Area-to-point Kriging in spatial hedonic pricing models

Abstract

This paper proposes a geostatistical hedonic price model in which the effects of location on house values are explicitly modeled. The proposed geostatistical approach, namely area-to-point Kriging with External Drift (A2PKED), can take into account spatial dependence and spatial heteroskedasticity, if they exist. Furthermore, this approach has significant implications in situations where exhaustive area-averaged housing price data are available in addition to a subset of individual housing price data. In the case study, we demonstrate that A2PKED substantially improves the quality of predictions using apartment sale transaction records that occurred in Seoul, South Korea, during 2003. The improvement is illustrated via a comparative analysis, where predicted values obtained from different models, including two traditional regression-based hedonic models and a point-support geostatistical model, are compared to those obtained from the A2PKED model.

Appendices

Appendix 1: A2PKED as a form of generalized linear regression

Given both n point support data and K area-averaged data, the A2PKED prediction of the unknown value at a location u _p (see Eq. 2 for the original form) can be rewritten in the form of a linear regression model with correlated residuals, i.e., a point prediction of mean response and the residual, as (Chilès and Delfiner 1999):

$$ \begin{aligned} {\hat z}({{\mathbf{u}}}_p) &= {\hat \mu}({{\mathbf{u}}}_p) + {\hat r}({{\mathbf{u}}}_p) \\ &= {\hat{\varvec{\upbeta}}}^T {{\mathbf{f}}}_p^{{\mathbf{u}}} + \left[\tilde{{\varvec{\upeta}}}_p^T {{\mathbf{r}}}_{{\mathbf{u}}} + \tilde{{\varvec{\lambda}}}_p^T {{\mathbf{r}}}_s \right]\\ &= \sum_{m=0}^M {\hat\upbeta}_m f_m({{\mathbf{u}}}_p) + \left[\sum_{i=1}^{n} \tilde{\eta}_p({{\mathbf{u}}}_i) r({{\mathbf{u}}}_i)+\sum_{k=1}^{K}\tilde{\lambda}_p(s_k)r(s_k)\right]\\ \end{aligned} $$

(9)

where $\tilde{{\varvec{\upeta}}}_p$ and $\tilde{{\varvec{\lambda}}}_p$ denote, respectively, the area-to-point Simple Kriging (A2PSK) weights for n point data residuals and the K areal data residuals.

The GLS estimator of regression coefficients ${\varvec{\upbeta}}$ can be obtained as a function of the spatial correlation model of the underlying residuals ${\varvec{\Upsigma}}_R$ and the design matrix F as (Cressie 1993, p 167):

$$ {\hat{\varvec{\upbeta}}} = ({{\mathbf{F}}}^T {\varvec{\Upsigma}}_R^{-1}{{\mathbf{F}}})^{-1}{{\mathbf{F}}}^T {\varvec{\Upsigma}}_R^{-1}{{\mathbf{z}}} $$

(10)

with

$$ {{\mathbf{F}}} = \left[\begin{array}{l} {{\mathbf{F}}}_{{\mathbf{u}}} \\ {{\mathbf{F}}}_s \\ \end{array}\right], \quad {\varvec{\Upsigma}}_R = \left[\begin{array}{ll} {\varvec{\Upsigma}}_{{{\mathbf{u}}}{{\mathbf{u}}}}^R & {\varvec{\Upsigma}}_{{{\mathbf{u}}}s}^R\\ {\varvec{\Upsigma}}_{s{{\mathbf{u}}}}^R & {\varvec{\Sigma}}_{ss}^R\\ \end{array}\right], \quad {{\mathbf{z}}} = \left[\begin{array}{l} {{\mathbf{z}}}_{{\mathbf{u}}}\\ {{\mathbf{z}}}_s\\ \end{array}\right] $$

(11)

Note that when ${\varvec{\Upsigma}}_R$ is diagonal, with constant entries along its main diagonal, the drift prediction obtained from Eq. 1 coincides with that of OLS. Again, the unique solution of estimate ${\hat{\varvec{\upbeta}}}$ requires the ((n + K) × (n + K)) matrix of variogram values of residuals ${\varvec{\Upsigma}}_R$ to be non-singular and the ((n + K) × (M + 1)) design matrix F to be of full column rank.

Once the drift component is predicted as ${\hat \mu}({{\mathbf{u}}}_p) = {\hat {\varvec{\upbeta}}}^T{{\mathbf{f}}}_p^{{\mathbf{u}}}$ using the GLS estimator of regression coefficients ${\hat{\varvec{\upbeta}}},$ we can predict unknown residual component ${\hat r}({{\mathbf{u}}}_p)$ at the prediction location u _p using A2PSK. The A2PSK prediction at u _p is a weighted linear combination of the n “point data” residuals ${{\mathbf{r}}}_{{\mathbf{u}}} = [r({{\mathbf{u}}}_i) = z({{\mathbf{u}}}_i) - \sum_{m=0}^M f_m({{\mathbf{u}}}_i){\hat \upbeta}_m,\;i = 1, \ldots, n]^T$ and the “areal data” residuals ${{\mathbf{r}}}_s = [r(s_k) = z(s_k) - \sum_{m=0}^M f_m(s_k){\hat{\upbeta}}_m, k = 1, \ldots, K]^T$ as:

$$ {\hat r}({{\mathbf{u}}}_p) = \tilde{{\varvec{\upeta}}}_p^T {{\mathbf{r}}}_{{\mathbf{u}}} + \tilde{{\varvec{\lambda}}}_p^T {{\mathbf{r}}}_s = \sum_{i=1}^{n} \tilde{\eta}_p({{\mathbf{u}}}_i) r({{\mathbf{u}}}_i) + \sum_{k=1}^{K} \tilde{\lambda}_p(s_k)r(s_k). $$

(12)

Here, the A2PSK weights $\tilde{{\varvec{\upeta}}}_p = [{\tilde{\eta}}_p({{\mathbf{u}}}_i), i=1, \ldots, n]^T$ for the n point data and the weight $\tilde{\varvec{\lambda}}_p = [\tilde \lambda_p(s_k), k = 1, \ldots, K]^T$ for the areal data are determined per solution of a A2PSK system similar to that of Eq. 4 as:

$$ \left[\begin{array}{ll} {\varvec{\Upsigma}}_{{{\mathbf{u}}}{{\mathbf{u}}}}^R & {\varvec{\Upsigma}}_{{{\mathbf{u}}}s}^R \\ {\varvec{\Upsigma}}_{s{{\mathbf{u}}}}^R & {\varvec{\Upsigma}}_{ss}^R\\ \end{array}\right] \left[\begin{array}{l} \tilde{{\varvec{\eta}}}_p \\ {\tilde{\varvec{\lambda}}}_p\\ \end{array}\right]= \left[\begin{array}{l} {\varvec{\sigma}}_p^{{\mathbf{u}}} \\ {\varvec{\sigma}}_p^s\\ \end{array}\right] $$

(13)

where the covariance model of residuals involved in above A2PSK system is assumed to be identical to that used in Eq. 4.

In summary, A2PKED is equivalent to optimum drift estimation followed by area-to-point simple Kriging of the residuals from this drift estimate, as if the mean were known. The A2PKED prediction error variance accounts for the fact that the drift is actually unknown but estimated.

Appendix 2: Estimation of a local drift coefficients in A2PKED

Consider the task of predicting the unknown value at location u _p using n point data and a subset of the areal data instead of all K such data. This modification of the original A2PKED system in Eq. 4 may be necessary when dealing with large data sets of point support as well as areal support or when statistical modeling of spatial variation, i.e., spatially varying regression model coefficients, is adopted.

In our case study, we include only one areal datum in which the prediction location u _p falls, which may affect the efficiency of the GLS estimator as well as the residual component. This change of the original system may yield some problems in the design matrix, particularly when dummy variables associated with areal data are included. However, this change does not destroy the desirable properties of Kriging prediction, such as unbiasedness and minimum prediction error variance. In what follows, we present the A2PKED prediction at location u _p as a linear regression form with correlated residuals based on n point data and a single areal datum:

$$ {\hat z}({{\mathbf{u}}}_p) = \sum_{m=0}^M {\hat \upbeta}_m({{\mathbf{u}}}_p) f_m({{\mathbf{u}}}_p) + \left[\sum_{i=1}^{n} \tilde{\eta}_p({{\mathbf{u}}}_i) r({{\mathbf{u}}}_i) + \tilde{\lambda}_p(s_k)r(s_k) \right] $$

(14)

where ${\tilde \eta}_p({{\mathbf{u}}}_i)$ and ${\tilde \lambda}_p(s_k)$ denote, respectively, the A2PSK weight assigned to the ith point support residual $r({{\mathbf{u}}}_p) = z({{\mathbf{u}}}_p)-{\hat \mu}({{\mathbf{u}}}_p)$ and the kth areal residual $r(s_k) = z(s_k) - {\hat \mu}(s_k)$ at the region s _k in which the prediction location u _p falls. Note that the A2PSK weights for point data and the single areal datum in Eq. 14 need to be updated at each prediction location as the areal datum associated with each prediction location is subject to change. This amounts to estimate a spatially varying (local) linear drift component whose GLS regression coefficients are constant within each neighborhood, but vary from one area to another. For example, the GLS regression coefficients ${\hat{\upbeta}}_m({{\mathbf{u}}}_p)$ with m = 0,…, M at location u _p are different from those ${\hat \upbeta}_m({{\mathbf{u}}}_{p'})$ at location u _p′ if u _p′ ∈ s _k′ for k ≠ k′.

Typically, the application of GLS to hedonic price models assumes that the relationship between house price and covariates is fixed over the study area. The proposed approach takes into account heteroskedasticity present in house prices so that the implicit price of housing attributes varies spatially by submarkets defined by areal units (Orford 2000).