Implicit House Prices: Variation over Time and Space in Spain

Abstract

Asking price has a fundamental influence on market behaviour and understanding how attributes affect the perceived price of housing. This paper employs a large database of houses from the Spanish housing market to estimate the role of attributes in asking price formation. STAR methodology and GLM with random effects are used to extract the price of attributes and the spatial and time pattern. Results show that the pricing of attributes varies by geographical region and over time with property size and economic and demographic attributes being the key variables explaining asking price formation.

1. Introduction

Research into house prices and heterogeneity in the housing market from both theoretical and empirical perspectives has centred on the application of hedonic techniques which consider that price is obtained from a combination of attributes reflecting household preferences. Hedonic methods have largely been used to estimate the price of housing attributes, calculate price indexes adjusted by quality and determine the shadow price of housing components. For example, single-family homes in the same neighbourhood are likely to share similar structural characteristics (Basu and Thibodeau, 1998), while the weight of each attribute, relative to the others, reflects purchaser preference or taste (Tse, 2002). It is argued that weights are invariant inside a homogeneous sub-market (Sun et al., 2005), but that weights change among sub-markets reflecting differences in demand preferences. In this context, space–state techniques are used in the estimation of the price of housing attributes to avoid spatial and time autocorrelation that is usually present in information obtained at the house (transaction) level and which may bias estimated parameters.

The aim of this paper is to estimate the price of attributes for houses in the Spanish market, to test differences among regions and assess how variation across urban areas affects house prices. A hedonic method is used to define the implicit prices of characteristics using asking prices, the supply side of the market. Asking prices encapsulate a set of housing characteristics that refer to both supplier features and buyer features with the residuals considered to capture information about sellers’ preferences. The use of asking or list price is not unusual in the housing literature; for instance, time on the market studies are often based on list price (Knight et al., 1998; Arnold, 1999; Anglin et al., 2003). There is also a body of literature which has analysed the search process and its impacts on transaction prices, comparing asking prices with selling or transaction prices (Genesove and Mayer, 1997; Harding et al., 2003). Pryce and Mason (2006) observe that indices based on sale prices carry transaction biases in that properties which trade form the respective samples with no cognisance paid to those which do not trade. Furthermore, McGreal et al. (2010) show that asking prices tend to be close to transaction prices and only deviate to any appreciable extent in rapidly rising or falling markets with asking prices lagging sale price in the up-cycle and the opposite effect in the down-cycle.

In assessing how implicit prices vary over time and space, the paper seeks to separate the relative importance of these two effects. The former captures the dynamics of change whereas the latter embraces different market and economic structures, household variation and sentiment. The paper extends the existing literature base by examining whether parameters change and tests the hypothesis that time and space effects do not modify the value of hedonic parameters. Controlling by time and space to reduce autocorrelation problems in hedonic studies has been previously discussed by authors such as Tse (2002), in this paper the particular research question tested is whether parameter weights are stable or change over time at an urban level and how dynamics across urban areas may modify the price assigned to attributes.

The paper is organised as follows. The second section reviews the literature on hedonic house price modelling and state–space models. Section 3 provides details of the database and variables used in the analysis and model implementation. Sections 4 and 5 present the results of the analytical component of the study; the former considers spatio-temporal autoregressive (STAR) models and the latter the general linear model (GLM). Conclusions stemming from the paper are discussed in section 6.

2. Literature Review: Hedonic Price Models

The literature on hedonic models is well established and used mostly to estimate quality adjusted house price indices (Rosen, 1974; Linneman, 1980; Haurin et al., 1991; Peek and Wilcox, 1991; Geltner, 1993; Adair et al., 1996; Clapp, 2004) or to test the impact of different characteristics on prices (Goodman and Thibodeau, 1995; Clapp and Giaccotto, 2002; Bourassa et al., 2005). The majority of these papers are based on the seminal work of Rosen (1974) who argued that the estimated hedonic price characteristics identify neither demand nor supply but are described by a joint envelope function containing both housing attribute groups. However, certain authors have criticised the adherence to hedonic models maintaining that they are characterised by econometric problems and thus provide limited accuracy in the estimation of house prices (Goodman and Thibodeau, 1995). This has raised questions concerning the ability to capture the full behaviour of house prices, in light of criticisms that hedonic models focus on internalising the dynamic evolution of the market (Case and Wachter, 2005).

The potential for bias in hedonic models, arising from spatial and temporal autocorrelation effects, results in inefficient estimated parameters with large errors (Anselin, 1999; Tse, 2002) while failure to correct for autoregressive processes could lead to strongly biased price indexes (Hwang and Quigley, 2010). Two further statistical problems identified in the literature can affect the estimation. First, attributes change among locations and market segments due to sub-market effects (Maclennan and Tu, 1996; Adair et al., 1996; Tu et al., 2007); and, secondly, highly correlated attributes produce multicolinearity which impedes robustness in regression estimated parameters. Furthermore, a lack of sufficient attributes creates an omitted variable problem in the model with residual correlation (Pace et al., 1998; Basu and Thibodeau, 1998; Des Rosiers et al., 2000). Thus, it is important to reduce the effect of autocorrelation by time and space.

The conventional linear hedonic model follows a functional form as in equation (1)

P h_{it} = α + X_{it} β + v_{it}

(1)

where, Ph = price of the house

$X = {x_{1}, x_{2}, x_{3} \dots x_{k}}$ is a set of k independent attributes; v = error term; and $α$ , $β$ are parameters to be estimated. In log form, $β$ represents the attribute’s shadow price.

With observations belonging to different groups (i) across years (t), the model produces efficient parameters and robust results when X and v are assumed to be independent with $Corr (X_{it}, v_{it}) = 0$ and $Cov (X_{it}, v_{it}) = 0$ and v a random error $~ N (0, σ^{2})$ . In the presence of spatial and time dependence, $Cov (X_{it,} v_{it,} X_{it - k}, v_{it - k}) \neq 0$ the error term includes information not captured by the model and introduces an autoregressive component in the residuals.

Three different forms of dependence create non-zero correlations adding unknown components to the residuals (Kuethe et al., 2008; Petersen, 2009; Tse, 2002). First, parameters that vary across housing attributes within a known and observed sample are the result of similarities among houses in a neighbourhood and differences between neighbourhoods, $X_{it} = μ_{i} + η_{it}$ . They are non stochastic and usually referred to as structural spatial dependences (Tse, 2002). The second arises from dependence on errors due to the impact of unobserved variables in the model $v_{it} = γ_{i} + ε_{it}$ . In both cases, $γ$ , $ε$ , $μ$ and $η$ are independent of each other, with a mean of zero and finite variance. Third is the dependence among the observations actual and lagged including a non-stationary autoregressive Ar(n) process. The latter occurs when data are observed on housing prices over time, that is $v_{it} = δ v_{it - 1} + ε_{it}$ .

Different econometric techniques have been developed to avoid the bias generated by autocorrelation and control for spatial dependence. In particular, STAR models have been utilised to avoid parameter mis-estimation. The STAR model (see Anselin, 1999; Anselin et al., 1992 and Pace et al., 1998) considers that both geographical and time dependences jointly affect housing attributes and could be parametrised in a matrix W which captures the spatial relations between each pair of observations (distance) as well as the time dependence.

The matrix W is a combination of the spatial distances between each pair of observation ( $d_{ij}$ ) and the time relation of each observation with their lagged values ( $d_{i, j - n}$ ), filtering for space and time (Pace et al., 1998). Most studies use different techniques to build the matrix, the most popular is based on distances between properties captured by GIS and time lag measures (Basu and Thibodeau, 1998). Each element of W matrix, $s_{ij}$ , should be the inverse of the distance between each pair of locations.

The STAR model is obtained by multiplying both sides of the equation by $(I - W)$ and has the following form (Pace et al., 1998)

(I - W) Y = (I - W) X β + ε

(2)

Then

Y = WY + X β + WX β + ε

(3)

As W contains the time-lag operator, this expression can be presented as

Y_{t} = Y_{t - 1} + W Y_{t} + X β + WX β + ε

(4)

Anselin (1999) defined four types of STAR models. The full STAR time–space dynamic model, which includes all forms of dependence, follows equation (5)

y_{it} = λ y_{i, t - 1} + ρ [W y_{t - 1}]_{i} + y [W y_{t}]_{i} + f (x) + ε_{it}

(5)

where, $f (x)$ are the regressors (which could be also lagged in time and/or space);¹ y is the dependent variable (housing prices); $ε$ the random errors; and W refers to the spatial dependence. This embraces the pure space recursive (where $λ, γ = 0$ and $ρ \neq 0$ ), the time–space recursive (where $γ = 0$ and $ρ, λ \neq 0$ ), and the time–space simultaneous model (with $ρ = 0, λ, γ \neq 0$ and W being the full spatio-temporal matrix).

The econometric method to estimate STAR models depends on the specification and in this respect there is general agreement in the literature that OLS is an imperfect tool. However, Anselin (1999) considered that OLS is a good estimator when a pure-space recursive model is estimated as errors are defined as independent and satisfy the asymptotic assumption of the classical regression model. Other options to estimate the model include two-stage spatial least squares (2SSLS), which achieves the consistency and asymptotic normality properties of the standard 2SLS, the generalised method of momentums (GMM) and non-parametric models.

Literature concerned with estimation of spatial-temporal effects in housing markets has embraced studies adopting varying econometric tools. For example, Bourassa et al. (1999) employed a sequential method, clustering the information using principal components analysis and estimating the hedonic price equation using a priori sub-markets. Clustering attributes in order to minimise collinearity in the hedonic estimation has been applied by Leishman (2009) using a random intercept model in the hedonic OLS specification of housing prices. Pace et al. (1998) employed maximum likelihood and Kuethe et al. (2008) used flexible least square FLSE methodology. Case et al. (2003) compared OLS, OLS with spatial trend and Dubin’s maximum likelihood while Caliman and di Bella (2011) used OLS in a spatio-temporal recursive model specification.

3. Data and Model Development

This paper is concerned with the variation of house prices in seven Spanish provinces² over the period from 1995 to 2010, a period which incorporates the global financial crisis (GFC) with Spain amongst those countries most affected by the GFC. The valuation database³ includes evidence from the whole of Spain but there is a strong regional presence in the provinces of Alicante, Valencia, Murcia, Castellón and the Balearic Islands, as well as significant activity in the two major provinces of Spain, Madrid and Barcelona. In total, the database includes 2 362 800 observations over the period considered.

The variables in the database are sub-divided into property variables (capturing structural attributes/building characteristics), neighbourhood and accessibility variables and socioeconomic characteristics. The dependent variable is the asking price of each property. In addition, the data include variables reflecting time and spatial characteristics. Table 1 provides a summary of the variables, their scaling and other characteristics.

Table 1.

Summary of variables

Spatio–time variable names $X_{ki}$		Scaling of variables	Description of variables
1	t	Time variable measured by years	Year
2	lPhask	Log of price in euros by square metres	Log of asking price
3	Urb_	1 = urban area dependent of other municipality; 2 = autonomous municipality; 3 = county capital;4 = province capital and national capital	Urban dependence is the space variable
		Urb1_ = dummy with Urb_ = 1; zero otherwise
		Urb2_ = dummy with Urb_ = 2; zero otherwise
		Urb3_ = dummy with Urb_ = 3; zero otherwise
		Urb4_ = dummy with Urb_ = 4; zero otherwise
4	Prov		Provinces
Structural/property variables
5	Constq	0 = very low, 6 = superior plus extras	Quality of construction
6	dtype_t3	dummy 3 = house with green area extracted from variable type	If private green areas = 1; 0 otherwise
7	ldwell	number of dwellings in logs	Number of dwellings in building
8	lage	age, log of number of years since the property was built, lage² = log of (age)²	Current age
9	View	0 = degradated area or interior house; 1 = narrow street; 2 = avenue or street; 3 = 2nd line to the sea or significant element in the city; 4 = facing the sea or square/garden; 5 = sea/significant location front (first line); 6 = exceptional	Quality of the views
10	lm2	Surface in square metres measured in logs	Surface (square metres)
11	llift	lift – number, measured in logs	Number of lifts
12	Orient	0.1 = north; 1 = north-west; 2 = west; 3 = south-west; 4 = north-east; 5 = east; 6 = south; 7 = south-east	Subject property’s orientation (north, south …)
13	lm2 other areas	Surface in square metres, measured in logs	Surface in non-covered areas
Neighbourhood variables
14	Qshop	0 = do not exist; 1 = very deficient; 2 = deficient; 3 = basic; 4 = acceptable; 5 = good; 6 = very good	Quality of trade and shopping facilities
15	Cons	ranged from 1 to 100	Area consolidation
16	Qlight	0 = does not exist; 1 = very deficient; 2 = deficient; 3 = acceptable; 4 = good; 5 = very good	Quality of light network in the neigbourhood
17	Qsport	0 = does not exist, 1 = only in city area, 2 = deficiency, 3 = limited, 4 = acceptable, 5 = all type of sport equipment	Quality of the sport facilities
18	Resarea	1 = first residence; 2 = mix between 1st and 2nd residence; 3 = 2nd residence; 4 = business and shoping area; 5 = industrial	Nature of area
		Resarea1 = dummy when resarea = 1
		Resarea2 = dummy when resarea = 2
19	Qroad	0 = do not exist; 1 = very deficient; 2 = deficient; 3 = acceptable; 4 = good; 5 = very good	Quality of the roads
20	Qhealth	0 = does not exist; 1 = only in city area; 2 = scarce; 3 = limited; 4 = sufficient; 5 = all types of health centres	Level of health assistance facilities
21	Qwater	0 = does not exist; 1 = deficient; 2 = acceptable; 3 = good; 4 = very good	Quality of water network supply
22	Qschool	0 = does not exist; 1 = only in city area; 2 = scarce; 3 = limited; 4 = sufficient; 5 = all types of school centres	Quality of school facilities
23	Urbanenv	1 = rural; 2 = suburban; 3 = urban	Type of urban area
24	Qleisure	0 = does not exist; 1 = only in city area; 2 = scarce; 3 = limited; 4 = sufficient; 5 = all typesof leisure centres	Quality of leisure facilities
Accessibility variables
25	Bus	0 = does not exist; 1 = only in city area; 2 = insufficient and scarce; 3 = sufficient; 4 = urban network; 5 = interurban network	Number of bus routes
26	Train	0 = does not exist; 1 = exists in proximity; 2 = exists in the municipality	Train stop available
27	Underg	0 = does not exist; 1 = exists in proximity; 2 = exists in the municipality	Underground stop
Socioeconomic variables
28	Ecoact	1 = agricultural; 2 = industry; 3 = tourism; 4 = services; 5 = multiple	Economic activity in the area
29	Income	Range from 1 = very low to 7 = very high	Income level
30	$Δ Pop$	0 = negative; 1 = slow or stable; 2 = positive	Population dynamics
31	Popdens	1 = low, 2 = medium, 3 = high	Population density in the neighbourhood

Asking price (the median price) over the period 1995–2010 for the seven provinces is illustrated in Figure 1.⁴ Differences are apparent between the seven provinces. For example, in Madrid the asking price distribution is distinctly different from most other provinces, although Barcelona shows a similar pattern from 2002. Differences are also apparent concerning when asking prices started to increase rapidly. In the case of Murcia, Valencia, Castellón, the Balearic Islands and Alicante, strong growth commenced in 1999, whereas in Madrid and Barcelona this is observed later, in 2001.

Figure 1.

Asking prices average: by year in euros.

The valuation database, as it contains observations at different points of time, is essentially a pool of data but also has the characteristics associated with panel data. A large number of different characteristics of the property are included, several of which are qualitative variables scaled from best to worst in an effort to capture the specific differences in these characteristics as part of the valuation, but raising the potential that the database contains a degree of endogeneity in similar attributes and non-independence.

Observations in the database are at a property-specific level with both spatial (location) and time dimensions in an appropriate format to utilise STAR models. However, the database is not geo-referenced, meaning that a conventional spatio-temporal matrix cannot be utilised. A further limitation is the lack of an exact date, with year being the only time reference. The database provides information about the location of each property and whether the comparable is located in a dependent municipality (social and economic) of a larger city, an autonomous city, a county capital or the province capital. These four levels describe an urban structure which is taken as the space reference for this paper and used to build the spatial matrix.

For the analysis, the time–space recursive functional form proposed by Anselin (1999) to estimate a pseudo spatio-temporal hedonic model is adopted. The model is stated in equation (6).

y_{it} = λ y_{i, t - 1} + ρ [W y_{i, t - 1}] + Γ (x_{ki, t}) + ε_{it}

(6a)

where, y_it is the element of the vector Y of the dependent variable (asking price). The independent variables are classified in four groups: socioeconomic attributes, structural/property, neighbourhood and accessibility attributes respectively, for each ‘i’. W is the pseudo spatio-temporal matrix; $λ$ and $ρ$ are the time and spatial parameters; $Γ$ contains the shadow prices of housing attributes as well as the spatial effects on attributes; and $ε$ the independent and random residuals.

It is assumed, following Dubé and Legros (2011), that time and spatial effects do not occur simultaneously allowing exploration of the separate effects of time and space on both asking prices and attributes with T being the time matrix and V being the space matrix ( $W = T + V$ )

y_{it} = λ y_{i, t - 1} + p [(V) y_{i, t - 1}] + Γ [(I + V) x_{ki, t}] + ε_{it}

(6b)

Then

\begin{matrix} y_{it} = λ y_{i, t - 1} + ρ [(V) y_{i, t - 1}] + β (X_{ki, t}) \\ + ϕ (V x_{ki, t}) + μ_{it} \end{matrix}

(7)

where, $ε$ is the spatial correlated vector of errors; and $μ$ is an uncorrelated and independent vector of errors; $ρ$ , $β$ , $λ$ and $ϕ$ are vectors of parameters to be estimated.

The V (spatial) matrix is not a conventional n×n distances matrix. In this case, it has 4 ×n dimensions where the four columns contain information about the type of urban location (variable Urb_). Each property is classified into four categories using a matrix V with the following form

V = \begin{matrix} s_{j 1} & s_{j 2} & s_{j 3} & s_{j 4} \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ .. & .. & .. & .. \\ 0 & 0 & 0 & 1 \end{matrix} \forall j = 1 \dots n

where, $s_{j 1} = 1$ when ${Urb}_{j -} = 1$ , 0 otherwise; $s_{j 2} = 1$ when ${Urb}_{j -} = 2$ , 0 otherwise; $s_{j 3} = 1$ when ${Urb}_{j -} = 3$ , 0 otherwise; and $s_{j 4} = 1$ when ${Urb}_{j -} = 4$ , 0 otherwise.

Such dimensions reflect the spatial dependence between urban areas—how they depend on each other administratively or for particular services (schools, health system, public services). Matrix T contains the date information in years. It is assumed that asking price data follow an AR(1) process, an assumption that is common in STAT models, and that T is used to calculate one lag on the dependent variable ( $Y_{t - 1}$ ).

The time matrix is represented by the pooled set of dummy variables by year with n x 16 dimensions. The W (spatio-temporal effect) is T+V, summing up the effects estimated separately in the model due to the matrix shapes. As only T+V effects are included, it is assumed that the effects have isotropic autocorrelations. While the final model may still include some level of space–time autocorrelation, in order to control for the existence of any remaining autocorrelation, the estimation minimises the standard errors in the model to ensure that dependence is not severe and parameters are robust.

4. Empirical Analysis and Modelling Space and Time–STM

The analysis seeks to identify the role of attributes in explaining the asking prices for properties over the period 1995–2010. Specific attention is paid to the role of demographic variables due to the population shock that occurred in Spain during the time-period covered by the analysis. The size of the database, the extent of the geographical coverage (seven provinces) and the time-series (16 years) provides added complexity. In this respect, the analysis isolates the space and time independent effects to allow for the different provinces and to capture variation, controlled by quality.

A time–space recursive model using 2SLS with spatial parameters, following Anselin (1999), is estimated using the pseudo time–space matrix (W).⁵ Time–space attributes by provinces are also estimated using maximum likelihood (ML) by including time–space interaction effects among the covariates (using equation (7)). The assumption is made, following Dubé and Legros (2011), that time and spatial effects do not occur simultaneously, thereby allowing exploration of the separate effects of time and space on both asking prices and attributes as previously discussed. This has the effect of linearising the functional form of the model as defined in equation (7). A linear relationship between the log of housing prices (y)⁶ and its characteristics $X β$ is assumed including three spatial components: time autocorrelation (λ), space–time autocorrelation ( $ρ$ ) and spatial lag $ϕ$ .

A number of variables are transformed into dummies such as type of building which generates a dummy variable for property quality. Age is also included with its squared function (in logs, lage and lage² ) to control for the non-linearity associated with this variable. Resarea captures whether the property is located in a prime residential area (resarea = 1), in a mixed neighbourhood (resarea = 2) or in a secondary area (resarea = 3). The variable is transformed into dummies to avoid collinearity, its parameter captures the effect of primary homes relative to a secondary location. All continuous variables are transformed into natural logs.

As the STM model includes the dependent variable in a log format, beta parameters represent pseudo elasticities in the log–log relationships. Thus each parameter is in effect the attribute’s shadow price representing how the owner values such attributes in the total asking price. The V*X parameter records the spatial effect of attributes by urban location, capturing how shadow prices change depending on property location. In the 2SLS estimation, categorical scaling, rather than a dummy specification, is selected to estimate the spatial effect on attributes.⁷ For the recursive term ( $lphask (t - 1)$ ), the time parameter (lambda) captures the effect of the lagged price on the current house price in the model reflecting the overall effect of prices in the previous period on the current price. The space-recursive parameter at each urban level ( $ur b_{j}_lphask (t - 1)$ ), that is $ρ (V y_{i, t - 1})$ is also calculated.

The results of the model are shown in Table 2.⁸ Two variables were discarded from the calculation: urbanenv (as more than 93 per cent of the observations were classified as being in an urban environment) and log age, as the variable log age squared (lage² ) was more efficient in capturing the effect of this parameter. Residuals of the model are small and bell-shaped suggesting that they are randomly distributed as white noise.

Table 2.

2SLS regression model in recursive space–time framework: STM model results $y_{it} = λ y_{i, t - 1} + ρ (V y_{i, t - 1}) + β x_{ki, t} + ϕ (V x_{ki, t}) + μ_{it}$

Dependent variable, y= lPhask	Full model			Alicante			Balearic Islands			Barcelona			Castellón			Madrid			Murcia			Valencia
Variables (X) by groups^a	$β$	t	significance	$β$	t	significance	$β$	t	significance	$β$	t	significance	$β$	t	significance	$β$	t	significance	$β$	t	significance	$β$	t	significance
Constant	2.37	152.4	***	1.875	20.2	***	0.98	5.0	***	2.50	10.0	***	2.361	30.42	***	2.90	29.6	***	2.62	104.0	***	2.95	116.5	***
Socioeconomic
Ecoact	0.04	74.9	***	0.011	16.1	***	−0.01	−2.8	***	−0.01	−5.0	***	−0.001	−0.29		0.04	4.0	***	0.00	4.5	***	0.00	1.6
$Δ pop$	−0.01	−12.7	***	0.061	29.9	***	0.01	3.2	***	0.09	1.1								−0.02	−0.8		0.18	3.1	***
Inc ome	0.03	30.5	***	0.027	21.9	***	0.32	3.5	***	0.06	26.3	***	0.492	2.86	***							0.04	32.8	***
Popdens	0.01	10.3	***	−0.004	−2.0		0.14	1.8		0.16	1.1		−0.68	−3.08	***				0.27	2.5	***	−0.12	−1.3
Neighbourhood
Cons	0.00	6.3	***	0.000	−0.2		0.00	4.8	***	0.00	1.8		0.00	1.699		0.00	−0.3		0.000	3.9	***	0.001	6.7	***
Qroad	−0.01	−2.6	***	−0.009	−1.4														−0.03	−5.6	***
Qwater	0.01	2.7	***	−0.020	−3.7	***	0.04	2.9	***	0.04	5.0	***	0.08	4.545	***	−0.02	−2.2	**	0.01	1.2		0.05	7.3	***
Qlight	−0.01	−3.0	***	0.009	1.6		−0.02	−4.6	***	−0.03	−5.9	***							0.01	2.6	***	−0.06	−13.9	***
Qshop	−0.01	−14.1	***	0.011	11.7	***	−0.12	−2.3	**	0.01	8.3	***	−0.007	−2.69	***	0.02	8.5	***	−0.07	−2.3	**
Qschool	0.01	5.1	***	0.006	2.6	***	0.00	1.0		−0.04	−7.0	***	−0.008	−1.15		0.00	−0.6		0.01	4.8	***	0.03	10.2	***
Qleisure	−0.02	−5.4	***	−0.022	−5.0	***
Qsport	0.04	14.9	***	0.008	1.7		0.02	6.1	***	0.01	3.1	***	0.026	3.85	***	0.01	0.9		0.00	0.7		0.00	1.6
Qhealth	−0.02	−10.7	***	0.004	1.3		−0.01	−2.3	**	0.03	4.4	***	−0.014	−1.68		0.01	2.2	**	0.00	−0.9		−0.03	−15.3	***
Resarea1	−0.14	−3.9	***	−0.006	−0.1		−0.03	−6.1	***	−0.37	−2.9	***	−0.089	−9.38	***				−0.07	−18.5	***	0.29	1.7
Resarea2	0.10	1.7		−0.069	−0.4											−0.41	−6.4	***
Accessibility
Bus	0.04	3.6	***	−0.046	−0.9		0.04	1.6											0.00	0.2		0.01	0.2
Train	0.05	2.8	***	−0.093	−1.8		−0.04	−0.5		0.13	2.3	**	0.253	1.64		0.44	2.3	**	−0.02	−0.5		−0.06	−0.9
Underg	0.06	2.8	***										0.01	0.71		−0.13	−1.7		0.00	0.4
Structural
Orient	0.01	30.8	***	0.002	7.7	***	0.01	17.1	***	0.01	18.7	***	0.003	2.71	***	0.00	0.4		0.00	11.6	***	0.00	4.8	***
View	0.02	34.3	***	0.021	27.1	***	0.02	17.3	***	0.01	10.4	***	0.028	11.76	***	0.00	−0.8		0.02	17.8	***	0.01	7.4	***
Constq	0.01	0.4		−0.154	−3.6	***										−0.12	−1.0		0.15	2.6	***	0.01	10.6	***
dType_t3	−0.01	−0.5		0.045	1.4		0.05	1.4		0.08	1.7								0.12	2.5	***	0.10	2.3	**
ldwell	−0.01	−3.7	***	−0.007	−2.3	**	−0.02	−5.4	***										0.01	3.7	***	0.00	0.8
llift	0.02	4.6	***	0.045	7.0	***	−0.01	−1.8		−0.01	−0.8		0.088	5.57	***	0.04	3.3	***	0.08	3.9	***	0.02	2.6	***
lage2	0.00	−0.2		0.064	1.4		−0.14	−2.8	***	−0.21	−1.5								−0.11	−2.5	***	−0.15	−3.2	***
lm2	0.54	9.1	***	0.525	2.7	***	0.11	0.5		0.58	3.1	***	0.783	4.86	***	0.73	48.8	***	0.44	2.5	***	0.73	5.5	***
lm2 other areas	0.06	47.7	***	0.056	28.1	***	0.05	20.1	***	0.06	27.3	***							0.05	22.6	***	0.04	21.9	***
	$λ$	t		$λ$	t		$λ$	t		$λ$	t					$λ$	t		$λ$	t		$λ$	t
Time parameters
lPhask(t-1)	0.26	97.3	***	0.446	15.7	***	0.55	6.8	***	0.17	2.4	**				0.15	2.9	***	0.17	1.9		−0.06	−1.3
	$ϕ$	t		$ϕ$	t		$ϕ$	t		$ϕ$	t		$ϕ$	t		$ϕ$	t		$ϕ$	t		$ϕ$	t
Space cross-effects
urb_income	0.01	33.0	***	0.013	29.9	***	−0.09	−2.8	***				−0.159	−2.78	***	0.02	25.7	***	0.02	31.1	***
urb_popdens	0.01	25.7	***	−0.002	−2.9	***	−0.02	−0.9		−0.05	−1.1		0.233	3.16	***	−0.01	−3.6	***	−0.09	−2.4	**	0.04	1.3
urb_pop	0.01	26.3	***	−0.041	−1.2					−0.01	−0.3		0.015	6.71	***	0.01	10.3	***	0.02	1.7		−0.05	−2.5	***
urb_consq	0.00	0.0		0.056	3.9	***	0.00	9.9	***	0.00	1.5		0.003	2.34	**	0.04	1.0		−0.04	−2.4	**
urb_qshop	0.00	−25.3	***	−0.001	−2.8	***	0.04	2.5	***										0.03	2.6	***	0.00	10.7	***
urb_resarea1	0.03	2.4	***	−0.020	−0.5					0.14	3.1	***				−0.10	−5.1	***				−0.12	−2.2	**
urb_resarea2	−0.03	−1.5		0.033	0.6		0.00	−1.2		0.01	2.7	***	0.005	1.36					0.00	−0.5		−0.01	−2.9	***
urb_train	−0.02	−2.7	***	0.033	1.9		0.01	0.4		−0.05	−2.3	**	−0.078	−1.52		−0.15	−2.4	***	0.01	0.7		0.02	0.9
urb_bus	−0.02	−3.9	***	0.013	0.7		−0.01	−1.7		0.00	0.0		−0.007	−4.78	***	0.00	−0.7		0.00	−0.1		0.00	−0.3
urb_underg	−0.01	−1.6		0.002	1.0		0.00	1.9		0.00	12.5	***				0.05	1.9					0.001	5.4	***
urb_lage2	0.00	−0.5		−0.024	−1.5		0.04	2.5	***	0.04	1.5		−0.010	−10.00	***	0.00	−5.2	***	0.03	2.1	**	0.04	2.7	***
urb_lm2	−0.04	−1.9		−0.081	−1.3		0.05	0.7		0.03	0.6		0.034	0.63					0.04	0.7		0.00	0.0
urb_dtype3	0.01	1.6		−0.009	−0.9		−0.01	−1.0		−0.02	−1.4		0.004	4.87	***	0.01	12.2	***	−0.04	−2.5	***	−0.03	−1.9
	$ρ$	t		$ρ$	t		$ρ$	t		$ρ$	t		$ρ$	t		$ρ$	t		$ρ$	t		$ρ$	t
Time lag-space effect
urb4_lphask(t-1)				0.008	11.3	***				0.00	−0.2		0.006	4.42	***	0.01	9.9	***	0.02	15.5	***	0.01	12.8	***
urb3_ lphask(t-1)	0.00	1.5		0.009	11.0	***	−0.01	−9.7	***	−0.01	−9.0	***	−0.001	−.806		0.00	−0.4		0.00	−0.4		−0.001	−8.7	***
urb2_ lphask(t-1)	0.00	−8.3	***				0.00	−3.9	***										0.00	2.8	***
urb1_ lphask(t-1)	−0.01	−18.7	***	−0.016	−14.9	***	0.00	−1.9		−0.01	−4.0	***	0.003	1.002		−0.04	−3.6	***				0.00	−0.8
Tests
R2	0.83			0.86			0.85			0.82			0.92			0.84			0.94			0.86
Adjusted R ²	0.83			0.86			0.85			0.82			0.92			0.84			0.94			0.86
Standard error	0.12			0.09			0.08			0.09			0.08			0.10			0.08			0.08
Residuals	954.7			158.47			43.27			89.17			8.19			35.16			57.87			105.4
F	5457			1883.0			746.8			1127			401.5			436.2			2589.7			1954
Sign p <0.01	0.00			0.00			0.00			0.00			0.00			0.00			0.00			0.00

^anon-standardised parameters.

Notes: *** p <0.01; ** p <0.05.

The model gives results within acceptable margins with small standard deviations and errors for all provinces (with the exception of the Balearic Islands). Parameters are significant at an aggregate level for the seven provinces, an observation supported by the provinces’ parameter values. The model also includes time dummy variables representing the price index.⁹

4.1 Socioeconomic Attributes

For the spatio-temporal model, all four attributes are statistically significant at the overall level and also for most provinces. Income is significant and positive in all models with a high power parameter in the Balearics and Castellón, between 0.3 and 0.5, but has lower impact ( $β$ = 0.03/0.06) in Alicante, Barcelona and Valencia. For Madrid and Murcia, this parameter is not statistically significant when isolated, but it is strongly significant at the space level ( $ϕ$ = 0.02) which suggests that the level of income differences affecting house price changes with spatial impact. Alicante also shows a space effect associated with income level, with higher incomes increasing house prices by 1.3 per cent relative to the weakest location. The two provinces (Balearic Islands and Castellón) with the strongest impact of income (at a city level) have negative parameters for neighbourhood income suggesting that the general income effect has a declining impact on house prices depending upon the urban hierarchy (−0.09 and −0.159 respectively). The two provinces concerned are of different character. The Balearic Islands have a large German population and the service sector is the main source of GDP. In contrast, Castellón, is very industrialised and export orientated (high-quality construction materials) with a medium-sized city as its economic centre. While both economies are very different regarding the source of wealth, there are some similarities in urban form characterised by a network of small –medium-sized centres providing proximity to services and goods—infrastructure that encourages wealthier residents to live in higher-quality environments outside the main cities.

Population attributes show the effect of migration and concentration arising from demand pressures with a positive effect on house prices. Two variables, density and changes in population, are included. Alicante ( $β$ = 0.061), the Balearics ( $β$ = 0.01) and Valencia ( $β$ = 0.18) show positive effects on house prices from net immigration with a 10 per cent net increase in immigration increasing house prices by 0.6 per cent, 0.1 per cent and 1.8 per cent respectively. The model does not capture such direct effects in Madrid, Barcelona, Murcia and Castellón, although there is evidence of such a relationship at the urban level for Madrid and Castellón. In both cases, the parameter $ϕ$ is positive and strongly significant ( $ϕ$ = 0.01 and 0.015 respectively) suggesting that house prices rise as immigration increases. The strong effect of immigration shown by the $ϕ$ parameter which, in Valencia, reverses at an urban level ( $ϕ$ =−0.05) suggesting that price impact, due to a rise of immigration, is 5 per cent higher in dependent areas (suburbs).

The results indicate that the reaction of prices to an increase in population density is not consistent. Although the overall model returns a positive and statistically significant parameter (location in dense areas are priced at a premium), at the urban level, half of the provinces have a negative and statistically significant parameter between house prices and population density. As the model controls for neighbourhood quality, the willingness to live in a high-quality dense urban area should yield a positive parameter reflecting the ability to pay for that location, while the desire for a less dense area should give a negative parameter. For example, density is strongly negative in Castellón ( $β$ =−0.68) but positive in Murcia ( $β$ = 0.27). Alicante, Madrid and Murcia each have a negative spatial parameter for population density ( $ϕ$ =−0.002, −0.01 and −0.09 respectively) suggesting that in their respective roles as capital cities, density acts to reduce house price. Barcelona and the Balearics are two provinces where population density does not affect house price. These results indicate different tastes related to the type of the urban area. There is also the added complexity of density at the neighbourhood level having an opposite sign. In the case of Castellón, density (as already noted) is highly negative, but at the neighbourhood level the opposite sign prevails ( $ϕ$ = 0.233) suggesting that the negative effect of density on house prices is reduced locally. In Murcia, density had a positive effect in the province-wide model, but the urban-level effect is negative ( $ϕ$ =−0.09) suggesting that the price premium implied by the former is reduced at the neighbourhood level.

4.2 Neighbourhood and Accessibility Attributes

These embrace a bundle of quality measures ranging from the extent of developed land to the quality of facilities. The analysis suggests that shopping has a strong impact in Madrid ( $β$ = 0.02), Barcelona ( $β$ = 0.01) and Alicante ( $β$ = 0.011). However, this variable is negative in the Balearics (−0.12), Castellón (−0.007) and Murcia (−0.07), implying that proximity to a shopping area reduces house price. The latter provinces have a shopping provision largely based on malls located outside the cities which would tend to support this negative interpretation. However, when space cross-effects are analysed, there is a positive effect in both the Balearics and Murcia with an increase in housing prices as quality shopping is located in higher-order neighbourhoods.

School quality has mainly a positive effect on house prices especially in Alicante, Murcia and Valencia, but is negative in Barcelona and has no clear effect in Madrid, the Balearics and Castellón. Sport facilities are statistically significant with a positive effect on prices in the Balearics, Barcelona and Castellón, but not in Madrid, Murcia or Valencia, while leisure facilities have a negative effect in the overall model. Health facilities have surprising results, with a negative effect on price in the overall model as well in the models for the Balearics and Valencia suggesting that such activities are perceived as diminishing price. However, in Barcelona and Madrid, the opposite effect is apparent with a positive and statistically significant impact on house price ( $β$ = 0.03 and 0.01 respectively).

The overall model and those at a province level notably for the Balearics, Barcelona, Castellón and Murcia indicate that properties located in a primary home area have a lower rate of price increase than those located in areas with a strong second-home market. Significantly, these provinces are located on the Mediterranean coast with a high incidence of second homes.

Accessibility measures the existence of bus, train and underground stations near the property. The overall model captures a strong significant and positive influence of the three transport modes on house prices ( $β$ = 0.4, 0.5 and 0.6 for bus, train and underground), but their significance at the province level is not linear. The results demonstrate no effect of bus stops on house price although a strong significant effect of train stops is apparent in Madrid ( $β$ = 0.44) and Barcelona ( $β$ = 0.13). Spatial parameters for accessibility show negative values suggesting that the closer the house is to a bus or train stop reduces house price when the house is located in higher-order urban areas. The results suggest that the effect of a station on house prices in Madrid and Barcelona reduces the closer a property is to the respective city.

4.3 Property/Structural Attributes

This group of attributes measures the quality of house characteristics. View and orientation are shown to have a positive effect on housing prices. View is especially relevant in Alicante and Castellón ( $β$ = 0.021 and 0.028 respectively) with their large vacation and second-home housing markets on the coast. This variable is also significant in the rest of the provinces, apart from Madrid.

The number of dwellings in the building (measuring density of construction) is negative and statistically significant in the overall model. A similar interpretation is apparent in Alicante and the Balearics ( $β$ =−0.007 and −0.02 respectively) suggesting that lower housing density is more valued in those provinces but not in the rest of the provinces. Quality of construction is statistically significant in Murcia ( $β$ = 0.15) and Valencia $β$ = 0.01). The number of lifts in the building has strong positive effect on house prices with parameters ranging from 0.02 to 0.088.

In all models, age is captured by the log of the squared term which is negative and statistically significant in Balearics ( $β$ = −0.14), Murcia ( $β$ =−0.11) and Valencia ( $β$ =−0.15). However, the space cross-effects analysis produces positive parameters for the Balearics, Madrid, Murcia and Valencia suggesting that older houses in the major urban areas increase their value with age due to location or historical value. Size of property (lm² ) has a positive effect on house prices as expected: the larger the house, the higher the price. This is shown by the positive and statistically significant beta parameter in all provinces. The strongest effect is in Castellón, Madrid and Valencia, with $β$ >0.7, while for Alicante, Barcelona and Murcia coefficients range between 0.44 and 0.58. However, in the Balearic Islands, size does not show an effect on price. Different parameters for size illustrate the distinct price reaction to a marginal increase in the area offered by a house. As a result of this effect, Castellón, Madrid and Valencia are higher priced than Alicante, Barcelona and Murcia.

4.4 Time and Space Effect

Time and time–space effects on prices are also apparent, the lambda coefficients show a strong effect and are statistically significant in the overall model ( $λ$ = 0.26) supporting the existence of the autoregressive model but not strongly, as the parameter is considerably lower than one. The time effect is more apparent in Alicante and the Balearics ( $λ$ = 0.44 and 0.55), but smaller in Madrid (0.15) and Barcelona (0.17) with no evidence apparent in the rest of the provinces. Autocorrelation presents similar patterns across space as reflected by coefficients for Alicante ( $ρ$ =−0.01 for urban level 1), the Balearics (negative close to zero for urban levels 1, 2 and 3), Barcelona ( $ρ$ =−0.01 for urban levels 1 and 3), Madrid ( $ρ$ =−0.04 for urban level 1) and Valencia ( $ρ$ =−0.001 for urban level 3). Murcia is the only province presenting a positive (zero) and statistically significant relationship at urban level 2. At urban level 4 (province capitals), the space-recursive parameter is positive in all provinces, although for Barcelona and the Balearics it is not statistically significant.

5. General Linear Model

This section considers space–time effects using the general linear model (GLM) taking space (Urb_) and time (year) as random variables and estimating the interaction between these variables and house price attributes (as covariates) using maximum likelihood. The general linear model is used to estimate the variability in asking prices (expressed in logs) in accounting for various housing attributes or covariates and the random effects of time and space. Random effects which are included as fixed effects (assume that all observations are independent of each other) are not appropriate for analysis in panel data with the potential for correlated data structures that occur for house characteristics. Random effects are used at time and space levels to account for correlations between interest rates and house prices. The combined effects between random factors and the covariates are included as random interactions to capture the linear relationship between a covariate and dependent variable change for different levels of random factors. In order to compare with time–space results, $lPhask (t - 1)$ is also included as a covariate. The model is defined as the following expression

lPhas k_{i} = α + Γ C_{i} + Ω Z_{i} + μ_{i}

(8)

where, $C_{i}$ is a matrix of 27 housing attributes plus the lagged log of asking prices; Z is a matrix of random terms = {t, urb, t * urb, $t * C_{i}$ , $urb * C_{i}$ }, being time-year (t) and urban dependence (Urb_).

The estimated effects are calculated against the last observed year (2010) and the higher urban level (level 4 referenced to capital cities) and interactions are defined using 25 out of 27 attributes.¹⁰ $Γ$ and $Ω$ represent the matrix of parameters to be estimated. In relation to the parameters there are 76 time and space references, 508 random interactions and 51 single effects (including the intercept). Table 3 shows the estimated parameters from the main effects in the model.¹¹

Table 3.

Two models: GLM with random effects by time and space and pseudo STAR 2SLS model (STM)

STM Model: attributes /parameters estimated (dependent variable: lPhask)				GLM with random effects: attributes/parameters estimated (dependent variable: lPhask)
Socioeconomic	$β$	t	Significance		$β$	t	Significance	Obs power
Constant	2.371	152.41	***
Ecoact	0.037	74.88	***	Ecoact	0.009	2.95	***	0.839
$Δ pop$	−0.012	−12.65	***	$Δ pop$	0.028	5.00	***	0.999
Income	0.027	30.49	***	Income	0.084	17.41	***	1.000
Popdens	0.015	10.30	***	Popdens	−0.014	−1.43		0.299
Neighbourhood
Cons	0.000	6.29	***	Cons	0.001	5.20	***	0.999
Qroad	−0.010	−2.65	***	Qroad	−0.048	−0.72		0.112
Qwater	0.009	2.67	***	Qwater	−0.096	−1.78		0.430
Qlight	−0.011	−3.00	***	Qlight	0.127	1.49		0.321
Qshop	−0.007	−14.05	***	Qshop	0.028	9.13	***	1.000
Qschool	0.007	5.09	***	Qschool	0.075	4.63	***	0.996
Qleisure	−0.015	−5.40	***	Qleisure	0.021	1.34		0.267
Qsport	0.044	14.92	***	Qsport	0.002	0.21		0.055
Qhealth	−0.016	−10.71	***	Qhealth	0.026	1.26		0.242
resarea1	−0.139	−3.86	***	resarea1	−0.112	−10.60	***	1.000
resarea2	0.101	1.65		resarea2	−0.080	−5.47	***	1.000
Accessibility
Bus	0.043	3.58	***	Bus	−0.006	−1.35		0.270
Train	0.051	2.78	***	Train	−0.011	−2.84	***	0.810
Underg	0.055	2.82	***	Underg	0.018	5.19	***	0.999
Structural/property
Orient	0.006	30.82	***	Orient	0.008	6.50	***	1.000
View	0.017	34.29	***	View	0.020	6.90	***	1.000
Constq	0.009	0.37		Constq	0.014	3.84	***	0.970
dtype_t3	−0.006	−0.45		dtype_t3	0.016	38.21	***	1.000
ldwell	−0.007	−3.74	***	ldwell	−0.006	−3.90	***	0.974
llift	0.017	4.58	***	llift	0.008	2.72	***	0.778
lage, lage2	−0.004	−0.18		lage, lage2	−0.059	−8.39	***	1.000
lm2	0.537	9.12	***	lm2	0.202	5.60	***	1.000
lm2 other areas	0.057	47.67	***	lm2 other areas	0.053	57.57	***	1.000
lPhask_t1	0.261	97.31	***	lPhask_t1	0.763	26.77	***	1.000
urb1_lphask(t-1)	−0.008	−18.73	***	urb1_lphask(t-1)	−0.076	−5.88	***	1.000
urb2_lphask(t-1)	−0.003	−8.28	***	urb2_lphask(t-1)	−0.010	−1.86		0.462
urb3_lphask(t-1)	0.001	1.51		urb3_lphask(t-1)	−0.035	−5.48	***	1.000
R ²	0.830			Variability due to the errors	561.415
Adjusted R ²	0.830
Error standards	0.116			R ²	0.50
Residual	954.723			F (test Levene)	32.40
F	5457.22			p <0.01	0.00
significance p <0.01	0

The results are broadly similar to the pseudo spatio-temporal model calculated before (STM) with most estimated parameters in GLM having the same sign. In general, the STM estimated parameters are at the lower bound of those in the GLM random model. The main differences in sign are found in neighbourhood attributes for which co-linearity is strong. The lack of significance of bus stops at the province level is reproduced in the GLM; similarly, the relevance of underground stations in the main capital cities (Madrid, Barcelona) is also captured by the random model. View, orientation, construction quality, number of dwellings in buildings and lifts, all have similar parameter values. The age effect is better captured by the GLM random effects model while the floor area tends to be overweighted in STM. The parameter accounting for the effect on house price of extra housing areas (patios, garden) is similar in both models. The recursive parameter ( $lPhask (t - 1)$ ) is stronger in the GLM model both with time and space interactions and with the same sign.

6. Conclusions

This paper evaluates the role of housing and related attributes in explaining asking prices in the Spanish market over a long period from 1995 to 2010. The paper utilises hedonic models to fit the pricing process and observes how the parameters change with time and space. In this respect, the paper makes an important contribution to the literature through its application to the Spanish market, of which there has been little previous analysis in the international literature, but more significantly adds to the knowledge base on how parameters can change over the property market cycle and spatially at a macro level by province. The former reflects the dynamics of the market and the latter captures the effect of different perceptions of the value of housing and housing-related parameters by region arising from different economic, social, cultural and household structure factors.

The modelling process uses two methodologies to estimate hedonic models obtaining shadow prices of housing attributes and controlling by space and time. The first method is based on STAR models; the second uses GLM to estimate the hedonic model including time and space as random factors and calculating interaction effects. By observing changes over time and space, the paper shows how perceptions of value influence different pricing models. In particular, the results illustrate that the structure of attribute values is stable among regions reflecting specific characteristics of their housing market. Weightings change with time suggesting that the perception of attribute values in price formation varies depending on the position in the housing cycle.

The results support the relevance of the income, population, accessibility and structural features in explaining house price and differences at a spatial level. The analysis also discovers how different regions shape their particular characteristics and the different role played by primary and second-home urban areas depending on location. It is shown that economic activity and diversity increase house prices in the regions and that population dynamism and density are important influences. The paper complements the literature regarding how a large database could incorporate bias in hedonic price indices if not controlled for time and space.

Footnotes

Acknowledgements

The authors wish to acknowledge TABIMED for access to the company’s valuation database.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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