Constructing Panel Price Indexes using Hedonic Methods: The ...

the notation, in the remainder of the paper we will omit the hats on the estimators ofthe price indexes.The main problem with the region-time dummy method is that it assumes thesame hedonic model and characteristics for each region-period, and that the shadowprices of the characteristics do not differ across region-periods. Also, it does not satisfytemporal fixity [see Hill (2004)]. Temporal fixity is the requirement that the resultsin a panel comparison for existing region-periods do not change when a new period isadded to the comparison. This is a very desirable property, since users of statistics,including government, generally do not like it when statistics are revised retrospectively.When a new period’s data is added to the panel, the region-time dummy method mustrecompute the shadow prices of all the characteristics. Clearly, this will change all theprice indexes P kt . These observations provide the main rationales for using hedonicimputation methods.(ii) Hedonic Imputation MethodsHedonic imputation methods impute prices for products that are missing in a particularregion-period thus allowing standard price index formulas to be used. Imputationmethods are recommended by Griliches (1990; p. 189), and have been used quite widely[for example by the US Census Bureau to construct its housing price index, and morerecently by Pakes (2003) and Benkard and Bajari (2003)].We will show that imputation methods can be implemented in a greater number ofways than has been previously realized. In particular, depending on how one proceeds,one can obtain a number of varieties of each index number formula.Suppose to begin with that we run the hedonic regression for each region-period ktseparately. The regression equations now are as follows:C∑ln p h kt = β c,kt zc,kt h + ɛ h kt, for h = 1, . . . , H kt . (2)c=1In other words, the regression in (2) is run only over the houses sold in region-period kt.H kt denotes the total number of houses sold in region-period kt. This approach allows6

the shadow prices of characteristics to vary across region-periods. Assuming that thelisted characteristics in all region-periods coincide, we can compute an imputed price ofa house actually sold in region-period kt in any of the other region-periods. For examplethe imputed price of house h in region-period kt if it were sold in region-period js iscomputed as follows:ln ˆp h js(z h kt) ≈C∑ˆβ c,js zc,kt, h for h = 1, . . . , H js , (3)c=1where ˆβ c,js denotes the OLS estimator of β c,js in (2). 5 The adjustment required in (3)to obtain an unbiased price index P kt depends on which price index formula in beingused and the assumptions regarding the distribution of the errors. The adjustment forTörqnvist is considerably easier than for Paasche, Laspeyres and Fisher [see AppendixA].Using these imputed prices it is then possible to compute matched bilateral priceindexes. In each of the price index formulas listed below, h = 1, . . . , H kt indexes thehouses actually sold in a particular region-period kt. Also, w h kt = p h kt/ ∑ H ktn=1 p n kt is theexpenditure share of house h in region-period kt. If the comparison is made over thehouses sold in region-period kt, then it is clear that imputed prices ˆp h js(z h kt) must beused for the other region. However, with regard to region-period kt itself we are facedwith a dilemma. We could use either the actual prices p h kt or imputed prices ˆp h kt(z h kt).The same is true for the expenditure shares. We could use the actual shares w h kt orimputed shares ŵ h kt (where ŵ h kt = ˆp h kt/ ∑ H ktn=1 ˆp n kt). As a result, we obtain four varieties ofthe Paasche and Laspeyres price index formulas.Paasche 1 : P P 1js,kt =Paasche 2 : P P 2js,kt =⎧⎨∑H ktw⎩ kthh=1⎧ [⎨∑H kt ˆphw h js (z h ] ⎫ −1⎬kt)⎩ kt=h=1p h ⎭kt[ ˆphjs (zkt)h ] ⎫ −1⎬ˆp h kt (zh kt ) =⎭5 Again it is not clear whether OLS is preferable to WLS.[ ∑Hkth=1 ph kt∑ Hkth=1 ph kt∑ Hkt(4)h=1 ˆph js(zkt h ),] [ ∑Hkth=1 ˆph kt(zkt) ] h∑ Hkth=1 ˆph js(z h kt )ph kt(5)7

[ ∑HktPaasche 3 :⎧ [⎨Pjs,kt P 3 ∑H kt ˆph= ŵ h js (z h ] ⎫ ] [−1∑Hkt⎬kt)h=1⎩ kt=kt h=1 ˆph kt(zkt) ] hh=1p h ⎭∑ Hktkth=1 ˆph js(zkt h )ˆph kt (zh kt ) (6)Paasche 4 :⎧ [⎨Pjs,kt P 4 ∑H kt ˆph= ŵ h js (z h ] ⎫ −1⎬∑ Hktkt)h=1⎩ kth=1ˆp h kt (zh kt ) = kt(zkt)h⎭∑ Hkth=1 ˆph js(zkt h ),(7)Laspeyres 1 :H js[ ∑ ˆphPjs,kt L1 = wjsh kt (z h ] ∑ Hjsjs) h=1= kt(zjs)hh=1p h ∑ Hjs,jsh=1 ph js(8)Laspeyres 2 :H js[ ∑ ˆphPjs,kt L2 = wjsh kt (z h ] ∑ Hjsjs)h=1=jsˆp h kt(zjs)hh=1ˆp h js(zjs)h [ ∑Hjs] [ ∑Hjsh=1 ph js h=1 ˆph js(zjs) ] h (9)Laspeyres 3 :H js[ ∑ ˆphPjs,kt L3 = ŵjsh kt (z h ] ∑ Hjsjs)h=1= js(zjs)ˆp h h kt(z hh=1p h [js)∑Hjs] [ ∑Hjsjsh=1 ph js h=1 ˆph js(zjs) ] h (10)Laspeyres 4 :H js[ ∑ ˆphPjs,kt L4 = ŵjsh kt (z h ] ∑ Hjsjs) h=1= kt(zjs)hh=1ˆp h js(zjs)h ∑ Hjsh=1 ˆph js(zjs) , h (11)In temporal comparisons, the presence of repeat sales further complicates matters. Forexample, in a temporal P1 index (i.e., P P 1ks,kt), we have a choice between using theimputed price ˆp h ks(z h kt) or the actual price p h ks for house h in period s if it was sold inboth periods s and t. The possibility of repeat sales effectively doubles the numberof varieties of Paasche and Laspeyres indexes above from four to eight. The correctprocedure with respect to repeat sales in the non-reference period is presumably totreat them in the same way as house sales in the reference period. That is, the actualprice p h kt for the repeat sale house should be used when actual prices p h ks are used in thereference period. On the other hand, the imputed price ˆp h kt(z h ks) for the repeat sale houseshould be used when imputed prices ˆp ks (z h ks) are used in the reference period. The sameprinciples apply mutatis mutandis for Laspeyres indexes. Hence with these adjustmentsin mind we can restrict attention to the four varieties of each index previously defined.The treatment of repeat sales is more of academic than practical significance since theyusually account for a small proportion of total sales in a temporal bilateral comparison.In a spatial comparison, this issue cannot even arise. More generally, however, in othermarkets where hedonic methods are applied, the treatment of models that are on salein both periods or regions can be of much greater practical significance.8

Four varieties of Fisher indexes can be derived by combining matched pairs ofPaasche and Laspeyres indexes. 6Fisher x :√Pjs,kt F x = Pjs,kt P x js,kt Lx (12)where x = 1, . . . , 4.Also of interest are weighted geometric means of the price relatives. Indexes of thisform are sometimes referred to as geometric-Paasche and geometric-Laspeyres indexes.Four varieties of each of these indexes can be derived in an analogous manner.⎧[H ktGeometric Paasche 1 : Pjs,kt GP 1 ∏ ⎨ p h ] ⎫ w hkt⎬ kt=⎩ ˆp h js(zkt h ) ⎭ , (13)h=1⎧⎨HGeometric Paasche 2 : Pjs,kt GP 2 ∏ kt=⎩h=1⎧[⎨HGeometric Paasche 3 : Pjs,kt GP 3 ∏ kt=⎩h=1⎧⎨HGeometric Paasche 4 : Pjs,kt GP 4 ∏ kt=Geometric Laspeyres 1 :Geometric Laspeyres 2 :Geometric Laspeyres 3 :Geometric Laspeyres 4 :⎩h=1⎧∏ ⎨⎩h=1⎧∏ ⎨⎩h=1⎧∏ ⎨⎩h=1⎧∏ ⎨⎩h=1H jsPjs,kt GL1 =H jsPjs,kt GL2 =H jsPjs,kt GL3 =H jsPjs,kt GL4 =[ ˆphkt (zkt)h ] ⎫ w hkt⎬ ˆp h js(zkt h ) ⎭ , (14)p h ktˆp h js(z h kt ) ]ŵhkt⎫⎬⎭ , (15)[ ⎫ˆphkt (zkt)h ]ŵhkt⎬ ˆp h js(zkt h ) ⎭ , (16)[ ˆphkt (zjs)h ] w h⎫js⎬ ⎭ , (17)p h js[ ˆphkt (zjs)h ] w h⎫js⎬ ⎭ , (18)ˆp h js(z h js)[ ⎫ˆphkt (zjs)h ]ŵh js⎬ ⎭ , (19)p h js[ ⎫ˆphkt (zjs)h ]ŵh js⎬ ⎭ , (20)ˆp h js(z h js)Four varieties of Törnqvist indexes are derived by combining matched pairs of geometric-Paasche and geometric Laspeyres indexes as follows:Törnqvist x :√Pjs,kt T x = Pjs,kt GP x js,kt GLx,(21)6 This total would rise to 16 if we allowed Fisher indexes to be constructed from unmatched pairs.9

where x = 1, . . . , 4.(iii) Characteristics Price Index MethodsCharacteristics price index methods compare the shadow prices of characteristicsrather than the prices of products (in our case houses) across region-periods [see Triplett(2004; pp. 56-65)]. This process is straightforward for the linear hedonic model wherethe β c,kt parameters are the shadow prices of the characteristics. For the semi-log model,the derivation of shadow prices is slightly more complicated. The β c,kt parametersmeasure the percentage contribution of a characteristic to the overall price, while whatis required is a shadow price in dollars. The shadow price of a characteristic (denotedby b c,kt ) is defined here as the geometric mean across all houses h = 1, . . . , H kt sold inregion-period kt of the partial derivative of price in (2) with respect to the characteristiccoefficient.b c,kt ≡H kt ∏h=1⎡⎣( ∂phkt∂z h c,kt) ⎤ 1/Hkt H∏kt [⎦ = β c,kt (phkt )kt]1/Hh=1Estimates of the shadow prices, therefore, can be derived from estimates of the ˆβ c,ktparameters. Again this can be done in two different ways depending on whether theactual or imputed prices are used.ˆbc,kt = ˆβH∏ktc,ktˆbc,kt = ˆβH∏ktc,kth=1h=1[ ](phkt ) 1/H kt , (22){ }[ˆphkt (zkt)] h 1/H kt . (23)Using this approach three more varieties of Paasche and Laspeyres price indexesare obtained for each shadow price formula. Here we only consider the indexes derivedusing the first shadow price formula.Paasche 5 : P P 5js,kt =∑ Cc=1 ∑ Hkth=1 ˆb c,kt z h c,kt∑ Cc=1 ∑ Hkth=1 ˆb c,js z h c,kt=Paasche 6 : P P 6js,kt =∑ Cc=1[ ˆbc,kt¯z C)] −1c,kt ∑(ˆbc,js∑ Cc=1= vcˆbc,js¯z kt , (24)c,kt c=1ˆbc,kt[ C ∑u ktcc=1(ˆbc,jsˆbc,kt)] −1, (25)10

Geometric Laspeyres 5 :P GL5js,kt =⎡⎤) ujscC∏⎢(ˆbc,kt ⎥⎣⎦ , (36)c=1ˆbc,jswhere ˆb c,kt and u ktcare defined in (22) and (??) respectively. 8 Another Törnqvist indexis obtained by taking the geometric mean of GP5 and GL5.3. Choosing a Bilateral Price Index FormulaA clear consensus has emerged in the price index literature that Fisher and Törnqvistshould be preferred to Paasche, Laspeyres, geometric-Paasche and geometric-Laspeyresindexes. For example, Diewert (2004) shows that Fisher and Törnqvist have superioraxiomatic properties, while Diewert (1976) shows that they have superior economicproperties (i.e., they are superlative). However, here we must choose not only betweenFisher and Törnqvist, but also between different varieties of Fisher and Törnqvist.Consider the first four varieties of Fisher formulas. One argument in favor of P F 1js,ktis that it does not use more imputations than are absolutely necessary, which in generalis presumably a good thing. Alternatively, it could be argued that to level the playingfield, P F 2js,kt should be preferred. For example, suppose that the price of observation h ishigher than predicted by the model (i.e., p h kt > ˆp h kt). This means that the ratio p h kt/ˆp h jswill be misleadingly high, since the “error” impacts only on the numerator. The presenceof omitted variables further strengthens the case for F 2 [see Appendix B]. If, however,imputations are used for prices in both region-periods, then why not for expenditureshares as well? This line of reasoning leads us to P F 4js,kt. Exactly analogous argumentsapply to Törnqvist indexes.As far as we are aware, only methods of the first andfourth variety have been used in practice (usually for Laspeyres indexes). Pakes (2003)for example uses P L1js,kt, while Benkard and Bajari (2003) prefer P L4js,kt. These formulasare compared empirically by van Mulligen (2003) for the case of computers. For PCs8 Also, û ktcand Laspeyres indexes.could be used instead of u ktc , which would generate another pair of geometric Paasche13

he finds that prices fell on average by 26.2 percent per year using L1 as compared with24.3 percent for L4.We lean towards F2 and T2 on the grounds that other things equal, actual datashould be preferred to imputations. This means that actual expenditure shares shouldbe used. For the case of prices, other things are not equal, since the use of the actualprice in the base region-period can introduce distortions into the price relatives [seeAppendix B]. This point has been made previously by Silver and Heravi (2001b). Forlarge outliers this distortion could have a serious impact on the imputed price indexes.For this reason we prefer F2 to F1, F3 and F4, and T2 to T1, T3 and T4.The choice between F2 and T2 is linked to the choice of hedonic equation. For thecase of the semi-log model used here, one attractive feature of T2 is that it is mucheasier to obtain an unbiased estimator of P T 2js,kt than for P F 2js,kt [again see Appendix A].T2 also has firm foundations in goods space and at the same time can be describedin characteristics space, allowing it to be decomposed to reveal the impact of eachcharacteristic. As noted above, T2 is also equivalent to F8.We do not consider characteristics price index methods that are defined in termsof the actual shadow prices (i.e., F5, F6, F7, T5 and T6). F5, F6 and F7 are not veryattractive since they cannot be easily decomposed by characteristic. While this can bedone for T5 and T6, unlike T2, these methods (and F5, F6 and F7) cannot be easilyinterpreted in goods spaceFor these reasons therefore we prefer T2. It should be noted, however, that Törnqvistindexes are potentially more sensitive to outliers. For example, one price observationin the vicinity of zero can cause havoc. Hence it is particularly important to screen thedata for large outliers when Törnqvist is used.Also of interest is the empirical significance of the choice between Fisher andTörnqvist and between different varieties of these indexes. In standard (i.e., nonhedonicand nonhousing) data sets, Fisher and Törnqvist usually approximate each otherclosely, and hence the choice between them is of little practical significance.In the14

housing context the standard rules of thumb no longer necessarily apply. We explorethis issue in the empirical application.So far we have assumed that the listed characteristics across region-periods coincide.Suppose now instead that an additional characteristic is listed in region-period kt whichis not listed in region-period js. This does not create any particular problem for thehedonic imputation method (although it does for characteristics price indexes). Itsimply implies that zc,js h = 0 for all h = 1, . . . , H js in (2). The reverse situation ismore problematic. Suppose that an additional characteristic is listed in region-periodjs which is not listed in kt. The problem now is that no price ˆβ c,kt is available for thischaracteristic, and hence as it stands the imputed price ˆp h js(zkt) h cannot be computed.In practice, it is not clear whether it is better to set ˆβ c,kt = 0 for this characteristic orto use ˆβ c,js instead.The problem of unmatched characteristics is more likely to arise in spatial thantemporal comparisons. For example, the harbor view characteristic for houses in Sydneyis relevant to some regions but not others. However, its relevance to a particular regiondoes not change over time. For this reason, we believe that hedonic imputation methodsare more reliable in a temporal than spatial context.4. Panel Methods(i) The EKS MethodOnce the comparison is extended to three or more region-periods we must confrontthe problem of internal consistency. None of the bilateral formulas considered aboveare transitive. That is P js,kt × P kt,lu ≠ P js,lu . A number of approaches have beensuggested in the literature for transitivizing bilateral indexes. Perhaps the simplest andmost intuitively appealing is the Eltetö-Köves-Szulc (EKS-Gini) formula [see Eltetöand Köves (1964), Szulc (1964) and Gini (1931)]. The EKS-Gini formula transitivizes15

ilateral indexes as follows:T∏ K∏P kt = [(P js,kt ) 1/(T K) ].s=1 j=1The EKS-Gini formula requires a bilateral comparison to be made between all possiblecombinations of region-periods in the panel. In our context, it has two main weaknesses.First, there may not be enough data to estimate the hedonic model in (2) separatelyfor every single region-period. Second, like the region-time dummy method it violatestemporal fixity. Hence below we consider some alternative approaches that resolve theseproblems.(ii) Spatial and temporal grouping methodsBy combining aspects of the region-time dummy and hedonic imputation methodsit is possible to improve on both. We refer to such methods as hybrid methods. Thefirst problem of the hedonic imputation method – that there may not be enough data toestimate the hedonic model for each region-period – can be overcome by first groupingthe region-periods, and estimating the model at the level of groups rather than individualregion-periods. Two relevant criteria that can be used in the selection of groups arefirst that there should be enough observations in each group to estimate the hedonicmodel. Second, there should be a priori reasons for believing that the shadow priceson the characteristics are the same for all region-periods in each group. The results foreach group can then be linked using hedonic imputation methods.Two natural ways of grouping the region-periods are either spatially or temporally.That is, each group could consist of region-periods belonging to the same period (i.e.,spatial grouping), or the same region (i.e., temporal grouping). A pertinent question atthis juncture is whether the shadow prices on the characteristics will be more similaracross a spatial or temporal grouping? In general, the answer to this question willdepend on the data set. For the case where the regions refer to a single city and theperiods are of six month duration (the case considered here), it probably makes moresense to assume that the shadow prices of the characteristics are constant across regions16

than across periods (thus indicating that spatial grouping is preferable). If, however,each region refers to a country, the assumption of fixed shadow prices across regionsbecomes less tenable.There are, however, other reasons for preferring spatial-grouping methods. First,temporal fixity can easily be imposed while maintaining a low temporal displacement.The temporal displacement of a particular bilateral spatial comparison, P jt,kt , within abroader panel comparison is measured by the time span of the data used to computeit. Ideally this should be zero (i.e., the bilateral spatial comparison should dependonly on data from that particular period). The temporal displacement of a wholepanel comparison is the maximum of the temporal displacements of all the bilateralspatial comparisons, P jt,kt , subsumed within it. The temporal displacement of a panelmethod applied to a data set containing T periods of data must lie between zero andT − 1. Ideally, the temporal displacement should be as low as possible. Althoughpanel methods exist that have a zero temporal displacement, these are not always usedsince the achievement of this goal can create other problems [see Hill (2004)]. Second,the bulk of the hedonic imputations in a spatial-grouping method will be temporal innature. This reduces the likelihood of mismatches of characteristics.Below we consider two versions of the spatial-grouping method. Two-stage spatialgrouping can be used when it can be reasonably assumed that in any given period,the shadow prices on the characteristics are the same across all regions. When thisassumption is considered inappropriate, three-stage spatial grouping should be used.(iii) The two-stage spatial-grouping methodThis method in the first stage estimates the region-dummy hedonic model separatelyfor each period. That is, T regression equations are estimated, one for each time periodt = 1, . . . , T .C∑K∑ln p h kt = β c,kt zc,kt h + δ κt d h κt + ε h kt, for h = 1, . . . , H kt , k = 1, . . . , K (37)c=1κ=1In the second stage, hedonic imputation methods are used to link the periods. Here17

we recommend using purely temporal bilateral comparisons between adjacent periodsfor a single region to link the first-stage results together. A graphical example of thismethod is provided in Figure 1 (with the 2010 postcode serving as the temporal linkingregion). There still remains the question of which region should be used? In our dataset there are 128 regions (each corresponding to a Sydney postcode), the use of each ofwhich as the link region will generate different results. One solution to this problem isto compute the results using in turn each region as the link, and then take a geometricmean of these 128 sets of results. This method has the attractive feature that it treatsall regions symmetrically. This is the approach followed in the empirical section below.Insert Figure 1 Here(iv) The three-stage spatial-grouping methodThe three-stage method differs from the two-stage method in that it splits thefirst-stage of the two-stage method into two parts. First, we group the regions intoR districts (where each district consists of a collection of regions) and run the regiondummyhedonic model separately on each district for each period. The regions in eachdistrict should be sufficiently homogeneous so as to make the assumption that they facethe same shadow prices on characteristics tenable. That is, R × T regression equationsare estimated, over all possible combinations of districts r = 1, . . . , R and time periodt = 1, . . . , T . K r in (38) denotes the number of regions in district r. The regions indistrict r are indexed by k = 1, . . . , K r .C∑ln p h kt = β c,kt z h ∑K rc,kt + δ κt d h κt + ε h kt, for h = 1, . . . , H kt , k = 1, . . . , K r . (38)c=1κ=1In the second stage, the hedonic-imputation method is used to make spatial comparisonsand hence link all the districts in a particular period together. Each pair of districts in aparticular period are compared bilaterally using T2. The EKS-Gini method can then beused to impose transitivity on these spatial price indexes. The criticisms of EKS-Ginidiscussed above no longer apply since it is being applied at the level of districts andonly in a spatial context. Using these EKS-Gini parities, overall spatial results for each18

period are obtained. The third stage is identical to the second stage of the two-stagemethod above. The three stage method is illustrated graphically in Figure 2. 9Insert Figure 2 HereOne additional attraction of the three-stage spatial grouping method is that whenEKS-Gini is used in conjunction with T2, it is then possible to decompose the priceindexes to determine the contribution of each characteristic.This decomposition ismost easily demonstrated for the limiting case where each district in stage 1 consists ofa single region (i.e., the region-time dummy is run separately for each region). In thiscase the spatial price indexes can be written as follows:⎡( ) ⎤ ⎧ˆP ktK∏ PT 2 1/K K∏ ⎨exp [ 1 ∑ Cc=1( ˆβ= ⎣⎦ 2c,kt − ˆβ c,κt )(¯z c,κt w + ¯z c,kt) ] ⎫ w ⎬≈ˆP ⎩ ∑ Cc=1 jt ( ˆβ c,jt − ˆβ c,κt )(¯z c,κt w + ¯z c,jt) ] w ⎭κ=1κt,ktP T 2κt,jt=⎩c=1κ=1exp [ 12⎧C∏ ⎨exp [ 1 ∑ Kκ=1( ˆβ 2c,kt − ˆβ c,κt )(¯z c,κt w + ¯z c,kt) ] ⎫ w ⎬∑ Kκ=1( ˆβ c,jt − ˆβ c,κt )(¯z c,κt w + ¯z c,jt) ] w ⎭ .exp [ 125. The Results and their InterpretationOur data set was obtained from Australian Property Monitors and consists of pricesand characteristics of houses sold in 128 postcodes (each containing at least 4000 privatedwellings) in Sydney for the years 2001, 2002 and 2003. Out of a total of 222,423observations (i.e. house sales), information on characteristics were available for only47,535 or 21.4 percent. Hence our hedonic analysis was restricted to these 47,535observations. It is not clear whether this biases the results in any way.The interpretation of spatial results is not entirely straightforward. It depends onexactly what question one is trying to answer. A distinction can be drawn between thephysical characteristics of a house (e.g., number of bedrooms, number of parking spaces,etc), and its locational characteristics (e.g., sea views, the local crime rate, distance to9 In principle it is not necessary that the groupings of regions in each district are the same from oneperiod to the next. Nevertheless, we would normally expect this to be the case since the criteria fordeciding which region is put in which district are unlikely to change much over time.19

nearest hospital and shopping center). If only physical characteristics are used as explanatoryvariables on the righthand side of the hedonic equation, then the resultingprice indexes will measure the price difference between identical houses in different postcodes.The price will vary across postcodes since each postcode has different locational(and perhaps unobserved physical) characteristics. For example, the price will tend tobe highest in postcodes with sea views, low crime rates, and that are close to hospitalsand shopping centers. If, however, locational characteristics are also included on therighthand side of the hedonic equation, the resulting price indexes will measure the pricedifference between identical houses with identical locational characteristics in differentpostcodes. These price differences are harder to explain. Either they imply that thereare unobserved characteristics (of the physical and/or locational variety) that have beenomitted from the righthand side of the hedonic equation, or that the housing market isnot in equilibrium (i.e., that quality-adjusted prices are higher in some postcodes thanothers and hence arbitrage opportunities exist).By contrast, in a temporal context, locational characteristics are of limited interestsince they will be similar for most houses in a particular postcode. This is onlytrue, however, because the postcodes cover relatively small areas. Whether locationalcharacteristics are included or not, therefore, should have little impact on the temporalprice indexes.In our data set all but four of the characteristics are of the physical variety. Theexceptions are city view, harbor view, waterfront and beachfront. Given our limitedaccess to locational characteristics, here we have simply included all characteristics inthe hedonic equation, and have not explored the impact on the results of excludinglocational characteristics.Region-period price indexes for the 128 postcodes at 6 month intervals are shownin Tables 1 and 2. The price indexes in Table 1 are computed using the region-timedummy method, while those in Table 2 are computed using the three-stage spatialgrouping method. Our postcodes are grouped according to a classification created by20

Residex as follows: 2000-2020, 2021-2036, 2037-2059, 2060-2069, 2070-2087, 2088-2091,2092-2109, 2092-2109, 2110-2126, 2127-2145, 2146-2159, 2160-2189, 2190-2200, 2201-2223, 2224-2249. This gives us a total of 15 spatial groups. The price level in Sydneycentral business district (postcode 2000) is normalized to 1 in 2001(1) in both tables.Referring to Table 1 we can deduce, for example, that according to the region-timedummy method, house prices in Edgecliff (postcode 2027) in 2003(2) were 41 percenthigher than in Sydney central business district (postcode 2000) in 2001(1).Insert Table 1 HereInsert Table 2 HereThe postcodes in both tables are ordered from most expensive to cheapest in2003(2). Also shown are the percentage price changes between 2001(1) and 2003(2).The rankings in Tables 1 and 2 are similar, but certainly not identical. The percentageprice changes differ quite significantly depending on the choice of method. For example,according to the region-time dummy method, prices rose by 29 percent in Mosman,while according to the three-stage spatial grouping method prices rose by 67 percent.Both methods, however, agree that prices have risen in every single postcode (except2155-Kellyville for the three-stage spatial grouping method). The average price increase(computed by taking the geometric mean of the price indexes in 2003(2) and dividingby the geometric mean for 2001(1)) was 51 percent according to the region-time dummymethod as compared with 48 percent for the three-stage spatial grouping method. Itis worth digressing for a moment to explain how the overall three-stage temporal priceindex is obtained. A Törnqvist price index is computed between each pair of adjacentperiods using the price data for each region-period in Table 2. The expenditure shareof each region-period simply equals its share of total expenditure in that period. TheseTörnqvist indices are then chained to obtain the overall temporal price index measuringthe price change from 2001(1) to 2003(2). By comparison, according to the official ABShouse price index prices in Sydney rose by 56 percent.The characteristics parameter estimates (i.e., β c ) derived from the region-time21

dummy method in equation (1) are provided in Table 3.These parameters can beinterpreted as measuring the average impact across all postcodes and periods in percentageterms of a characteristic on the price of a house. 10 For example, the parameterestimate for air conditioning is 0.0766, implying that the presence of air conditioningon average acts to push up the price of a house by 7.66 percent. Some of these parametersare intuitively more plausible than others. Nevertheless, on the whole the resultslook reasonable. The largest parameter estimates are obtained for the beachfront andwaterfront characteristics which act to increase price by 31 and 45 percent respectively.Insert Table 3 HereAs was discussed earlier, one attraction of combining EKS-Gini with T2 is that itallows the price indexes to be decomposed by characteristics. Table 4 provides two examplesof such decompositions. The first decomposes spatial comparisons between theSydney (2000) and Edgecliff (2027) postcodes in each period. The second decomposestemporal comparisons for the Sydney (2000) postcode. The characteristics are decomposedinto four groups, denoted by C1, C2, C3 and C4. From the spatial comparisonsit can be seen that most of the price difference between Sydney and Edgecliff in eachperiod is attributable to C4. C4 can be interpreted as a brand effect, which capturesthat part of the price change that cannot be attributed to higher prices of the C1, C2,and C3 groupings of characteristics. From the temporal comparison it can be seen thatthe rise in house prices in Sydney (2000) over the period 2001(1) to 2003(2) can becompletely attributed to a rise in the price of the C1 characteristics. As can be seen,such decompositions can aid the interpretation of the results. 11Insert Table 4 Here10 One potentially problematic characteristic is the age of a house. Age may be problematic since itsimpact depends on how well the house is maintained. This is not an issue that we have to confront inthis paper since age was not one of the characteristics we had available in our data set.11 It might make intuitively more sense if the land area characteristics were relocated from C2 to C1.We will probably do this in due course. Also, we plan to extend the empirical comparison to includesome geospatial data.22

6. ConclusionWe have shown how hedonic imputation methods can be used to construct bilateralprice indexes in a number of different ways. We favor the approach that leads to theTörnqvist 2 (T2) index. In a multilateral spatial context, it is often desirable to groupregions into districts to increase the degrees of freedom when estimating the hedonicimputation regression equation. This suggests a two stage hybrid procedure where aregion-dummy regression equation is estimated for each district, and then the EKS-Gini is applied to the T2 price indexes at the level of individual regions, so as to linkthe districts together. In a panel comparison, due to concerns over temporal fixity, werecommend a three-stage procedure. The first two stages proceed as before. In thethird stage, temporal T2 indexes for a single region are used to link the spatial resultstogether. Each region k = 1, . . . , K is used in turn to link the spatial results together,and then we take a geometric mean of these K sets of results. This hybrid approachcombines the best features of the hedonic imputation and region-dummy method, andis more flexible and reliable than simply applying the region-time dummy method tothe whole panel.This methodology can be applied in any context where hedonic price indexes arebeing constructed. However, here we have illustrated the methodology for the caseof house prices in Sydney. Using this approach we are able to construct consistenttemporal and spatial house price indexes for 128 postcodes in Sydney over a three yearperiod.Appendix A: Bias from Logarithmic Hedonic FunctionsIn this paper we take the following logarithmic model as our estimable hedonicequation:ln p h kt = zktβ h kt + ε h kt, h = 1, . . . , H kt , (39)where zkt h is a 1 × C vector and β kt a C × 1 vector (C denotes the number of characteristics).This equation is estimated using OLS under the assumption that the errors are23

independently and identically normally distributed:ε kt ∼ N(0, σktI 2 Hkt ), (40)where ε kt is a H kt ×1 vector, with each element ε h kt, and I Hkt denotes the identity matrixof dimension H kt by H kt .The form of equation (39) poses a problem because what we require from the hedonicregression is an estimate of the price p h kt not the logarithm of price. In this appendix weexplore the impact of ‘undoing’ the logarithmic transformation, and the possible biasesthat may result. In the discussion below we make use of the fact that the momentgenerating function of a normally distributed random vector, x ∼ N(µ, Σ), has thefollowing form.(M x (t) = E[exp(t T x)] = exp t T µ + 1 )2 tT Σt(41)We begin with a discussion of the bias that arises from simply estimating a priceusing the hedonic function. For notational convenience let us denote ykt h = ln p h kt anddetermine the expectation of exp(ŷkt), h in order to identify the bias in the transformation.E{exp[ŷkt(z h kt)]} h = E[exp(zkt h ˆβ kt )]= E{exp[zkt(Z h ktZ T kt ) −1 Zkty T kt ]}= E{exp[zkt(Z h ktZ T kt ) −1 Zkt(Z T kt β kt + ε kt )]}= exp(zktβ h kt )E{exp[zkt(Z h ktZ T kt ) −1 Zktε T kt ]} (42)We can use the moment generating function in (41) to determine the expectation in(42) by setting x = ε kt , t T = zkt(Z h ktZ T kt ) −1 Zkt T and by using (40) we have µ = 0 andΣ kt = σ 2 ktI Hkt .E{exp[z h kt(Z T ktZ kt ) −1 Z T ktε kt ]} = exp[ σ2kt2 zh kt(Z T ktZ kt ) −1 (z h kt) T ](43)As can be seen the straightforward application of the exponential transformation tothe estimated parameter values leads to an upward bias reflected in (43). The biasis upward as the leverage term, G h kt = zkt(Z h ktZ T kt ) −1 (zkt) h T , in (43) must be a positive24

number. This is because (Z T ktZ kt ) and hence (Z T ktZ kt ) −1 are positive definite matrices,which in turn implies that z h kt(Z T ktZ kt ) −1 (z h kt) T > 0. The term G h kt will lie between 1/Nand 1. It follows, therefore, thatE[ˆp h kt(z h kt)] = E{exp[ŷ h kt(z h kt)]} = exp(z h ktβ kt + σ 2 ktG h kt/2) > exp(z h ktβ kt ).Let us now consider the biases of our price indexes. As shown in the text theprimary price index we use, Tornqvist 2 (T 2), can be written in the form shown in (44)where ¯z js,kt is an average characteristics vector.ˆP js,kt = exp[¯z js,kt ( ˆβ kt − ˆβ js )] (44)This expression has much the same form as (42) above. The application of analogoustechniques yields the following:E( ˆP js,kt ) = E{exp[¯z js,kt ( ˆβ kt − ˆβ js )]}= exp[¯z js,kt (β kt − β js )] ×E{exp[¯z js,kt (Z T ktZ kt ) −1 Z T ktε kt − ¯z js,kt (Z T jsZ js ) −1 Z T jsε js ]} (45)Again using the moment generating function now with, x = (ε T js, ε T kt) T , µ = 0,t TΣ =⎛⎜⎝⎞σjsI 2 Hjs00 Hjs ,H ktσktI 2 ⎟⎠ ,= (¯z js,kt (Z T ktZ kt ) −1 Z T kt, −¯z js,kt (Z T jsZ js ) −1 Z T js), we obtain the following expression forthe expectation in (44):E( ˆP js,kt ) = exp[¯z js,kt (β kt − β js )] ×[( ) ( σ2exp js (¯zjs,kt (Z T2jsZ js ) −1¯z ) σjs,ktT 2+kt2) (¯zjs,kt (Z T ktZ kt ) −1¯z T js,kt) ] (46) .The influence statistics ¯z js,kt (Z T ktZ kt ) −1¯z T js,kt and ¯z js,kt (Z T jsZ js ) −1¯z T js,kt will again be positiveleading the index to be upwardly biased.25

We have also discussed the time-region dummy hedonic model in the paper and itis an approach which has been greatly used in the literature. The results on bias abovecan be easily applied to this special case where the parameter vector of (39) becomesβ kt = (β T , δkt) T T where a region-period specific dummy variable is include with all otherparameters shared between region-periods. Note that a dummy variable is excludedfor one period, say region-period js, in order to avoid perfect collinearity between thevariables. The price index can be calculated as the difference in the estimated prices forsome reference characteristics vector z ∗ relative to region-period js. The expectationof this price index is shown below.E{exp[ŷ kt (z ∗ ) − ŷ js (z ∗ )]} = E{exp[z ∗ (β kt − β js )]} (47)The only difference between the parameter vectors is in the last term so thatE{exp[z ∗ ( ˆβ kt − ˆβ js )]} = E{exp[z ∗ (0, ..., 0, ˆδ kt ) T ]}, (48)We can rewrite (48) in the following form:E{exp[z ∗ (0, ..., 0, ˆδ kt ) T ]} = E{exp[z ∗∗ ( ˆβ 1 , ..., ˆβ M , ˆδ kt ) T ]} (49)Here we have essentially ‘exchanged’ the parameter and characteristics vectors anddefined, z ∗∗ = (0, ..., 0, 1). It can be seen that (49) is very much of the form discussedin (42). Using (43) it is straightforward to calculate the bias in this index.E{exp[ŷ kt (z kt ) − ŷ js (z js )]} = exp(δ kt ) ×{ [( ) ]}σ2E exp (0, ..., 0, 1)(Z T Z) −1 (0, ..., 0, 1) T2[= exp δ kt + Var(δ ]kt), (50)2where Z denotes the following pooled matrix:Z =⎛⎜⎝⎞Z js 0 Hjs ⎟26⎠ .

Again the estimator is upwardly biased by the exponent of half the variance of theregion-period dummy variable.Given this discussion we need to decide how best to account for this bias. Withregard to the price indexes we have shown above thatE( ˆP js,kt ) = exp[¯z js,kt (β kt − β js )] ×[( ) ( σ2exp js (¯zjs,kt (Z T2jsZ js ) −1¯z ) σjs,ktT 2+kt2) (¯zjs,kt (Z T ktZ kt ) −1¯z T js,kt) ] (51) .The second term on the right hand side of (51) represents the bias in the price index.A natural approach to accounting for this bias [following Kennedy (1981)] is simply touse estimated values in the place of true values and subtract the bias. This gives thefollowing bias-adjusted price index:˜P js,kt = exp(¯z js,kt ( ˆβ kt − ˆβ js ))×[ ( ) ( ) ˆσ2 (zjs,kt )exp − jsˆσ(Z T2jsZ js ) −1 zjs,ktT 2 (zjs,kt ) ]+kt(Z T2ktZ kt ) −1 zjs,ktT . (52)However, it has been pointed out that this estimator is still biased, as a result of theconvexity of the exponential function and the randomness of the estimator ˆσ 2 . It hasbeen shown by Goldberger (1968, p. 465, 469) [whose results were also used by Giles(1982) and more recently van Garderen and Shah (2002)] that for an arbitrary constantc and a random variable w where vw/σ 2 ∼ χ 2 v thatE[F (w; v, c)] = exp(cσ 2 ) (53)where∞∑F (w; v, c) = f j (cw) j /j!j=0f j = (v/2)j Γ(v/2)Γ[(v/2) + j)] . (54)In our context, we define two versions of the F function, F (w js ; v js , c js ) and F (w kt ; v kt , c kt ),where w js = ˆσ 2 js, w kt = ˆσ 2 kt, v js and v kt denote the degrees of freedom in the hedonicregressions for region-periods js and kt respectively, c js = − 1 2 [¯z js,kt(Z T jsZ js ) −1¯z T js,kt], andc kt = 1 2 [¯z js,kt(Z T ktZ kt ) −1¯z T js,kt].27

An immediate implication of (53) and (54) and the fact that ˆβ js , ˆβ kt , ˆσ js 2 and ˆσ kt2are independent, is that the price index ˇP js,kt below is an unbiased estimator of theprice index of interest (in fact this is the Minimum Variance Unbiased Estimator as itis a function of the jointly sufficient statistics).ˇP js,kt = exp[¯z js,kt ( ˆβ kt − ˆβ js )] × F (w js ; v js , c js ) × F (w kt ; v kt , c kt ) (55)E( ˇP js,kt ) = exp[¯z js,kt (β kt − β js )]The Goldberger approach provides a ready method for obtaining an unbiased estimateof the desired price indexes. One of the problems with this approach is thecomputations required. The price index ˇP is calculated as a function of known parameters.However, we must also evaluate F (w; v, c) at the relevant point. The priceindex estimator ˜P is somewhat easier to estimate as it does not require us to evaluateF (w; v, c). What is also of interest is that lim v→∞ F (w; v, c) = exp(cw). This can beseen by rearranging the function.F (w; v, c) ===∞∑j=0∞∑j=0∞∑j=0(v/2) j Γ(v/2) (cw) jΓ[(v/2) + j] j!(v/2) j [(v/2) − 1]! (cw) j[(v/2) + j − 1]! j!(v/2) jΦ(j, v)(cw) j, (56)j!whereΦ(j, v) = ∏ ji=1[(v/2) + j − i] for j > 0= 1 for j = 0.The result now follows from the fact that[ (v/2)j ]lim = 1, for j = 0, 1, . . . , ∞.v→∞ Φ(j, v)This leaves the final term in the sum in (56) which is simply the exponential functionfor cw.lim F (w; v, c) = ∑ ∞ (cw) jv→∞j=0j!= exp(cw) (57)28

This means that lim v→∞ ˜Pjs,kt = ˇP js,kt .Appendix B: Omitted Variables and the Estimation of Price Relatives inHedonic ModelsA common situation encountered by the hedonic imputation method is where theprice of a house h is available in one region-period, js, but not in another, kt. In thiscase a Laspeyres price index P L js,kt must use the estimate ˆp h kt from the hedonic regression.It is not clear, however, whether we should use the estimate ˆp h js or the actual price p h jsin the Laspeyres formula.To help shed light on this issue here we distinguish between a 1 × C A vector ofobservable characteristics for house h (denoted by z h,Ajs ), and a 1 × C B vector of unobservablecharacteristics (denoted by z h,Bjs ). C A and C B denote, respectively, the numberof observable and omitted characteristics. It follows that β A js and β A kt below are C A × 1vectors, and β B js and β B kt are C B × 1 vectors. We assume that equations (58) and (59)represent the true data generating process.yjs h = z h,Ajs βjs A + z h,Bjs βjs B + ε h js, h = 1, . . . , H js , (58)ykt h = z h,Akt βkt A + z h,Bkt βkt B + ε h kt, h = 1, . . . , H kt , (59)where y h js = ln p h js and y h kt = ln p h kt. Estimates of y h js and y h kt based on the vector ofobservable characteristics z A,hjs , are obtained from this model as follows: 12ŷjs h = z A,hjsˆβ js, A h = 1, . . . , H js , (60)ŷkt h = z A,hjsˆβ kt, A h = 1, . . . , H js , (61)where ˆβ A js and ˆβ A kt denote the OLS estimators of β A js and β A kt, respectively, obtained fromthe restricted model (excluding the z B,hjscharacteristics).12 Here we abstract from issues relating to the exponential transformation - which are discussed inAppendix A.29

Suppose to begin with there are no omitted variables (i.e., there are no z B characteristics).It follows thatŷ h kt − y h js = (ŷ h kt − ŷ h js) + ˆε h js. (62)The argument in favor of using ŷ h js in preference to y h js rests on the fact that the actualprice includes ‘noise’. Although the estimated residuals sum to zero (i.e., ∑ H jsh=1 ˆεh js = 0),individual residuals could depart quite significantly from zero, resulting in misleadingprice relatives.We now turn to the more interesting and realistic case where there are omittedvariables. ˘y h js and ˘y h kt denote estimates of y h js and y h kt obtained from the true model:˘y h js = z A,hjs˘y h kt = z A,hjswhere ˘β A js, ˘β B js, ˘β A kt and ˘β B kt denote OLS estimators.˘β js A + z B,hjs˘β js, B (63)˘β kt A + z B,hjs˘β kt, B (64)Consider first the case where we use the actual prices in region-period js and theestimated prices in region-period kt. This gives the following estimates of the log pricerelatives:ŷ h kt − y h js = (ŷ h kt − ˘y h js) − ˘ε h js = z A,hjsˆβ A kt − z A,hjs˘β js A − z B,hjs˘β js B − ˘ε h js. (65)The alternative approach is to use predictions from the models in both periods.ŷ h kt − ŷ h js = z A,hjsˆβ kt A − z A,hjsˆβ js A (66)Comparing the difference between the estimates of the log price relatives from eachapproach with those from the true model we obtain the following: 13(˘y h kt − ˘y h js) − (ŷ h kt − y h js) = z A,hjs( ˘β kt A − ˆβ kt) A + z B,hjs˘β kt B + ˘ε h js, (67)13 Whether we use (˘ykt h − ˘yh js ) or (˘yh kt − yh js ) as our point of reference on the left hand side of (67) and(68) is of little consequence to the discussion below. It simply determines which of the two equationscontains an error term, ˘ε h js . 30

(˘y kt h − ˘y js) h − (ŷkt h − ŷjs) h = z A,hjs ( ˘β kt A − ˆβ kt) A + z B,hjs ( ˘β kt B − ˘β js) B − z A,hjs ( ˘β js A − ˆβ js). A (68)|(˘y h kt − ˘y h js) − (ŷ h kt − ŷ h js)| will usually tend to be smaller than |(˘y h kt − ˘y h js) − (ŷ h kt − y h js)|when the bias in the parameter estimates is reasonably stable over time, thus implyingthat˘β A kt − ˆβ A kt ≈ ˘β A js − ˆβ A js.Hence these terms approximately cancel each other out in equation (68). In addition,the parameter estimates themselves should also be reasonably stable, implying that˘β B kt ≈ ˘β B js.It follows that |(˘y h kt − ˘y h js) − (ŷ h kt − ŷ h js)| should be quite small.cancellations occur in equation (67).No such equivalentIt is also interesting to rewrite the difference between ˆβ A js and ˘β A js as follows [see forexample Greene (2000), p. 334]:ˆβ A js − ˘β A js = [ (Z A js) T Z A js] −1(ZAjs ) T Z B js ˘β B js, (69)where Z A js and Z B js denote characteristics matrices of dimensions H js ×C A and H js ×C B ,respectively. Substituting (69) into (67) and (68), these equations can be rewritten asfollows:(˘y h kt − ˘y h js) − (ŷ h kt − y h js) =The vector z h,Bjsthe actual z h,Bjs{z B,hjs(˘y h kt − ˘y h js) − (ŷ h kt − ŷ h js) ={− z B,hjs− z A,h [ ] }js (ZAkt ) T ZktA −1(ZAkt ) T ZktB ˘βB kt + ˘ε h js, (70){z B,hjs− z A,h [ ] }js (ZAkt ) T ZktA −1(ZAkt ) T ZktB ˘βB kt− z A,h [ ] }js (ZAjs ) T ZjsA −1(ZAjs ) T ZjsB ˘βB js . (71)− z h,A [ ] −1js (ZAkt ) T ZktA (ZAkt ) T Zkt B in (70) represents the difference betweenvector and the ‘estimated’ vector from the restricted model and thecharacteristics matrix Z B kt. The size of the error (i.e., |(˘y h kt − ˘y h js) − (ŷ h kt − ŷ h js)|) will dependon the correlation between Z A js and Z B js, and on the magnitude of ˘β B js. Analogousarguments apply for equation (71).31

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FIGURE 1. — TWO STAGE SPATIAL GROUPING METHOD WITH 2010 ASTHE LINKING POSTCODE2000 ✎ 2009 2010 2011 2016 2018 2021 2022 2023 2024 2026 2027 2029 2030 2031☞2003(2) 2003(2)✍✌✎☞2003(1) 2003(1)✍✌✎☞2002(2) 2002(2)✍✌✎☞2002(1) 2002(1)✍✌✎☞2001(2) 2001(2)✍✌✎☞2001(1) 2001(1)✍✌2000 2009 2010 2011 2016 2018 2021 2022 2023 2024 2026 2027 2029 2030 2031FIGURE 2. — THREE STAGE SPATIAL GROUPING METHOD WITH 2010 ASTHE LINKING POSTCODE2000 ✎ 2009 2010 2011 2016☞2018 ✎ 2021 2022 2023 2024☞2026 ✎ 2027 2029 2030 2031☞2003(2) 2003(2)✍✌✍✌✍✌✎☞✎☞✎☞2003(1) 2003(1)✍✌✍✌✍✌✎☞✎☞✎☞2002(2) 2002(2)✍✌✍✌✍✌✎☞✎☞✎☞2002(1) 2002(1)✍✌✍✌✍✌✎☞✎☞✎☞2001(2) 2001(2)✍✌✍✌✍✌✎☞✎☞✎☞2001(1) 2001(1)✍✌✍✌✍✌2000 2009 2010 2011 2016 2018 2021 2022 2023 2024 2026 2027 2029 2030 2031

Table 1. Region-Time Dummy Hedonic Method: Price Indexes byRegion-Period [Sydney 2001(1) = 1]Sydney Postcode 2001(1) 2001(2) 2002(1) 2002(2) 2003(1) 2003(2) % Change2027 Edgecliff 1.17 1.26 1.43 1.50 1.61 1.65 +412000 Sydney 1.00 1.08 1.14 1.26 1.26 1.34 +342088 Mosman 1.01 1.05 1.13 1.18 1.26 1.31 +292023 Bellevue Hill 0.97 1.00 1.20 1.20 1.22 1.30 +342030 Vaucluse 0.99 1.03 1.16 1.27 1.25 1.30 +312095 Manly 0.91 1.01 1.10 1.09 1.24 1.29 +412029 Rose Bay 1.00 0.95 1.09 1.16 1.27 1.27 +272024 Waverley 0.82 0.88 0.97 0.96 1.09 1.21 +482021 Paddington 0.86 0.91 0.99 1.04 1.08 1.17 +372011 Elizabeth Bay 0.89 0.96 0.99 1.06 1.05 1.16 +302026 Bondi 0.79 0.88 0.96 0.98 1.02 1.16 +462034 Coogee 0.78 0.86 0.94 1.01 1.05 1.15 +482060 North Sydney 0.85 0.91 1.01 0.97 1.06 1.14 +352090 Cremorne 0.83 0.95 1.00 1.03 1.01 1.13 +362022 Bondi Junction 0.73 0.84 0.93 0.95 1.07 1.10 +502089 Neutral Bay 0.83 0.92 1.00 1.02 1.05 1.06 +272096 Curl Curl 0.67 0.69 0.84 0.89 0.87 1.05 +562031 Randwick 0.73 0.80 0.86 0.96 0.98 1.04 +422093 Balgowlah 0.65 0.70 0.75 0.88 0.96 1.00 +552009 Pyrmont 0.65 0.66 0.81 1.04 1.04 1.00 +532041 Balmain 0.72 0.81 0.85 0.94 0.97 0.99 +372068 Willoughby 0.70 0.71 0.80 0.85 0.92 0.97 +402065 Crows Nest 0.71 0.78 0.80 0.86 0.90 0.96 +352033 Kensington 0.69 0.73 0.86 0.82 0.98 0.93 +352230 Cronulla 0.62 0.70 0.71 0.81 0.82 0.92 +482064 Artarmon 0.64 0.68 0.80 0.84 0.86 0.91 +422047 Drummoyne 0.71 0.76 0.81 0.85 0.87 0.91 +292069 Roseville 0.67 0.70 0.74 0.88 0.85 0.91 +352010 Surry Hills 0.68 0.75 0.80 0.86 0.87 0.91 +342066 Lane Cove 0.64 0.69 0.75 0.83 0.82 0.89 +392035 Maroubra 0.63 0.69 0.78 0.85 0.88 0.89 +422032 Kingsford 0.60 0.71 0.79 0.84 0.80 0.88 +472037 Glebe 0.62 0.71 0.71 0.82 0.78 0.88 +422135 Strathfield 0.60 0.64 0.70 0.77 0.80 0.87 +462111 Boronia Park 0.50 0.61 0.73 0.73 0.75 0.87 +722046 Abbotsford 0.60 0.66 0.71 0.77 0.79 0.86 +442067 Chatswood 0.63 0.69 0.72 0.77 0.81 0.85 +362101 Narrabeen 0.57 0.57 0.64 0.82 0.76 0.83 +482070 Lindfield 0.61 0.66 0.69 0.78 0.85 0.82 +352097 Collaroy 0.60 0.63 0.67 0.84 0.87 0.82 +382219 Sans Souci 0.53 0.59 0.63 0.74 0.80 0.81 +542137 Concord 0.51 0.60 0.66 0.70 0.74 0.79 +552224 Sylvania 0.43 0.46 0.52 0.73 0.78 0.79 +832134 Burwood 0.54 0.61 0.66 0.75 0.72 0.78 +432221 Blakehurst 0.52 0.55 0.57 0.71 0.69 0.78 +512042 Newtown 0.52 0.60 0.65 0.68 0.68 0.78 +492040 Leichhardt 0.53 0.58 0.64 0.69 0.73 0.77 +462099 Dee Why 0.55 0.62 0.62 0.81 0.80 0.77 +412229 Caringbah 0.48 0.52 0.59 0.66 0.67 0.75 +562107 Avalon 0.56 0.60 0.68 0.72 0.91 0.75 +332036 La Perouse 0.51 0.54 0.62 0.67 0.71 0.74 +44

2073 Pymble 0.52 0.57 0.59 0.62 0.64 0.74 +412016 Redfern 0.55 0.60 0.63 0.68 0.72 0.73 +322049 Lewisham 0.47 0.55 0.58 0.63 0.66 0.72 +552018 Eastlakes 0.47 0.57 0.62 0.61 0.68 0.72 +532228 Miranda 0.44 0.44 0.49 0.58 0.59 0.71 +612100 Brookvale 0.42 0.51 0.61 0.64 0.60 0.70 +662217 Kogarah 0.49 0.56 0.60 0.65 0.69 0.70 +422216 Rockdale 0.47 0.51 0.57 0.61 0.63 0.70 +492227 Gymea 0.41 0.45 0.49 0.58 0.54 0.69 +692203 Dulwich Hill 0.44 0.50 0.57 0.61 0.65 0.69 +562121 Epping 0.44 0.50 0.54 0.60 0.64 0.69 +562131 Ashfield 0.44 0.51 0.57 0.61 0.64 0.69 +552112 Ryde 0.43 0.50 0.56 0.60 0.62 0.68 +582223 Mortdale 0.44 0.48 0.50 0.60 0.62 0.67 +522171 Cecil Hills 0.34 0.33 0.35 0.44 0.51 0.67 +992204 Marrickville 0.43 0.46 0.51 0.55 0.58 0.66 +552122 Eastwood 0.43 0.49 0.52 0.58 0.63 0.66 +542220 Hurstville 0.39 0.46 0.51 0.57 0.58 0.66 +702076 Normanhurst 0.46 0.49 0.54 0.60 0.59 0.65 +412074 Turramurra 0.50 0.52 0.56 0.64 0.64 0.65 +312206 Earlwood 0.44 0.50 0.56 0.59 0.59 0.65 +502087 Forestville 0.41 0.46 0.53 0.55 0.64 0.65 +582075 St Ives 0.44 0.49 0.55 0.57 0.59 0.63 +422114 Denistone 0.40 0.46 0.50 0.58 0.58 0.63 +582113 Macquarie Park 0.41 0.45 0.51 0.60 0.60 0.63 +542086 Frenchs Forest 0.42 0.46 0.55 0.58 0.61 0.62 +482193 Canterbury 0.40 0.47 0.55 0.59 0.61 0.62 +572207 Bexley 0.40 0.43 0.48 0.53 0.55 0.62 +572218 Allawah 0.43 0.47 0.50 0.54 0.56 0.62 +442208 Kingsgrove 0.39 0.43 0.49 0.56 0.57 0.60 +542133 Croydon Park 0.41 0.45 0.49 0.54 0.55 0.60 +452226 Bonnet Bay 0.36 0.40 0.45 0.51 0.47 0.59 +672194 Campsie 0.36 0.41 0.48 0.50 0.56 0.59 +662222 Penshurst 0.38 0.44 0.49 0.55 0.58 0.59 +542120 Pennant Hills 0.35 0.40 0.42 0.48 0.50 0.58 +662150 Parramatta 0.37 0.41 0.44 0.52 0.46 0.57 +552209 Beverly Hills 0.40 0.39 0.45 0.48 0.55 0.56 +402234 Lucas Heights 0.36 0.39 0.43 0.47 0.49 0.56 +542232 Sutherland 0.35 0.41 0.44 0.47 0.50 0.55 +552154 Castle Hill 0.38 0.39 0.43 0.44 0.49 0.55 +432210 Peakhurst 0.35 0.40 0.41 0.47 0.51 0.53 +512126 Cherrybrook 0.32 0.35 0.40 0.47 0.48 0.52 +632118 Carlingford 0.35 0.38 0.42 0.47 0.48 0.52 +492117 Dundas 0.34 0.37 0.40 0.47 0.48 0.52 +542213 East Hills 0.31 0.34 0.41 0.43 0.49 0.52 +662125 West Pennant Hills 0.41 0.42 0.45 0.51 0.53 0.52 +262151 North Parramatta 0.36 0.37 0.43 0.44 0.51 0.52 +442192 Belmore 0.31 0.37 0.45 0.47 0.50 0.51 +632200 Bankstown 0.30 0.32 0.35 0.41 0.44 0.51 +702077 Hornsby 0.36 0.42 0.45 0.50 0.50 0.50 +392211 Padstow 0.30 0.32 0.38 0.43 0.51 0.50 +682141 Lidcombe 0.31 0.29 0.39 0.41 0.47 0.48 +572233 Waterfall 0.30 0.34 0.36 0.47 0.48 0.48 +592196 Punchbowl 0.28 0.31 0.40 0.38 0.42 0.48 +722212 Revesby 0.28 0.30 0.36 0.39 0.41 0.48 +712153 Baulkham Hills 0.32 0.32 0.37 0.41 0.43 0.47 +50

2144 Auburn 0.27 0.32 0.36 0.42 0.47 0.47 +722190 Greenacre 0.29 0.32 0.36 0.42 0.44 0.45 +572199 Yagoona 0.25 0.28 0.38 0.37 0.34 0.45 +792155 Kellyville 0.39 0.39 0.33 0.38 0.38 0.44 +142195 Lakemba 0.28 0.31 0.37 0.38 0.41 0.44 +582160 Merrylands 0.27 0.29 0.39 0.41 0.45 0.43 +622162 Chester Hill 0.25 0.28 0.33 0.37 0.36 0.43 +732145 Westmead 0.28 0.31 0.33 0.39 0.40 0.42 +482142 Granville 0.26 0.29 0.34 0.37 0.39 0.42 +632161 Guildford 0.27 0.26 0.30 0.38 0.37 0.41 +522176 Abbotsbury 0.23 0.28 0.30 0.33 0.36 0.38 +632164 Smithfield 0.21 0.25 0.27 0.28 0.34 0.38 +812170 Liverpool 0.22 0.25 0.28 0.31 0.32 0.37 +732146 Toongabbie 0.24 0.25 0.29 0.36 0.35 0.37 +552165 Fairfield 0.20 0.23 0.27 0.32 0.33 0.37 +852163 Villawood 0.19 0.21 0.26 0.28 0.34 0.36 +922166 Cabramatta 0.19 0.22 0.25 0.30 0.33 0.36 +942177 Bonnyrigg 0.24 0.21 0.23 0.27 0.36 0.35 +492148 Blacktown 0.22 0.24 0.26 0.29 0.31 0.35 +612147 Seven Hills 0.23 0.25 0.29 0.30 0.32 0.34 +472168 Ashcroft 0.18 0.22 0.23 0.25 0.26 0.31 +71

Table 2. Three-Stage Spatial Grouping Method: Price Indexes byRegion-Period [Sydney 2001(1) = 1]Sydney Postcode 2001(1) 2001(2) 2002(1) 2002(2) 2003(1) 2003(2) % Change2027 Edgecliff 1.44 1.54 1.80 2.00 2.09 2.18 +522088 Mosman 1.10 1.33 1.38 1.48 1.69 1.84 +672090 Cremorne 0.92 1.21 1.27 1.33 1.41 1.66 +802023 Bellevue Hill 1.15 1.19 1.40 1.55 1.54 1.65 +442029 Rose Bay 1.19 1.15 1.32 1.46 1.52 1.64 +382000 Sydney 1.00 1.14 1.20 1.26 1.17 1.62 +622089 Neutral Bay 0.99 1.20 1.29 1.39 1.42 1.60 +622095 Manly 1.00 1.07 1.13 1.20 1.55 1.60 +592024 Waverley 1.02 1.10 1.19 1.30 1.40 1.55 +522030 Vaucluse 1.15 1.22 1.35 1.54 1.52 1.55 +352026 Bondi 0.99 1.09 1.20 1.32 1.33 1.53 +542034 Coogee 0.97 1.05 1.17 1.37 1.36 1.47 +512021 Paddington 1.04 1.13 1.22 1.34 1.41 1.47 +422022 Bondi Junction 0.89 1.02 1.15 1.22 1.36 1.40 +582031 Randwick 0.90 0.98 1.07 1.27 1.26 1.36 +512011 Elizabeth Bay 1.08 1.06 1.07 1.09 1.07 1.34 +242060 North Sydney 0.96 0.99 1.06 1.09 1.19 1.33 +392093 Balgowlah 0.76 0.80 0.87 1.06 1.20 1.29 +692096 Curl Curl 0.78 0.74 0.92 1.09 1.10 1.28 +642033 Kensington 0.82 0.85 1.05 1.06 1.21 1.17 +422009 Pyrmont 0.72 0.74 0.90 1.11 1.07 1.15 +592041 Balmain 0.77 0.93 0.96 1.08 1.07 1.14 +492035 Maroubra 0.76 0.83 0.95 1.09 1.10 1.12 +482065 Crows Nest 0.80 0.86 0.87 0.99 1.01 1.11 +392010 Surry Hills 0.86 0.89 0.91 0.95 0.93 1.09 +272032 Kingsford 0.72 0.86 0.95 1.08 1.04 1.09 +512068 Willoughby 0.80 0.81 0.86 0.96 1.00 1.09 +372047 Drummoyne 0.75 0.80 0.90 0.94 0.94 1.07 +432064 Artarmon 0.75 0.76 0.87 0.99 0.98 1.07 +432230 Cronulla 0.70 0.79 0.72 1.02 0.87 1.04 +482101 Narrabeen 0.71 0.67 0.74 1.00 0.92 1.02 +432097 Collaroy 0.71 0.73 0.76 1.01 1.10 1.01 +422135 Strathfield 0.64 0.70 0.82 0.91 0.97 1.01 +592037 Glebe 0.66 0.80 0.78 0.90 0.83 1.00 +522069 Roseville 0.77 0.76 0.80 1.00 0.97 1.00 +292046 Abbotsford 0.65 0.72 0.80 0.86 0.84 0.99 +532066 Lane Cove 0.74 0.76 0.81 0.94 0.94 0.99 +342111 Boronia Park 0.52 0.64 0.85 0.77 0.82 0.97 +882067 Chatswood 0.73 0.77 0.78 0.88 0.91 0.96 +322107 Avalon 0.71 0.73 0.78 0.91 1.17 0.94 +322099 Dee Why 0.63 0.67 0.70 0.96 0.99 0.94 +492137 Concord 0.57 0.66 0.77 0.81 0.88 0.94 +642134 Burwood 0.60 0.66 0.79 0.87 0.92 0.93 +552070 Lindfield 0.70 0.73 0.76 0.93 0.94 0.92 +322224 Sylvania 0.49 0.56 0.59 0.90 0.84 0.91 +862016 Redfern 0.72 0.72 0.74 0.74 0.80 0.90 +252040 Leichhardt 0.58 0.66 0.74 0.79 0.80 0.89 +532100 Brookvale 0.55 0.61 0.76 0.79 0.73 0.89 +622036 La Perouse 0.60 0.63 0.74 0.83 0.84 0.88 +462219 Sans Souci 0.61 0.66 0.72 0.79 0.85 0.86 +412229 Caringbah 0.57 0.63 0.66 0.85 0.70 0.86 +51

2042 Newtown 0.55 0.67 0.72 0.78 0.73 0.85 +552073 Pymble 0.59 0.65 0.66 0.74 0.72 0.84 +412049 Lewisham 0.51 0.63 0.66 0.73 0.69 0.83 +632018 Eastlakes 0.53 0.68 0.72 0.69 0.80 0.83 +552228 Miranda 0.52 0.52 0.60 0.78 0.62 0.83 +592131 Ashfield 0.49 0.56 0.68 0.71 0.80 0.83 +702171 Cecil Hills 0.40 0.39 0.35 0.51 0.59 0.82 +1042221 Blakehurst 0.58 0.62 0.66 0.80 0.74 0.81 +402087 Forestville 0.48 0.53 0.62 0.65 0.75 0.80 +652227 Gymea 0.55 0.54 0.59 0.75 0.57 0.79 +422121 Epping 0.49 0.55 0.65 0.64 0.71 0.78 +612122 Eastwood 0.48 0.53 0.63 0.61 0.72 0.77 +612217 Kogarah 0.55 0.62 0.68 0.70 0.75 0.75 +352074 Turramurra 0.58 0.59 0.63 0.76 0.73 0.75 +282112 Ryde 0.46 0.54 0.65 0.62 0.69 0.74 +612076 Normanhurst 0.53 0.55 0.61 0.73 0.66 0.73 +392075 St Ives 0.53 0.58 0.64 0.71 0.68 0.73 +382216 Rockdale 0.53 0.57 0.61 0.64 0.67 0.73 +382203 Dulwich Hill 0.51 0.55 0.64 0.64 0.67 0.72 +412113 Macquarie Park 0.46 0.51 0.63 0.63 0.67 0.72 +552223 Mortdale 0.51 0.54 0.58 0.65 0.67 0.71 +392220 Hurstville 0.46 0.52 0.58 0.62 0.64 0.70 +522086 Frenchs Forest 0.50 0.52 0.63 0.67 0.66 0.70 +412204 Marrickville 0.49 0.51 0.57 0.58 0.60 0.70 +422206 Earlwood 0.51 0.56 0.64 0.63 0.63 0.69 +342114 Denistone 0.42 0.49 0.59 0.60 0.64 0.69 +622133 Croydon Park 0.46 0.50 0.58 0.62 0.69 0.68 +472218 Allawah 0.51 0.53 0.57 0.60 0.61 0.66 +292234 Lucas Heights 0.43 0.49 0.52 0.60 0.54 0.66 +532207 Bexley 0.46 0.49 0.55 0.56 0.59 0.66 +422226 Bonnet Bay 0.42 0.50 0.54 0.67 0.50 0.66 +572120 Pennant Hills 0.40 0.45 0.51 0.51 0.57 0.65 +632193 Canterbury 0.46 0.55 0.68 0.60 0.66 0.65 +402194 Campsie 0.42 0.48 0.57 0.54 0.60 0.64 +532118 Carlingford 0.41 0.44 0.52 0.53 0.55 0.63 +552232 Sutherland 0.39 0.51 0.55 0.61 0.52 0.63 +612208 Kingsgrove 0.46 0.50 0.55 0.61 0.60 0.63 +372126 Cherrybrook 0.37 0.40 0.50 0.51 0.55 0.61 +682222 Penshurst 0.44 0.50 0.54 0.58 0.61 0.61 +382125 West Pennant Hills 0.45 0.48 0.57 0.58 0.61 0.60 +332117 Dundas 0.37 0.42 0.49 0.50 0.54 0.59 +622154 Castle Hill 0.48 0.48 0.55 0.52 0.73 0.59 +222209 Beverly Hills 0.47 0.44 0.52 0.52 0.58 0.58 +232150 Parramatta 0.39 0.44 0.47 0.58 0.60 0.56 +422192 Belmore 0.36 0.45 0.55 0.52 0.53 0.56 +572210 Peakhurst 0.42 0.45 0.47 0.53 0.55 0.56 +342077 Hornsby 0.39 0.44 0.46 0.56 0.50 0.55 +402141 Lidcombe 0.34 0.33 0.47 0.46 0.58 0.55 +592200 Bankstown 0.34 0.37 0.44 0.44 0.45 0.54 +592144 Auburn 0.31 0.36 0.43 0.49 0.58 0.54 +772151 North Parramatta 0.47 0.44 0.55 0.51 0.70 0.54 +162233 Waterfall 0.36 0.43 0.45 0.62 0.52 0.54 +502196 Punchbowl 0.31 0.37 0.51 0.43 0.44 0.53 +692155 Kellyville 0.56 0.46 0.44 0.45 0.53 0.53 -52213 East Hills 0.37 0.39 0.45 0.47 0.51 0.53 +422176 Abbotsbury 0.28 0.34 0.31 0.37 0.45 0.51 +80

2160 Merrylands 0.30 0.35 0.39 0.46 0.54 0.51 +682190 Greenacre 0.33 0.39 0.45 0.46 0.46 0.51 +562153 Baulkham Hills 0.40 0.39 0.46 0.47 0.63 0.51 +262161 Guildford 0.32 0.32 0.31 0.42 0.43 0.51 +572211 Padstow 0.36 0.37 0.44 0.45 0.53 0.51 +392212 Revesby 0.33 0.34 0.42 0.43 0.44 0.50 +502162 Chester Hill 0.29 0.32 0.33 0.41 0.40 0.50 +712199 Yagoona 0.30 0.33 0.48 0.40 0.37 0.49 +642170 Liverpool 0.26 0.30 0.28 0.35 0.37 0.48 +882195 Lakemba 0.33 0.35 0.47 0.41 0.43 0.48 +442142 Granville 0.29 0.33 0.41 0.43 0.46 0.47 +612164 Smithfield 0.25 0.30 0.28 0.31 0.41 0.47 +912145 Westmead 0.32 0.34 0.40 0.44 0.48 0.46 +452165 Fairfield 0.23 0.28 0.27 0.35 0.38 0.45 +1022177 Bonnyrigg 0.28 0.27 0.28 0.32 0.40 0.44 +592166 Cabramatta 0.22 0.24 0.26 0.33 0.37 0.44 +1052163 Villawood 0.22 0.21 0.28 0.30 0.38 0.43 +992168 Ashcroft 0.21 0.26 0.23 0.29 0.31 0.41 +972146 Toongabbie 0.34 0.31 0.33 0.41 0.50 0.39 +172148 Blacktown 0.27 0.28 0.31 0.35 0.43 0.37 +362147 Seven Hills 0.31 0.29 0.32 0.36 0.47 0.36 +16

Table 3. Region-Time Dummy Model: Summary of Regression ResultsCharacteristic Parameters Standard Errors T-StatisticsIntercept 13.8372 0.0654 211.68cottage -0.0373 0.0061 -6.15duplex -0.1438 0.0160 -8.98house 0.0000 --- ---semi -0.1390 0.0065 -21.35terrace -0.1312 0.0071 -18.54townhouse -0.3106 0.0081 -38.26unit -0.4922 0.0048 -103.35villa -0.2511 0.0135 -18.61bedroom = 1 -0.5174 0.0068 -76.60bedroom = 2 -0.1794 0.0035 -51.23bedroom = 3 0.0000 --- ---bedroom = 4 0.1264 0.0038 33.41bedroom = 5 0.2089 0.0065 32.30bedroom >= 6 0.2568 0.0131 19.64bathroom = 1 -0.0656 0.0034 -19.22bathroom = 2 0.0000 --- ---bathroom = 3 0.1026 0.0063 16.18bathroom >= 4 0.2459 0.0126 19.50land area (500 - 999 sqm) 0.0576 0.0037 15.62land area (1000 - 1499 sqm) 0.2010 0.0071 28.31land area (1500 - 1999 sqm) 0.2674 0.0139 19.19land area (2000 - 2499 sqm) 0.3479 0.0212 16.40land area (2500 - 2999 sqm) 0.5467 0.0413 13.23land area (3000 - 3499 sqm) 0.2967 0.0468 6.35land area (3500 - 4000 sqm) 0.2408 0.0538 4.47air conditioning 0.0766 0.0051 14.90alarm system 0.0521 0.0102 5.09balcony -0.0132 0.0038 -3.48bbq -0.0372 0.0110 -3.39brick construction 0.0108 0.0074 1.47cellar 0.1043 0.0102 10.20conservatory 0.1054 0.0589 1.79courtyard -0.0100 0.0047 -2.12deck 0.0253 0.0062 4.08ensuite 0.0648 0.0047 13.84fire place 0.0307 0.0048 6.37garden 0.0160 0.0057 2.82ground floor -0.0788 0.0255 -3.09gym 0.0875 0.0149 5.85heating 0.0385 0.0173 2.22library 0.1346 0.0372 3.62lock up garage 0.0450 0.0031 14.59loft 0.0063 0.0636 0.10office 0.0846 0.0205 4.13polished floor -0.0047 0.0059 -0.80pool 0.0754 0.0039 19.11rear lane access 0.0016 0.0074 0.22rumpus room 0.0130 0.0066 1.96

sandstone 0.0775 0.0619 1.25sauna 0.0821 0.0343 2.39secure parking 0.0487 0.0029 16.50self contained 0.0756 0.0094 8.03separate dining 0.0037 0.0037 1.00strata -0.0850 0.0102 -8.35study 0.0588 0.0043 13.83sunroom 0.0178 0.0049 3.60tennis court 0.2386 0.0162 14.73timber floor -0.0078 0.0063 -1.24timber kitchen -0.0147 0.0131 -1.12top floor 0.0863 0.0124 6.94unrenovated -0.0074 0.0083 -0.89walk in wardrobe 0.0372 0.0091 4.08weatherboard 0.0014 0.0208 0.07beachfront 0.3138 0.0604 5.20city views 0.0484 0.0144 3.35harbour views 0.2176 0.0048 45.00waterfront 0.4515 0.0127 35.62Regression StatisticsNumber of regressors in model 831Number of parameters in model 832Error degrees of freedom 46,470R-squared 0.7801

Table 4. Decompositions of Spatial and Temporal Results byCharacteristics for the Three-Stage Spatial Grouping MethodSpatial comparisons between Sydney (2000) and Edgecliff (2027)2001(1) 2001(2) 2002(1) 2002(2) 2003(1) 2003(2)C1 1.105 1.010 0.947 1.047 1.191 0.904C2 0.942 0.984 1.075 1.033 0.996 1.010C3 0.995 0.994 0.998 0.989 1.011 0.989C4 1.387 1.360 1.473 1.485 1.480 1.490Total 1.437 1.344 1.495 1.587 1.776 1.345Temporal comparisons for Sydney (2000)2001(1) 2001(2) 2002(1) 2002(2) 2003(1) 2003(2)C1 1.000 1.161 1.299 1.327 1.197 1.664C2 1.000 0.976 0.921 0.940 0.979 0.964C3 1.000 1.009 1.006 1.008 1.002 1.012C4 1.000 1.000 1.000 1.000 1.000 1.000Total 1.000 1.143 1.204 1.258 1.174 1.624Characteristics groupingsC1 /* dwelling type */unit terrace studio /* house */ semi cottage townhouse duplex villa/* number of bedrooms */bed_0 bed_1 bed_2 /* bed_3 */ bed_4 bed_5 bed_gteq_6 /* *//* number of bathrooms */bath_1 /* bath_2 */ bath_3 bath_gteq_4 /* */C2 /* land area - using discrete jumps */ courtyard rumpus_roomarea_500_999 deck sandstonearea_1000_1499 ensuite saunaarea_1500_1999 fire_place secure_parkingarea_2000_2499 garden self_containedarea_2500_2999 greenhouse separate_diningarea_3000_3499 ground_floor strataarea_3500_4000 gym study/* other characteristics */ heating sunroomair_con library tennis_courtalarm_system lock_up_garage timber_floorbalcony loft timber_kitchenbbq office top_floorbrick_const polished_floor unrenovatedcellar pool walk_in_wardrobeconservatory rear_lane_access weatherboardC3 beachfront C4 All the Time-Region Dummy Variablescity_viewsharbour_viewswaterfront