D ISEÑO Y EJECUCIÓN DE LOS C ASOS DE P RUEBA DE C AJA N EGRA

This study extends the spatio-temporal filtering process proposed by Pace, Barry, Clapp and Rodriquez (1998) to a two order spatio-temporal autoregressive model. The rational behind is due to the unique characteristic of multi-unit residential market. We argue that in a multi-unit residential market, the spatial process is different from that of a single family market, which has been extensively covered in US literature such as in Pace, Barry, Clapp and Rodriquez (1998) and Pace, Barry, Gilley and Sirmans (2000). In the multi-unit residential market, there are two kinds of spatial effects which may cause spatial autocorrelations among the housing prices. The first is named as the building effect. It refers to the effect of the unique characteristics of every building, such as the quality of design and layout, orientation and view, and distance from the main road. Building effects can differ from one building to another within the same condominium project. In this study, such effect is captured by a first order filtering process. The second is the conventional neighborhood effect, which encompasses location, distance to amenities and so on and is captured by a second order filtering process in the model proposed in this study.

These two kinds of spatial effects may have different influences on the price of an individual housing unit. For example, the units in two buildings which have the same structural characteristics and are very close to each other may fetch different prices. Units

Chapter 4 Research Methods

__________________________________________________________________________________ 56 in the building that is farther from the main road are likely to fetch a higher price because of the reduction in traffic noise. Therefore multi-unit residential properties in the same neighborhood could still exhibit relative location advantages which produce an irregular price pattern within the neighborhood that should be captured.

The existence of building effect implies that there may be potential irregularity among the prices of housing units in different buildings but are geographically close to each other, due to the unique location characteristics of each building. The irregularity may not be efficiently captured by the traditional neighborhood effect. In the context of single family market, since no two housing units would be in exactly the same location, such building effect doesn’t exist. Therefore, the STAR model proposed by Pace, Barry, Clapp and Rodriquez (1998) removes the autocorrelation caused by the neighborhood effect only. In a multi-unit residential market, however, transaction prices of housing units in the same building would include information about the building effect. So, instead of capturing spatial autocorrelation by one spatial weight matrix S as in Pace, Barry, Clapp and Rodriquez (1998), we split S into two separate spatial weight matrices, W1 and W2 and thus extending the STAR model into a two order one.

W1 refers to the building effect. Practically, in our data, each condominium has a few building blocks and every building has one postcode that corresponds to a pair of unique coordinates. The distance between the units is the linear distance between the buildings computed by these coordinates. Therefore the distance between the units in the same building is zero (attribute to building effect matrix). The building effect is captured by the

Chapter 4 Research Methods

__________________________________________________________________________________ 57 prior transacted units in the same building. However, if there is no prior transaction in the same building, it is substitued by the earlier transactions in the first nearest building.

W2 is the neighborhood effect matrix, similar as defined by Pace, Barry, Clapp and Rodriquez (1998). It is worth to mention that, W1 and W2 may be collinear as W1 may capture certain degree of neighborhood effect. Therefore if there are not enough neighborhood variations, W2 may be insignificant empirically.

With this treatment, equation (4.6) in section 4.2 is extended as equation (4.7).

) (

)

(I −W = I −φ_W1W₁ −φ_W2W₂ −φ_TT −φ_W1TW₁T−φ_W2TW₂T −φ_TW1TW₁−φ_TW2TW₂ (4.7) Where are the building effect and the neighborhood effect matrices; T is the temporal matrix as before; W

2

Equation (4.5) can be re-written as equation (4.8) by substituting equation (4.7) into (4.5) and rearranging:

Where, are the parameters, indicating

2

spatio-temporal lags of independent variables. The spatio-temporal lags of dependent

variable are W .

Chapter 4 Research Methods

__________________________________________________________________________________ 58 There are two points to note here. First, obviously, there may be multicollinearity problem in equation (4.8). However, in case of large sample as this study, the impacts of multicollinearity problem can be reduced significantly. Second, the lagged dependent variables may create estimation bias if the OLS method is used and the sample size is small. In large samples, however, it will still produce consistent and asymptotically efficient estimates provided that the residuals are not autocorrelated (which is supported by two order spatial-temporal filtering process).

The difference between equation (4.7) and equation (4.6) is that in equation (4.7), two separate spatial weight matrices W1 and W2 are used instead of one neighborhood effect matrix S in the STAR model. Therefore, equation (4.8), derived from equations (4.7) and (4.5) is named as a two order spatio-temporal autoregressive model (2STAR).

When thinking about spatio-temporal interactive process, it is not sure about whether building effect and neighborhood effect may have different interactive processes with time, and hence, we specify another kind of spatio-temporal interactive process by adding the assumption of (φ_W1TW₁T +φ_W2TW₂T)=(φ_STST) and (φ_TW1TW₁+φ_TW2TW₂)=(φ_TSTS) . Therefore, a equation (4.9) can be derived by considering the interaction of spatio-temporal effects using the combined spatial weight matrix (S) only.

ε

Chapter 4 Research Methods

__________________________________________________________________________________ 59 Where,

β

β

β

β

β

2

1 φ β

W1 ,φ_W2β ,φ_Tβ , β

φ_ST , φ_TSβ separately. In next chapter, the results estimated against equations (4.8) and (4.9), as well as the original STAR model in equation (4.5) are reported and compared.