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Despite the use o f a spatial smoothing estimator, the possibility remains that a variety o f features o f each property’s location are not captured by the model. Since these features are held in common by properties in close proximity, it is possible that the regression residuals will exhibit spatial correlation.

As described in Section 3 o f Chapter 3, the presence o f spatial correlation in regression residuals ensures that OLS estimation will return inefficient estimates o f the model parameters and biased estimates o f the parameter’s standard errors.

Over recent years, the existence o f spatial autocorrelation has received a great deal o f attention in the hedonic literature (e.g. Dubin, 1992; Can, 1992; Pace and Gilley,

1997; Basu and Thibodeau, 1998; Bell and Bockstael, 2000). Under the assumption that the spatial processes can be modelled as a nuisance process, researchers have tended to focus on the spatial error dependence (SED) model. In this context the SED model is given by;

f = Z / ? ° + £ (26)

where

s = pW s + u (27)

and F , Z are defined as before and can be replaced by F , Z for the spatial smoothing estimator. is the K x 1 vector o f “true” parameters and e is the [N x 1] vector o f random error terms with mean zero. The nature of the spatial error dependence is defined by equation (27). Here W is an N x N weighting matrix, p is the error dependence parameter to be estimated and u is the usual N x 1 vector o f random error terms with expected value zero and variance-covariance matrix <fl.

Notice that p = 0 implies e - u and there is no spatial dependence in the data.

Rearranging (27) we find that;

e = ( l - p f V ) ~ 'u (28)

which indicates that the error terms s have a non-spherical variance-covariance matrix cr2 ( i - pW )~l ( i - p W ’)~x. Further, the error in the SED model can be seen to be made up o f two parts; a purely random element and an element containing a weighted sum o f the errors on nearby properties. The association between one property and another is contained in the weighting matrix, W. As in Chapter 4, the diagonal elements o f the weighting matrix are zero, whilst the off-diagonal elements that represent the potential spatial dependence between observations, are non-zero only if properties are closer than some predetermined distance, d. Further we adopt a binary weights matrix in which the Wj/h element o f W is initially set to one if the ith

a n d /h property are located within d metres o f each other, otherwise that element is set to zero.

Clearly, the choice o f d is important. Here we follow a similar line o f logic to that used to determine the spatial smoothing bandwidth b. First we note that enumeration districts (EDs), the smallest spatial unit at which census data is available, are defined so as to isolate regions with relatively homogenous characteristics. The size o f EDs, therefore, provides a guide to the spatial area over which localised similarities between properties and their inhabitants are likely to hold. As such we choose to define d for each market segment as the average radius of EDs inhabited by members o f that market segment . Since EDs vary in size across the cityscape, the value o f d as listed in Table 3, also differs across market segments, averaging around 250m.

Table 3: Characteristics o f the spatial weights matrices

Characteristics of the Spatial Weights

1. Low Income White 257 1,484 8.8 31 44

2. Low Income Ethnic 222 1,016 13.2 34 18

3. Young FTB 246 1,523 16.97 64 55

4. Middle Income White 236 1,362 5.54 22 105

5. Middle Income Ethnic 211 1,058 5.44 21 101

6. High Income Large 230 424 2.00 10 122

7. High Income Small 266 2,341 11.70 36 45

8. High Income Medium 321 1,433 10.92 47 45

Following normal procedure, W is row standardised such that each row’s elements are made to sum to one. When W is row standardised, the product We equals

32 In particular, we set the d 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 ED.

and has an intuitive interpretation; it is simply a vector o f weighted averages o f the errors o f neighbouring observations. As Bell and Bockstael (2000) point out, row standardisation is undertaken to simplify estimation o f the model.

There is usually no underlying economic story supporting the procedure. Moreover, the spatial dependence parameter p estimated on a row standardised weights matrix must be interpreted with caution. In particular, p in this case is not directly equivalent to an autocorrelation coefficient.

The characteristics o f the weights matrices constructed for each of the eight market segments are detailed in o f Table 3.Even with a relatively restrictive approximately 250 metre cut-off, the majority o f properties are associated with other properties in the same market segment. In market segment 2, for example, only 18 properties out o f the 1,016 observations were further than 222 metres from another property in the sample. On average in this market segment, each property was located within 222m o f 13 other properties in the sample, with at least one observation within 222m of 34 other properties in the sample. Notice that the number o f associations in market segment 6 is somewhat lower than in the other market segments. One explanation o f this observation is that properties in the affluent suburbs are more greatly dispersed than those in the other market segments.

The SED model can be estimated using maximum likelihood (ML) techniques in which the u vector is assumed to follow a multivariate normal distribution. However, for large samples this may be computationally prohibitive. Instead we follow Bell and Bockstael (2000) and use the generalised moments (GM) estimator developed by Kelejian and Prucha (1999). As Bell and Bockstael (2000) describe, whilst this estimator may not be as efficient as the ML estimator it possesses two advantages.

First, the calculation o f the estimator is fairly straightforward even with extremely large samples. And second, the GM estimator is consistent even when the error terms u are not normal.

The GM estimator is based on the somewhat weaker assumption that u are distributed //Z )(o , a2). As Kelejian and Prucha (1 9 9 9 ) show, this assumption allows us to construct the following three moment conditions;

— u ' W W u

N N T r( W W ) (29)

N ^{u'W 'u} = 0

where the third equality results from the fact that the diagonal elements o f W are set to zero.

O f course, the error term u is unobservable from a regression Y on Z . Rather, we must rewrite the moment conditions in (29) in terms o f e. Using (28) we get;

± - g' ( l - p w j { l - p W ) e

- U '( / - p W 'j W TV ( i - p w ) e ?— T r(W W ) (30)

- U '( i - p fV ^ TV'(l - p w ) e ⁼ 0

Under our assumptions, OLS estimation o f our two models (20 & 25) will provide consistent estimates o f the error terms e and we label these £. To simplify notation we follow Bell and Bockstael (2000) and denote s = Ws and £ = W W e. Thus from (30) we can build the following three-equation system;

GN\ p , p \ a 2\ - g N = v„ (p,< r2)

where the data vectors C/ v and gn are defined as;

(31)

Gn =

■ ₂ - - 1

1

' 1

— ££ --- £ £ — £f£

N N N

2 r.,?

— £ £ ² r.,p.

— £ £ — T rtiV W ) and g N = — s'e

N N N N

2 / *. — 1 c,.*.

0 1^{1 Al .}-

[££ + ££) --- £ £ — ££

\ / N ^_

and v ^ p jC r2) is a 1 x 3 vector o f residuals dependent on the parameters p and o2.

The system o f equations in (31) can be solved by nonlinear least squares (NLS) in which the parameter estimates p and a 2 are defined as those values that minimise the sum o f square residuals; v N (p, cr2) v N (p, a 2).

Armed with a consistent estimate o f the spatial correlation parameter, p , the PL models (20 & 25) can be re-estimated using feasible generalised least squares (FGLS). Accordingly, the spatial constants model can be estimated by;

P

( , . ^{v - 1 \ ~ !}

S C -S A

Z '[ { l - pW ') ( / - p H 'jj Z Z ' { ( l - p w j ( l - p W ) ) Y (32)

where p SA are estimates o f the parameters o f a PLSC model accounting for spatial autocorrelation. Similarly, the spatial smoothing model can be estimated by;

P

where p ss SA are estimates o f the parameters o f a PLSS model accounting for spatial autocorrelation.33

The estimators in (32) and (33) are calculated using code written by the author in the Gauss programming language. The calculations are made feasible even in relatively large sample sizes through the use o f sparse matrix commands that take advantage o f the relatively large number o f zero elements in the weights matrix W.

33 New residuals could be estimated using f? s or f? SSA and the new solution for p recovered in order to iterate the FGLS estimator. However, such a procedure is not relevant for large samples and this approach is not followed here.