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DEVELOPING A METHODOLOGY TO CAPTURE LAND VALUE UPLIFT AROUND TRANSPORT FACILITIES
APPENDIX B: PREDICTING RENT USING GEOGRAPHICALLY WEIGHTED REGRESSION TECHNIQUES AND HEDONIC PRICING
1 Geographically weighted Regression (GWR)
1.1 Introduction
Geographically Weighted Regression is a technique for exploratory spatial data analysis. There are, perhaps, thousands of examples of the use of multiple regression modelling in geographical enquiry. Typically these will involve estimating the relationship between one variable and a set of predictor variables for a collection of geographical entities (often a set of points, or zones). As an illustration, we might have a model with two predictor variables:
1.2 GWR and housing market
The housing market can be accurately portrayed as a set of distinct but interrelated submarkets that encompass dwellings differentiated by one or several alternative dimensions. These submarkets arise due to the joint nature of structural and locational attributes. Accordingly, attributes and sub-market locational features are essential ingredients in predicting prices.
It is essential that a large geographical area be divided into realistic submarkets or neighbourhoods to enable the model to more accurately reflect the influence of location. This form of stratification has the advantage of being tailored to local supply and demand factors that can vary substantially across a region. In terms of mass appraisal modelling the primary technique utilised is multiple regression analysis. This traditional econometric approach has been used to regress the housing value as a function of various structural, accessibility and neighbourhood attributes of dwellings. Analysing these characteristics the latter two are strongly related to location. This therefore gives rise to a reclassification of these characteristics between locational and structural (Olmo, 1995). The estimated coefficients provide hedonic prices or implicit marginal prices of the attributes considered.
To incorporate a spatial element within what is typically non-spatial modelling technique requires the proper specification of the spatial regressor. Generally speaking it is possible to derive individual models for each discrete sub-market or alternatively to employ an overall model encompassing several neighbourhoods, where each neighbourhood enters into the model as a dummy variable. The application of separate models for stratified homogeneous subsets induces a problem of sample size, which could result in statistically unsound and biased results. The use of multiple models can create circumstances when adjacent properties are grouped into different neighbourhoods and valued with reference to different models. Alternatively, the approach of using dummy neighbourhood variables to reflect the influence of location does have an intuitive appeal, but presupposes that the affect of that location is uniform across all properties within a particular neighbourhood. In addition, this form of delineation can be construed as static, which results in ignoring the potential effects of spatial trends. Since location is usually the most important variable in real estate value, it is imperative to account for locational variation before attempting to derive at definitive market coefficients for individual property attributes, given that such variables as age, plot size are highly correlated with location.
A multiple regression analysis that deliberately has no independent variables representing location (so-called, location blind multiple regression analysis) can be used to produce residual value in the rental data that is not attributed to any non-location variables.
As a general concept, location rent is the part of rent value that give rise to spatial differences in rents that cannot be explained by intervening variables and are assumed to be a premium for location. This normally means that alternative variables that could explain rent differentials have been controlled in some way as far as possible. The remaining differential in rent that cannot be explained by the systematic influence of any other factor is thought to be location rent. The extend to which it is possible to determine location rent is highly dependent on the availability and quality of data for alternative factors. The residual from a location blind multiple regression analysis can be used to crate a proxy for location rent.
2 Hedonic pricing
2.1 Introduction
The hedonic price regression continues to be a popular method for modelling quality-differentiated goods, with common applications to housing, to recover marginal values for public goods, and to fast-changing products like electronics, to compute quality-adjusted price indices. Because of its intuitive appeal and ease of implementation, the model's popularity continues despite misgivings about its strong assumptions about the continuity of choices available to households in the amenity space.
The hedonic model treats differentiated products as bundles of their underlying attributes z (houses as bundles of rooms, lots, neighbourhood characteristics; cars as bundles of horsepower, handling, safety; and so forth). Households bid on the products based on their demand for the amenities offered, sorting themselves by preferred types. This results in an equilibrium price function, p( z), where prices are functions of the underlying attributes and determined by the supply and demand for those attributes. It also implies that the slope of the hedonic price function at a point is equal to the marginal willingness to pay of households located at that point. If households are all identical, it is exactly equal to households' bid s (Freeman 1974); if households are heterogeneous, it is an upper envelope to their bids.
The hedonic model has been used both to forecast prices based on characteristics (using the price function) and to estimate implicit prices of amenities (using its derivative). Both approaches have been used to adjust price indices for quality change and new goods and to evaluate changes in public-good amenities for benefit cost analysis. In applications to price indices, the derivative of the price function can be used to adjust prices for the change in quality, or the price function can be used to impute what the price would have been if quality had remained constant. In applications to benefit cost analysis, the derivative of the price function recovers marginal values, while movement along the price function can serve as a bound for non-marginal values (Bartik 1988, Kanemoto 1988).
As is well known, the hedonic model requires some strong simplifying assumptions. In particular, the production of products must be convex in amenities, so that any continuous quantity is available, even when conditioned on other quantities. As the industrial organization literature has frequently pointed out, this means there is no space for the creation of new differentiated commodities. Or, to the extent it is used as an approximation when this condition is not met, the hedonic model implies that any products that do "fill in" the product space have no value (eg, Trajtenberg, 1990). In some cases, this problem has motivated economists to use discrete choice models to estimate quality-adjusted price indices that account for new varieties in lieu of hedonics (Trajtenberg, 1990; Nevo, 2003).
Recent work in public economics has begun to emphasize the importance of discreteness or discontinuity in exogenous choice sets. For example, Sieg et al. (2003) use a sorting model to recover values when households first sort into discrete communities defined by bundles of public goods, but then conditional on community can endogenously choose houses of continuously available sizes. Others have used more traditional discrete choice models for individual houses (Banzhaf, 2002a; Chattopadhay, 2000; Palmquist and Israngkura, 1999).
Four points of caution need to be made when interpreting exogenous amenities that are seemingly continuous. First, it is not enough that the amenity be distributed over a continuous support; the distribution itself must be continuous, for any atoms in the density of products will lead to clumping of potentially heterogeneous households at that point. Second, even if it can be thought of as being generated from a continuous distribution, any finite sample of houses will not be. Third, even if the marginal density of the amenity of interest is continuously distributed, if the joint distribution with other amenities is not then it may cause difficulty for estimating hedonics in practice. Although households should be sorted by their tastes for the continuous amenity after conditioning on the discrete amenities, conventional functional forms and even semi-parametric models may have grave difficulties identifying the price effects. Fourth, and perhaps more practically, amenities such as air pollution, while generated by a continuous process, are often modelled as entering preferences in a discrete form.
2.2 Hedonic price index
The hedonic price index is based on the idea that the rents of dwellings are determined in the market on the basis of their characteristics. The rents of dwellings rented in the market and the characteristics of those dwellings provide a basis for forming a so-called hedonic price model linking together the rents and the characteristics of dwellings. The price model applied is a weighted regression model, where the rent per square metre is explained by the various characteristics of the dwellings concerned. By including in the price model dummy variables relating to the time of renting, it is possible to estimate the trends in housing rents once the impact of the other variables in the model has been standardised. The coefficients of time dummy variables describe the ratio of rents for similar dwellings at different points of time. It is assumed that the variation observed in the rents per square meter is due to three main factors: differences in the individual characteristics of dwellings, the effect of the location and residential area of the dwelling, and the time at which the tenancy agreement was signed. The model applied can be written in general form as follows (StatFin, 2003):
The parameters of the regression model are derived by minimising the weighted residual sum of squares
where the dependent variable log( P iat) is the logarithmic rent per square metre, and log( P iat) is the estimated value of this. The weights w iat are derived from the current sampling frame.
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