Identifying the Extent of the Urban Fringe and Its Impact on Agricultural Land Values

II. Modeling Urban Influence

Capozza and Helsley (1989) developed a stylized model in which land values are determined by four components: (1) the capitalized value of agricultural returns, (2) the value of expected future rent increases caused by population growth in the urban area, (3) the value of accessibility to the city center, and (4) the cost of development conversion. The model provides the motivation for a number of empirical studies on urban influence on farmland values (Tsoodle, Featherstone, and Golden 2007; Plantinga, Lubowski, and Stavins 2002; Plantinga and Miller 2001). As shown in Figure 1, the capitalized value of agricultural returns—the agricultural land value—and the cost of conversion are assumed to be invariant to location and in perpetuity. The value of accessibility and value of expected future rent increases, however, are dependent on location of the land relative to the distance to the central business district (CBD) or urban center.

According to the stylized model, the total value of land at any given distance from the CBD is the sum of these four components. It can be seen that land values decline as one moves farther from the city center. The decline is a result of the decreasing value of accessibility or, alternatively, an increase in the costs of transportation. At the edge of the urban area, z*, a structural break occurs where the costs of conversion is greater than urban land use returns, and beyond this point, land is principally allocated to agricultural production. Capozza and Helsley (1990, 191) state that “under certainty [and time-invariant agricultural returns] land is converted when its rent in urban use equals the agricultural rent forgone plus the opportunity cost of the capital needed to convert the land.” However, beyond z*, the potential for conversion still provides a premium above the agricultural returns. Thus, even in primarily rural areas, farmland values are often greater than their agricultural use value alone (Anderson 2012; Kuethe, Ifft, and Morehart 2011).

Although the model forms the basis for a number of studies, it requires several limiting assumptions. For example, it assumes that capital is durable, that landowners have perfect foresight, and that land exists on a featureless plain. As a result, the model has been refined to account for uncertainty (Capozza and Sick 1994), irreversibility of development (Capozza and Li 1994), and the influence of multiple urban areas (Plantinga and Miller 2001). In addition, a growing literature has expanded the understanding of land conversion premium and developments away from the urban-fringe boundary in the form of exurban development (e.g., see Newburn and Berck 2011, 2006). The assumption of a featureless plain has also been relaxed. For example, Wu and Plantinga (2003) show how public open space can result in discontinuous development.

III. Empirical Model

The hedonic price method has become the standard empirical approach for modeling the determinants of agricultural land values (Dillard et al. 2013; Tsoodle, Featherstone, and Golden 2007; Huang et al. 2006). The method posits that the value of a parcel of farmland is determined by the value of its characteristics (Rosen 1974). The hedonic price model takes the form

[1]

where y is an n × 1 vector of farmland prices, X is an n × k matrix of the quantifiable characteristics of each parcel, β is a set of unknown parameters, and ε is the regression error, typically assumed to follow a white noise process with mean zero and a constant variance.

A number of studies explore the price impacts of urban areas by including measures of urban influence in quantifiable characteristics in [1] (Dillard et al. 2013; Huang et al. 2006). Other studies augment the specification of [1] to reflect the influence of urban areas on farmland values. Tsoodle, Featherstone, and Golden (2007), for example, develop an empirical model that blends both the traditional hedonic price approach and Capozza and Helsley's (1989) model of urban influence. The model includes both a set of traditional control variables, such as parcel size and land quality measures, and a negative exponential function based on the distance to urban areas.

We present an alternative approach for estimating a hedonic price model that captures urban influence as outlined by Capozza and Helsley (1989). In contrast to Tsoodle, Featherstone, and Golden (2007), our empirical specification allows the coefficients on the quantifiable characteristics to vary between the urban fringe and rural regimes instead of holding these price impacts constant. Our empirical model is adapted from the analysis of nonlinear time series. In a typical autoregressive time-series process, data are generated by previous observations and a set of fixed parameters. However, under structural instability, the data-generating process changes, and data are generated under multiple sets of parameters throughout the observed time series. The time series literature provides a strong foundation for modeling data generated under multiple regimes, and our model similarly includes the presence of two regimes linked by a smooth transition. However, in our case, the regimes are marked by physical location instead of temporal ordering. Theory suggests that the farmland values in the urban fringe are determined by both agricultural characteristics and (potential) returns of redevelopment to urban use, such as residential housing. The urban fringe is therefore marked by a mix of urban and agricultural land use activities (including discontinuous urban development). Yet, in rural areas beyond the urban fringe, farmland values are principally determined by returns from agricultural production. Thus, one can test whether coefficients associated with the value of expected future rent increases are statistically different from zero within the urban-fringe regime and insignificant in the rural regime. Further, one can test whether the marginal impacts of urban characteristics are statistically different between the urban fringe and rural areas.

The structural instability in the determination of farmland values is analog, in many ways, to the concept of structural change in time series analysis. Structural change in a time series framework refers to the long-term and widespread change of the fundamental structure or data-generating process of economic variables. This is contrasted by shortterm or microscale changes. This leads to nonlinear changes in observed data series or the data-generating parameters. One popular method to account for nonlinearity in parameter values and the identification of structural breaks in time series is the family of smooth transition autoregressive (STAR) models (Teräsvirta 1994). The hedonic price function [1] can be expressed following the family of smooth transition regime-switching models, to which STAR models belong, as

[2]

where β₁and β₂are two vectors of unknown, G is the so-called transition function, s is the transition variable, o is the Hadamard or element-by-element product, and ε is the same as previously defined in [1]. In the smooth transition model approach, the parameters β₁and β₂are allowed to be different from one another (nonlinear). The relationship between parameters β₁ and β₂ is guided by the transition function, G, which is bounded by zero and one. In our case, the transition variable, s, is defined as the distance from major urban areas. The transition function, G(s), can therefore be assimilated to a continuous dummy variable that guides the transition from the urban fringe to rural areas. In its most general form, the logistic function displays two clearly defined plateaus (when its value approaches zero or one) connected by a continuous S-shaped transition, which makes it the ideal candidate for our empirical framework. The logistic transition function G takes the form

[3]

where γ is the smoothness parameter, c is the location parameter, and σ_s is the standard deviation of the transition variable, s. The location parameter, c, determines the inflection point of the transition. With respect to our study, the smoothness parameter represents the rate of decay in the potential conversion premium or the decline in urban influence.

The regime switching model defined in [2] is typically referred to as the “cumulative” specification because it includes both the transition function G and its complement (1 - G). Additionally, the logistic transition function G defined in [3] tends to zero (one) as the transition variable, s, defined as the distance to the edge of nearest large urban area, approaches zero (its maximum). Thus, prices are generated under a linear combination of β₁ and β₂, with the weight shifting from β₁ to β₂ as farmland is located further out in the hinterland. In other words, the price response to a change in the quantifiable characteristics of a parcel at any location s is equal to β × [1 - G(s)]+ β₂ × G(s).

By allowing prices to be determined under different coefficients in urban-fringe and rural areas, the model is able to address structural instability in market values for land attributed to “spatial heterogeneity.” Anselin (1988) defines spatial heterogeneity as a special form of structural instability in which the spatial structure of a process provides the basis of its specification. The concept of spatial heterogeneity is widely recognized in microanalysis of real estate values in the form of market segmentation or submarkets (Goodman 1981; Bourassa, Cantoni, and Hoesli 2007). In this study, the urban fringe and rural areas serve as distinct submarkets for agricultural land, and each regime is marked by different marginal impacts of parcel characteristics.

In addition to capturing spatial heterogeneity in the determination of farmland prices, our model also corrects for potential spatial dependence in the regression residuals. Within the context of land value models, Anselin and Lozano-Garcia (2009) argue that accounting for spatial dependence in the regression residuals is required because of the high potential for spatially related omitted or unobservable variables. Failure to control for spatial error autocorrelation when present results in statistical inefficiency (Mueller and Loomis 2008). Spatial econometric methods have previously been applied in a number of recent studies of farmland values, including those of Hardie, Narayan, and Gardner (2001), Huang et al. (2006), and Dillard et al. (2013).

The resulting specification with spatially dependent residuals takes the form

[4a]

[4b]

where X_L comprises annual and regional dummy variables that are assumed a priori to enter the model linearly, X_NL contains parcel characteristics associated with both urban and agricultural returns, u is a spatially autocorrelated residual vector, I is an n ×n identity matrix, and W is an n × n spatial weights matrix that defines the relevant neighborhood of each observation.

The model provides a novel contribution to modeling farmland valuation in two ways. First, the model allows the marginal impact of returns from urban land use activities and agricultural production to vary according to spatial regimes, or more specifically, the model parameters are allowed to vary nonlinearly over space. The differentiation between our model and other analyses of submarkets is that, instead of modeling the differences in prices based on locations, we model how the determinants of land prices behave differently over space. Further, in contrast to Tsoode, Featherstone, and Golden's (2007) model, the price impacts of the parcel characteristics vary over space as a function of distance to major metropolitan areas.

Second, the empirical method does not impose a priori restrictions on the position and the speed of the transition between the two regimes. Spatial heterogeneity, as it is commonly defined, can be either discrete or continuous (Anselin 1988). Discrete spatial heterogeneity is associated with abrupt changes in economic variables, typically related to administrative boundaries or natural barriers like lakes or deserts. Continuous spatial heterogeneity is a more gradual process, such as varied land use patterns associated with exurban development. Our method nests the potential for either form. The larger the γ parameter, the more instantaneous the transition, up to the point where it behaves like a traditional discrete dummy variable interacted with the other regressors. As γ takes smaller values and approaches zero, the logistic transition function tends to be increasingly linear, and the two separate regimes become less and less distinguishable. When γ = 0, the model reduces to a simple linear regression model in which the estimated parameters are a linear combination of β₁ and β₂ with equal weights G =(1 - G) = ½. In nonlinear time series nomenclature, these are referred to as instantaneous versus gradual changes. Further, the steepness of the transition in conjunction with its estimated position determines the size, or extent, of the regimes. Our method allows us to estimate the extent of the urban fringe based on observed price behavior instead of placing an ad hoc definition of urban-fringe and rural regime. In other words, our model lets the data identify the spatial regimes.

The empirical model [4a] and [4b] is estimated through maximization of the following log likelihood function:

[5]

where B = (I - λW), u = y - X_Lα - X_NLβ1^o [1 - G(s | γ,c)] - X_NLβ₂ o G(s | γ,c), and Ω is the error variance-covariance matrix, with Ω = σ² I under the assumption of homoske-dasticity.

Estimation is carried out in R (R Core Team 2013) by numerical optimization using a quasi-Newton method with box constraints (Byrd et al. 1995). Specifically, nonsingularity of matrix B is ensured by restricting the spatial autocorrelation parameter, λ, to the interval where ω_min and ω_max are the minimum and maximum eigenvalues of the spatial weight matrix W.

In our case, the spatial weights matrix is a 10 nearest-neighbors, inverse-distance, row-normalized matrix. Under this definition, the “neighborhood” is defined as the 10 closest observations, each of which is weighted according to geographic separation such that closer observations carry a greater impact. The matrix is normalized so that the sum of each row is equal to one. Thus, the values contained in Wu represent the distance-weighted “average” of the residuals for the 10 nearest neighbors for each observation. When the spatial weights matrix is normalized in this manner, it implies that ω_max = 1, and ω_min is defined by the most negative purely real eigenvalue (LeSage and Pace 2009). Given W, ω_min = -0.997. The parameter space for λ over which optimization is performed is therefore (-0.997,1). Parameters γ and c of the transition function G are restricted to be strictly positive. The Hessian used to derive the variance-covariance matrix of the estimated parameters is differentiated numerically during the optimization process.

Spatial adaptations of smooth transition regime switching models have been previously suggested by a number of researchers. Pede (2009) developed a similar model in which the transition variable is the spatially weighted average (or spatial lag) of an exogenous control variable, and it is estimated through maximum likelihood. Lebreton (2005) also developed a spatial version of the STAR model by incorporating spatial autocorrelation in the transition function, and Gress (2004), Basile and Gress (2005), and Basile (2008) presented a nonparametric version of the spatial STAR process.

We validate the choice of the smooth transition spatial error model (ST-SEM) in [4a] and [4b] by formally testing whether the assumptions of nonlinearity and spatially correlated errors are warranted. Testing for the null hypothesis of no spatial error autocorrelation, that is, λ = 0, whether in the presence of nonlinearity or not, is performed using likelihood ratio (LR) tests. Testing the null hypothesis of linearity, that is, H₀: γ =0 or β₁= β₂, is not straightforward because standard test statistics do not apply. We follow Luukkonen, Saikkonen, and Teräsvirta (1988) who, in the context of time series, proposed a testing strategy based on the linearization of the transition function using a third-order Taylor expansion. The parameter γ does not appear in the resulting auxiliary regression, and testing for nonlinearity becomes a simpler matter of testing a set of linear restrictions. Using this approach, we use LR tests to test the ST-SEM and the aspatial smooth transition model against the simple linear model. Systematic pairwise testing¹ suggests that the ST-SEM is the appropriate model specification, given our empirical framework.