Glosario de Términos del Modelo de Dominio

Up to this point, our statistical analysis has ignored the spatial organisation o f the data. In effect, we have assumed that the observations o f property sales are independent such that we can glean no information on the selling price o f a property from the selling price o f other properties. O f course, this is hardly likely to be the case. Properties that are located near to each other in space are also likely to share common environmental, accessibility, neighbourhood and perhaps even structural characteristics. Even once we account for the values o f known covariates, omitted variables are likely to induce spatial correlation amongst property prices. Since we know the spatial distribution o f properties, it is possible to model this unaccounted spatial variation. Here we contrast two different approaches.

Spatial Constants Model (PLSC)

The simpler o f these two approaches is to partition the Birmingham cityscape into a patchwork o f areas. Ideally these areas should reflect regions o f relative physical and socioeconomic homogeneity. Here we choose to use the political boundaries that define the 39 electoral wards in the City o f Birmingham (see Figure 2). As such the covariate data is augmented with a set o f spatial constants indicating location in one o f these 39 wards.

Hedonic functions for each market segment are estimated by selecting one ward as the baseline and including spatial constants for the remaining wards in the covariate vector Zi. The parameters o f the PL model with spatial constants (PLSC) are estimated by applying Equation (20). Parameters for the spatial constants identify the average difference in property prices between each ward and the baseline ward once the influence o f all the other covariates has been accounted for.

Spatial Smoothing Model (PLSS)

O f course, including a set o f spatial constants to capture similarities in the prices of closely located properties is relatively crude. For example, this model assumes that a property on the boundary o f a ward has more in common with the other properties in its own ward than properties in its direct vicinity but on the other side o f the ward boundary.

Figure 2: Wards of the City of Birmingham

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An alternative approach has been suggested by Gibbons and Machin (2002) and Gibbons (2002). Their smooth spatial effects (SSE) estimator is a simple extension of the PL model described above.

The location o f a property can be expressed by the vector ci = ( c u, c2i ) where cu is an easting and c2i is a northing. Gibbons and Machin (2002) define their SSE estimator as a PL model in which the regressors to be modelled nonparametrically,

jc„ are simply the property locations, ct. Roughly speaking, the smooth spatial effects estimator strips location-specific determinants o f property prices out o f the regression equation.

Here we extend the Gibbons and Machin model by including both locational and property characterisitcs data in the nonparametric part o f the PL model. We call this a PL model with Spatial Sm oothing (PLSS). The PLSS model reformulates (5) as;

In/} = z,P + 9(*„e,.)+ e,. (21)

where jt,- is the vector o f variables listed in Table 1. Similarly, (6) is reformulated as;

In P, - £[ln P | x ,, c, ] = ( z, - E[z \ x ,, c, ] )fi + e, (22)

The quantities E \ \ n P \ x i,ci] and E[z | ,ci] are once again estimated using non­

parametric kernel regression according to;

Z r K X x i ~ x i ) K t (cy - c>)

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Z K X x i ~ x , ) K X ci - ci)

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where, following Machin and Gibbons (2001), K b( • ) is a bivariate normal kernel such that;

K-Xcj - c, ) = X z exp( ~ \ ( cj ~ C‘)B '(ci _c')] 2 n _\ <24)

and B is a 2 x 2 bandwidth matrix in which the diagonal elements take the values b2 and the off diagonals are zero. Choosing the bivariate normal as the kernel function allows properties closer to property i to be more important in determining the conditional mean functions. Similarly, using a diagonal bandwidth matrix with constant diagonal elements ensures that properties equidistant from property i exert identical influence in the calculation o f the conditional means.

Once again we can simplify notation by expressing the data in differences from the spatially smoothed conditional expectations; y i, = ln^ = ln/^ - E [\n P \ x itc^\ and

zt = Zj - E [z | Jtj. Stacking the y . data to form the N x 1 matrix Y and the

Zj vectors to form the N x k matrix Z allows the parameters o f the PLSS model to be estimated according to;

jj55 = {z'z)~'z'Y

Again, an important issue is the choice o f bandwidth parameters h and b. As previously we choose h through cross-validation and allow for an adaptive kernel. In contrast, we impose a predetermined value on the spatial smoothing bandwidth, b.

This decision is motivated by a number o f factors;

• In practical terms, adequate estimation o f the conditional expectations in (23) can only be achieved if we ensure that the area o f spatial smoothing is large enough to corral a sufficient quantity o f data points.

• The primary objective o f this research is to investigate the impact o f noise pollution on property prices. For rail and road traffic, this is a relatively localised phenomenon; noise environments can change markedly over tens o f metres and will almost certainly differ over hundreds o f metres. To ensure that we can identify these localised phenomena, spatial smoothing o f the regression data must operate at a somewhat larger geographical scale.30

A suitable scale o f spatial smoothing was adjudged to be an area roughly equal to the area covered by a ward. Clearly, choosing such a scale allows an interesting comparison between the results spatial constants model and the spatial smoothing model. As a result, the spatial smoothing bandwidth, b, is set to a distance roughly equal to the radius o f a ward31. The bandwidths reported in Table 2 differ across market segments reflecting differences in the size o f the wards from which properties in that market segment are drawn.

30 In contrast, the use of spatial smoothing (or for that matter spatial constants) will tend to reduce the ability of the model to pick out the influence on property prices of less localised spatial phenomena such as aircraft noise.

31 In particular, we set the bandwidth for each market segment as half the average of the 20% longest distances separating properties in that market segment that are located in the same ward.

Table 2: Spatial smoothing bandwidths

Market Segment Bandwidth (m)

1. Low Income White 1,744

2. Low Income Ethnic 1,484

3. Young FTB 1,652

4. Middle Income White 2,073 5. Middle Income Ethnic 1,738 6. High Income Large 1,981

7. High Income Small 2,274

8. High Income Medium 2,397

A priori the PLSS model is considered the better specification of the HPF. In contrast to the PLSC model, the spatial smoothing model considers information drawn from all properties in the environs o f a property in adjusting for location specific variation in property prices. Further, it attaches greater weight to more proximate properties than more distant properties. Finally, including location as an argument in the nonparametric part o f the PL model introduces substantially greater flexibility to the modelling o f unaccounted locational factors than constraining these effects to a handful o f parametric constants.