retained in the dataset but are not included directly in the hedonic regression equations.\footnote{Geographic coordinates are measured using Mesh Block centroids for the training sample and property-level geocoded coordinates for the evaluation sample.}
Instead, latitude and longitude are used after prediction to identify nearby prior transactions and construct local prediction-error adjustments. This adjustment is applied to the model prediction rather than entering the OLS estimation equation itself.\\
For private-treaty transactions, Table~\ref{tab:ols_summary_stats_private_treaty} shows that the global hedonic benchmark captures several standard price gradients. Bedrooms, bathrooms and parking are all positively associated with sale prices, indicating that larger and better-equipped dwellings command higher transaction prices. Land size also enters positively and significantly, suggesting that additional land area is capitalised into private-treaty prices once other structural and locational characteristics are controlled for. Suburb-level median rent is strongly positive, consistent with higher-sale-price properties being located in stronger local rental markets and more desirable neighbourhoods. 

\begin{table}[H]
\centering
\caption{\small Private-treaty sales: OLS coefficient estimates from the model fitted to the 2023--2024 training sample. The dependent variable is log sale price. The table reports coefficient estimates, conventional standard errors, $t$-statistics and estimation-sample diagnostics for the unadjusted global OLS benchmark. *, ** and *** denote significance at the 10\%, 5\% and 1\% levels, respectively. Locally adjusted 2025 out-of-sample prediction performance is reported separately in Table~\ref{tab:prediction_metrics_ols_private_auction}.}

\label{tab:ols_summary_stats_private_treaty}
\small
\renewcommand{\arraystretch}{1.25}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
\textbf{Variable} & \textbf{Estimate} & \textbf{Std. Error} & \textbf{$t$-stat.} & \textbf{$p$-value} \\
\midrule
Intercept & 4.6200*** & 0.0547 & 84.534 & 0.0000 \\
Bedrooms & 0.0652*** & 0.0020 & 33.278 & 0.0000 \\
Bathrooms & 0.1358*** & 0.0025 & 53.985 & 0.0000 \\
Parking & 0.0154*** & 0.0016 & 9.844 & 0.0000 \\
$\ln(\text{Land size})$ & 0.1692*** & 0.0027 & 63.332 & 0.0000 \\
$\ln(\text{Median rent})$ & 1.2145*** & 0.0087 & 139.654 & 0.0000 \\
Distance to transport & -0.0033*** & 0.0002 & -16.670 & 0.0000 \\
Distance to CBD & -0.0021*** & 0.0000 & -47.863 & 0.0000 \\
Autumn & 0.0014 & 0.0036 & 0.399 & 0.6903 \\
Winter & -0.0007 & 0.0037 & -0.200 & 0.8416 \\
Summer & -0.0095** & 0.0038 & -2.531 & 0.0114 \\
\midrule
Season indicators & \multicolumn{4}{c}{Yes} \\
Suburb fixed effects & \multicolumn{4}{c}{Yes} \\
\midrule
Adjusted $R^{2}$ & \multicolumn{4}{r}{0.4906} \\
Akaike information criterion (AIC) & \multicolumn{4}{r}{13,950.66} \\
Bayesian information criterion (BIC) & \multicolumn{4}{r}{14,055.37} \\
\bottomrule
\end{tabular*}
\end{table}

The locational coefficients have the expected signs: distance to public transport and distance to the Melbourne CBD are both negative and highly significant, indicating that prices decline as properties become less accessible to transport amenities and farther from central Melbourne. The seasonal indicators are generally weak, although summer transactions are associated with slightly lower prices. The adjusted $R^2$ of 0.491 indicates that the global OLS model captures broad price variation in the training sample, but the model remains a single global benchmark and cannot fully capture local variation in Melbourne housing prices. This motivates the spatially adaptive GWR benchmark, which allows pricing relationships to vary locally across Melbourne rather than imposing one common set of coefficients.\footnote{Median rent is measured at the suburb level and matched to each transaction by suburb and year. Because the preferred specification does not include suburb fixed effects, median rent is used as an observable local-market control rather than being separately identified from within-suburb variation after fixed effects.}

For auction transactions, Table~\ref{tab:ols_summary_stats_auction} shows a broadly similar hedonic structure, although the estimated gradients differ from the private-treaty segment. Bedrooms and bathrooms are positively and highly significantly associated with sale prices, while parking is weakly negative and only marginally significant. Land size enters positively and significantly, indicating that larger lots are capitalised into auction prices after controlling for other property and location characteristics. 

\begin{table}[H]
\centering
\caption{\small Auction sales: OLS coefficient estimates from the model fitted to the 2023--2024 training sample. The dependent variable is log sale price. The table reports coefficient estimates, conventional standard errors, $t$-statistics and estimation-sample diagnostics for the unadjusted global OLS benchmark. *, ** and *** denote significance at the 10\%, 5\% and 1\% levels, respectively. Locally adjusted 2025 out-of-sample prediction performance is reported separately in Table~\ref{tab:prediction_metrics_ols_private_auction}.}

\label{tab:ols_summary_stats_auction}
\small
\renewcommand{\arraystretch}{1.25}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
\textbf{Variable} & \textbf{Estimate} & \textbf{Std. Error} & \textbf{$t$-stat.} & \textbf{$p$-value} \\
\midrule
Intercept & 4.1155*** & 0.0931 & 44.193 & 0.0000 \\
Bedrooms & 0.0590*** & 0.0030 & 19.434 & 0.0000 \\
Bathrooms & 0.1215*** & 0.0036 & 33.643 & 0.0000 \\
Parking & -0.0040* & 0.0024 & -1.684 & 0.0922 \\
$\ln(\text{Land size})$ & 0.1550*** & 0.0047 & 33.316 & 0.0000 \\
$\ln(\text{Median rent})$ & 1.3740*** & 0.0140 & 98.400 & 0.0000 \\
Distance to transport & -0.0037*** & 0.0006 & -6.177 & 0.0000 \\
Distance to CBD & -0.0101*** & 0.0002 & -50.036 & 0.0000 \\
Autumn & -0.0053 & 0.0054 & -0.978 & 0.3280 \\
Winter & -0.0056 & 0.0055 & -1.014 & 0.3107 \\
Summer & -0.0135** & 0.0060 & -2.243 & 0.0249 \\
\midrule 
Season indicators & \multicolumn{4}{c}{Yes} \\
Suburb fixed effects & \multicolumn{4}{c}{Yes} \\
\midrule
Adjusted $R^{2}$ & \multicolumn{4}{r}{0.5008} \\
Akaike information criterion (AIC) & \multicolumn{4}{r}{7,745.53} \\
Bayesian information criterion (BIC) & \multicolumn{4}{r}{7,840.52} \\
\bottomrule
\end{tabular*}


\end{table}

Suburb-level median rent is also strongly positive, suggesting that auction prices are higher in stronger local rental markets and more desirable neighbourhoods. The locational gradients have the expected signs: distance to public transport and distance to the Melbourne CBD are both negative and statistically significant, implying that auction prices decline as properties become less accessible to transport amenities and farther from central Melbourne. Seasonal effects are limited, although summer transactions are associated with slightly lower prices. The adjusted $R^2$ of 0.501 indicates that the global OLS benchmark captures broad price variation in the auction training sample, but the model remains deliberately transparent and global. The remaining spatial heterogeneity in housing prices motivates the use of the GWR benchmark.

\subsubsection{Locally adjusted OLS predictions}

The 2025 prediction metrics in Table~\ref{tab:prediction_metrics_ols_private_auction} are based on locally adjusted OLS predictions, not the unadjusted fitted values summarised in Tables~\ref{tab:ols_summary_stats_private_treaty} and \ref{tab:ols_summary_stats_auction}. For each 2025 property $i$ in sale segment $m$, the global OLS model first produces an unadjusted log-price prediction $\widehat{y}^{OLS}_{im}$. We then apply a real-time comparable-sales correction based on prior completed 2025 transactions from the same sale segment. For each prior comparable transaction $j$, the log prediction error is
\[
e_j
=
\log(P_j)-\widehat{y}^{OLS}_{jm},
\]
where $\widehat{y}^{OLS}_{jm}$ is generated using the segment-specific OLS model estimated only on the 2023--2024 training sample.

For private-treaty properties, the comparable set first searches for prior sales in the same suburb and with the same bedroom count within the previous 60 days, with a 3 km spatial fallback. For auction properties, the comparable set first searches for prior sales in the same suburb and with the same bedroom count within the previous 90 days, with a 5 km spatial fallback. Only transactions with sale dates preceding the sale date of property $i$ are used. The local adjustment is the median prediction error among the selected comparables:
\[
\widetilde{e}^{OLS}_{i}
=
\operatorname{median}\{e_j:j\in \mathcal{C}^{OLS}_i\}.
\]
If no qualifying prior comparable is available, the adjustment is set to zero. The adjusted OLS prediction is then
\[
\widehat{y}^{adj,OLS}_{im}
=
\widehat{y}^{OLS}_{im}
+
\widetilde{e}^{OLS}_{i},
\qquad
\widehat{P}^{adj,OLS}_{im}
=
\exp\left(\widehat{y}^{adj,OLS}_{im}\right).
\]
Thus, Table~\ref{tab:prediction_metrics_ols_private_auction} evaluates the out-of-sample accuracy of the locally adjusted OLS predictions used in the guide-classification exercise.

\begin{table}[H]
\centering
\caption{\small
Out-of-sample prediction accuracy for the 2025 test dataset based on the locally adjusted global OLS hedonic pricing model. The table reports standard forecast evaluation metrics measuring prediction errors in both absolute dollar terms and relative percentage terms. Prediction accuracy is reported separately for auction and private-treaty transactions in the 2025 evaluation sample. Absolute error P50 is the median absolute difference between realised and predicted transaction prices. MAPE is the median absolute percentage error. Log error is defined as $\log(P^{\mathrm{actual}})-\log(P^{\mathrm{predicted}})$.}
\label{tab:prediction_metrics_ols_private_auction}
\small
\renewcommand{\arraystretch}{1.2}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}l r}
\toprule
\textbf{Metric} & \textbf{Value} \\
\midrule
\multicolumn{2}{l}{\textbf{Panel A: Auction}} \\
\midrule
Absolute error P50 (median absolute error, AUD) & 138,599.47 \\
Median Absolute Percentage Error (\%) & 13.60 \\
Median log error & -0.0256 \\
Median absolute log error & 0.1348 \\

\midrule
\multicolumn{2}{l}{\textbf{Panel B: Private-treaty sales}} \\
\midrule
Absolute error P50 (median absolute error, AUD) & 90,245.51 \\
Median Absolute Percentage Error (\%) & 10.98 \\
Median log error & -0.0095 \\
Median absolute log error & 0.1086 \\

\bottomrule
\end{tabular*}
\end{table}

Table~\ref{tab:prediction_metrics_ols_private_auction} reports the 2025 out-of-sample accuracy of the locally adjusted global OLS benchmark. These metrics should be distinguished from the estimation-sample diagnostics reported in Tables~\ref{tab:ols_summary_stats_private_treaty} and \ref{tab:ols_summary_stats_auction}.\footnote{Log error is defined as $\log(P^{\mathrm{actual}})-\log(P^{\mathrm{predicted}})$.}

The private-treaty model performs somewhat better in typical error terms. The median absolute error is AUD 90,246 and the median absolute percentage error is 10.98\%, both below the corresponding auction errors.  Overall, the locally adjusted OLS model provides a useful transparent benchmark, but its prediction errors remain large enough to motivate the spatially adaptive GWR benchmark.
 
 %%%%%
 Table~\ref{tab:ols_agent_classification_private_auction} compares realised 2025 sale prices with two alternative price ranges: the advertised agent range and the OLS-implied valuation range. The OLS range is not intended to be a formal statistical prediction interval. It is a model-implied guide range with the same 10\% upper-to-lower width used in the classification exercise. The table therefore provides a transparent comparison between the public guide supplied to buyers and a benchmark guide constructed from observed housing characteristics and local-market information.\footnote{The 10\% upper-to-lower valuation width is chosen to align the model-implied valuation interval with the institutional logic of advertised price guides. Victorian underquoting regulation requires agents to justify price guidance using comparable sales and local market information, and policy discussion of underquoting commonly treats a 10\% guide range or a sale price more than 10\% above the guide as an economically meaningful threshold. The purpose of the 10\% interval in this paper is therefore not to define legal underquoting, but to place the model-implied benchmark on a comparable scale to the advertised guide range observed by buyers. This also ensures that the OLS and GWR benchmarks are evaluated using the same proportional interval width.}

 For private-treaty sales, 56.05\% of realised prices fall within the advertised agent range, while 19.62\% are above the advertised maximum and 24.33\% are below the advertised minimum. This relatively balanced pattern is consistent with the role of the private-treaty guide as a bargaining range. In contrast, the OLS-implied range shows greater dispersion, with 24.86\% of sales falling within the predicted range, 34.65\% above it, and 40.49\% below it. This pattern reflects both the limitations of a global hedonic benchmark and the deliberately narrow 10\% range imposed around the predicted price.

The auction segment shows a much sharper asymmetry in the advertised guide comparison. Around 65.72\% of auction sales sell above the advertised maximum, while only 8.01\% sell below the advertised minimum. This pattern is consistent with the idea that auction guides often sit below eventual transaction prices and may help attract bidder participation before the auction. By contrast, the OLS-implied auction range classifies 19.01\% of sales inside the predicted range, 36.08\% above it and 44.91\% below it. The OLS benchmark therefore does not reproduce the strong upward asymmetry observed in advertised auction guides. However, the listing-to-sale gap alone is not sufficient to identify underquoting, since final auction prices also reflect competition among bidders and market conditions at the time of sale. The OLS-implied range provides a useful transparent benchmark comparison, but its global structure and remaining prediction errors motivate the use of GWR as the main local valuation benchmark.
 %%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{\small The table classifies realised 2025 sale prices relative to advertised agent guides and OLS-implied guide ranges. The OLS-implied range is constructed around the predicted price $\widehat{P}$. The lower and upper bounds are $\widehat{P}/\sqrt{1.10}$ and $\widehat{P}\sqrt{1.10}$, so the ratio of the upper bound to the lower bound is 1.10. Percentages are calculated within each sale segment.}
\label{tab:ols_agent_classification_private_auction}
\small
\renewcommand{\arraystretch}{1.3}

\begin{tabular*}{0.90\textwidth}{@{\extracolsep{\fill}}l l r r}
\toprule
\textbf{Benchmark} & \textbf{Category} & \textbf{Count} & \textbf{Percent (\%)} \\
\midrule

\multicolumn{4}{l}{\textbf{Panel A: Private-treaty sales}} \\
\midrule
Agent guide 
& Price between listing minimum and maximum 
& 1{,}488 & 56.05 \\

Agent guide 
& Price above listing maximum 
& 521 & 19.62 \\

Agent guide 
& Price below listing minimum 
& 646 & 24.33 \\

OLS
& Price between predicted minimum and maximum 
& 660 & 24.86 \\

OLS
& Price above predicted maximum 
& 920 & 34.65 \\

OLS
& Price below predicted minimum 
& 1{,}075 & 40.49 \\

\midrule
\multicolumn{4}{l}{\textbf{Panel B: Auction}} \\
\midrule
Agent guide 
& Price between listing minimum and maximum 
& 351 & 26.27 \\

Agent guide 
& Price above listing maximum 
& 878 & 65.72 \\

Agent guide 
& Price below listing minimum 
& 107 & 8.01 \\

OLS
& Price between predicted minimum and maximum 
& 254 & 19.01 \\

OLS
& Price above predicted maximum 
& 482 & 36.08 \\

OLS
& Price below predicted minimum 
& 600 & 44.91 \\

\bottomrule
\end{tabular*}
\end{table}


\subsection{Geographically weighted regression (GWR)}

To allow for spatial heterogeneity in housing price determinants, we next estimate a geographically weighted regression (GWR) model, which extends the hedonic framework by allowing the marginal effects of housing attributes to vary across space \citep{brunsdon1996gwr,fotheringham2002gwr,fotheringham2009}. Unlike the fixed-effects hedonic model, which imposes common marginal effects across all suburbs, the GWR framework permits the pricing of structural and locational characteristics to differ across local housing submarkets. This flexibility is particularly relevant in Melbourne, where the value of housing attributes may vary substantially across locations.
The model is estimated on completed 2023--2024 transactions and then applied out of sample to the 2025 evaluation sample. As in the fixed-effects hedonic analysis, the estimation is conducted separately within each sale segment. The auction model is estimated using auction transactions only, while the private-treaty model is estimated using private-treaty transactions only. The method-of-sale variable is therefore used to define the estimation sample rather than being included as a regressor.

The GWR specification is
\begin{equation}
\label{eq:gwr_general}
\ln(\text{SalePrice}_i)
=
\beta_0(x_i,y_i)
+
\mathbf{X}_i^{\prime}\boldsymbol{\beta}(x_i,y_i)
+
v_i,
\end{equation}
where $(x_i,y_i)$ denotes the geographic coordinates of property $i$, and $\boldsymbol{\beta}(x_i,y_i)$ is a vector of location-specific coefficients. Unlike the fixed-effects hedonic model, which imposes a common set of marginal effects across all locations, the GWR model allows the relationship between property attributes and sale prices to vary spatially.

The covariate vector $\mathbf{X}_i$ contains the same parsimonious core variables used in the revised hedonic benchmark. These include structural characteristics, local market conditions, accessibility measures, seasonal indicators and land-use controls. Specifically, the structural and locational variables are bedrooms, bathrooms, parking, log land size, log suburb-level median rent, distance to public transport and distance to the Melbourne CBD. Seasonal indicators are included for autumn, winter and summer, with spring as the omitted category. Land-use indicators are included where available, with rare categories grouped into an ``Other'' category to improve estimation stability. 
The fitted log prices are exponentiated to obtain model-implied expected property values for the 2025 evaluation sample. These predicted values are then used to construct model-implied valuation intervals, which are compared with advertised listing ranges and realised transaction prices in the underquoting analysis.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Construction of the GWR valuation interval.}

Because the underquoting classification depends on the model-implied valuation bounds, we construct the GWR interval explicitly in five steps. Let $i$ denote a 2025 evaluation property in sale segment $m \in \{A,PT\}$, where $A$ denotes auction and $PT$ denotes private-treaty sales. Let $\widehat{y}^{GWR}_{im}$ be the GWR prediction for log sale price, obtained from the segment-specific GWR model estimated on 2023--2024 completed transactions. The corresponding price-level prediction is
\[
\widehat{P}^{GWR}_{im}
=
\exp\left(\widehat{y}^{GWR}_{im}\right).
\]

First, we construct a raw model-implied guide interval around the GWR prediction using the same 10\% upper-to-lower width used for the hedonic benchmark. Define
\[
h=\frac{1}{2}\log(1.10).
\]
The initial GWR bounds are
\[
\underline{\widehat{P}}^{0,GWR}_{im}
=
\exp\left(\widehat{y}^{GWR}_{im}-h\right),
\qquad
\overline{\widehat{P}}^{0,GWR}_{im}
=
\exp\left(\widehat{y}^{GWR}_{im}+h\right).
\]
Equivalently,
\[
\underline{\widehat{P}}^{0,GWR}_{im}
=
\frac{\widehat{P}^{GWR}_{im}}{\sqrt{1.10}},
\qquad
\overline{\widehat{P}}^{0,GWR}_{im}
=
\widehat{P}^{GWR}_{im}\sqrt{1.10},
\]
so that
\[
\frac{\overline{\widehat{P}}^{0,GWR}_{im}}
{\underline{\widehat{P}}^{0,GWR}_{im}}
=
1.10.
\]

Second, we construct a local real-time prediction-error adjustment using only prior completed sales.\footnote{The comparable pool consists only of completed 2025 evaluation-sample transactions whose sale dates precede $t_i$. For each comparable transaction $j$, the associated prediction is generated out of sample using the segment-specific GWR model estimated only on the 2023--2024 training data. The realised price of comparable transaction $j$ is therefore not used in estimating its own GWR prediction. This ensures that the local prediction-error adjustment uses only information that would have been available before the sale of property $i$ and avoids in-sample residuals from the GWR training period.} For each 2025 property $i$ with sale date $t_i$, we first search for comparable transactions $j$ from the same sale segment, same suburb and same bedroom count, with sale dates satisfying
\[
0 < t_i-t_j \leq 90 \ \text{days}.
\]
For each comparable sale $j$, define the log prediction error
\[
e_j
=
\log(P_j)-\log(\widehat{P}^{GWR}_{jm}),
\]
where $P_j$ is the realised sale price and $\widehat{P}^{GWR}_{jm}$ is the GWR prediction for that prior transaction. Thus, only information that would have been available before the 2025 sale of property $i$ is used.

Third, if no same-suburb and same-bedroom prior comparable is available within the 90-day window, we use a spatial fallback rule.\footnote{The 90-day window and 3-km fallback radius are chosen to approximate the type of recent and local comparable-sales information available to agents and buyers at the time of listing, while preserving enough observations for a property-level real-time adjustment. As a robustness check, we re-estimate the main underquoting classification using alternative interval widths and comparable-sales windows. The auction--private-treaty ranking remains unchanged.} The comparable set is expanded to prior transactions from the same sale segment and same bedroom count located within 3 km of property $i$, again requiring
\[
0 < t_i-t_j \leq 90 \ \text{days}.
\]
The adjustment is based on all prior transactions satisfying the relevant rule; the number of comparable sales is therefore allowed to vary by property. If no prior comparable is available even after the 3 km fallback, no local adjustment is applied and the adjustment term is set to zero.

Fourth, the local adjustment for property $i$ is the median prior log prediction error:
\[
\widetilde{e}_i
=
\operatorname{median}\{e_j:j\in \mathcal{C}_i\},
\]
where $\mathcal{C}_i$ denotes the selected comparable set. If $\mathcal{C}_i$ is empty, $\widetilde{e}_i=0$. The adjusted GWR prediction is then
\[
\widehat{y}^{adj,GWR}_{im}
=
\widehat{y}^{GWR}_{im}
+
\widetilde{e}_i,
\]
or, equivalently in price levels,
\[
\widehat{P}^{adj,GWR}_{im}
=
\widehat{P}^{GWR}_{im}
\exp(\widetilde{e}_i).
\]
Thus, the comparable-sales correction shifts the GWR midpoint upward or downward according to recent local forecast errors.

Fifth, the final GWR valuation interval is constructed around the adjusted GWR prediction using the same 10\% upper-to-lower width:
\[
\underline{\widehat{P}}^{GWR}_{im}
=
\exp\left(\widehat{y}^{adj,GWR}_{im}-h\right),
\qquad
\overline{\widehat{P}}^{GWR}_{im}
=
\exp\left(\widehat{y}^{adj,GWR}_{im}+h\right).
\]
Equivalently,
\[
\underline{\widehat{P}}^{GWR}_{im}
=
\frac{\widehat{P}^{adj,GWR}_{im}}{\sqrt{1.10}},
\qquad
\overline{\widehat{P}}^{GWR}_{im}
=
\widehat{P}^{adj,GWR}_{im}\sqrt{1.10}.
\]
The proportional width of the GWR interval is therefore fixed at 10\% for every property, while the dollar width varies with the adjusted predicted price. Advertised guide prices are not used in estimating the GWR model, constructing the local error adjustment, or forming the GWR valuation interval.\footnote{The adjustment is based on all prior transactions satisfying the relevant rule; the number of comparable sales is therefore allowed to vary by property.}
%%%%%%%%%%%%%%%%%%%%
The GWR-based underquoting indicator is then defined as
\[
U^{GWR}_i
=
\mathbf{1}
\left\{
P_i >
\overline{L}_i
\ \text{and}\
P_i >
\underline{\widehat{P}}^{GWR}_{im}
\right\},
\]
where $P_i$ is the realised sale price, $\overline{L}_i$ is the advertised upper guide, and $\underline{\widehat{P}}^{GWR}_{im}$ is the comparable-adjusted lower bound of the GWR valuation interval. This screened classification does not identify a property as underquoted solely because its realised sale price exceeds the advertised range. It additionally requires the realised price to exceed the lower bound of the guide-independent model-implied valuation interval, thereby reducing reliance on the guide-to-sale price gap alone.
%%%%%%%%%%%%%%%%%%%%
Table~\ref{tab:gwr_summary_stats_local_auction_k25} shows substantial spatial variation in the auction pricing equation. The GWR coefficients should not be interpreted as average marginal effects because they vary locally across Melbourne. The table is therefore best interpreted as evidence on the distribution of local pricing relationships rather than as a direct comparison of individual coefficients.

For bedrooms, the median local coefficient is 0.0498, with an interquartile range from -0.0030 to 0.1112. For bathrooms, the median is 0.1195, with an interquartile range from 0.0521 to 0.1944. These estimates indicate that the price gradients associated with dwelling characteristics are generally positive but vary across local housing submarkets. Land size also displays a positive local gradient. The median coefficient on $\ln(\text{Land size})$ is 0.2614, with an interquartile range from 0.1179 to 0.3997, suggesting that land area is capitalised into auction prices, but with different intensity across locations. Local market conditions, measured by $\ln(\text{Median rent})$, also vary across space, with a median coefficient of 0.0626 and an interquartile range from -0.4807 to 0.5997.

Distance-related effects also vary across locations. The median coefficient on distance to transport is 0.0076, with an interquartile range from -0.2246 to 0.2316, while the median coefficient on distance to the CBD is -0.0202, with an interquartile range from -0.2248 to 0.1968. These patterns show that accessibility gradients are not constant across Melbourne, which is precisely the type of spatial heterogeneity that a global OLS model cannot capture. Overall, the table supports the use of a spatially adaptive benchmark, but the wide coefficient ranges mean that the GWR model should be judged mainly by its out-of-sample prediction performance and its role as a local valuation benchmark.
%%%%%%%

\begin{table}[H]
\centering
\caption{\small AUCTION: Distribution of local coefficient estimates from the manual GWR model estimated on the 2023--2024 training sample. The table reports the cross-sectional distribution of location-specific parameter estimates and global diagnostic statistics for the spatially varying coefficient specification. AIC, and BIC are computed from the manual GWR Gaussian likelihood using the effective number of parameters.}
\label{tab:gwr_summary_stats_local_auction_k25}
\small
\renewcommand{\arraystretch}{1.25}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrrrr}
\toprule
\textbf{Variable} & \textbf{Min} & \textbf{1st Qu.} & \textbf{Median} & \textbf{Mean} & \textbf{3rd Qu.} & \textbf{Max} \\
\midrule

Intercept
& -143.6859 & 5.4767 & 11.4816 & 11.5588 & 17.3866 & 150.5031 \\

Bedrooms
& -0.6233 & -0.0030 & 0.0498 & 0.0617 & 0.1112 & 4.4280 \\

Bathrooms
& -3.4001 & 0.0521 & 0.1195 & 0.1690 & 0.1944 & 13.5481 \\

Parking
& -2.6667 & -0.0245 & 0.0148 & 0.0385 & 0.0597 & 11.3312 \\

$\ln(\text{Land size})$
& -2.9039 & 0.1179 & 0.2614 & 0.2549 & 0.3997 & 3.8010 \\

$\ln(\text{Median rent})$
& -20.9810 & -0.4807 & 0.0626 & 0.0425 & 0.5997 & 11.4836 \\

Distance to transport
& -7.9318 & -0.2246 & 0.0076 & -0.0080 & 0.2316 & 5.6806 \\

Distance to CBD
& -5.3529 & -0.2248 & -0.0202 & -0.0056 & 0.1968 & 7.8289 \\

Autumn
& -5.5000 & -0.0820 & -0.0035 & -0.0034 & 0.0748 & 5.9738 \\

Winter
& -5.4481 & -0.0819 & 0.0012 & -0.0010 & 0.0827 & 4.1038 \\

Summer
& -1.8180 & -0.0948 & -0.0069 & -0.0083 & 0.0798 & 4.1006 \\

\midrule
Season indicators & \multicolumn{6}{c}{Yes} \\
Suburb fixed effects & \multicolumn{6}{c}{Yes} \\
\midrule
Adjusted $R^{2}$ & \multicolumn{6}{r}{0.9255} \\
Akaike information criterion (AIC) & \multicolumn{6}{r}{-27,561.23} \\
Bayesian information criterion (BIC) & \multicolumn{6}{r}{88,600.89} \\
Median absolute error (AUD) & \multicolumn{6}{r}{30,240.27} \\
\bottomrule
\end{tabular*}
\end{table}
%%%%%%%%%%%%%%
Table~\ref{tab:gwr_summary_stats_local_private_treaty_k25} reports a similarly strong degree of local heterogeneity for private-treaty transactions. The median bedroom coefficient is positive at 0.0461, while the median bathroom coefficient is larger at 0.1133, again indicating that bathroom capacity is more strongly priced than bedrooms across many local markets. The land-size coefficient has a positive median of 0.2491, but the first and third quartiles are 0.1133 and 0.3888, showing that the value of land size is not uniform across Melbourne. In some locations larger land parcels are rewarded more strongly, while in others the estimated premium is weaker once local market conditions and other dwelling characteristics are controlled for. The median-rent coefficient is positive at 0.1404, suggesting that local rental-market strength remains informative for private-treaty prices, although the wide range of estimates points to substantial neighbourhood variation. Distance to the CBD has a small positive median coefficient of 0.0193, with an interquartile range from -0.2588 to 0.3164, reinforcing that the accessibility gradient is spatially uneven rather than constant across Melbourne.

%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{\small Private-treaty sALES: Distribution of local coefficient estimates from the manual GWR model estimated on the 2023--2024 training sample. The table reports the cross-sectional distribution of location-specific parameter estimates and global diagnostic statistics for the spatially varying coefficient specification. AIC and BIC are computed from the manual GWR Gaussian likelihood using the effective number of parameters.}
\label{tab:gwr_summary_stats_local_private_treaty_k25}
\small
\renewcommand{\arraystretch}{1.25}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrrrr}
\toprule
\textbf{Variable} & \textbf{Min} & \textbf{1st Qu.} & \textbf{Median} & \textbf{Mean} & \textbf{3rd Qu.} & \textbf{Max} \\
\midrule

Intercept
& -158.0759 & 0.0466 & 7.8053 & 8.9038 & 16.6105 & 302.9506 \\

Bedrooms
& -5.2716 & 0.0013 & 0.0461 & 0.0636 & 0.1043 & 14.0843 \\

Bathrooms
& -105.7962 & 0.0390 & 0.1133 & 0.2749 & 0.1989 & 43.2256 \\

Parking
& -20.0371 & -0.0178 & 0.0169 & 0.0685 & 0.0561 & 37.4105 \\

$\ln(\text{Land size})$
& -3.2855 & 0.1133 & 0.2491 & 0.2492 & 0.3888 & 15.3800 \\

$\ln(\text{Median rent})$
& -73.2336 & -0.2634 & 0.1404 & 0.2266 & 0.6621 & 44.1714 \\

Distance to transport
& -17.3489 & -0.2860 & 0.0160 & -0.0040 & 0.2881 & 16.1177 \\

Distance to CBD
& -16.1698 & -0.2588 & 0.0193 & 0.0338 & 0.3164 & 15.8689 \\

Autumn
& -4.5488 & -0.0675 & -0.0025 & -0.0022 & 0.0564 & 9.0875 \\

Winter
& -9.5567 & -0.0642 & 0.0000 & 0.0005 & 0.0657 & 7.7584 \\

Summer
& -9.0641 & -0.0742 & -0.0049 & -0.0049 & 0.0605 & 9.0167 \\

\midrule 
Season indicators & \multicolumn{6}{c}{Yes} \\ 
Suburb fixed effects & \multicolumn{6}{c}{Yes} \\
\midrule
Adjusted $R^{2}$ & \multicolumn{6}{r}{0.9173} \\
Akaike information criterion (AIC) & \multicolumn{6}{r}{-59,558.70} \\ 
Bayesian information criterion (BIC) & \multicolumn{6}{r}{216,141.63} \\
Median absolute error (AUD) & \multicolumn{6}{r}{19,769.13} \\
\bottomrule
\end{tabular*}
\end{table}
%%

%%%%%

%%%
 The model diagnostics show a strong in-sample fit, with an adjusted $R^2$ of 0.9173 and median absolute percentage error of 2.52\%. However, as with the auction model, the very large coefficient ranges imply that the GWR results should be evaluated mainly through out-of-sample prediction and guide-classification performance.
%%%
Table~\ref{tab:gwr_prediction_metrics_auction_private} shows that the GWR model materially improves the typical prediction error relative to the global OLS benchmark. For auction transactions, the median absolute error is AUD 72,305, meaning that for the typical auction property the GWR-predicted value is within about AUD 72,000 of the realised sale price. This is a substantial reduction compared with the OLS auction median absolute error reported earlier. The median absolute percentage error is also low at 7.05\%, indicating that the model provides a reasonably tight valuation benchmark for the median auction transaction. The median log error is close to zero at $-0.0110$, suggesting little systematic median overprediction or underprediction.\\
For private-treaty transactions, the typical prediction error is similarly favourable. The median absolute error is AUD 60,346 and the median absolute percentage error is 7.03\%. These values indicate that the GWR model gives a practically useful estimate of market value for the central part of the private-treaty distribution. The median log error of $-0.0075$ is very small, again suggesting that the model is not materially biased at the median.  Therefore, the GWR model is strongest as a median-based local valuation benchmark, rather than as a model that eliminates all extreme forecast errors.
%%%%%%%%
\begin{table}[H]
\centering
\caption{\small
Out-of-sample prediction accuracy for the 2025 test dataset based on the GWR model. 
The table reports forecast evaluation metrics measuring prediction errors in both absolute dollar terms and relative percentage terms. 
Absolute error P50 denotes the median absolute prediction error and reflects the typical deviation between predicted and realised transaction prices.}
\label{tab:gwr_prediction_metrics_auction_private}
\small
\renewcommand{\arraystretch}{1.2}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}l r}
\toprule
\textbf{Metric} & \textbf{Value} \\
\midrule

\multicolumn{2}{l}{\textbf{Panel A: Auction}} \\
\midrule
Absolute error P50 (median absolute error, AUD) & 72,305.36 \\
Median Absolute Percentage Error (\%) & 7.05 \\
Median log error & -0.0110 \\
Median absolute log error & 0.0713 \\

\midrule
\multicolumn{2}{l}{\textbf{Panel B: Private-treaty sales}} \\
\midrule
Absolute error P50 (median absolute error, AUD) & 60,346.11 \\
Median Absolute Percentage Error (\%) & 7.03 \\
Median log error & -0.0075 \\
Median absolute log error & 0.0701 \\

\bottomrule
\end{tabular*}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Table~\ref{tab:gwr_agent_classification_private_auction} compares realised 2025 sale prices with two ranges: the advertised agent guide and the GWR-implied valuation range. The table is central to the price-guidance analysis because the GWR range is constructed from observed property characteristics and local pricing relationships, rather than from the selling agent's advertised guide.

The sharpest contrast appears in the auction panel. Under the advertised agent guide, only 26.27\% of auction prices fall within the quoted range, while 65.72\% sell above the listed maximum and only 8.01\% fall below the listed minimum. This is not simply evidence of greater dispersion around the guide; the errors are strongly asymmetric and concentrated above the advertised maximum. The GWR-implied range gives a more balanced comparison: 42.44\% of auction prices fall inside the predicted range, 22.38\% are above the predicted maximum, and 35.18\% are below the predicted minimum. Although the GWR model does not eliminate prediction error, its classifications are not systematically concentrated below realised auction prices in the same way as the advertised agent guides.

For private-treaty sales, the advertised guide is much more closely aligned with realised transaction prices. In this segment, 56.05\% of prices fall inside the advertised range, 19.62\% are above the listed maximum, and 24.33\% are below the listed minimum. This pattern is consistent with the interpretation that private-treaty guides operate partly as bargaining ranges: they anchor buyer offers and seller expectations, and therefore remain more closely connected to the eventual transaction price. The GWR classification is more dispersed, with 41.28\% inside the predicted range, 29.30\% above, and 29.42\% below, but it does not display the same strong upward asymmetry observed for auction agent guides.
The evidence is consistent with the comparative predictions of the framework in Proposition~\ref{pro4}, which predicts that final transaction prices should be less likely to fall within the quoted range in auction campaigns than in private-treaty sales. The relevant comparison is the advertised agent guide. Auction prices are far less likely to remain inside the quoted range than private-treaty prices, and they are much more likely to exceed the advertised maximum. This is precisely the pattern implied by the participation-role mechanism: in auctions, a lower guide can attract bidder interest before the competitive process begins, while the final price is subsequently determined through bidding. In private-treaty sales, by contrast, the guide is more disciplined by its bargaining and information role.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{\small
Classification of realised transaction prices relative to GWR-implied valuation bounds and advertised listing price ranges for the 2025 test dataset.}
\label{tab:gwr_agent_classification_private_auction}
\small
\renewcommand{\arraystretch}{1.15}

\begin{tabular*}{0.90\textwidth}{@{\extracolsep{\fill}}l l r r}
\toprule
\textbf{Benchmark} & \textbf{Category} & \textbf{Count} & \textbf{Percent (\%)} \\
\midrule

\multicolumn{4}{l}{\textbf{Panel A: Private-treaty sales}} \\
\midrule
Agent guide 
& Price between listing minimum and maximum 
& 1{,}488 & 56.05 \\

Agent guide 
& Price above listing maximum 
& 521 & 19.62 \\

Agent guide 
& Price below listing minimum 
& 646 & 24.33 \\

GWR
& Price between predicted minimum and maximum 
& 1{,}096 & 41.28 \\

GWR
& Price above predicted maximum 
& 778 & 29.30 \\

GWR
& Price below predicted minimum 
& 781 & 29.42 \\

\midrule
\multicolumn{4}{l}{\textbf{Panel B: Auction}} \\
\midrule
Agent guide 
& Price between listing minimum and maximum 
& 351 & 26.27 \\

Agent guide 
& Price above listing maximum 
& 878 & 65.72 \\

Agent guide 
& Price below listing minimum 
& 107 & 8.01 \\

GWR 
& Price between predicted minimum and maximum 
& 567 & 42.44 \\

GWR
& Price above predicted maximum 
& 299 & 22.38 \\

GWR 
& Price below predicted minimum 
& 470 & 35.18 \\

\bottomrule
\end{tabular*}
\end{table}
This subsection examines the type of properties classified as underquoted under the conservative GWR-based rule. The aim is to see whether underquoting is concentrated in particular dwelling types rather than being evenly distributed across all properties. The analysis focuses on bedrooms, bathrooms and parking, which are the most visible housing attributes in buyer search and price expectations.
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[H]
\centering
\caption{\small Log-based diagnostics for advertised guide distortion. The table reports how often model-implied valuation bounds exceed advertised listing bounds under the fixed-effects hedonic and GWR models, separately for private-treaty and auction transactions. Percentages are calculated within each panel using observations with valid advertised and model-implied bounds. The private-treaty and auction panels contain 2,655 and 1,336 observations, respectively.}

\label{tab:log_underquoting_index}
\small
\renewcommand{\arraystretch}{1.2}

\begin{tabular*}{0.95\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
\textbf{Condition} 
& \multicolumn{2}{c}{\textbf{Hedonic model}} 
& \multicolumn{2}{c}{\textbf{GWR model}} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
& \textbf{Count} & \textbf{Percent (\%)} 
& \textbf{Count} & \textbf{Percent (\%)} \\
\midrule

\multicolumn{5}{l}{\textbf{Panel A: Private-treaty sales} ($N = 2,655$)} \\
\midrule
$\log(\widehat{P}_{\max}) > \log(\text{List}_{\max})$
& 1{,}524 & 57.40
& 1{,}512 & 56.95 \\

$\log(\widehat{P}_{\min}) > \log(\text{List}_{\min})$
& 1{,}293 & 48.70
& 1{,}173 & 44.18 \\

\midrule
\multicolumn{5}{l}{\textbf{Panel B: Auction} ($N = 1,336$)} \\
\midrule
$\log(\widehat{P}_{\max}) > \log(\text{List}_{\max})$
& 932 & 69.76
& 1{,}038 & 77.69 \\

$\log(\widehat{P}_{\min}) > \log(\text{List}_{\min})$
& 888 & 66.47
& 983 & 73.58 \\

\bottomrule
\end{tabular*}


\end{table}
Table~\ref{tab:log_underquoting_index} compares advertised guide bounds with model-implied valuation bounds. The results show that advertised prices are frequently below the values implied by both valuation models, but the pattern is considerably stronger for auctions. Under the hedonic benchmark, the model-implied upper bound exceeds the advertised upper guide in 69.76\% of auction cases, compared with 57.40\% for private-treaty sales. The GWR comparison is even sharper: 77.69\% for auctions versus 56.95\% for private-treaty sales. The lower-bound comparison gives the same conclusion. Under the hedonic benchmark, the model-implied lower bound exceeds the advertised lower guide in 66.47\% of auction cases, compared with 48.70\% for private-treaty sales. Under GWR, the corresponding figures are 73.58\% and 44.18\%, respectively. Thus, auction guides are more frequently positioned below model-implied valuation bounds across both valuation benchmarks.

These results are consistent with the proposed mechanism for Proposition~\ref{pro2}, which predicts that auction guides are more strongly distorted downward than private-treaty guides. The important feature of Table~\ref{tab:log_underquoting_index} is that the comparison is made against independently estimated valuation intervals rather than realised prices alone. This matters because realised auction prices can be affected by bidder competition after the guide is announced. By contrast, the model-implied bounds are constructed from observed property characteristics, local market conditions and spatial information, without using the advertised guide as an explanatory variable. The fact that model-implied bounds exceed advertised bounds much more often in auctions indicates that auction guides are systematically positioned lower relative to market-consistent values. This pattern is consistent with the economic mechanism in Proposition~\ref{pro2}: in auctions, the guide has a stronger participation role, so lowering it can attract more potential bidders and intensify competition. In private-treaty sales, by contrast, the guide is more disciplined by its bargaining and information role.
Before using the GWR predictions in the guide-classification analysis, we examine the local estimation diagnostics to assess spatial coverage, local collinearity and out-of-sample stability. The preferred manual GWR specification uses an adaptive nearest-neighbour bandwidth with $k=25$ and an adaptive bisquare kernel. Thus, the bandwidth is not a fixed kilometre radius; it is the realised spatial radius required to include the local neighbours around each prediction point.  

The realised bandwidths remain local in scale. For private-treaty predictions, the median bandwidth is 0.51 km, with 90th and 95th percentiles of 0.91 km and 1.03 km, respectively. For auction predictions, the corresponding values are 0.69 km, 1.27 km and 1.64 km. Local condition numbers are generally moderate, with medians of 8.26 for private-treaty sales and 7.16 for auctions. Condition numbers exceed 30 in 11.19\% and 5.69\% of local regressions, respectively. The maximum-VIF diagnostics indicate that some local collinearity remains, especially for distance-to-transport and distance-to-CBD variables, so the local coefficients are not interpreted as structural marginal effects. Instead, the GWR is used as a local valuation benchmark.

Out-of-sample performance is strong at the median. The auction GWR produces a median absolute log error of 0.0713, while the private-treaty GWR has a median absolute log error of 0.0701. The empirical analysis emphasises median prediction accuracy, valuation-bound comparisons and guide-classification rates, rather than mean predictions or individual local coefficients. Overall, these diagnostics indicate that the preferred GWR specification provides a predominantly local and reasonably well-conditioned valuation benchmark for the subsequent classification analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dwelling characteristics of GWR-identified underquoted properties}

This subsection examines whether properties identified as underquoted under the conservative GWR-based rule share common observable dwelling characteristics. The purpose is not to estimate the causal effects of bedrooms, bathrooms or parking on underquoting, but to describe the housing segment in which the model-based underquoting signal occurs most frequently. A high share of three- or four-bedroom properties among the identified cases does not necessarily imply that these dwellings have a higher probability of underquoting, as it may partly reflect their prevalence in Melbourne's detached-house market. The results should therefore be interpreted as compositional evidence about the properties identified by the classification rule.
Table~\ref{tab:property_characteristics_gwr_auction} reports the dwelling characteristics of auction properties classified as underquoted. Most identified cases are conventional family-sized houses. Three-bedroom properties account for 44.99\% of the underquoted auction sample, while four-bedroom properties account for 40.07\%. Together, these groups represent 85.06\% of the identified auction cases. The bathroom and parking distributions exhibit a similar pattern: 58.91\% of the properties have two bathrooms and 61.97\% have two parking spaces. Thus, the auction underquoting signal is not concentrated primarily among very small dwellings or unusually large luxury properties. Instead, most identified cases occur among standard suburban houses likely to attract a broad pool of family-home buyers. This composition is consistent with the proposed participation mechanism, although it should not be interpreted as evidence that these dwelling types have higher conditional underquoting rates.

\begin{table}[H]
\centering
\caption{\small
Distribution of dwelling characteristics for auction properties classified as underquoted under the conservative GWR-based identification rule. Percentages are calculated within the underquoted auction sample.}
\label{tab:property_characteristics_gwr_auction}
\small
\renewcommand{\arraystretch}{1.2}

\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}l l r r}
\toprule
\textbf{Category} & \textbf{Value} & \textbf{Count} & \textbf{Percent (\%)} \\
\midrule

\multicolumn{4}{l}{\textit{Bedrooms}} \\
\midrule
Beds & 3 & 265 & 44.99 \\
Beds & 4 & 236 & 40.07 \\
Beds & 5 & 59 & 10.02 \\
Beds & 2 & 23 & 3.90 \\
Beds & 6 & 5 & 0.85 \\
Beds & 7 & 1 & 0.17 \\

\midrule
\multicolumn{4}{l}{\textit{Bathrooms}} \\
\midrule
Baths & 2 & 347 & 58.91 \\
Baths & 1 & 192 & 32.60 \\
Baths & 3 & 47 & 7.98 \\
Baths & 4 & 3 & 0.51 \\

\midrule
\multicolumn{4}{l}{\textit{Parking}} \\
\midrule
Parking & 2 & 365 & 61.97 \\
Parking & 1 & 85 & 14.43 \\
Parking & 4 & 63 & 10.70 \\
Parking & 3 & 59 & 10.02 \\
Parking & 5 & 9 & 1.53 \\
Parking & 6 & 8 & 1.36 \\

\midrule
\multicolumn{4}{l}{\textit{Method of Sale}} \\
\midrule
Method of sale & Sold at auction & 589 & 100.00 \\

\bottomrule
\end{tabular*}
\end{table}
Table~\ref{tab:property_characteristics_gwr_private_treaty} reports the corresponding distribution for private-treaty properties. The composition is broadly similar to that of the auction sample. Three-bedroom dwellings account for 49.06\% of the identified private-treaty cases, while four-bedroom dwellings account for 40.75\%. Together, these categories represent 89.81\% of the private-treaty cases. Two-bathroom properties account for 64.34\% of the sample, and properties with two parking spaces account for 66.76\%. These results similarly indicate that the identified private-treaty cases are predominantly standard suburban family homes rather than atypical properties. As with the auction results, however, the percentages describe the composition of the identified sample and do not establish that these property types have higher underquoting rates relative to their representation in the full private-treaty sample.

\begin{table}[H]
\centering
\caption{\small
Distribution of dwelling characteristics for private-treaty properties classified as underquoted under the conservative GWR-based identification rule. Percentages are calculated within the underquoted private-treaty sample.}
\label{tab:property_characteristics_gwr_private_treaty}
\small
\renewcommand{\arraystretch}{1.2}

\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}l l r r}
\toprule
\textbf{Category} & \textbf{Value} & \textbf{Count} & \textbf{Percent (\%)} \\
\midrule

\multicolumn{4}{l}{\textit{Bedrooms}} \\
\midrule
Beds & 3 & 183 & 49.06 \\
Beds & 4 & 152 & 40.75 \\
Beds & 5 & 19 & 5.09 \\
Beds & 2 & 13 & 3.49 \\
Beds & 6 & 5 & 1.34 \\
Beds & 1 & 1 & 0.27 \\

\midrule
\multicolumn{4}{l}{\textit{Bathrooms}} \\
\midrule
Baths & 2 & 240 & 64.34 \\
Baths & 1 & 109 & 29.22 \\
Baths & 3 & 22 & 5.90 \\
Baths & 4 & 2 & 0.54 \\

\midrule
\multicolumn{4}{l}{\textit{Parking Spaces}} \\
\midrule
Parking & 2 & 249 & 66.76 \\
Parking & 1 & 47 & 12.60 \\
Parking & 3 & 35 & 9.38 \\
Parking & 4 & 32 & 8.58 \\
Parking & 5 & 5 & 1.34 \\
Parking & 6 & 5 & 1.34 \\

\midrule
\multicolumn{4}{l}{\textit{Method of Sale}} \\
\midrule
Method of sale & Private-treaty  sales & 373 & 100.00 \\

\bottomrule
\end{tabular*}
\end{table}

The incidence estimates are consistent with Proposition~\ref{pro5}, which predicts that underquoting is more prevalent in auction campaigns than in private-treaty sales. Under the conservative GWR-based rule, 589 of the 1,336 auction transactions are classified as underquoted, corresponding to 44.09\% of the auction evaluation sample. The corresponding figure for private-treaty sales is 373 of 2,655 transactions, or 14.05\%. The estimated incidence is therefore more than three times higher for auctions. Tables~\ref{tab:property_characteristics_gwr_auction} and~\ref{tab:property_characteristics_gwr_private_treaty} complement this comparison by describing the characteristics of the identified properties within each mechanism. Because classification requires the realised price to exceed both the advertised upper guide and the lower bound of the GWR-implied valuation interval, the measure reduces reliance on the guide-to-sale price gap alone. The substantially higher auction incidence is consistent with the proposed participation mechanism, under which lower guides may attract additional bidders and intensify competition, while private-treaty guides remain more closely connected to bargaining and negotiation.

\subsubsection{Formal comparison across sales mechanisms.}

To assess whether the auction--private-treaty differences are statistically meaningful rather than reflecting descriptive variation in sample proportions, we complement the classification evidence with formal binary-outcome tests. Let $Y_i$ denote a binary guide outcome for property $i$. For the principal test, $Y_i$ is the conservative GWR-based underquoting indicator, defined as
\[
Y_i^{GWR}
=
\mathbf{1}
\left\{
P_i>\overline{L}_i
\ \text{and}\
P_i>\underline{\widehat{P}}^{GWR}_i
\right\},
\]
where $P_i$ is the realised sale price, $\overline{L}_i$ is the advertised upper guide, and $\underline{\widehat{P}}^{GWR}_i$ is the lower bound of the GWR-implied valuation interval. We also construct indicators for whether the sale price falls inside, above or below the advertised guide and whether the GWR-implied valuation bounds exceed the corresponding advertised bounds. For each outcome, we estimate the linear probability model
\[
Y_i
=
\alpha
+
\beta Auction_i
+
X_i'\gamma
+
\varepsilon_i,
\]
where $Auction_i$ equals one for auction transactions and zero for private-treaty transactions. The vector $X_i$ contains property, location, season and land-use controls, and standard errors are clustered at the suburb level. The coefficient $\beta$ measures the regression-adjusted conditional difference between auction and private-treaty transactions; it should not be interpreted as a causal effect of selecting the auction mechanism.
Table~\ref{tab:formal_mechanism_tests_gwr} reports the results. The conservative GWR-based underquoting incidence is 44.09\% for auctions and 14.05\% for private-treaty sales, producing a raw difference of 30.04 percentage points. The difference is highly statistically significant, with a 95\% confidence interval of [27.07, 33.01] and a two-sample proportion-test $p$-value below 0.001. After controlling for observable property, location, season and land-use characteristics, the estimated auction--private-treaty difference remains 28.74 percentage points and is statistically significant at the 1\% level using suburb-clustered standard errors. Thus, the substantially higher auction incidence cannot be explained solely by differences in the observable composition of the two samples. The advertised-guide outcomes reinforce this conclusion. After controlling for observable characteristics, auction sale prices are 25.43 percentage points less likely to fall within the advertised guide and 42.40 percentage points more likely to exceed its upper bound. 



\begin{table}[H]
\centering
\caption{\small
Formal tests of auction--private-treaty differences in binary guide outcomes. Raw percentages are calculated using 1,336 auction transactions and 2,655 private-treaty transactions. The conservative GWR-based underquoting indicator equals one when the realised sale price exceeds both the advertised upper guide and the lower bound of the GWR-implied valuation interval. Adjusted differences are percentage-point coefficients from linear probability models containing property, location, season and land-use controls. Standard errors are clustered at the suburb level.}
\label{tab:formal_mechanism_tests_gwr}
\small
\renewcommand{\arraystretch}{1.45}

\begin{threeparttable}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{@{}lrrrrrrrr@{}}
\toprule
\textbf{Outcome}
& \makecell{\textbf{Private}\\\textbf{(\%)}}
& \makecell{\textbf{Auction}\\\textbf{(\%)}}
& \makecell{\textbf{Raw}\\\textbf{diff.}}
& \textbf{95\% CI}
& \makecell{\textbf{Prop.}\\\boldmath{$p$}}
& \makecell{\textbf{Adj.}\\\textbf{diff.}}
& \makecell{\textbf{Clust.}\\\textbf{SE}}
& \makecell{\textbf{Adj.}\\\boldmath{$p$}} \\
\midrule

\makecell[l]{Conservative GWR-based\\underquoted indicator}
& 14.05 & 44.09 & 30.04 & [27.07, 33.01] & $<0.001$ & 28.74 & 1.85 & $<0.001$ \\

\makecell[l]{Sale price inside\\advertised agent guide}
& 56.05 & 26.27 & -29.77 & [-32.80, -26.75] & $<0.001$ & -25.43 & 2.27 & $<0.001$ \\

\makecell[l]{Sale price above\\advertised agent maximum}
& 19.62 & 65.72 & 46.10 & [43.14, 49.06] & $<0.001$ & 42.40 & 2.19 & $<0.001$ \\

\makecell[l]{Sale price below\\advertised agent minimum}
& 24.33 & 8.01 & -16.32 & [-18.51, -14.14] & $<0.001$ & -16.98 & 1.64 & $<0.001$ \\

\makecell[l]{GWR upper bound exceeds\\advertised upper bound}
& 56.95 & 77.69 & 20.75 & [17.82, 23.67] & $<0.001$ & 22.00 & 1.99 & $<0.001$ \\

\makecell[l]{GWR lower bound exceeds\\advertised lower bound}
& 44.18 & 73.58 & 29.40 & [26.37, 32.42] & $<0.001$ & 30.04 & 1.90 & $<0.001$ \\

\bottomrule
\end{tabular}
\end{adjustbox}
\end{threeparttable}
\end{table}
%%%%%%%

The GWR-bound comparisons produce a similar pattern. The GWR-implied upper valuation bound exceeds the advertised upper guide for 77.69\% of auction properties, compared with 56.95\% of private-treaty properties. The corresponding lower-bound percentages are 73.58\% and 44.18\%, respectively. Overall, the results show economically large and statistically precise differences across sales mechanisms that remain after regression adjustment and are consistent with the comparative predictions of the theoretical framework.
%%%%%%
Figure~\ref{fig:millpark_gwr_guidance} provides a suburb-level visual summary of the GWR-based price guidance for the auction market. Panel (a) shows that the median GWR-implied midpoint is close to the median realised suburb-level sale price, while the median agent midpoint lies lower. This pattern is consistent with the earlier classification results, where advertised auction guides were frequently below realised prices. Panel (b) gives a more detailed suburb-by-suburb comparison. For many high-activity auction suburbs, the GWR-implied interval is positioned closer to the realised median price than the advertised agent interval. The difference is especially visible in suburbs where the black point lies above, or near the top of, the agent guide range but is more aligned with the GWR-implied range. This is useful because the GWR guide is constructed from observed property characteristics and local spatial information, not from the selling agent's advertised range. It therefore provides a more systematic benchmark for assessing whether auction price guidance is low relative to local market valuations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\centering

\begin{subfigure}{0.95\textwidth}
    \centering
    \includegraphics[width=0.85\textwidth]{auction_gwr_Ar.pdf}
    \caption{Distribution of suburb-level median realised auction prices.}
    \label{fig:millpark_gwr_distribution}
\end{subfigure}

\vspace{0.5cm}

\begin{subfigure}{0.95\textwidth}
    \centering
    \includegraphics[width=0.85\textwidth]{auction_gwrComparison_Ar.pdf}
    \caption{Suburb-level comparison of agent and GWR-implied intervals.}
    \label{fig:millpark_gwr_interval}
\end{subfigure}

\caption{\small
GWR-implied price guidance for auction transactions in the 2025 evaluation sample. Panel (a) shows the distribution of suburb-level median realised auction prices for suburbs with at least five auction transactions. The vertical reference lines report the median realised sale price, the median midpoint of the advertised agent range, and the median midpoint of the GWR-implied valuation range. Panel (b) compares advertised agent intervals with GWR-implied valuation intervals for the 30 suburbs with the largest number of auction transactions. Each row represents one suburb; the black point denotes the suburb-level median realised sale price. }
\label{fig:millpark_gwr_guidance}
\end{figure}


\section{Conclusions}\label{SecDiscussion}

This paper studies underquoting as a problem of price guidance rather than simply as a gap between the advertised price and the final sale price. The central argument is that advertised guides perform different economic roles across sales mechanisms. In auctions, the guide can be used to attract attention, increase inspections and expand the potential bidder pool, while the reserve price protects the seller from a low outcome. In private-treaty sales, the guide is more closely tied to the seller's asking position and the negotiation process, making misleadingly low guidance more costly.

The theoretical framework formalises this distinction by treating the guide as both a participation device and an information signal. It predicts that auction guides should be more strongly distorted downward, auction sale prices should be less likely to fall within the quoted range, and underquoting should be more prevalent in auction campaigns. The empirical evidence from Melbourne detached-house transactions is consistent with these predictions. Only 26.27\% of auction prices fall within the advertised range, compared with 56.05\% of private-treaty prices, while 65.72\% of auction properties sell above the advertised upper guide.

The model-based evidence reinforces this interpretation. The GWR-implied valuation range produces a more balanced distribution of realised prices than the advertised auction guide, suggesting that the strong upward asymmetry in auction outcomes is specific to agent guidance rather than valuation uncertainty alone. Under the conservative GWR-based rule, 44.09\% of auction transactions and 14.05\% of private-treaty transactions are classified as underquoted, indicating that the underquoting signal is substantially stronger in auctions.

These findings have practical implications for price-guidance governance. Model-implied valuation benchmarks can help regulators, agencies and listing platforms identify campaigns where advertised guides appear materially low relative to local market values. Such tools would not replace legal judgment, but they can support earlier monitoring, compliance review and more targeted enforcement. Overall, the paper provides a theoretical interpretation and empirical measurement framework for understanding underquoting in regulated housing markets.
%%%%%%%%%%%%%%%%%

\bibliographystyle{econ}
\bibliography{references}

\end{document}

