# RECORD 5170d50eab56ddf40c03eaf3e11552023cf9142d
# TITLE: House Price Models for Banking and Insurance Applications: The Impact of Property Characteristics
# YEAR: 2017.0  VENUE: nan  DOI: 10.1515/apjri-2017-0003
# AUTHORS: Adam W. Shao; Katja Hanewald; M. Sherris
# PAGES: 28  CANDIDATE TABLE PAGES (1-based): [7, 9, 10, 11, 12, 13]
# ABSTRACT: Abstract House price indices are needed to assess house price risk in households’ portfolio allocation decisions and in many housing-related financial products such as reverse mortgages, mortgage insurance and real estate derivatives. This paper first introduces nine widely-used house price models to the insurance, risk management and actuarial literature and provides new evidence on the relative performance of these models. We then show how portfolio-level house price indices for properties with specific physical and locational characteristics can be constructed for these different models. All analyses are based on a large dataset of individual property transactions in Sydney, Australia, for the period 1971-2011. The unrestricted hedonic model and a hybrid hedonic repeat-sales model provide a good model fit and reliable portfolio-level house price indices. Our results are important for banks, insurers and investors that have exposure to house price risks.


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House Price Models for Banking and Insurance Applications:
The Impact of Property Characteristics
Author:
Shao, Adam W; Hanewald, K; Sherris, M
Publication details:
Asia-Pacific Journal of Risk and Insurance
v. 12
Chapter No. 1
2153-3792 (ISSN)
Publication Date:
2018-01
Publisher DOI:
https://doi.org/10.1515/apjri-2017-0003
License:
https://creativecommons.org/licenses/by-nc-nd/4.0/
Link to license to see what you are allowed to do with this resource.
Downloaded from http://hdl.handle.net/1959.4/unsworks_51760 in https://
unsworks.unsw.edu.au on 2026-10-20


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man, 1978; Rosen, 1974). Hedonic models can be divided into restricted hedonic models
where the coeﬃcients of all property characteristics are assumed to be time-invariant
during the observation period and unrestricted hedonic models which allow the implicit
prices of property characteristics to change over time (e.g., Hansen, 2009; Triplett, 2006).
Both model variants are included in our analysis.
The age of a property is an important factor to include in hedonic models. Its eﬀect
on price can be decomposed into two components: a depreciation eﬀect, resulting from
physical deterioration of the house, and a vintage eﬀect, which captures changes in pref-
erences for certain building styles (Goodman and Thibodeau, 1995). We allow for both
depreciation and vintage eﬀects in the hedonic models and in the hybrid model as follows:
(i) We include a quadratic function of the dwelling age to estimate the depreciation eﬀect.
(ii) We include dummy variables indicating the decade in which the property was built to
capture the vintage eﬀect. The depreciation eﬀect can be negative if the property value
is improved through renovation and maintenance.
We estimate the following restricted hedonic model:
Vit = α + βt + X′
iγ + εit,
(2.1)
where Vit is the log price of individual property i at time t, α is the intercept, βt is a
time-varying coeﬃcient, Xi is a vector of K property characteristics, γ is a K × 1 vector
of coeﬃcients and εit is a white-noise disturbance term.
The estimated coeﬃcients ˆβt for the time dummy variables are used to construct a market-
wide house price index:2
HPIt = 100 exp

ˆβt

,
t > 0.
(2.2)
The unrestricted hedonic model we use includes interaction terms of time dummy
2The regression-based house price models are semi-log. The house price indices speciﬁed in Equa-
tions (2.2), (2.4), and (2.15) can be thought of as approximations to the median house price (Malpezzi
et al., 1998). This speciﬁcation is used in order to allow for comparisons with the two median methods.
The house price index at the base year is scaled to 100.
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variables and house characteristics to allow for time-varying implicit prices of property
characteristics, in a parsimonious and ﬂexible approach. The model is given by:
Vit = α + βt + X′
iγ + X′
i∆t + εit,
(2.3)
where ∆t is a time-varying K ×1 vector of coeﬃcients capturing the interactions between
the time dummy variables and the property characteristics.
A market-wide house price index at time t is calculated as:
HPIt = 100 exp
 
ˆβt +
K
X
u=1
ˆδutX0
u
!
,
t > 0,
(2.4)
where ˆβt is the estimated value of βt in Equations (2.3), X0
u denotes the average value of
the uth house characteristic in the base year, ˆδut is the estimated value of the uth element
of ∆t, and K is the number of property characteristics in the regression.3
2.3
Repeat-Sales Models
The choice of explanatory variables is critical in hedonic models and omission of import-
ant variables can result in biased estimates. Repeat-sales models aim to avoid the bias
introduced by an incorrectly speciﬁed hedonic model and take advantage of information
contained in repeated transactions of the same property (Bailey et al., 1963; Case et al.,
1991). The method focuses on properties that are transacted at least twice during the
sample period, assuming that property characteristics remain unchanged between transac-
tions. Repeat-sales models can suﬀer from a selection bias if properties that are transacted
multiple times diﬀer from properties that are transacted only once. We include a standard
repeat-sales model, an age-adjusted repeat-sales model and a weighted repeat-sales model
in our model comparison.
3Since property characteristics can change through time it is necessary to specify a reference period
to construct the indices. The Laspeyres index uses a base year, the Paasche index uses the most recent
year, and the Fisher index uses an average measure during the observation period (see, e.g., Hill and
Melser, 2008). We use the Laspeyres’ method in the unrestricted hedonic model and in the hybrid model.
It is assumed in the repeat-sales models that property characteristics do not change over time, so ﬁxing
quality measures for the reference period is not needed. All indices have a base-year value of 100.
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The standard repeat-sales model assumes that property characteristics remain con-
stant over time and that the implicit prices of property characteristics are time-invariant
(Bailey et al., 1963; Jansen et al., 2007; Wang and Zorn, 1997). We remove observations
where multiple transactions are recorded for the same property within one calendar year
in all repeat-sales models as these indicate renovations and other changes of property
characteristics (Clapp and Giaccotto, 1999). To derive the standard repeat-sales model
from the hedonic model, a speciﬁcation error ηi is introduced to the restricted hedonic
regression for an individual house in Equation (2.1):
Vit = α + βt + X′
iγ + ηi + ξit,
(2.5)
E(ξit) = 0, E(ηi) = 0 ,
E(ξ2
it) = σ2
ξ, E(η2
i ) = σ2
η ,
E(ξitξjs) = 0 if (i −j)2 + (t −s)2 ̸= 0 ,
E(ηiηj) = 0 if i ̸= j,
E(ηiξit) = 0.
(2.6)
Because the speciﬁcation error ηi is uncorrelated with the white noise term ξit, the variance
of ηi + ξit is σ2
η + σ2
ξ. Diﬀerencing Equation (2.5) gives the repeat-sales model:
Vjt −Vjs = βt −βs + ξjt −ξjs,
(2.7)
where s and t (s < t) are the times for a pair of consecutive sales of property j.
We also estimate an age-adjusted repeat-sales model which accounts for depreciation
of houses over time (McMillen, 2003; Shimizu et al., 2010).
We include a quadratic
function of the dwelling age in Equation (2.7) to account for the non-linear relationship
between a property’s age and value observed in the data:
Vjt −Vjs = τ1(Ajt −Ajs) + τ2(A2
jt −A2
js) + βt −βs + ξjt −ξjs,
(2.8)
where Ajt is the dwelling age of property j at time t. The diﬀerence between a property’s
age at two consecutive sales is identical to the interval between the two sales (i.e. Ajt −
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Ajs = t−s) and that causes collinearity between (Ajt−Ajs) and the time dummy variables
(βt−βs). We therefore remove the term Ajt−Ajs and estimate the following age-adjusted
repeat-sales model:
Vjt −Vjs = τ(A2
t −A2
s) + βt −βs + ξjt −ξjs.
(2.9)
The standard repeat-sales model assumes that disturbance terms are uncorrelated over
time and across individual properties and have a constant variance. Case and Shiller
(1987) found that the variance can depend on the time interval between sales and pro-
posed a weighted repeat-sales model where heteroskedasticity of the residuals in the
standard repeat-sales model is allowed for by assuming that the residuals’ variance de-
pends on the time intervals between consecutive sales. Their method requires a three-stage
regression. In the ﬁrst stage, the standard repeat-sales regression is used; in the second
stage, the squared residuals from the ﬁrst stage are regressed against the time intervals
between sales; ﬁnally the repeat-sales regression is performed after dividing each equation
by the square root of the variance estimates from the second stage.
We include the additional assumption in Equation (2.7) for the disturbance term:
E
 (ξjt −ξjs)2
= θ0 + θ1(t −s),
(2.10)
where θ0 and θ1 are coeﬃcients that are estimated by regressing the squared residuals
from Equation (2.7) against an intercept term and (t −s). The estimated coeﬃcients,
denoted by ˆθ0 and ˆθ1, are then used in a weighted linear regression of Equation (2.7) with
weights equal to

ˆθ0 + ˆθ1(t −s)
−1
2.
Based on each repeat-sales model a market-wide house price index can be constructed
using the estimated coeﬃcients ˆβt for the time dummy variables:
HPIt = 100 exp

ˆβt

,
t > 0.
(2.11)
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2.4
Hybrid Models
Hybrid models have been developed to combine the advantages of hedonic and repeat-
sales models. They incorporate data for single-sale and repeat-sale transactions and aim
to overcome the speciﬁcation error in hedonic models and the selection bias in repeat-sales
models. A ﬁrst hybrid model was developed by Case and Quigley (1991) and improved
by Quigley (1995) by introducing an explicit error structure. We apply the model by
Quigley (1995), which was also used in Wallace and Meese (1997), Jones (2010), and Goh
et al. (2012) (see Table 1). The model is derived from the unrestricted hedonic model
given in Equation (2.3) by introducing a random error term ηi to account for potential
speciﬁcation errors:
Vit = α + βt + X′
iγ + X′
i∆t + ηi + ξit,
(2.12)
where the assumptions on the two error terms are the same as in Equation (2.6). In the
following, we use εit to denote the sum ηi +ξit. By diﬀerencing Equation (2.12) we obtain
a repeat-sales regression equation for properties with multiple transactions:
Vit −Vis = βt −βs + X′
i (∆t −∆s) + ξit −ξis.
(2.13)
The hybrid model is estimated as a stack of three equations taking into account the cov-
ariance structure: Equation (2.12) for properties with single transactions, Equation (2.12)
again for properties with multiple transactions but excluding the last sales to avoid col-
linearity, and Equation (2.13) for properties with multiple transactions.
The covariance matrix of this hybrid model is (Fogarty and Jones, 2011; Jones, 2010):
Σ =






σ2
εIN−J
0
0
0
σ2
εIJ
−σ2
ξIJ
0
−σ2
ξIJ
2σ2
ξIJ






,
(2.14)
where N is the total number of observations for all houses, J is the number of pairs of
observations on houses with repeated sales, and IN−J and IJ denote (N −J) × (N −J)
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and J × J identity matrices respectively.
Estimation of the hybrid model requires two steps. The ﬁrst step is required to estimate
the variance of the error terms in Equation (2.14). To do this, Equation (2.12) is estimated
for all sales which gives an estimate for variance of εit, denoted by ˆσ2
ε. The estimated
variance of εit is ˆσ2
ε =
 1
N−M−T−1
 NP
i=1
ˆε2
it, where M is the number of exogenous variables
used, and T is the number of years. Then Equation (2.13) is estimated using repeat-
sales data to estimate the variance of ξit, denoted by ˆσ2
ξ. The estimated variance of ξit is
ˆσ2
ξ = 1
2
 1
J−T−2

JP
j=1
(ˆξjt −ˆξjs)2, which is used to determine the estimated covariance matrix
in Equation (2.14).
The second step performs a general linear regression on the three stacked equations using
the estimated covariance structure. This requires a Cholesky decomposition of Σ−1 to
obtain a linear transformation matrix. Since the dimension of Σ−1 is very high, it is not
feasible to calculate the Cholesky decomposition of Σ−1 directly. Instead, a method that
utilises the diagonal block-wise matrices is adopted (Fogarty and Jones, 2011).
The hybrid model has the same functional form and explanatory variables as the un-
restricted hedonic model and diﬀers only in the assumptions for the disturbance terms.
Therefore, a market-wide house price index is constructed in the same way as in the
unrestricted hedonic model (see Section 2.2).
2.5
Portfolio-Level House Price Indices
Non-Regression-Based Models: Portfolio-level mean and median indices can be con-
structed by calculating the respective index for each property portfolio stratiﬁed by the
property characteristics of interest. The mix-adjusted median method proposed by Prasad
and Richards (2008) can be applied to construct portfolio-level house price indices by us-
ing additional property characteristics as classiﬁcation variables. For example, an index
for houses with three bedrooms can be constructed by stratifying the sample of three-
bedroom houses using location variables or the number of bathrooms. The index is then
computed as the equally-weighted average of the median house prices in each stratiﬁca-
tion.
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Restricted Hedonic and Repeat-Sales Models: Portfolio-level house price indices
based on these models can be constructed by estimating the models separately for houses
with the speciﬁed characteristics.
Unrestricted Hedonic and Hybrid Models: Portfolio-level indices based on the
non-regression-based, the restricted hedonic and the diﬀerent repeat-sales models are
constructed by estimating separate models, each using only the subsample of houses with
the selected speciﬁc property characteristics. To construct accurate house price indices
a large number of observations is required. By contrast, portfolio-level indices based on
the unrestricted hedonic model and the hybrid model are constructed by estimating one
model and using the full data set, which ensures more precise estimates even for ﬁnely
stratiﬁed portfolios. The index for a portfolio of properties with speciﬁed characteristics
Xρ
u is given by as:
HPIρ
t = 100 exp
 
ˆβt +
K
X
u=1
ˆδutXρ
u
!
,
t > 0.
(2.15)
Another advantage of the unrestricted hedonic model and the hybrid model is that the
portfolio-level house price indices can be easily linked to the market-wide house price
index. Combining Equations (2.4) and (2.15), the time-t price index for properties with
speciﬁed characteristics HPIρ
t is linked to the market-wide house price index HPIt using:
HPIρ
t = HPIt exp
 K
X
u=1
ˆδut(Xρ
u −X0
u)
!
.
(2.16)
3
Data
The Sydney-based company Residex Pty Ltd provided two proprietary data sets. The
ﬁrst data set contains 810,407 observations on sales prices and detailed property char-
acteristics for houses in the Sydney Statistical Division over the period 1951 to 2011.
This data set only includes sales prices and characteristics at the most recent sales. Re-
corded characteristics include building year, land area, number of bedrooms, number of
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bathrooms, number of garages, geographical location (latitude and longitude)4, and sales
date. We calculate the following variables based on the data provided:
• We include distance to the Central Business District (CBD) as an indicator of a
property’s location. This variable has been included in a number of studies that
compare models describing housing submarkets (e.g., Bourassa et al., 2007, 1999;
Leishman et al., 2013).
We calculate the distance to CBD of a property as its
distance to the Sydney General Post Oﬃce (GPO) at No. 1 Martin Place using the
formula used in Google Maps.5
• The dwelling age at the date of transaction is calculated as the diﬀerence between
the sales year and the building year. There are observations where the building year
is more recent than the sales year because the property was pre-sold. The dwelling
age is set to zero in these cases.
• Decade dummy variables are also created that indicate the decade in which the
house was built: before 1960, in the 1960’s, ... , in the 2000’s, and between 2010
and 2011.
The second data set contains 1,685,419 observations on sales prices and contract dates,
which are used as proxies for sales dates, for properties with multiple sales.
We merged the two data sets containing property characteristics and repeat-sales using a
unique property identity number. The merged data set comprises 1,716,922 observations.
To produce a reliable dataset for the analysis we remove the following observations:
(1) any observation that has incomplete entries (73,853 observations),
(2) any house with the number of bedroom, bathroom, or garages of -1 as this value
implies a missing or abnormal value in the raw data (48,422 observations),
(3) any observation with a sales date before 1971 because this data is sparse and dis-
continuous (1,657 observations),
4The majority (92%) of the geographical coordinates are recorded at the street level.
5See http://www.movable-type.co.uk/scripts/latlong.html
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