1# RECORD 5170d50eab56ddf40c03eaf3e11552023cf9142d2# TITLE: House Price Models for Banking and Insurance Applications: The Impact of Property Characteristics3# YEAR: 2017.0 VENUE: nan DOI: 10.1515/apjri-2017-00034# AUTHORS: Adam W. Shao; Katja Hanewald; M. Sherris5# PAGES: 28 CANDIDATE TABLE PAGES (1-based): [7, 9, 10, 11, 12, 13]6# 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.789===== PAGE 1 (score 0, numeric-paren 0, table-words 0, var-words 0) =====1011House Price Models for Banking and Insurance Applications:12The Impact of Property Characteristics13Author:14Shao, Adam W; Hanewald, K; Sherris, M15Publication details:16Asia-Pacific Journal of Risk and Insurance17v. 1218Chapter No. 1192153-3792 (ISSN)20Publication Date:212018-0122Publisher DOI:23https://doi.org/10.1515/apjri-2017-000324License:25https://creativecommons.org/licenses/by-nc-nd/4.0/26Link to license to see what you are allowed to do with this resource.27Downloaded from http://hdl.handle.net/1959.4/unsworks_51760 in https://28unsworks.unsw.edu.au on 2026-10-20293031===== PAGE 7 (score 10, numeric-paren 5, table-words 0, var-words 5) =====3233man, 1978; Rosen, 1974). Hedonic models can be divided into restricted hedonic models34where the coefficients of all property characteristics are assumed to be time-invariant35during the observation period and unrestricted hedonic models which allow the implicit36prices of property characteristics to change over time (e.g., Hansen, 2009; Triplett, 2006).37Both model variants are included in our analysis.38The age of a property is an important factor to include in hedonic models. Its effect39on price can be decomposed into two components: a depreciation effect, resulting from40physical deterioration of the house, and a vintage effect, which captures changes in pref-41erences for certain building styles (Goodman and Thibodeau, 1995). We allow for both42depreciation and vintage effects in the hedonic models and in the hybrid model as follows:43(i) We include a quadratic function of the dwelling age to estimate the depreciation effect.44(ii) We include dummy variables indicating the decade in which the property was built to45capture the vintage effect. The depreciation effect can be negative if the property value46is improved through renovation and maintenance.47We estimate the following restricted hedonic model:48Vit = α + βt + X′49iγ + εit,50(2.1)51where Vit is the log price of individual property i at time t, α is the intercept, βt is a52time-varying coefficient, Xi is a vector of K property characteristics, γ is a K × 1 vector53of coefficients and εit is a white-noise disturbance term.54The estimated coefficients ˆβt for the time dummy variables are used to construct a market-55wide house price index:256HPIt = 100 exp5758ˆβt5960,61t > 0.62(2.2)63The unrestricted hedonic model we use includes interaction terms of time dummy642The regression-based house price models are semi-log. The house price indices specified in Equa-65tions (2.2), (2.4), and (2.15) can be thought of as approximations to the median house price (Malpezzi66et al., 1998). This specification is used in order to allow for comparisons with the two median methods.67The house price index at the base year is scaled to 100.686697071===== PAGE 8 (score 4, numeric-paren 3, table-words 0, var-words 1) =====7273variables and house characteristics to allow for time-varying implicit prices of property74characteristics, in a parsimonious and flexible approach. The model is given by:75Vit = α + βt + X′76iγ + X′77i∆t + εit,78(2.3)79where ∆t is a time-varying K ×1 vector of coefficients capturing the interactions between80the time dummy variables and the property characteristics.81A market-wide house price index at time t is calculated as:82HPIt = 100 exp83 84ˆβt +85K86X87u=188ˆδutX089u90!91,92t > 0,93(2.4)94where ˆβt is the estimated value of βt in Equations (2.3), X095u denotes the average value of96the uth house characteristic in the base year, ˆδut is the estimated value of the uth element97of ∆t, and K is the number of property characteristics in the regression.3982.399Repeat-Sales Models100The choice of explanatory variables is critical in hedonic models and omission of import-101ant variables can result in biased estimates. Repeat-sales models aim to avoid the bias102introduced by an incorrectly specified hedonic model and take advantage of information103contained in repeated transactions of the same property (Bailey et al., 1963; Case et al.,1041991). The method focuses on properties that are transacted at least twice during the105sample period, assuming that property characteristics remain unchanged between transac-106tions. Repeat-sales models can suffer from a selection bias if properties that are transacted107multiple times differ from properties that are transacted only once. We include a standard108repeat-sales model, an age-adjusted repeat-sales model and a weighted repeat-sales model109in our model comparison.1103Since property characteristics can change through time it is necessary to specify a reference period111to construct the indices. The Laspeyres index uses a base year, the Paasche index uses the most recent112year, and the Fisher index uses an average measure during the observation period (see, e.g., Hill and113Melser, 2008). We use the Laspeyres’ method in the unrestricted hedonic model and in the hybrid model.114It is assumed in the repeat-sales models that property characteristics do not change over time, so fixing115quality measures for the reference period is not needed. All indices have a base-year value of 100.1167117118119===== PAGE 9 (score 11, numeric-paren 7, table-words 1, var-words 2) =====120121The standard repeat-sales model assumes that property characteristics remain con-122stant over time and that the implicit prices of property characteristics are time-invariant123(Bailey et al., 1963; Jansen et al., 2007; Wang and Zorn, 1997). We remove observations124where multiple transactions are recorded for the same property within one calendar year125in all repeat-sales models as these indicate renovations and other changes of property126characteristics (Clapp and Giaccotto, 1999). To derive the standard repeat-sales model127from the hedonic model, a specification error ηi is introduced to the restricted hedonic128regression for an individual house in Equation (2.1):129Vit = α + βt + X′130iγ + ηi + ξit,131(2.5)132E(ξit) = 0, E(ηi) = 0 ,133E(ξ2134it) = σ2135ξ, E(η2136i ) = σ2137η ,138E(ξitξjs) = 0 if (i −j)2 + (t −s)2 ̸= 0 ,139E(ηiηj) = 0 if i ̸= j,140E(ηiξit) = 0.141(2.6)142Because the specification error ηi is uncorrelated with the white noise term ξit, the variance143of ηi + ξit is σ2144η + σ2145ξ. Differencing Equation (2.5) gives the repeat-sales model:146Vjt −Vjs = βt −βs + ξjt −ξjs,147(2.7)148where s and t (s < t) are the times for a pair of consecutive sales of property j.149We also estimate an age-adjusted repeat-sales model which accounts for depreciation150of houses over time (McMillen, 2003; Shimizu et al., 2010).151We include a quadratic152function of the dwelling age in Equation (2.7) to account for the non-linear relationship153between a property’s age and value observed in the data:154Vjt −Vjs = τ1(Ajt −Ajs) + τ2(A2155jt −A2156js) + βt −βs + ξjt −ξjs,157(2.8)158where Ajt is the dwelling age of property j at time t. The difference between a property’s159age at two consecutive sales is identical to the interval between the two sales (i.e. Ajt −1608161162163===== PAGE 10 (score 7, numeric-paren 6, table-words 0, var-words 1) =====164165Ajs = t−s) and that causes collinearity between (Ajt−Ajs) and the time dummy variables166(βt−βs). We therefore remove the term Ajt−Ajs and estimate the following age-adjusted167repeat-sales model:168Vjt −Vjs = τ(A2169t −A2170s) + βt −βs + ξjt −ξjs.171(2.9)172The standard repeat-sales model assumes that disturbance terms are uncorrelated over173time and across individual properties and have a constant variance. Case and Shiller174(1987) found that the variance can depend on the time interval between sales and pro-175posed a weighted repeat-sales model where heteroskedasticity of the residuals in the176standard repeat-sales model is allowed for by assuming that the residuals’ variance de-177pends on the time intervals between consecutive sales. Their method requires a three-stage178regression. In the first stage, the standard repeat-sales regression is used; in the second179stage, the squared residuals from the first stage are regressed against the time intervals180between sales; finally the repeat-sales regression is performed after dividing each equation181by the square root of the variance estimates from the second stage.182We include the additional assumption in Equation (2.7) for the disturbance term:183E184 (ξjt −ξjs)2185= θ0 + θ1(t −s),186(2.10)187where θ0 and θ1 are coefficients that are estimated by regressing the squared residuals188from Equation (2.7) against an intercept term and (t −s). The estimated coefficients,189denoted by ˆθ0 and ˆθ1, are then used in a weighted linear regression of Equation (2.7) with190weights equal to191192ˆθ0 + ˆθ1(t −s)193−11942.195Based on each repeat-sales model a market-wide house price index can be constructed196using the estimated coefficients ˆβt for the time dummy variables:197HPIt = 100 exp198199ˆβt200201,202t > 0.203(2.11)2049205206207===== PAGE 11 (score 15, numeric-paren 9, table-words 3, var-words 0) =====2082092.4210Hybrid Models211Hybrid models have been developed to combine the advantages of hedonic and repeat-212sales models. They incorporate data for single-sale and repeat-sale transactions and aim213to overcome the specification error in hedonic models and the selection bias in repeat-sales214models. A first hybrid model was developed by Case and Quigley (1991) and improved215by Quigley (1995) by introducing an explicit error structure. We apply the model by216Quigley (1995), which was also used in Wallace and Meese (1997), Jones (2010), and Goh217et al. (2012) (see Table 1). The model is derived from the unrestricted hedonic model218given in Equation (2.3) by introducing a random error term ηi to account for potential219specification errors:220Vit = α + βt + X′221iγ + X′222i∆t + ηi + ξit,223(2.12)224where the assumptions on the two error terms are the same as in Equation (2.6). In the225following, we use εit to denote the sum ηi +ξit. By differencing Equation (2.12) we obtain226a repeat-sales regression equation for properties with multiple transactions:227Vit −Vis = βt −βs + X′228i (∆t −∆s) + ξit −ξis.229(2.13)230The hybrid model is estimated as a stack of three equations taking into account the cov-231ariance structure: Equation (2.12) for properties with single transactions, Equation (2.12)232again for properties with multiple transactions but excluding the last sales to avoid col-233linearity, and Equation (2.13) for properties with multiple transactions.234The covariance matrix of this hybrid model is (Fogarty and Jones, 2011; Jones, 2010):235Σ =236237238239240241242σ2243εIN−J244024502460247σ2248εIJ249−σ2250ξIJ2510252−σ2253ξIJ2542σ2255ξIJ256257258259260261262,263(2.14)264where N is the total number of observations for all houses, J is the number of pairs of265observations on houses with repeated sales, and IN−J and IJ denote (N −J) × (N −J)26610267268269===== PAGE 12 (score 9, numeric-paren 4, table-words 0, var-words 5) =====270271and J × J identity matrices respectively.272Estimation of the hybrid model requires two steps. The first step is required to estimate273the variance of the error terms in Equation (2.14). To do this, Equation (2.12) is estimated274for all sales which gives an estimate for variance of εit, denoted by ˆσ2275ε. The estimated276variance of εit is ˆσ2277ε =278 1279N−M−T−1280 NP281i=1282ˆε2283it, where M is the number of exogenous variables284used, and T is the number of years. Then Equation (2.13) is estimated using repeat-285sales data to estimate the variance of ξit, denoted by ˆσ2286ξ. The estimated variance of ξit is287ˆσ2288ξ = 12892290 1291J−T−2292293JP294j=1295(ˆξjt −ˆξjs)2, which is used to determine the estimated covariance matrix296in Equation (2.14).297The second step performs a general linear regression on the three stacked equations using298the estimated covariance structure. This requires a Cholesky decomposition of Σ−1 to299obtain a linear transformation matrix. Since the dimension of Σ−1 is very high, it is not300feasible to calculate the Cholesky decomposition of Σ−1 directly. Instead, a method that301utilises the diagonal block-wise matrices is adopted (Fogarty and Jones, 2011).302The hybrid model has the same functional form and explanatory variables as the un-303restricted hedonic model and differs only in the assumptions for the disturbance terms.304Therefore, a market-wide house price index is constructed in the same way as in the305unrestricted hedonic model (see Section 2.2).3062.5307Portfolio-Level House Price Indices308Non-Regression-Based Models: Portfolio-level mean and median indices can be con-309structed by calculating the respective index for each property portfolio stratified by the310property characteristics of interest. The mix-adjusted median method proposed by Prasad311and Richards (2008) can be applied to construct portfolio-level house price indices by us-312ing additional property characteristics as classification variables. For example, an index313for houses with three bedrooms can be constructed by stratifying the sample of three-314bedroom houses using location variables or the number of bathrooms. The index is then315computed as the equally-weighted average of the median house prices in each stratifica-316tion.31711318319320===== PAGE 13 (score 13, numeric-paren 4, table-words 2, var-words 5) =====321322Restricted Hedonic and Repeat-Sales Models: Portfolio-level house price indices323based on these models can be constructed by estimating the models separately for houses324with the specified characteristics.325Unrestricted Hedonic and Hybrid Models: Portfolio-level indices based on the326non-regression-based, the restricted hedonic and the different repeat-sales models are327constructed by estimating separate models, each using only the subsample of houses with328the selected specific property characteristics. To construct accurate house price indices329a large number of observations is required. By contrast, portfolio-level indices based on330the unrestricted hedonic model and the hybrid model are constructed by estimating one331model and using the full data set, which ensures more precise estimates even for finely332stratified portfolios. The index for a portfolio of properties with specified characteristics333Xρ334u is given by as:335HPIρ336t = 100 exp337 338ˆβt +339K340X341u=1342ˆδutXρ343u344!345,346t > 0.347(2.15)348Another advantage of the unrestricted hedonic model and the hybrid model is that the349portfolio-level house price indices can be easily linked to the market-wide house price350index. Combining Equations (2.4) and (2.15), the time-t price index for properties with351specified characteristics HPIρ352t is linked to the market-wide house price index HPIt using:353HPIρ354t = HPIt exp355 K356X357u=1358ˆδut(Xρ359u −X0360u)361!362.363(2.16)3643365Data366The Sydney-based company Residex Pty Ltd provided two proprietary data sets. The367first data set contains 810,407 observations on sales prices and detailed property char-368acteristics for houses in the Sydney Statistical Division over the period 1951 to 2011.369This data set only includes sales prices and characteristics at the most recent sales. Re-370corded characteristics include building year, land area, number of bedrooms, number of37112372373374===== PAGE 14 (score 21, numeric-paren 0, table-words 7, var-words 7) =====375376bathrooms, number of garages, geographical location (latitude and longitude)4, and sales377date. We calculate the following variables based on the data provided:378• We include distance to the Central Business District (CBD) as an indicator of a379property’s location. This variable has been included in a number of studies that380compare models describing housing submarkets (e.g., Bourassa et al., 2007, 1999;381Leishman et al., 2013).382We calculate the distance to CBD of a property as its383distance to the Sydney General Post Office (GPO) at No. 1 Martin Place using the384formula used in Google Maps.5385• The dwelling age at the date of transaction is calculated as the difference between386the sales year and the building year. There are observations where the building year387is more recent than the sales year because the property was pre-sold. The dwelling388age is set to zero in these cases.389• Decade dummy variables are also created that indicate the decade in which the390house was built: before 1960, in the 1960’s, ... , in the 2000’s, and between 2010391and 2011.392The second data set contains 1,685,419 observations on sales prices and contract dates,393which are used as proxies for sales dates, for properties with multiple sales.394We merged the two data sets containing property characteristics and repeat-sales using a395unique property identity number. The merged data set comprises 1,716,922 observations.396To produce a reliable dataset for the analysis we remove the following observations:397(1) any observation that has incomplete entries (73,853 observations),398(2) any house with the number of bedroom, bathroom, or garages of -1 as this value399implies a missing or abnormal value in the raw data (48,422 observations),400(3) any observation with a sales date before 1971 because this data is sparse and dis-401continuous (1,657 observations),4024The majority (92%) of the geographical coordinates are recorded at the street level.4035See http://www.movable-type.co.uk/scripts/latlong.html40413405