# Hedonic regression and attribute-shares in house prices

Given a house with price $$P$$ and attributes $$h_1,\dots,h_n$$, I want to estimate how much each attribute costs as a percentage of house price $$P$$. In other words, if we express house price as the sum of the costs of individual attributes $$$$P=c_0+c_1h_1+\cdots+c_nh_n,$$$$ where $$c_0$$ is the "baseline cost" of a house and $$c_1,\dots,c_n$$ are costs of housing attributes $$1,\dots,n$$, I'm interested in computing $$$$p_i=\frac{c_ih_i}{P},\quad i=1,\dots,n,$$$$ where $$p_i$$ can be interpreted as the cost share of attribute $$i$$ of a house with price $$P$$.

Suppose I have data on $$P$$ and $$h_i$$'s. The natural way is to run a hedonic regression to estimate the $$c_i$$'s. However, I face the following difficulty:

Some of the estimated $$c_i$$'s may be negative, which leads to the corresponding $$p_i$$'s also being negative. For example, an $$h_i$$ could be distance to work, which reduces house price as the property is located farther and farther away from work. This creates a problem for interpreting $$p_i$$ as a cost share since it's less than zero.

Is there a standard method in the literature to deal with the problem of negative coefficients in hedonic regression? Any reference or tips would be greatly appreciated.

What you are proposing is an unusual application of hedonic regression in respect of house prices. The method is most commonly used for one of two purposes:

a) To explain and perhaps predict relative house prices.

b) To estimate the value of characteristics of houses and their neighbourhoods, especially those which are hard to value in other ways such as non-market environmental goods.

When the method is used for either of these purposes, negative parameters are unlikely to be viewed as a problem. Taking one recent study as an illustration, Mora-Garcia et al (2019) estimate hedonic regressions with negative parameters on various regressors (see Table 6, pp 15ff). Their discussion (p 25) notes the negative parameters without concern (focusing rather on the consistency of the negative signs between different models). When the aim of a hedonic study is to estimate the effect on house prices of an evironmental ‘bad’ such as air pollution, a negative parameter on the pollution regressor is to be expected. Komarova (2009), for example, found negative parameters for several air pollutants in Moscow (see Table III p 323 and Interpretation p 324).

It’s only your particular aim of analysing house prices into positive cost shares that makes negative parameters a problem. There isn’t a standard way of dealing with the problem because for most studies it isn’t a problem.

There are however some ideas in the literature that may be helpful. In the case of distance to work (and generally of distance to any amenity), an alternative is to use some measure of accessibility with the property of associating greater ease of access with a larger regressor value. Heyman et al (2019) review various measures of accessibility used in hedonic studies. In particular their Equation (3) in section 3.3.2 contains a term of the form $$e^{-c}$$, where $$c$$ is cost of travel between zones, giving that measure the desired property. Franklin et al (2003) present a rather sophisticated index of accessibility to work, also with the desired property, that takes account of multiple work locations with different numbers of employees (see Equation 2 p 5).

A general approach that could be used for any regressor that is found to have a negative parameter would be to derive from it a measure defined so as to vary in the opposite direction. Suppose for example that age of property $$A_i$$ was found to be significant and to have a negative parameter, and the highest age in the sample was 100 years. One could simply replace it by a regressor defined as $$100-A_i$$. Or one could define a scale of discrete points, say 5 for 0-10 years, 4 for 10-25 years, and so on. Trying out different measures of this kind might have the additional benefit of improving the fit of the regression.

• Thanks very much for the explanation and references! Much appreciated! Commented Mar 6, 2020 at 4:54