33  The Hedonic Housing Price Model

Many data science tutorials use the example of predicting housing sales prices in regression modeling exercises. Many spatial analysis tutorials also use housing price models to help illustrate the concept of spatial autocorrelation and the use of formal models to ivestigate spatial structure. Nearly all of the examples in modern curricula, however, focus almost exclusively on the modeling frameworks themselves, with little background on the theory of why we apply these models in the first place (save that they can accomodate spatial features). This is understandable, because covering both urban economic theory in addition to the statistical analysis of spatial data is a lot of ground to cover. But it is also unfortunate, because the lack of theoretical background can lead to confusion regarding which modeling strategies are appropriate in different situations.

The reason housing price models are ubiquitous in regression modeling curricula is because of their classical use in urban economics and regional science to help uncover the determinants of housing market pricing. That may sound like a tautology, but unlike many modern examples, the goal of housing price modeling in the social sciences is almost never to predict the selling price of a home. Rather, it is to understand how public policies or other exogenous shocks to a housing market may affect consumers’ willingness to pay for certain features of the urban fabric, like access to jobs, clean air, or other place-based policies (Harrison & Rubinfeld, 1978; Neumark & Simpson, 2015; Reynolds & Rohlin, 2014; Won Kim et al., 2003).

Thus, although estimating them as such requires considerable data and forethought into the modeling structure, housing price models in urban economics were conceived in a causal inference framework. As such, the application of spatial econometric regression modeling is not simply to treat geography as an additional “feature” that helps predict variation in prices, but to formalize explicitly and explore the process(es) of spatial spillover, and to ensure that any unobserved spatial process does not bias estimates of the “true” effect under study.

This is an underemphasized point in the era of kaggle competitions and Deep Learning chatbots, because the goals of data science/ML and social science are, quite often, fundamentally distinct. Whereas data science in industry is more commonly focused on prediction (e.g. a real estate company is more interested in predicting the selling price of a home than they are about whether the price is more affected by proximity to restaurants or a regional minimum-wage law, because the former is valuable to users of their website), the reverse is generally true in policy analysis and the social
sciences.

A major reason for estimating hedonic prices and inverse demand functions is to be able to measure the benefits of changes in the level of environmental amenities. Briefly, a household’s marginal benefit for a small improvement in amenities is its marginal willingness to pay-as estimated by the marginal implicit price it faces. For a non-marginal change the benefit is approximated by the area under the inverse demand curve for the change in question. And aggregate benefits for an urban area are found by summing the relevant household measures across all households

– Freeman III (1979)

The logic of housing price regression models originates from Rosen (1974) and the theory of hedonic pricing in implicit markets1. The concept holds that a housing unit is a bundle of goods, rather than a single item; that is, a housing unit represents several consumption choices at once: a location decision that defines access to education systems, employment markets, environmental externalities and so on, as well as a piece of architecture, with size, quality, amenities, and maintenance characteristics. The selling price of a home is a combination of all these attributes that we can only observe in aggregate. To recover the implicit prices of a housing unit’s constituent components, we can follow Rosen’s approach (extended by Harrison).

Following Harrison & Rubinfeld (1978), the hedonic price model assumes consumers maximize a utility function is defined as

\[U(x,h)\]
subject to the budget constraint: \[Y = x+p(h)\] Where
\(x\) = quantity of composite private goods, whose price is set equal to one
\(h = (h_1,\dots,h_n)\) is a bundle of housing attributes, including accessibility, structure and neighborhood characteristics, and air pollution concentrations
\(y\) = annual money income,
\(p(h)\) = housing (or hedonic) price function

Then, we can use regression models of various forms to approximate \(p(h)\), according to the following assumptions:

  1. All consumers accurately perceive the characteristics represented by the vector \(h\) at every location.
  2. There is sufficient variation in \(h\) so that the function \(p(h)\) is continuous, with continuous first and second partial derivatives.
  3. The market is in short-run equilibrium.
  4. Spatial variations in housing characteristics are capitalized into differentials in housing prices.

The first assumption is a big one, and it’s false in many cases. Real estate agents, prior inequalities, and other market forces absolutely ensure that some groups of consumers have better knowledge of \(h\) than others (and this knowledge will vary imperfectly across space as well). Large real estate firms that buy and flip homes nationwide certainly have better knowledge of the housing market than a first-time homebuyer with a median income. Like most economic assumptions about perfect markets, we can swallow this large grain of salt and proceed, but it is worth remembering that because everyone’s utility function is different, the hedonic approach is designed to approximate the consumption of the ‘marginal consumer’. Exactly who the ‘marginal consumer’ represents is an open question.

The second assumption is essentially the law of large numbers; we need to observe good amount of variation before the signal is reliably detectable. The third assumption follows loosely from Tiebout (1956), another seminal contribution in urban economics and location theory, holding that people “vote with their feet,” by sorting into neighborhoods that provide the greatest utility to them. This assumption is subject to the same critiques as the first. Perfect freedom of mobility is not a realistic assumption, especially for lower income strata, and the correlation between housing unit quality and neighborhood attributes mean that it’s unlikely that the housing market provides a perfect range of options that allow consumers to optimize perfectly. In the country’s best school district, it’s unlikely to find a housing unit that is small and cheaply constructed, even if there are consumers who would happily trade size and perceived quality for access to the best schools (DeLuca et al., 2024). Our view of the optimization is limited because swaths of the population may remain “stuck in place,” (Sharkey, 2008, 2013; Wilson, 1987).

Nonetheless, empirical evidence for the Tiebout hypotheis is strong, suggesting that ‘short run equilibrium’ can be achievived as long as location sorting remains viable for some part of the housing market. There are parallels here to Schelling’s work showing small preferences can yield large changes in regional patterns. The geography of opportunity may well be defined by Tiebout equilibrium–according to the neighborhood tastes of the affluent (tieback to Sampson work on importance of concentration of affluence vs historical focus on concentration of poverty)

“hedonic price theories of housing are most useful in providing a conceptual basis for the determination of a temporary equilibrium in which supply is fixed.” Arnott (1987)

The fourth assumption is the fundamental logic behind the hedonic model; the combination of physical qualities, location attributes, and other externalities affecting a housing unit are capitalized into its price (note that unlike Harrison & Rubinfeld (1978), in the formulation above we assume transportation costs are also part of the housing function rather than a separate term). An interesting question at the frontier is when the capitalization signal for certain amenities can be detected (for example, if a new transit station is planned, when do rents in the nearby apartments rise? Does the station need to be open, or does speculation cause rents to rise when the station is announced?). If the amenity has not been realized yet, will renters pay the premium for an anticipated good?

The hedonic approach was conceived as a structural model, designed to uncover the determinants of housing demand based on principles of economic theory (Koopmans, 1949; Nevo & Whinston, 2010). Rosen (1974) postulated that under equilibrium conditions, the bundle of housing goods can be decomposed into its constituent parts. The structural components rely on two critical assumptions: 1. that the housing market is in short-run equilibrium (i.e. a tieboutian process is at work) and (relatedly) 2. that each consumer individually maximizes her utility function. Under those two assumptions, the market clears and we can observe revealed preferences and willingness to pay for different housing attributes. In practice, many scholars want to estimate hedonic models to understand the effect of some policy change on home prices, but in these cases identification issues are rife, especially if home prices can affect one another.

33.1 Spatial Hedonics

Spatial data help solve some inherent difficulties in identifying the hedonic model. First, as Won Kim et al. (2003) describe, the traditional hedonic model is biased in the presence of spatial processes such as land use regulations or environmental quality

“The traditional hedonic property value model does not capture these induced [spatial] effects. It is customary in traditional models to include exogenous variables in the neighborhood characteristics category that try to explain why some neighborhoods tend to have higher or lower housing prices than other neighborhoods. Such a specification cannot capture spatial price effects that are generated by a change in a neighborhood’s housing characteristics (such as a shift in environmental quality). Therefore, a traditional hedonic property value model may lead to a biased or at least imprecise estimate of the benefits of a housing characteristic change if these induced effects are present”

– Won Kim et al. (2003)

Second, as Bartik (1987) describes, OLS estimates will be biased because quantity and consumption are endogenous, so an instrumental variable approach is necessary to provide accurate estimates. Modern spatial econometrics help solve both of these problems simultaneously. “Spatial lag models” which include endogenous spillovers in the \(y\) variable use instrumental variable regression (two-stage least squares) with \(WX\) variables as instruments to identify the parameters of interest (Anselin, 2011; Arraiz et al., 2009; Kelejian & Prucha, 1999). This follows Bartik’s logic of using cities or neighborhoods as plausible instruments while simulateneously providing insight into spillovers and capturing unobserved spatial effects.

Finally, the housing market is a clear example where spatial autoregressive processes are both conceptually intuitive and substantively interesting. Houses are sold at different times and their value is often unclear until the point of sale. As such, each sale in the market sends a signal about the value of homes in that neighborhood; there is a clear process of spatial spillover, as other nearby homes are revalued according to the nearby sale.2

In this specification of HPF, the value of a house at any location is dependent on its counterparts at nearby locations in addition to its structural and neighborhood attributes. The hypothesized spatial dependence among residential structures is determined by W which is specified in an a priori fashion. The coefficient p measures the absolute price impact of nearby houses on the price of a particular house. As argued in Can (1990), this conceptualization corresponds more closely with the actual workings of the real estate institution in urban housing markets. A realtor will appraise a house given the price history of houses in the immediate vicinity in addition to other substantive characteristics. At the same time, home owners will initiate or forego certain improvements based on the anticipated return on their investment considering housing prices in the immediate area.

– Can (1992)

Spatial econometric methods have been developed explicitly for use in contexts such as hedonic price modeling because they provide for efficient and unbiased estimation of coefficients when property sales are spatially interactive (Anselin & Lozano-Gracia, 2008; Can, 1992; Comber & Arribas-Bel, 2017; Diao, 2015; Diao et al., 2017; Dubé et al., 2014; Dubin, 1992; Kim et al., 2020; Steimetz, 2010; Won Kim et al., 2003). For example spatial econometric models provide an avenue for studying the relationship between housing characteristics and sales prices, even when nearby home sales have an endogenous influence on prices. These same properties provide an ideal opportunity for understanding how to value land and its features in a hedonic modeling framework when land and improvement characteristics would otherwise be difficult to separate.

Spatial econometrics offer important solutions to some identification problems, but also occupy a nebulous space in the econometric modeling world. Technically, spatial econometrics models are structural because they require a priori specification of a spatial weights matrix that defines the feedback structure among observations, and they should be guided by a strong theoretical basis for which DGP is at play (Gibbons & Overman, 2012; McMillen, 2012). But they are not fully structural because there is little theory guiding the appropriate specification of \(W\) (Corrado & Fingleton, 2012), and it is often hard to specify with certainty whether residual autocorrelation is caused by unobserved variables or processes of spatial spillover. The spillover term can never be known in practice, and differentiating spillover from spatially autocorrelated errors is exceedingly difficult in practice, so the inclusion of a global spillover parameter needs to be governed by theory (LeSage, 2014b; LeSage & Pace, 2014). That is, we might alternatively view the autoregressive term as a structural parameter.

“The main point is that the model should correspond to the workings of the spatial economies investigated, and that the consequences for their ulterior use – or uses – should be explicitly considered.”

– Paelinck (2007)

33.2 Model Specifications with Spatial Effects

In a seminal contribution, LeSage & Pace (2009) differentiate between two scales of spatial dependence (local vs global), and two substantive varieties (spillover vs diffusion). Spillover occurs in a situation where “changes to explanatory variables in region \(i\) impact the dependent variable values in region \(j\)” (LeSage & Pace, 2014, p. 1537), whereas diffusion occurs when a shock to region \(i\) affects the disturbances of regions that neighbor observation \(i\). Local effects occur when spillover or diffusion only falls on the immediate neighbors of unit \(i\), whereas global effects occur when spillover/diffusion also falls onto neighbors-of-neighbors (including back to unit \(i\) itself in a feedback loop). Local and global spillovers and diffusions are incorporated into regression models using a variety of specifications discussed below. In all cases, \(W\) refers to the spatial connectivity graph (spatial weights matrix) the specifies the “neighborhood” of each unit.

33.2.1 SLX Model

The SLX (“Spatial Lag of X”) model is the simplest spatial model, and it does not require any specialized estimation techniques other than OLS. This model allows for local spillovers, and assumes the price of a unit is a function of the unit’s exogenous characteristics, the exogenous characteristics of other units in the neighborhood, and a random error term. \[ y = \beta X + \theta WX + \epsilon \] The SLX model is nice because it can incorporate local spillovers and requires only OLS estimation of the coefficients (which are interpretable as usual). This is the general starting place for incorporating spatial effects (Halleck Vega & Elhorst, 2015). In this model, \(\beta\) measures how changes in the property itself affect its selling price (the “direct effect”), while “the coefficient \(\theta\) measures how changes in neighboring properties’ characteristics impact the value of a typical property (on average over the sample).” (LeSage & Pace, 2014, p. 1541) (the “indirect effect”).

33.2.2 Spatial Lag Model

The Spatial Lag Model (sometimes called the spatial/simultaneous autoregressive or SAR model) allows for global spillovers in the dependent variable. Here we assume the price of a unit is a function of neighboring prices, exogenous characteristics, and random error. We use the spatial lag model and the spatial Durbin model (explained below) when endogenous spillovers are of interest (or when there is reason to believe there are feedback mechanisms in the dependent variable). In the case of home prices, there is no solid consensus among urban economists about whether prices demonstrate global spillover, but there is sound rationale for intuiting such process via price speculation and game theoretic behavior (LeSage, 2014b; LeSage & Pace, 2014). The simple fact that a home sells for a large price tag nearby sends information that my location may have increased in value. That is, the price itself, apart from any other characteristics of nearby properties has a spillover effect in its own right. The spillover term in this case is pure space, and we can consider it part of the land value.

\[ y = \rho Wy +\beta X + \epsilon \]

By definition, this is no longer a linear model because \(y\) is partially endogenous. We can still estimate the average \(\beta\)s, but it becomes more difficult to explain this model. The coefficients are no longer interpretable as normal regression coefficients (because the model is non-linear), so proper interpretation of the results requires computation of the marginal effects (and dispersion estimates thereof) (Elhorst, 2010; LeSage & Pace, 2009). For this reason, it can also be difficult to generate predicted values. Computation of marginal effects is possible in spatial econometric packages in Python, R, MatLab and Stata, but discussion of marginal effects is still not widespread in the literature (LeSage & Pace, 2014).

33.2.3 Spatial Error Model

The spatial error model (SEM) does not assume that prices are endogenous, but rather that an unobserved spatial process causes spatial correlation among units. This is a model of global diffusion, and effectively treats spatial structure as a nuisance to be filtered away. The SEM is sometimes applied as a way of controlling for spatially-correlated omitted variables (though it is not effective at removing OVB, since it controls only for a specific spatial structure, so it is discouraged from that perspective).

\[ \begin{gathered} y = \beta X + u,\\ u = \lambda Wu + \epsilon \end{gathered} \]

The spatial error model is appropriate when you do not expect a “real” endogenous spillover process and you intuit that any residual spatial autocorrelation is a result of some higher-order geographic process. In those cases, this is an attractive model because (a) the coefficients can be interpreted as usual, and (b) it is straightforward to extract predicted values. The downside of this model is that if there is an endogenous spillover process, then the cumulative effects may be improperly estimated because they do not account for feedback effects that can only accrue through the \(\rho WY\) term.

33.3 Including Multiple Spatial Effects

The spatial “Durbin” models combine the SLX specification with either the spatial lag or the spatial error specifications. This allows differentiation of the “direct” spillovers from the exogenous characteristics of nearby observations as well as the “indirect” spillover that accumulates through the autoregressive process. This allows us to parse the differences between local spillovers in each exogenous variable (\(\theta WX\)) from the global effects induced by the autoregressive process (in either the dependent variable or the error). The two Durbin specifications are the recommended models for applied work because they provide the greatest generality while maintaining identifiability for all parameters of interest.

33.3.1 Spatial Durbin Model

The Durbin model (SDM) extends the spatial lag model by also allowing for local spillovers in the exogenous variables. Here, price is a function of (endogenous) neighboring prices, exogenous characteristics of the unit itself, exogenous characteristics of nearby units, and random error. The Durbin model includes both local and global spillovers, and in many cases is the “ideal” specification from an applied hedonic perspective.

\[ y = \rho Wy +\beta X + \theta WX + \epsilon \]

As with the Spatial Lag Model, the SDM is non-linear and requires estimation of marginal effects (and estimates of their dispersion) to properly interpret the coefficients (and their significance). A key benefit of this model over the spatial lag model is that the spillover effects can have a different sign than the direct effects (Elhorst, 2010; LeSage & Pace, 2014). For example having a tall building on a parcel may increase its value, but being surrounded by tall buildings could decrease its value. This is the major improvement of including the \(\theta WX\) terms, however, the coefficients still cannot be interpreted directly as spillovers (as in the SLX and SDEM models), but instead the model requires computation of marginal effects for understanding “direct” and “indirect” (or spillover) effects.

33.3.2 Spatial Durbin Error Model

The spatial Durbin error model (SDEM) allows for local spillovers and a global error diffusion process. Thus, it includes the exogenous from nearby observations and a spatially correlated error term (but no endogenous spillover in the dependent variable). The Durbin Error model is the recommended approach when (a) there is no substantive interest in feedback effects or endogenous spillover and (b) there is no reason to suspect that (prices, in this case) could be autoregressive (if they are, then SDEM would be misspecified and SDM should be fit instead) (LeSage, 2014b).

\[ \begin{gathered} y = \beta X + \theta WX + u, \\ u = \lambda Wu + \epsilon \end{gathered} \]

Thus, the SDEM is an attractive model from an interpretive perspective because it does not require computation of marginal effects; there is no \(y\) variable on the right-hand side, so the model is still linear, and the coefficients have the usual interpretation. While the SDEM may be unfamiliar to people outside spatial econometrics, its parameters are straightforward compared to SAR or SDM models. As LeSage & Pace (2009, p. 1541) describe, “For the SDEM and SLX models, the coefficients in the vector \(\theta\) represent local spillovers, since there is an impact only on immediately neighboring observations. We note that estimates from these two models should be similar, but in the face of spatial dependence in the disturbances, SDEM model estimates should be more efficient.” The dispersion estimates for SDEM can be estimated readily by standard spatial econometric software.

33.4 Others

Apart from these five model specifications, there are other common spatial econometric model specifications, as well as other ways to incorporate spatial relationships into a regression framework. In general, these approaches are less appropriate for land value modeling due to reasons discussed briefly below.

33.4.1 General Nesting Specification and Manski Model

Technically, you can combine some of the models described above, i.e., to allow for endogenous spillover and spatially-correlated error (the “general nesting specification” that includes both \(\rho\) and \(\lambda\)), or all spatial terms under the sun (the “Manski Model”). While the GNS was once a recommended practice, it is now discouraged because those models become inefficient and the effects become inseparable (Elhorst, 2010; LeSage, 2014b; LeSage & Pace, 2018; LeSage & Pace, 2009) (and the Manski model cannot be estimated). Instead, the recommended advice for applied work is to adopt either the spatial Durbin model, (SDM) or the spatial Durbin error model (SDEM), depending on whether endogenous spillovers are expected or not (LeSage & Pace, 2014).

33.4.2 Geographically-Weighted Regression (GWR)

flowchart LR
    choice{Should I use GWR?}
    answer(No)
    choice --> answer
Figure 33.1: To Use GWR or Not

This statement will draw lines in the sand among camps with competing views on spatial analysis–including my buddies (Fotheringham & Oshan, 2016; Oshan et al., 2020). But unless you are Dan McMillen and absolutely know what you are doing (McMillen, 2012; McMillen & McDonald, 1997; McMillen & Redfearn, 2010), my view is you probably should not be using Geographically-Weighted Regression (GWR). While there are clear negative consequences from the blind application of spatial econometric methods (Gibbons & Overman, 2012; McMillen, 2003, 2010), from an empirical perspective, the misapplication of GWR is by far more widespread. GWR is a very useful spatial analytical tool, but its inclusion into software packages like ArcGIS have given people the impression that it, too, is a panacea for all issues spatial. And while researchers may be misinterpreting the results from spatial econometric models, they are almost certainly misinterpreting the results from GWR.

GWR is an approach to spatial analysis that differs significantly from spatial econometric modeling. Whereas spatial econometrics uses a specific hypothesized structure of \(W\) to test for a formal DGP, the GWR approach emphasizes flexibility over statistical inference or generalizability3. It is a special case of non-parametric locally-weighted regression and, thus, has no underlying conceptual statistical model, and no appropriate way to interpret its coefficients–it is just an overfit model of a specific dataset, not a DGP. The method works by fitting many local regressions instead of a single global regression. That is, for each observation in the dataset, the approach is to select some nearby observations (say 10) and fit a unique regression for that point, then proceed to the next point until all points are exhausted.

This allows the coefficients to vary over space (i.e each observation), so it is a nice way of exploring spatial heterogeneity, and can also lead to some improved predictive power over OLS (because the coefficients are adaptive to local conditions). But because the data points are re-used repeatedly (and there is no underlying conceptual model) the coefficients from GWR are not interpretable like those from OLS, and do not represent the marginal change of X on Y (Wheeler & Calder, 2007; Wolf et al., 2017). GWR models are also unable to capture processes of spatial spillover when these effects are the substantive interest of study.

This has earned GWR many critiques in the spatial analysis literature, because it is commonly misinterpreted (Griffith, 2008; Páez et al., 2011; Wheeler & Calder, 2007). Indeed, Comber et al. (2022) (a group of GWR’s most prominent scholars) describe that, “as a rule, spatial effects via a [spatial econometric model] should be preferred [to GWR] due to its stronger inferential properties (e.g., see LeSage and Pace 2009). This is because inference in any GWR model is somewhat compromised by there being no-one single model, but a collection of models re-using sample data at multiple locations. This entails that a valid probability model is unavailable with GWR, making inference biased and problematic.” In other words, “GWR is more appropriately viewed as an exploratory approach and not a formal model to infer parameter nonstationarity. This view conflicts with the broad application of GWR as an inferential method” (Wheeler, 2014, p. 1443).

In the context of land-value modeling, GWR can be particularly problematic because it also suffers from issues of multicollinearity. Indeed, an early motivation for exploring the inferential properties of GWR is hedonic modeling for real-estate valuation (Bárcena et al., 2014). Here, prior work has shown that GWR results should be interpreted with extreme caution because multicollinearity is a natural expectation given that “houses close in space to house are usually similar in their typology, square footage, age, etc. Local design matrices are usually very poorly conditioned, which makes the effects of the regressors difficult or impossible to disentangle” (Bárcena et al., 2014, p. 443). Together, these issues make GWR a very unattractive method for trying to understand land value (Wheeler, 2007; Wheeler & Calder, 2007; Wheeler & Tiefelsdorf, 2005).

33.5 Estimating Land Value

Many questions in urban studies revolve specifically around the question of land value.

33.5.1 Exogenous and Endogenous Spillovers

If we take the Spatial Durbin Model as our point of departure, then lets assume we have two broad categories of variables related to land (\(L\)) and improvements (\(I\)). Then we can stylize the SDM model slightly using the decomposition approach from the land-value literature. Here, the goal is to fit a hedonic model where we differentiate the land components from the built-improvement components. Spatial econometric models provide a unique opportunity to parse these components because the endogenous relationship between land and improvement values can be decomposed with greater clarity (Can & Megbolugbe, 1997). Given a decomposed spatial Durbin model:

\[ y = \alpha + \rho Wy + \beta L + \delta I + \theta WL_j + \gamma WI_j+ \epsilon, \] this specification holds that for a given parcel of land, the selling price \(y\) is a function of

  • \(\alpha\) an intercept
  • \(\rho WY\) the selling prices of nearby parcels
  • \(\beta L\) the land characteristics of the parcel itself
  • \(\delta I\) the improvement characteristics of the parcel itself
  • \(\theta WL_j\) the land characteristics of nearby parcels
  • \(\gamma WI_j\) the improvement characteristics of nearby parcels
  • and \(\epsilon\) a random error term

Fitting this model to all observations in our dataset (including developed and undeveloped), should be a conceptual match for the DGP we expect to govern the hedonic pricing model (in my opinion, anyway). That is, if you were to use this model to predict the price of an undeveloped unit (out of sample), the only term that gets zeroed out is \(\delta I\) because there is no development on the parcel itself. But through the rest of the variables, we should have captured both the exogenous and endogenous spillovers inherent in the pricing model because we’re still accounting for features of the nearby developed and undeveloped parcels (as well as the general spillover in land prices). This view is consistent with classic contributions to hedonic price modeling by Can (1992, p. 456) who defines a generic hedonic function where

“the value of a house at any location is dependent on its counterparts at nearby locations in addition to its structural and neighborhood attributes. The hypothesized spatial dependence among residential structures is determined by W which is specified in an a priori fashion. The coefficient \(\rho\) measures the absolute price impact of nearby houses on the price of a particular house. As argued in Can (1990), this conceptualization corresponds more closely with the actual workings of the real estate institution in urban housing markets. A realtor will appraise a house given the price history of houses in the immediate vicinity in addition to other substantive characteristics. At the same time, home owners will initiate or forego certain improvements based on the anticipated return on their investment considering housing prices in the immediate area.”

Obviously, the \(\rho Wy\) term cannot be decomposed into land and improvement values, it’s just the effect of nearby sales prices. In the context of land value modeling, I think the decomposition of that term is actually immaterial. What \(\rho WY\) does, effectively, is change the way coefficients at one observation propagate through the system to affect others. For example, adding a new brick facade to my house may raise my neighbor’s property directly because they are now next to a new amenity, which raises the price of their house. My neighbor also gets a small boost simply because they now sit next to a more expensive property.

That is, my neighbor benefits from two positive externalities: the aesthetic improvement and the more expensive property next door (Anselin, 2003). Then, because of price spillovers, I have indirectly increased my neighbor’s neighbor’s house (second order neighbor), albeit by a smaller magnitude, because my second-order neighbor now abuts a more expensive property (my neighbor). Those spillover effects propagate the system through spatial interaction, i.e. proximity to one another, which is ultimately a land (and infrastructure connectivity) effect.

Here it is reasonable to swap “land” for “neighborhood”. Since we have so many vacant observations in the dataset, including a dummy for “developed” also means that the intercept refers to the average price of undeveloped land. And since we’ve already accounted for both exogenous and endogenous spillovers through the \(\rho WY\) and \(\theta WX\) terms, the intercept is no longer contaminated by endogeneity issues about the average cost with improvements and land mixed together. Here, \(\alpha\) is an unbiased estimate (exclusively) of the average cost of an undeveloped parcel.

33.5.2 Exogenous Spillovers and Endogenous Error

Alternatively, we could take the decomposed spatial durbin error model as a point of departure:

\[ \begin{gathered} y = \alpha + \beta L + \delta I + \theta WL_j + \gamma WI_j + u, \\ u = \lambda Wu + \epsilon, \end{gathered} \] which holds that, for a given parcel of land, the selling price \(y\) is a function of

  • \(\alpha\) an intercept
  • \(\beta L\) the land characteristics of the parcel itself
  • \(\delta I\) the improvement characteristics of the parcel itself
  • \(\theta WL\) the land characteristics of nearby parcels
  • \(\gamma WI\) the improvement characteristics of nearby parcels
  • \(\lambda Wu\) a spatially correlated error term
  • and \(\epsilon\) a random error term

This model should also yield a “conceptually accurate” valuation of undeveloped parcels, because the intercept should (again) refer exclusively to average land value (assuming we include a ‘developed’ dummy), and the characteristics of nearby developed and undeveloped parcels are taken into account. Our expectation with this model is that the endogenous relationship between land and improvement values is partially disentangled by allowing the characteristics of nearby developments (and undeveloped land) to impact the price of an undeveloped parcel. There is no need to calculate marginal effects for this model and the coefficients are interpretable as usual.

In such a case, the value for land is estimated by

\[land_i = \alpha + \beta L + \theta WL_j + \gamma WI_j + u\]

That is, the only variable (set) to ignore are the characteristics related to the improvements on the parcel itself (i.e. we set \(\delta=0\)). The price of the land at location \(i\) is still influenced by the improvement characteristics of other parcels nearby

:::

  1. Interestingly, the concept of the hedonic model was arguably foreshadowed by Hansen (1959) in his seminal work on accessibility and land use: “The immediate value of the relationships described in this paper is that it will be possible to isolate and examine empirically the effect of other factors on land development, such as income, zoning, taxes, and land costs. The results of such studies would provide the planner with a clearer understanding of the metropolitan community and of the effectiveness of land controls.”↩︎

  2. Gibbons & Overman (2012) argue this is a critical logic error in the application of spatial lag models because it allows future events to influence the past, however, LeSage & Pace (2009) argue that the ‘simultaneity’ in the models has an implicit temporal dimension. You could also make a ‘rational expectations’ argument that the future can influence the past in some housing contexts (Can, 1992; Lowry, 1960)↩︎

  3. There’s still an ongoing debate about this. Many folks are critical of spatial econometrics. McMillen says, essentially, that because you can never know the true DGP in an applied context, you should just abandon structural assumptions altogether and use GWR because it is flexible and explicit about lacking a-priori knowledge (McMillen, 2012). In many other cases, though, spatial econometric models are preceisely the tool for the job, because there are good theoretical grounds to expect spillovers exist. On this side of the argument, LeSage says, essentially, the best course is to adopt a Bayesian perspective and treat models as though they offer competing sources of evidence (LeSage, 2014a), in which case Bayesian fit metrics offer methods for choosing which model is most likely.↩︎