Abstract
The influence of buyer and personal relationship characteristics on agricultural land prices has received little attention in the literature. Recently, Perry and Robison (2001) used a restricted nonlinear specification to model the influence of personal relationships on the implicit prices of land characteristics. In this paper a more flexible alternative model is proposed. An approach to general model specification and model selection is presented. The results indicate that buyer characteristics and personal relationships exert nonuniform effects on the implicit prices of land characteristics. Our results support the hypothesis that social capital affects the terms of trade in the land market. (JEL C14)
I. Introduction
Social capital theory supports the notion that personal relationships between buyers and sellers may affect the price of agricultural land. Following Robison, Schmid, and Siles (2002), social capital can be defined as “a person’s or group’s sympathy or sense of obligation for another person or group.” It is argued that social capital may be treated in the same manner as the more conventional forms of capital, since social relationships may act as substitutes for physical inputs. When an individual is the object of another person’s sympathy, that individual gains social capital; this may mean receiving preferential treatment, since a sympathetic individual responds not only to his or her own incentives but also to the consequences of his or her actions on the other individual (Robison, Myers, and Siles 1999). The presence of social capital may therefore affect the terms of trade. The spatial fixity of agricultural land means that social capital is likely to be particularly important in the agricultural land market. The fact that agricultural land is fixed in terms of geographical space means that it cannot be moved from one location to another.
Past cross-sectional studies of the agricultural land market have primarily focused on the relationship between land price and land characteristics, using the hedonic pricing technique, and have largely excluded buyer and personal relationship characteristics. According to hedonic theory (Rosen 1974) the price of a heterogeneous good is a function of only those characteristics that describe the good itself. This automatically precludes the inclusion of the buyer or personal relationship characteristics within the hedonic regression equation. In the original hedonic framework the estimated coefficients of the hedonic function represent the implicit prices of each of the hedonic land characteristics, which when multiplied by the quantity of each characteristic sum to yield the total price of a unit of agricultural land. However, given that the implicit prices of characteristics may depend on shifters of buyers’ utility functions, some studies now adopt a broader approach by entering buyers’ characteristics within the main hedonic function. Schmid and Robison (1995), Siles et al. (2000), and Robison, Myers, and Siles (2002) develop a convincing theoretical model illustrating how social capital can affect the terms of trade for agricultural land. Tsoodle, Golden, and Featherstone (2006) present an extensive overview of these underlying theoretical arguments, and we will not discuss them here.
Nevertheless very few studies have examined the influence of buyer or personal relationship characteristics on agricultural land prices. The manner in which buyers’ characteristics are included in the hedonic function has differed from study to study. For example, Chicoine (1981), Dunford, Marti, and Mittelhammer (1985), Long (1995), and Tsoodle, Golden, and Featherstone (2006) estimated regression functions where land price per acre is specified as a function of buyer characteristics (among other factors) and the intrinsic characteristics of the land. The incorporation of buyers’ characteristics within the hedonic function as straightforward independent variables means that the coefficients of the intrinsic characteristics no longer represent implicit prices but merely indicate the extent each variable explains the variation in the price per acre. Perry and Robison 2001 seek to preserve that (original) property of the hedonic model where the sum of the implicit prices times the quantity of (only) the intrinsic land characteristics yields the total price of a unit of agricultural land. They proposed a specification in which personal relationship characteristics and other conditioning variables modify the valuation of the entire set of agricultural land characteristics in the hedonic price function. The Perry and Robison (2001) specification still ensures that the sum of the implicit prices times the quantity of the intrinsic land characteristics yields the total price of a unit of agricultural land. However, the Perry and Robison (2001) specification is quite restrictive, as it assumes that the conditioning variables uniformly modify the parameters of the hedonic function. This paper proposes a more flexible nonparametric model.
II. Conceptual Model
The empirical model presented by Perry and Robison 2001 can be written as follows:
[1]where P is the price per acre, fi() represents some (possibly nonlinear) functions, xi represents the individual hedonic land characteristics, zj represents factors (mainly social capital or personal attribute variables) that shift or condition the hedonic land values, gj represents the coefficients of the shift variables, n is the number of land characteristics, and m is the number of shift or conditioning variables. Perry and Robison (2001) propose parametric specification for the unknown functions fi(), but hereafter we will not impose such parametric restrictions and will consider any possible functions. Hence, unless specified otherwise, we will use f(·) as a generic term for function (i.e., denoting any function). The structure of the above model implies that the shift or conditioning variables exert the same effect on the implicit price for each of the land characteristics in the hedonic function. This is more clearly seen when equation [1] is rewritten as follows:
[2]It is clear from [2] that the term 
“scales” the implicit price of each hedonic characteristic in the same way. Although, the Perry and Robison (2001) model is parsimonious, it is unrealistic to expect that all the chosen shift variables will affect all the hedonic land characteristics and that they will do so in the same way. It might even be expected that conditioning variables could have positive effects on some of the hedonic land characteristics and negative effects on others.
The Perry and Robison (2001) approach can be generalized as follows:
[3]where P is the land price, xi represents the individual hedonic land characteristics that combine to yield the property’s overall value, n is the number of land characteristics, and z = {z1, z2, zm} is the information set containing the buyer or personal relationship and other (conditioning) variables. In contrast to equation [2] the hedonic functions with regard to the main attributes are not uniformly scaled, but generally conditioned upon the set of secondary characteristics. From equation [3] it can be seen that the land price (P) is the sum of the implicit prices times the quantities of the main hedonic attributes (xi). To make this explicit, consider a linear version of the hedonic functions. In this case the quantities of the hedonic attributes are the variables, while their coefficients are the implicit prices of these attributes. However, these implicit prices are modified by the conditioning variables (z). [fi(xi|z)]=xi is then the implicit price for the ith characteristic conditioned on the secondary variable, z. In a more general nonparametric setting the implicit prices will be given by the influence of the hedonic attributes on the price.
Kostov, Patton, and McErlean (2008) propose an even more general formulation that does not impose additivity on the hedonic pricing function. In addition to the purely practical considerations in the sense that additivity facilitates estimation in nonparametric setting by helping to avoid the curse of dimensionality when smaller data sets are employed, however, additivity is a desirable property of the hedonic models. If they are not additive with regard to the hedonic attributes, then the implicit price interpretation of the hedonic model breaks down.
The model represented in equation [3] is nevertheless still very general. We may restrict it to obtain a flexible, yet tractable formulation. Such a restricted version is presented in equation [4] as follows:
[4]where P is the price per acre, fi() represents the form of the hedonic function, xi represents the individual hedonic land characteristics that combine to yield the property’s overall value, zj represents buyer and/or personal relationship characteristics that shift the hedonic land values, n is the number of land characteristics, and m is the number of shift variables. We can view [4] as a restricted version of [3] in which the hedonic functions are additive, not only with regard to the main hedonic attributes (x), but also with respect to the secondary characteristics (z). Such an additional restriction is useful for analytical purposes when we are interested in identifying and separating the potential impact of the secondary variables on the implicit prices of the hedonic attributes. The additive formulation represented in equation [4] allows buyer and personal relationship characteristics, zj, to exert separate and different effects on each of the implicit prices of the land characteristics in the hedonic function. Thus, the model represented by equation [4] is much more general and flexible than the Perry and Robison (2001) model. Moreover, like the Perry and Robison model, it incorporates the influence of factors other than the hedonic land characteristics in such a way that they modify the implicit prices. As a result, summing the implicit prices times the quantities of the hedonic land characteristics continues to yield the total price of a unit of agricultural land. The only difference is that the implicit price for any of the hedonic attributes is represented as a sum of the conditional with regard to the secondary characteristics effects of the hedonic attributes; that is, the implicit price of the ith hedonic attribute is 
.
The proposed specification has an added benefit in allowing one to test whether the secondary characteristics affect the terms of trade. To clarify this let us consider a parametric regression model where the secondary characteristics are included as additional regressors in the hedonic model. Statistical significance of the secondary variables in such a model measures partial correlations. Consider market sorting in the standard hedonic model. Since buyers only participate in different parts of the market, the price they pay will obviously be correlated with their characteristics. Then significance of the secondary variables is consistent with the case of market sorting. Therefore this correlation does not imply that these characteristics impact on the implicit price of the hedonic attributes. One can in such cases employ graphical analysis of the data distribution in order to ascertain whether market sorting could be causing such results. The Perry and Robison (2001) approach essentially assumes that the secondary characteristics scale the pricing function. Thus, although the implicit price interpretation is maintained, the main difference from the straightforward inclusion of these variables in the hedonic regression is that a multiplicative, rather than an additive, formulation of the partial coefficients is employed. Therefore funding significant secondary variables in such formulation is again consistent with the case of market sorting. Therefore such formulation alone cannot be used to infer that buyers’ characteristics affect the terms of trade. In equation [4] we explicitly specify these terms of trade. We essentially allow them to vary with the secondary variables. To explain how this overcomes the problem in the previous approaches, lets us assume that best-quality land is traded among close friends only. Then the implicit price for this best-quality land would then be estimated only from the close friends’ transactions. In this extreme case, if there are no transactions for other categories of buyers, the estimated impact of these other categories on the implicit price for the best land will be zero. The effects of close friends on the implicit price of the best-quality land would also be zero, because being a close friend would not in this case entitle one to any different price (close friends are the market for this type of land). Thus in estimating equation [4] one would only find significant fi(xi|zj) for those zj that participate in the submarket for land with hedonic characteristics xi and compete on this submarket with buyers having different characteristics, so that they really obtain a price discount or pay a price premium for this type of hedonic attribute. Patton, McErlean, and Kostov (2003) proposed a parametric specification that can be considered as a restricted specific form of equation [4]. As explained above, in such (or any other) parametric specification the restrictions implied by the parametric form could result in spurious inference.
The hedonic model is, however, based on the premises of perfect competition. The Perry and Robison (2001) approach explicitly allows for imperfect competition. This could raise questions about whether a standard hedonic type of approach is applicable to such cases. Bajari and Benkard (2005) show that when demand is driven by the hedonic model, the function mapping characteristics to prices can be identified, regardless of the form of competition. This means that estimating the implicit prices does not depend on the assumption of perfect competition. Therefore we can use a hedonic price model in which the implicit prices of the hedonic attributes vary with the secondary characteristics. Nesheim (2002) established that nonlinearity is a robust feature of a hedonic economy with social interactions. Ekeland, Heckman, and Nesheim (2004) prove that this nonlinearity is a generic property of equilibrium in the hedonic model. Therefore a nonlinear specification, as postulated in equation [4], has strong theoretical support.
Note, however, that in its present form equation [4] is still too general. Thus one may wish to explore possible restrictions. Typically these would involve omitting some of the secondary characteristics as modifiers of the implicit prices of the main hedonic attributes. This would be an interesting inferential result. It would not be infeasible that some of these secondary characteristics have differential impact. For example, they could only affect (and modify) the implicit prices of some but not other hedonic attributes. There are two reasons one may wish to explore such restrictions. The first is purely technical. With the moderate-size sample typically available for empirical research, any reduction in model complexity increases estimation efficiency. The other reason is that one may wish to explore different competing models. Note for example that the Perry and Robison (2001) approach can be viewed as a particular case of the specification we outline in equation [4]. Therefore testing restriction leads us to implicitly test for the correct specification. The latter could be typically expected to lie somewhere between the parsimonious, yet restrictive in many ways, model implied by the Perry and Robison approach on one hand, and the general yet too complex specification outlined in equation [4]. Nonetheless if we want to avoid spurious results, we need to start with the general model implied by [4] and impose only those restrictions that are supported by the data set. In this way we can avoid compromising the identification requirements of the model.
III. Data
The data for this study were collected from buyers of agricultural land, using a mailed survey. The names and addresses of buyers of agricultural land were obtained from the Valuation and Lands Agency, which collects information on all agricultural land transactions in Northern Ireland. Using this information the survey targeted the entire population of agricultural land transactions throughout Northern Ireland between September 1996 and June 1999 and was implemented using Dilman’s “total design method” (Dillman 1978). Of the 1,314 questionnaires sent out, 395 were returned, and of these 160 were excluded from the subsequent analysis because the transactions referred to land that was purchased for nonagricultural purposes or information on key variables was missing. This yielded 235 useable transactions relating to parcels of agricultural land that were purchased for agricultural purposes.
The Northern Ireland agricultural land market is particularly well suited for an investigation of the potential impact of buyers’ and sellers’ characteristics on land prices. Agriculture is a relatively important sector in the Northern Ireland economy and is characterized by a large number of small and medium-size farms. In 1997 there were 32,118 farms with an average size of 33.3 ha. Agriculture in Northern Ireland is undergoing a restructuring process. In 2007 the number of farms reduced to 26,146, with an increased average area of 38.8 ha (DARD 2008). Land markets play an important role in this restructuring process, allowing for land amalgamation. Yet agriculture contributes 1.3% of the gross value added and 3.4% of employment, compared to 0.5% and 1.4% overall for the United Kingdom. These figures actually diminish the importance of agriculture for Northern Ireland, because they do not properly account for the difference in land quality and the nature of the farming enterprises. Most of the agricultural land in Northern Ireland is grazing land. Good-quality arable land is rare. As a result, cattle-based grazing enterprises (beef and dairy) dominate this country’s agriculture. Therefore the farms are in fact smaller than the land area alone suggests. DARD (2008) classifies over three-quarters of Northern Ireland farms as “very small.” In view of the small farm size it should come as no surprise that parttime farming is commonplace within Northern Ireland, with 42.8% of the total number of farmers classified as part-time (DARD 2001). Part-time farmers have a secondary source of income, and for some, their farming activities could be described as a hobby. Furthermore the nature of farming requires farmers’ continuous presence. Consequently, most farmers tend to live on the farm. This can be contrasted to the situation in some regions in the United States and Canada, where farming (e.g., cereal farming) can be seasonal and farmers may live far away from their farm. As a result, potential purchasers of agricultural land tend to live in close proximity to the parcel for sale. Thus, the buyer and seller are often neighbors who have formed a personal relationship. An additional channel for personal interaction is the seasonal conacre rental system prevalent in Northern Ireland (and the Republic of Ireland). The conacre system accounts for over 90% of land rentals. Under this system, land is rented for a maximum period of 11 months. In other countries land rental is based on longer-term leasing. The conacre system creates repeated interactions between potential buyers and sellers. Although rental is not guaranteed to be renewed, this often is the case. Therefore the fact that the buyer and the seller have been in repeated interaction involving the annual rental price negotiation process leads to forming a particular relationship. This relationship may, on one hand, involve trust but may, on the other hand, result in the buyer obtaining better knowledge of the characteristics of the land parcel. In the first case, the emotional attachment of sellers to land and farming may lead them to consider favorably people who previously rented the land, since they have shown they can take care of it; in other words, the buyers can obtain social capital and thus obtain a price discount. In the latter case, the superior knowledge of the land characteristics could help buyers avoid overbidding for it, resulting yet again in lower on average price. There is, however, another option, in which the buyers may attach additional value to a particular land parcel because of the ease of amalgamating it with their current holding, in which case one may expect them to be prepared to pay a higher price.
With parsimoniousness in mind, the number of variables used in the empirical analysis is kept to three hedonic land characteristics (x variables) and four buyer characteristic/personal relationship variables (z variables). The three hedonic land characteristics chosen are based on previous studies (Xu, Mittelhammer, and Barkley 1993; Elad, Clifton, and Eerson 1994; Kennedy et al. 1997; Patton and McErlean 2003) and are acreage (ACREAGE), measured in number of acres; land-quality score (SCORE),1 measured using values close to 1 to represent good-quality land and values close to 7 to represent poor-quality land; and distance to nearest urban area (DISURA), measured in meters (computed using GIS). The land-quality score is the variable most closely representing the productive capacity of agricultural land.
The dependent variable was computed by deducting nonland items in the transaction (e.g., residence, machinery, and dairy quota) from the total sales price to yield a pure land price. Price per acre was obtained by dividing the pure land value by adjusted acreage (total acreage of the parcel minus acreage of residence, if appropriate). Price per acre was deflated using a retail price index because of the time span over which it was collected. Variables such as SCORE and DISURA were computed using secondary data and making use of GIS.
The four buyer characteristic/personal relationship variables are indicator for family transactions (FAMILY = 1 where the transaction is between family members, 0 otherwise); distance between purchased parcel and buyer’s existing farm/residence (BUYDIS), measured in miles (0 = neighbors parcel); rent prior to purchase (RENTPP = 1 where the purchaser rented the land prior to purchase, 0 otherwise); and part-time farmer (PARTTIME = 1 where the purchaser is part-time, 0 otherwise).
BUYDIS may appear as an imprecise proxy for measuring personal relationship. Note, however, that due to the small size of Northern Ireland and the settled nature of farming (most farmers living on the farm), it can approximate the frequency of their interactions. Festinger, Schacter, and Back (1950) argue that people who are spatially far apart are less likely to form social connections. Glaeser and Sacerdote (2000) find that spatial proximity encourages investment in social capital. Another reason for inclusion of this variable is the fact that Northern Ireland is a divided society. The Protestant and Catholic communities (where these are used to denote cultural background rather than a conventional religious denomination) share a great deal of mistrust in each other. In other words there is a negative social capital between members of the two communities. Unfortunately, asking for such sensitive information is difficult in a postal survey and very likely to greatly reduce the response rate. Furthermore, even if this were not the case, the sellers also would have needed to be surveyed. In such circumstances a variable that can proxy such relationships would be useful. Anecdotal evidence suggests that in Northern Ireland 90% of people live in an area where 90% of the inhabitants are from the same religious denomination. Given the small distances, in particular the largest distance between a buyer and a parcel (and therefore seller) being only 10 miles (see Table 1), one could expect that at least for the smaller BUYDIS values, the corresponding transactions would involve members of the same community. Note, however, that the imprecise nature of this variable explicitly solicits general estimation methods and therefore justifies the nonparametric modeling strategy we adopt here. Finally, the use of a variable based on spatial distance is useful in econometric modeling because in this context such a variable will be unequivocally exogenous. When the types of relationships have to specified by some of the transaction parties after the transaction itself, the mode of elicitation can often endogenize their classification, resulting in general in a lack of model identification and other fundamental causality questions, as discussed by Durlauf (2002).
Summary Statistics
Summary statistics on the variables included within the model are provided in Table 1. Since the focus of this paper is on the influence of buyers’ characteristics, the differences in the summary statistics across different categories of buyers are presented in Table 2.
Summary Statistics for Different Groups
We observe a similar distribution of land quality between family and nonfamily transactions. Nevertheless the average per acre price for family transactions is, as expected, significantly lower. Taking into account that the family transactions are, in this particular sample, closer to urban areas, this demonstrates that family relationship affects the price of agricultural land. One of the objectives of this study is to uncover how, exactly, this impact is transferred, namely, the implicit prices of which hedonic attributes are affected and to what extent.
There is no noteworthy difference between the transactions involving part-time farmers as opposed to full-time farmers. Part-time farmers seem more often to buy land that is further from urban centers, but it is unclear whether this is a sample effect or a true underlying characteristic of these transactions. Interestingly the largest land parcel was bought by a farmer classified as part-time. Since it is a rather large parcel, namely 296 acres, which is larger than the average farm size in Northern Ireland, this shows a certain commitment to farming, and one would expect that this particular farmer would become involved full-time in agriculture. With this clarification in mind, part-time farmers generally buy smaller land plots, which is understandable given that some of their resources are devoted to additional income sources.
Farmers who rented the land parcel prior to purchase do buy smaller parcels. This makes sense because prior renting of the parcel implies that it is more or less a part of the buyer’s farming operation. Furthermore parcels that are rented under the conacre system are expected to be smaller than if a longer-term renting system were operating. These buyers pay a smaller per acre price compared to arm’s-length purchasers. This in general suggests, tentatively, that renting prior to purchase entitles the buyer to a price discount. Note, however, that these buyers buy parcels that are in general further away from urban areas. The latter characteristic in particular would in general be expected to lower the per acre price. In addition to this, in our sample these buyers live, on average, closer to the parcel than buyers who did not rent the land previously. This is, of course, to be expected, but since the distance between the buyer and the parcel is also accounting, however imperfectly, for the general social relationship between the buyer and the seller, this is also expected in general to lower the price, if exercising any effect. Thus the subsample of buyers who rented prior to purchase could be expected to be characterized by a lower per acre price even if this particular conacre renting relationship did not exist. Therefore it would be interesting to investigate whether renting prior to purchase does contribute to the above price discount, in which case it could be considered a social capital type of variable. The alternative would be that ceteris paribus renting prior to purchase increases the desirability of this particular plot, and buyers are prepared to offer more for it. In this case such a price premium would offset part of the possible discounts associated with the larger distance to urban area and the shorter distance between the buyer and the land plot.
IV. Methodology
Econometrically we need to estimate the general model defined by [4]. Numerous methods allow us to do so. Note, however, that here we also want to simplify the general model implied by equation [4]. In particular we want to perform both variable selection, in the sense of which secondary variables to include as effect modifiers, and model selection, in discriminating between potential linear and nonparametric effects. In other words, we need to perform a model selection in a semiparametric setting. In what follows I separately describe the model selection approach and the estimation of confidence intervals for the estimated effects.
We will consider an additive model where the response y is an additive function of the predictors. Thus we can denote
[5]The conditional expectation of y expressed in terms of the covariates in the functions fi defines a generic representation of different types of covariate effects. In this way our representation is similar to the structured additive regression models considered by Fahrmeir, Kneib, and Lang (2004).
In particular we can have the following types of functions:
Linear components f (x)=flinear(x)=xβ
Nonparametric, smooth components f (x)=fsmooth(x), where fsmooth(·) is an appropriately defined smoother
Varying coefficients terms f (x)=xfj(z)
Our conceptual model (equation [4]) can be viewed as consisting of the sum of varying coefficient terms, since representing f (x|z)=xfj(z) allows the effect of the main hedonic attribute x to depend on the secondary characteristic z, where for notational convenience we omit the corresponding subscripts. Thus including only this type of function is equivalent to equation [4]. Note that the other two types of functions we consider can be viewed as particular cases of the varying coefficient terms. For example, when fj(z)=[f(x)]/x, that is, when the effect of the main hedonic attribute (and therefore its implicit price) does not vary with the secondary characteristic, we obtain a nonparametric smooth component. Furthermore, linear effects are a particular case of the nonparametric components. If one is interested in the Perry and Robison (2001) specification, it is obtained when fj(z) is constant across all hedonic attributes, namely, when fk(z)=fl (z) for any k and l.
Recently Kneib, Hothorn, and Tutz (2009) suggested component-wise boosting to specifically deal with the issues of variable and model selection. Their approach can be viewed as an extension of the earlier work of Bühlmann and Yu (2003) and Tutz and Binder (2006).
I will briefly introduce the idea of the boosting algorithm. Then the componentwise version of boosting will be discussed in relation to the components used in this application.
In general, boosting can be viewed as a functional gradient descent method that minimizes the constrained empirical risk function
where wi are some weights, and ρ(·) is some suitable (in practice this means convex and differentiable) loss function. To simplify the discussion, from now on we will implicitly assume equal weights. Typical examples for a loss function would be the log-likelihood function or the L2 norm (sum of squared residuals). Note that classical estimators essentially solve the same optimization problem. The main difference is that they apply a specific algorithm that is typically applicable only to a given class of models specified by the underlying functions f (x). Therefore we may think of the boosting approach as providing a general approach to model estimation. The general idea of the boosting algorithm is to minimize the empirical risk with regard to η.
To explain the boosting algorithm, let us assume a given type of underlying function (base-learner) f. Lets us further simplify the matter and assume the L2 norm for the empirical risk function ρ(y,η)=(y−η)2.
The boosting algorithm is initialized by an initial value for η, for example, η0. This implies an initial evaluation for the underlying function 
. Typically we start with an offset term set to the unconditional mean of the response variable. Then it iteratively goes through the following steps:
Compute the negative gradient of the empirical risk function evaluated at the current function estimate (ηm for every step from m = 1, . . .)
Use the above calculated negative gradients (sometimes called residuals, because with the L2 norm empirical risk and linear model they do coincide with the current regression residuals) to fit the underlying function
.Update
for a given step size ν. The algorithm iterates between Steps 1 and 3 until a maximum number of iterations is reached.
As the generic description of the algorithm demonstrates, the boosting algorithm constructs iteratively η (i.e., all functions fi(x)) by pursuing iterative approximate steepest descent in function space, calculated using the adopted empirical risk function.
It is a simple algorithm. With an L2 empirical risk function it essentially does an iterative least-squares fitting of the residuals for a linear model. For the type of base-learners considered in this paper, the algorithm will do penalized least-squares calculations. The approach is also flexible because it can be applied to a wide range of alternative loss functions. This could be of concern when “robust” versions are required (see Lutz, Kalisch, and Bühlmann 2008).
Since in this case we have different base-learners corresponding to different underlying functions, and we effectively want to choose among these different candidates, a component-wise version of the algorithm is applied. This means that we have separate functions fj for j = 1, 2, . . ., k, and we specify the corresponding separate learners gj.
Then in Step 2 we simply chose the bestfitting component-wise learner
which leads to this particular base-learner being the only one updated in Step 3, that is, 
, while 
for 
, where the first subscript denotes the base-learner and the second one is the iteration counter.
In simple terms, in the component-wise boosting we fit base-learners. Only one of the different learners is selected for updating. If we assume for simplicity that the functional forms are given, selecting a base-learner corresponds to selecting a covariate. In this case the selected covariate is the one that gives the smallest residual sum of squares, that is, the variable that gives the largest contribution to the fit. After the algorithm has run for the maximum number of iterations, some of the base-learners will have never been selected for updating, which means their final evaluations are zero. In this way the algorithm may be used for variable selection. Similarly, if we include different base-learners for the same covariate, the selection procedure will, in effect, do a model selection.
Bühlmann 2006 provides detailed discussion of L2 boosting and component-wise linear fitting. It is worth noting that the algorithm is essentially the same as the matching pursuit algorithm used in signal processing (e.g., see Mallat and Zhang 1993). I now discuss in some detail the base-learners for the nonparametric effects and the varying coefficient terms.
For nonparametric effects we essentially need to construct a univariate smoother. A popular choice for such a smoother is the B-spline basis (see Wood 2006 for a detailed discussion of splines). Then the unknown nonparametric effect is simply a linear combination of the basis B-spline functions 
, where l is their degree. In other words we have
If instead of x we consider the evaluations of the basis function, combined in a design matrix X, then we essentially have a linear regression model that can be easily estimated by least-squares methods. Since the number of basis functions defines the smoothness of the estimated functions, it is an important parameter. Choosing a large number of such basis functions will result in overfitting, while a too small number will result in insufficient smoothing. To avoid this problem Eilers and Marx (1996) suggested penalized splines. The idea of penalized splines is very simple. A smoothness penalty is added to the least-squares criterion. Then the estimation with regard to the design matrix becomes penalized least squares.
Since we are using squared residuals to fit the unknown parametric effects, a penalty that is based on the squared derivatives of this unknown function would be convenient. The latter can, however, be approximated using the differences of the ordered sequence of regression coefficients β = (β1,β2,…,βK). Since we are using the B-splines to approximate the unknown function, we need the dth-order derivative of the B-splines. The latter is, however, in practice determined by the dth-order difference of the regression coefficients. The above reasoning leads to the following penalty term:
where Δd is the dth-order differencing operator. For compactness and notational convenience, the penalty term can be rewritten in the following quadratic form: λP(β)=λβ′Kβ, with the penalty matrix K = D′dDd where Dd is a dth-order difference matrix (with the appropriate dimensions). This allows us to write the penalized least squares in the following convenient form: 
, where u is the running residual.
Consequently the corresponding base-learner can be represented in terms of the following smoother matrix (Hastie and Tibshirani 1990): Sλ = X(X′X+λK)−1X′ as f (x)=Sλu.
So far we have not discussed how to chose the smoothing parameter λ. If λ were to be too large, it would bias the selection process toward choosing nonparametric effects over parametric effects, due to their additional flexibility. Note, however, that since we are also performing model selection, ideally we would like to select smoothing parameters in such a way that the nonparametric effects of different covariates are comparable in terms of their complexity. Think of the nonparametric effect as a big linear (with respect to the evaluations of the basis functions) model. Using this analogy to linear models, Hastie and Tibshirani (1990) suggested using the trace of the smoother matrix Sλ as equivalent degrees of freedom. Since degrees of freedom are a general measure of complexity, this provides a means for comparing the smoothness for different types of nonparametric effects.
Since the computation of the equivalent degrees of freedom is based on a generalization based on a linear model, it is clear that when we obtain a limiting linear model (i.e., when we set λ=0), the results will be the same as for the linear model (in this case the number of parameters describing the spline, i.e., K). Positive values of the smoothing parameter λ reduce the effective degrees of freedom, because they effectively reduce the number of parameters. If the degree of the spline is at least as large as the order of the difference penalty, a (d − 1)th order polynomial in x will remain unpenalized by the dth-order difference penalty. This means that for difference penalties exceeding 1, we cannot make the degrees of freedom equal 1 and thus make the nonparametric effect comparable in terms of complexity to a parametric effect. (For d = 1, it is possible to obtain this as a limiting case when λ grows to infinity [see Hothorn and Bühlmann 2006]).
Fahrmeir, Kneib, and Lang (2004) have suggested modification of the parameterization of the penalized spline in a mixed models context, but the same idea can be easily applied here. This reparameterization is
[6]It essentially splits the function f(x) into a parametric part, expressed by the (d−1)th order polynomial that remains unpenalized, and a nonparametric part (the deviation from this polynomial fcentered(x)).
This reparameterization allows us to interpret the polynomial part as a linear effect and thus use parametric base-learners for them. The deviation part is fully penalized. This allows the choice of smoothing parameter that makes the deviation part have exactly one degree of freedom and thus makes it comparable (in complexity) to the parametric effects. Since this reparameterization nests within itself both parametric and nonparametric effects, it allows the boosting algorithm in addition to the variable selection task to actually test the nonparametric model against a parametric specification. If none of the components in [6] is selected, then the variable in question has no effect. If only (some of) the parametric components are selected, then no nonparametric part is necessary. Technically the component-wise algorithm does model and variable selection implicitly. This means that all model terms that are defined initially stay within the model. The ones that have not been selected are, however, characterized by zero contribution and thus can be omitted from the model specification.
Varying coefficient terms (Hastie and Tibshirani 1993) represent interactions between variables in the following form: x f (z). In other words, in analogy to the linear regression, the fixed coefficient of x is replaced by a flexible function depending on another variable z. Note that the conventional nonparametric terms could be considered as a special case of the varying coefficients model since we can write fsmooth(x)=x f(x).
Varying coefficients models also allow us to include separate effects for a continuous covariate x in subgroups defined by an indicator variable z. In this case we can use the following formulation:
Although the second effect in this representation can be formally considered as a random coefficient effect of z, the overall representation represents a segmentation by the indicator variable z. Thus if z = 0, the effects of x will be fsmooth,1(x), while when z = 1, the resulting effect will be fsmooth,1(x)+fsmooth,2(x). In this way fsmooth,2(x) can be viewed as the additional (deviation) effects attributable to the group defined by z = 1.
The construction of the base-learner for varying coefficient terms x f (z) proceeds similarly to the nonparametric effects. The only difference is that the design matrix Z that contains the evaluation of the B-spline basis functions needs to be premultiplied by a diagonal matrix containing the values of x. In formal terms this produces diag(x1,x2,…,x1)Zβ=Xβ, that is, the row-wise scaled matrix X, as defined above, could be considered as an alternative “design matrix” for the varying coefficient terms. Then incorporating a penalty term, we obtain the same type of expression for the base-learner: Sλ=X(X′X+λK)−1X′, with regard to the new design matrix X.
For practical implementation, we need two further choices: to select the optimal number of iterations and to choose the updating factor ν (also referred to as the step-length factor or shrinkage factor). The former can be estimated via cross-validation. A value of 1 seems like a natural choice for the latter, but following Friedman (2001), most applications use smaller values. Friedman (2001) showed empirically that that small values of ν perform better and that the boosting procedure is not very sensitive to a whole range of small values of ν. Here we will use ν = 0.5, which is within the standard range of values between 0.1 and 0.5 typically used in boosting applications. We essentially chose a value closer to the upper end of the standard range, due to the aim of our application. We essentially need to do boosting for model and variable selection purposes. While the value of the updating factor does not matter for standard boosting algorithms where the purpose is to estimate the relationship, there are some differences when component-wise boosting is concerned. Using a larger value for ν essentially facilitates this by preventing small contributions of some variables that may turn out to be statistically insignificant in component-wise boosting. In principle, one may proceed with smaller values for ν and then use a standard approach to eliminate the statistically insignificant terms, but this would involve one additional step in the follow-up estimation procedure.
The boosting procedure provides us parameter estimates, but not confidence bounds for these estimates. For inferential purposes it is desirable to have such confidence intervals. Once the final model has been selected, however, various flexible nonparametric methods exist that may be used to obtain consistent estimates, including confidence intervals for a broad class of nonlinear relationships. The confidence intervals presented in this paper are estimated using Markov chain Monte Carlo methods, designed to replicate the specification used in the boosting algorithm. More details on the exact procedures are available from the authors upon request.
V. Results
The boosting procedure implicitly chose the following final model:
[7]To simplify the notation in equation [7], we pool together the penalized and unpenalized parts of the nonparametric effects into a single term. For the same reason, although we did treat them separately in the boosting application, from now on we will present them together. Note that the final model (equation [7]) is much simpler that the general model (equation [4]) but nevertheless is more general than the Perry and Robison (2001) approach, in that the impact of the secondary variables is nonuniform. In particular, the secondary characteristics do not modify the implicit price of every single hedonic attribute. Most secondary characteristics affect the implicit price of the land-quality score, which is the main productive attribute of agricultural land. PARTTIME affects the implicit price of ACREAGE, but not SCORE. Similarly, none of the other secondary characteristics (FAMILY, RENTPP, and BUYDIS) affects either ACREAGE or DISURA.
The estimated effects, together with their 90% confidence intervals, are presented in Figures 1–6.
Effect of Acreage
Effect of Part-Time on Acreage
Effect of Family on Land-Quality Score
Effect of Rent Prior to Purchase on Landquality Score
Effect of Buyer’s Distance on Landquality Score
Effect of Distance to Urban Area
Acreage itself does not significantly affect the price. Figure 1 suggests tentatively the possibility of lower acreages attracting higher prices, but this effect is not statistically significant. It is unclear whether the insignificance of this effect is due to the small sample employed in this study or simply that such an effect does not exist in the Northern Ireland agricultural land market.
For very small acreages, part-time farmers pay a higher price than full-time farmers (Figure 2). For larger acreages the price part-time farmers pay is essentially the same as the price paid by full-time farmers. Thus we obtain some evidence that part-time and full-time farmers value the characteristics of land differently. Part-time farmers are in principle able to cross-subsidize their farming activities from their other sources of income, which means that they may value other (e.g., recreational) characteristics of farming and land. In addition to that, parttime farmers may value larger acreages less highly than full-time farmers since time constraints restrict the number of acres they can realistically farm. Full-time farmers, on the other hand, may be willing to pay more for acreage since farming represents their main source of income.
Except for the part-time farming effect for smaller acreages, all other secondary variables modify the implicit price of the land-quality score, which is the variable measuring the productive capacity of agricultural land.
Family transactions command a lower price when the land quality is relatively low (Figure 3). Note, however, that this is not so for prime-quality land. The variable FAMILY is a personal relationship variable because family members will share a personal relationship. Thus it is to be expected that family transactions for almost all types of agricultural land take place on a discounted basis. For the scarce prime-quality land, however, this is no longer the case. This is an interesting effect that shows that the perception of scarcity and rarity conditions even family transactions. Prime-quality land is so scarce and valuable that even family relations do not cause it to be transacted on a discounted basis.
Renting prior to purchase ensures the buyer receives a price discount. This discount, however, exists only for the lowest-quality land (Figure 4). Since better-quality land is relatively scarce in Northern Ireland, it is not susceptible to personal relationships effects, and its price is determined by competitive market forces. Note, however, that most data in our sample are for land that is of lower quality than grade 3, and therefore although the effect is pronounced only for lower-quality land, it is actually well defined. This result suggests that the variable RENTPP is classified as a personal relationship characteristic. In spite of the short-term basis of the conacre rental system in Northern Ireland, where land is rented for up to 11 months, the buyer and seller may develop a social relationship during the rental period, and thus, the buyer may receive a favorable price. Yet, again, as is the case for family transactions, better-quality land does not attract such a price discount. Furthermore, since this personal relationship is not as strong as family relationships, the discount applies to a much more limited range of land grades, namely, only to the lowest quality grade.
The impact of RENTPP should nevertheless be treated with some caution. In other studies such a variable may be a buyer characteristic rather than a personal relationship characteristic. In such cases buyers who rented the parcel of land prior to purchase may be willing to pay a premium, as the land represents an important part of their business and its availability presents a once-in-a-lifetime opportunity to obtain it.
The distance between the buyer and the seller, which is used to proxy for their personal relationship, has a significant effect on the land price. Buyers living close by manage to obtain a significant price discount (Figure 5). Note, however, that although as the distance between buyer and seller increases, so does the price, but only up to a point. Once a significant spatial (and therefore personal) distance is established, a saturation point is reached in the sense that the relationship has already become impersonal. Due to the small size of Northern Ireland, this saturation level is observed only for a small part of our sample. One could, however, expect that for larger samples and different settings, such an effect would be more clearly distinguishable. Thus our results suggest that in this study, BUYDIS serves as a proxy for the closeness of the personal relationship, with low distances denoting close personal relationships and high distances denoting weak personal relationships.
As with RENTPP, in other settings BUYDIS could be a buyer characteristic since the distance between the buyer’s existing farm/residence and the purchased parcel may affect price due to factors related to ease of amalgamation. It is easier to assimilate neighboring farms within the existing farm than those further away, as resources may be shared and transportation costs are lower. If BUYDIS captures the impact of ease of amalgamation, the relationship between BUYDIS and land price will be negative. The small size of Northern Ireland and the relative importance of farming for its economy lead to an emotional attachment to farming and land that in this cases makes RENTPP and BUYDIS social capital variables. In other countries and regions they may be interpreted as buyer’s characteristics and thus have a different impact.
Finally, as expected, when the distance to urban areas increases, the land price decreases (Figure 6). Note, however, that the slope of the effect is quite steep for land closer to urban areas but flattens as the location becomes farther away. Thus the effect exists only when the land is sufficiently close to an urban area and essentially disappears when the distance is increased beyond a certain saturation point.
VI. Conclusions
The methodology employed in this paper builds on a study by Perry and Robison (2001) in which they propose a model specification that allows personal relationship characteristics and other conditioning variables to modify the valuation of the entire set of agricultural land characteristics in the hedonic price function. However, the Perry and Robison specification is quite restrictive, as it assumes that the conditioning variables uniformly modify the parameters of the hedonic function. This paper proposes a more flexible model as an alternative that permits personal relationship and buyer/seller characteristics to exert different individual effects on the implicit prices for each of the land characteristics in the hedonic function. A more parsimonious specification can be obtained by imposing some restrictions on the general model. These restrictions identify the manner in which the buyer and personal relationship characteristics interact with the main hedonic characteristics.
The methodology employed in this paper provides strong evidence to support the hypothesis that personal relationship and buyer characteristics exert a nonuniform impact on the implicit prices of land characteristics within the hedonic land price function. Generally, the results show that underlying personal relationship characteristics influence the land-quality characteristic of the hedonic function. Land quality is the characteristic most associated with the income generating capacity of the land and is an important determinant of land price. The finding that personal relationship characteristics affect land price agrees with the findings of Perry and Robison (2001). Buyer and personal relationship characteristics, therefore, do appear to be important in the agricultural land market. These factors are less likely to play a significant role in commodity markets for homogeneous products (with large numbers of buyers and sellers) or in markets where there is less likelihood of prolonged contact between potential buyers and sellers.
There are some variables such as renting prior to purchase and distance between buyer and seller that in this particular study are identified as social capital variables, since our results suggest that they measure the strength of personal relationship. In different settings, however, these variables may be pure buyer’s characteristics.
Footnotes
The author is senior lecturer, International Finance, Division of Accounting and Finance, Lancashire Business School, University of Central Lancashire.
↵1 The SCORE values capture the innate quality of land and are based directly on the agricultural land classification system used in the United Kingdom (and Northern Ireland). This system grades agricultural land on a scale according to the how limiting factors, such as climate, topography, and soil, impinge on the agricultural productivity of land. Therefore, this scale reflects the productivity of the land. However, the information was not available to allow us to check if there was direct and linear equivalence between the score and rates of productivity. Faux and Perry (1999) argue that this equivalence is important, otherwise misleading results can be created. GIS procedures were used to apportion the total acreage of each parcel of agricultural land among the different classifications in order to derive an average score for each parcel.
