A Spatial Econometric Approach to the Economics of Site-Specific

A Spatial Econometric Approach to the Economics of Site-Specific Nitrogen

Management in Corn Production

Luc Anselin (a), Rodolfo Bongiovanni (b) and Jess Lowenberg-DeBoer (c)

ABSTRACT Spatial technologies such as GPS and GIS increasingly form the basis for site-specific management in crop production. This paper assesses the contribution of an explicit spatial econometric methodology in the estimation of crop yield functions that are used to optimize fertilizer application. The specific case study is for Nitrogen (N) application to corn production in Argentina, where the implementation of variable rate technology (VRT) requires methods that use inexpensive information and that focus on the inputs and variability common to Argentine growing areas. The objective of the paper is to assess the economic value of the application of spatial regression analysis to yield monitor data as a means to optimize variable rate fertilizer strategies. The data in the case study are from on-farm trials with a uniform N rate along strips and a randomized complete block design to estimate site-specific crop response functions. Spatial autocorrelation and spatial heterogeneity are taken into account in regression estimation of N response functions by landscape position, in the form of both a spatial autoregressive error structure and groupwise heteroskedasticity. Both uniform rate and VRT returns are computed from a partial budget model. The results suggest that N response differs significantly by landscape position, and that VRA for N may be modestly profitable depending on the VRT fee level Profitability depends crucially on the model specification used, with all spatial models consistently suggesting profitability, whereas the non-spatial models do not. Keywords: spatial econometrics, spatial autocorrelation, precision agriculture, site-

specific nitrogen management, variable rate, corn, Argentina. (a) Professor, Department of Agricultural and Consumer Economics, Senior Research Professor, Regional Economics

Applications Laboratory (REAL), University of Illinois, Urbana-Champaign, 326 Mumford Hall, MC-710, 1301 W. Gregory Drive, Urbana, IL 61801. Phone (217) 333-7608. Email: [email protected]

(b) Ph.D. candidate, Department of Agricultural Economics, Purdue University. Researcher, National Institute for

Agricultural Technology (INTA), 5988 Manfredi, Córdoba, Argentina. Phone and Fax +54 (3572) 493039. Email: [email protected]

(c) Professor, Department of Agricultural Economics, Director, Site-Specific Management Center, Purdue University,

1145 Krannert Building, West Lafayette, IN 47907-1145. Phone: (765) 494-4230. Fax: (765) 494-9176. Email: [email protected]

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Contact Author Jess Lowenberg-DeBoer Department of Agricultural Economics Purdue University 1145 Krannert Building West Lafayette, IN 47907-1145. Phone: (765) 494-4230. Fax: (765) 494-9176. Email: <[email protected]>

A Spatial Econometric Approach to the Economics of Site-Specific Nitrogen

Management in Corn Production

INTRODUCTION

Technologies based on computerized geographic information and global

positioning systems (GPS) are transforming large-scale commercial agriculture throughout

the world. This technology is often labeled “precision agriculture” and has given new life

to the old idea of site-specific management by reducing the cost of information acquisition

and variable rate input application.

This paper assesses the contribution of an explicit spatial econometric methodology

in the estimation of crop yield functions that are used to optimize fertilizer application. The

specific case study is for Nitrogen (N) application to corn production in Argentina, where

the implementation of variable rate technology (VRT) requires methods that use

inexpensive information and that focus on the inputs and variability common to Argentine

growing areas.

The general principles underlying site-specific management are transferable from

place to place, but the fine-tuning of production systems is necessarily region-specific,

because soils, climate and economic conditions vary. In the particular case of Argentine

producers and agribusiness companies investigated here, some special problems pertain to

the adaptation of precision agriculture to local conditions. While yield monitoring in

Argentina has followed a similar adoption path to that in North America, variable rate

application of inputs has not been widely used because of the high cost of soil sampling

combined with relatively low fertilizer use.

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Consequently, it may be argued that a greater adoption of variable rate technology

(VRT) in Argentine conditions will depend on the development of management tools that

use inexpensive information, such as yield maps, topographical maps, satellite images,

aerial photographs and eventually remote sensing and soil sensors. This contrasts to the

U.S. practice of heavy reliance on grid soil sampling, which is still very expensive in

Argentina, due to a lack of economies of scale. For example, through the 1990s, the cost of

commercial laboratory analysis of soil samples in Argentina ranged from $40 to $70 per

sample, compared to around $6 per sample in the U.S. Two new laboratories using modern

technology recently opened (one in Buenos Aires city and the other one in Pergamino,

province of Buenos Aires) that offer soil testing at a cost of about $25 per sample. Even

with the new, lower cost soil testing facilities, the type of intensive grid or soil type

sampling routinely used in North America remains prohibitively expensive in a typical

Argentine corn production setting.

The objectives of this paper are threefold: 1) to assess the contribution of spatial

econometric methods in the estimation of low cost specifications for models of the site-

specific crop N response from yield monitor data, needed to optimize variable rate fertilizer

strategies; 2) to estimate the profits for site-specific N management using the crop

responses estimated for both spatial and non-spatial models; and 3) to compare profits from

site-specific N management using crop response functions with uniform rate management

and spatial management strategies.

The empirical application utilizes yield monitor data from an on-farm trial in

southern Córdoba Province in Argentina, with a specific focus on Nitrogen, the most

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commonly used fertilizer by corn farmers in Argentina. A spatial econometric methodology

is applied in the estimation of site-specific crop response functions that contain low cost

independent variables, such as landscape position and topography.

In the remainder of the paper, first some background is provided on the economics

of site-specific fertilizer management in general, and the application of spatial models in

particular. This is followed by an overview of the methodology and data used in the paper.

Results are presented next, with a special emphasis on the sensitivity of the economic

analysis to econometric methodology and the “cost of a wrong decision”. The paper closes

with some concluding remarks and suggestions for application of the methodology by

producers and crop consultants.

BACKGROUND

Site-specific fertilizer application is not a new idea. In the U.S., the first extension

recommendations on intensive soil sampling and variable rate fertilizer application

appeared as early as 1929 (Linsley and Bauer, 1929). The recent resurgence of interest in

the idea can be linked to the availability of global positioning systems (GPS) and

information technology (IT) which lower the cost of information acquisition and VRT

implementation dramatically. VRT fertilizer application was the earliest commercially

available precision agriculture service in the U.S. Currently, about 50% of the

approximately 7500 retail fertilizer dealers in the U.S. Midwest offer the service (Whipker

and Akridge, 2001). In contrast, in Argentina, only ten VRT fertilizer applicators (out of

about 200 fertilization services providers, and out of about 1500 pesticides applicators)

were being used in 2001 (Bragachini, 2001).

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In the U.S., VRT fertilizer application is a common practice among producers of

higher value field crops, such as sugar beets. However, for crops such as corn and

soybeans, doubts remain about its profitability (Lowenberg-DeBoer and Swinton, 1997). A

recent review by Swinton and Lowenberg-DeBoer (1998) considers a number of studies of

the profitability of site-specific N, phosphorus (P) and potassium (K) fertilizer application

that relies on intensive soil sampling. Their general finding is that VRT fertilizer

application is often profitable for higher value field crops, but seldom profitable for

extensive dryland crops like wheat and barley. For maize and soybeans, the returns from

VRT fertilizer application often fail to cover the added costs of implementing the

technology.

Some other recent assessments of the profitability of VRT fertilizer application

include Lowenberg-DeBoer and Aghib (1999), Bongiovanni and Lowenberg-DeBoer

(1998) and Finck (1998). Lowenberg-DeBoer and Aghib (1999) use on-farm trial data from

the eastern Corn Belt to show that VRT of P and K just about covers costs as a stand-alone

practice, and that it may have potential to reduce risks. Bongiovanni and Lowenberg-

DeBoer (1998) demonstrate that a VRT lime application is modestly profitable in the

Eastern Cornbelt. Finck (1998) found yield increases for corn grown in an integrated site-

specific management system in on-farm trials on the Sauder farm in central Illinois, which

combined VRT of N, P, K, lime, and plant population.

The recent resurgence in interest in site-specific management has led to many

alternatives to soil sampling as the basis for VRT application, as evidenced by the large

number of contributions to the Proceedings of the International Conferences on Precision

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Agriculture held in Minnesota (Robert, Rust and Larson, 1993, 1995, 1997, 1999, 2001).

For example, the 1997 and 1999 Proceedings contain as much as 52 papers focusing on N

management. In addition to soil sampling, proposed sources of spatial information to

guide N application include aerial photography and satellite remote sensing (Franzen et al.,

1998; Wood, Thomas and Taylor, 1998; McCann et al., 1996, Blackmer and White, 1996),

landscape position (Nolan et al., 1998; Solohub, Van Kessel and Pennock, 1996),

topography (Pennock et al., 1998; Hollands, 1996), yield and grain protein maps (Long,

Engel and Carlson, 1998; Blackmer and White, 1996), soil type (Fenton and Lauterbach,

1998) and chlorophyll sensors (Blackmer, Schlepers and Meyers, 1996).

In spite of this large interest and the prevalence of yield monitors for the last ten

years, it has remained difficult to link yields to crop conditions and to clearly establish the

profitability of VRT fertilizer application. Two possible explanations may be suggested for

this problem. On the one hand, the whole field recommendations on which variable rate

technology (VRT) applications are based do not reflect site-specific response differences.

As noted by Pan et al. (1997), current university and industry N recommendations in North

America may not be very useful for site-specific management because they are broad

compromises intended to be used regionally. N is spatially and temporally dynamic and its

availability to the plant at any one location and time depends on many factors, including

organic matter in the soil, previous crop, manure applications, recent temperature and

rainfall patterns, and leaching losses. Regression models for yield response will therefore

unlikely be properly specified, which may cause problems of omitted variable bias, long

known in applied econometrics (e.g., Griliches 1957). Since the coefficient estimates on

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managed input variables are used to develop management recommendations, statistical bias

may cause costly errors (Swinton et al., 2001).

A second, and arguably more important explanation suggests that the failure to

properly incorporate the spatial structure of the data will affect both estimation of site-

specific response functions as well as profitability analysis. More specifically, in the

recently formulated economic model of Bullock and Bullock (2000), crop yields are a

function of managed inputs, unmanaged site characteristics and unmanaged weather

phenomena. Site characteristics represent an important group of variables that tends to be

omitted from agronomic yield response models. In addition, these variables will tend to

show distinct spatial patterns, which, when omitted from the model, will induce spatially

correlated error structures. Increasingly, it has been argued that an explicit spatial

econometric methodology should be followed to address these issues (Anselin 2001a,

Florax et al. 2001, Swinton et al, 2001). Before outlining the particular approach taken in

this paper, it is useful to consider some other recent applications of spatial models in this

context.

Spatial Models. Regression crop response functions have the advantage of fitting

easily into the traditional crop production economics decision model (Heady and Dillon,

1961; Dillon and Anderson, 1990). This also extends to site-specific management, as

demonstrated by Lowenberg-DeBoer and Boehlje (1996). To date, most crop response

models used in this process are estimated by classical regression techniques, such as

ordinary least squares (OLS). Overall, these estimates have yielded mixed results (see for

instance, Khakural, Robert and Huggins, 1998; Coelho, Doran and Schlepers, 1998;

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Mallarino, Hinz and Oyarzábal, 1996). However, it is important to note that these studies

ignore any spatial patterns that may be present in the data. As demonstrated by Kessler and

Lowenberg-DeBoer (1998), spatial correlation of regression residuals may be important in

models for yield monitor data. As is well known (Anselin, 1988), ignoring such

autocorrelation will yield OLS estimates that are inefficient and will bias the standard

errors, t-test statistics and measures of fit, rendering standard statistical inference

unreliable.

While the application of explicit spatial econometric methods has recently shown a

tremendous increase in the social sciences in general and economics in particular

(Goodchild et al., 2000; Anselin, 2001a,b), to date, there have been only a small number of

studies that employed spatial regression analysis in the study of yield monitor data.

Some studies applied time series techniques to the spatial domain. For example,

Wendroth et al. (1998) used state-space analysis to remove spatial autocorrelation in a

study of wheat yield data for a field in northeast Germany. Similar to this paper, VRT-N

was compared to homogeneous application, but estimation was limited to a standard

analysis of variance. Moreover, unlike the design followed here, the empirical setup

employed in Wendroth et al. (1998) was not based on different treatment rates for N.

An early example of an explicit spatial perspective on crop yield modeling is Long

et al. (1992), in which some of the basic methodological issues were outlined, although no

application was presented. More recently, Long (1998) applied spatial autoregressive

response modeling (referred to below as a spatial lag model) to estimating the relationship

between site-specific wheat yields and selected terrain variables within a field. Both a

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regression model and analysis of variance were employed, with a focus on how the model

estimates varied with the scale of the unit of observation (the modifiable areal unit

problem). Error spatial autocorrelation is not considered, nor are the estimates part of an

evaluation of the economics of wheat production.

Two other recent papers are similar in spirit to the approach taken here, although

they differ in important respects both substantively and methodologically. In Florax et al

(2001), several spatial econometric techniques similar to the ones employed in the current

paper are reviewed and applied to millet yield functions for an experimental plot in the

Sahel. As in Long (1998), a spatial lag specification is employed, but groupwise

heteroskedasticity is introduced as well. The yield function employed by Florax et al

(2001) follows a general Cobb-Douglas specification and is not particularly geared to the

evaluation of VRT technologies.

Hurley et al. (2001) use field trial data for different N rates in southern Minnesota

to assess the value of different sources of information in guiding variable rate application

in corn. Similar to the specification used in this paper, the yield model is also a quadratic

specification in N. However, spatial autocorrelation and spatial heterogeneity are

approached in a different way. In the Hurley et al. (2001) paper, spatial heterogeneity is

modeled as random coefficient variation, whereas spatial residual autocorrelation is

specified to follow a spherical correlogram (based on notions from geostatistics). A three-

step estimation procedure is outlined, which does not allow for simultaneous inference of

all the parameters in the model. In contrast, the approach followed in this paper integrates

the treatment of both spatial autocorrelation and heterogeneity, using the spatial

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econometric framework and methods outlined in Anselin (1988). This results in a

specification search that allows for both spatially lagged dependent variables as well as

spatial error autocorrelation, and when necessary, includes models for spatial

heterogeneity.

METHODOLOGY

The approach taken in this paper is based on a spatial econometric methodology

(Anselin 1988) to carry out statistical inference for the response function in the yield

model. The motivation for this choice is three-fold. First, it allows for an explicit

accounting for the effects of spatial autocorrelation, due to spillovers, externalities or other

imperfections in model and measurement that show a spatial structure. This is approached

by carrying out specification tests for spatial autocorrelation and estimating models that

incorporate spatial autocorrelation where appropriate. Second, one of the objectives is to

implement a low-cost technique that allows evidence of spatial heterogeneity in the field to

be exploited in order to yield more efficient parameter estimates. Specifically, the degree

is assessed to which different landscape positions affect the magnitude, significance and

sign of the estimated coefficients in the model. This is implemented by testing for (spatial)

heteroskedasticity and by estimating models that incorporate structural change in the form

of spatial regimes (different coefficients in spatial subsets of the data) that match the

landscape positions. Third, the extent is assessed to which the indications for economic

decisions with respect to uniform or VRT application are affected by the choice of the

estimation and model specification. Specifically, interest focuses on measuring the impact

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of the increased precision obtained through the use of spatial econometric techniques in an

economic analysis based on a partial budgeting tool.

Spatial Econometric Models. In general terms, spatial autocorrelation may be

considered as the extension of serial correlation to the two-dimensional landscape (the

classic treatment is Cliff and Ord 1981). As such, it has received growing attention in the

economic modeling of natural resources and environmental factors (for recent reviews, see

Anselin and Bera 1998, Anselin 2001a, b). Spatial autocorrelation will be incorporated in

the regression model in two basic ways, following the taxonomy outlined in Anselin

(1988). In one model, the autocorrelation is limited to the error term in a regression model,

the so-called spatial error model. In the other, the spatial autocorrelation pertains to the

dependent variable (y) in the model itself and is referred to as a spatial lag model. Next,

each specification is considered more closely.

Formally, and using the notation of Anselin (1988), a spatial error model can be

expressed as:

y = Xβ + u with u = λWu + ε

where y is a vector (n by 1) of observations on the dependent variable, X an n by K matrix

of observations on the explanatory variables, and u an error term that follows a spatial

autoregressive (SAR) specification with autoregressive coefficient λ. In the spatial

autoregression, the vector of errors is expressed as a sum of a vector of innovation terms

(ε) and a so-called spatially lagged error, Wu. The latter boils down to a weighted average

of errors in the neighboring locations. The selection of neighbors is carried out through the

n by n spatial weights matrix W. Each row of this matrix contains non-zero elements for

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the columns corresponding to “neighbors.” Typically, the definition of neighbors is based

on geographical criteria, such as sharing a common border or being within a given distance,

although extensions to economic distance have been suggested as well (see Anselin and

Bera, 1998 for an overview of the relevant econometric issues). By convention, the

diagonal elements of W are set to zero (implying that locations are not neighbors of

themselves). Also, in practice, the rows of the weights matrix are rescaled such that the

sum of the row elements equals one. In the study of corn yield, the data from yield

monitors is usually collected in a regular layout that can be converted to a grid system in a

straightforward manner, and therefore the contiguity structure is easily derived. Typically,

either a “rook” structure (four neighbors to each cell, corresponding to north, south, east

and west) or a “queen” structure (eight neighbors to each cell) is employed (see Anselin

1988 for details).

While OLS estimates remain unbiased in the presence of spatial error

autocorrelation, their efficiency is affected, and inference based on the usual t-tests and R2

measures of fit can be misleading.

In a spatial lag model (or mixed regressive, spatial autoregressive model) a

spatially lagged dependent variable term Wy is introduced on the right-hand side (RHS) of

the regression equation:

y = ρWy + Xβ + u

where ρ is the autoregressive coefficient, and the other notation is as before. The presence

of Wy in the model induces endogeneity, which has two important consequences. First, the

spatial lag is correlated with the error term, which leads to “simultaneous equation bias”

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and precludes OLS from being consistent. Secondly, the endogeneity yields a spatial

multiplier effect that expresses how changes in one location affect all other locations.

Formally, after removing all endogenous elements on the RHS:

y = (I – ρW)-1Xβ + (I – ρW)-1u.

In other words, the value of y at a location is not only determined by the X at that

location, but also by the X at all other locations in the system, suitably adjusted by a

distance decay factor (for a more extensive treatment of the interpretation of spatial

multipliers in the context of these models, see Anselin 2001d).

A spatial regime model (Anselin 1990) allows for different parameter values

(including the error variance) in discrete but spatially contiguous subsets of the data. In the

current context, natural candidates for the specification of the regimes are the four

topographies that can be easily (and at low cost) distinguished in the field.

In the empirical case study, the selection of the final specification will be based on

pragmatic grounds. The corn response to N will be estimated as quadratic specification by

landscape position. With i indexing the landscape and j indicating the location within this

landscape:

Yieldij = αi + βi Nij + γi Nij2 +εij,

where Yieldij is the corn yield (from a yield monitor with GPS), and Nij is the N rate. This

specification allows for the estimation of topography effects on the level, αi (difference

from the mean) as well as interaction terms between the topography areas and N (βi) and

N2 (γi).

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Starting with this basic model, a spatial specification search will be carried out,

using standard OLS estimation and applying a series of specification tests to assess whether

there is evidence of spatial autocorrelation, and, if so, which alternative (lag or error) is

suggested by the data. The specific tests employed are Lagrange Multiplier (LM) statistics

for each of the alternatives (see Anselin 2001c, for a recent review). If appropriate, the

spatial models are estimated by means of maximum likelihood or method of moments

techniques (e.g., Anselin 1988, Kelejian and Prucha 1998, 1999). Variation of the response

function by landscape position will be assessed by a significance test on the coefficients of

the dummy variables. Since a dummy variable constraint is imposed to allow for the

interpretation of these coefficients as the difference from the mean, this provides a direct

indication of spatial structural instability. If necessary, a spatial regimes model can be

combined with a spatial autoregressive model in a straightforward fashion, which allows

for tests on the spatial homogeneity of coefficients that are properly corrected for the

presence of spatial autocorrelation, so-called “spatial” Chow tests (Anselin 1990).

All estimation and specification tests will be carried out by means of the

SpaceStatTM software package for spatial data analysis (Anselin 1999).

Profitability of VRT-N. The profitability of variable rate application of N will be

assessed relative to a VRT fee of $6 per ha. Net returns will be computed for VRT

application for N by landscape position, uniform application, and other strategies. The

input into the optimization procedure are the parameter estimates obtained from the

regression model (spatial and non-spatial).

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The optimal level N by landscape position is computed in the standard fashion

using ordinary calculus (Dillon and Anderson, 1990). Net returns over fertilizer cost, VRT

application fee, added non-N fertilizer costs for maintenance, and extra harvest and

handling costs are taken into account. These are expected returns, so prices and costs are

projected to be the best estimate of future expected levels. Seed, weed control, and

equipment costs are assumed to be the same everywhere in the field, so there is no reason

to deduct them (Boehlje and Eidman, 1984). The average return for the field will be

estimated as the weighted sum of returns in each landscape area, where the weights are the

proportion of area in that landscape position. The returns from site-specific management

(SSM) by landscape position will be compared to the returns for uniform applications at

the level recommended by university fertilization strategies for the area (e.g., Castillo et al.,

1998). The specific focus is on the degree to which the returns for N by landscape position

are “on average” higher than those of the commonly used uniform rate strategies.

The economic analysis will be performed using the partial budgeting tool (Boehlje

and Eidman, 1984), which determines whether the added benefits outweigh the added

variable costs in a typical year. Net returns from N will be calculated using marginal

analysis, which states that when the value of the increased yield from added N equals the

cost of applying one additional unit, profit is maximized; or when the marginal value

product equals the marginal factor cost (MVP = MFC). Profit maximizing N rates are

considered because they are the alternative to “agronomic rates” when response curve

information is available. It should be noted that the economic calculations pertain to

expected values, so that error structures will be averaged out.

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Finally, the maximization of expected profit can be expressed as:

Max [ ] ( )[ ]∑=

−++=4

1

2 *****i

iNiiiiici NrNNPEAreaE γβαπ

where: E = Expectation operator π = Total net returns over N fertilizer ($ ha-1) Areai = Proportion of area i (i = 1,…,4) i = Landscape area: 1=Low East, 2= Slope East, 3=Hilltop, 4=Slope West Pc = Price of corn ($6.85 per quintal) αi = Intercept estimate from the spatial autoregressive model. βi = Linear coefficient estimate γi = Quadratic coefficient estimate Ni = Quantity of elemental N applied in area i rN = Price of elemental N, plus interest for 6 months at 15% annual interest

rate. DATA

N response data was collected from strip trials at the “Las Rosas” farm, located at

63º 50’ 50” of longitude W and 33º 03’ 04” of latitude S in the Río Cuarto area, Córdoba

Province, Argentina, for the 1998-99 crop season.

The experimental design for the trials is a complete block strip trial that includes at

least three different types of soils in terms of landscape (hilltop, slope, and low). The strips

are wider or equal to the corn header width, with zero N application used as the control,

and five other rates of elemental N: 29, 53, 66, 106 and 131.5 Kg ha-1 (Figure 1).

<INSERT FIGURE 1 ABOUT HERE>

The N rate is held constant in each strip, across the four topography zones in which

the field trial was divided. The highest N rate applied is higher than the expected yield

maximizing level. The field was divided into three blocks, and within each block,

treatments are randomized. The source for N is urea.

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Data was collected with a standard AgLeaderTM yield monitor, a geo-positioned

device located on the harvester combine that measures and records crop yields on-the-go.

Yield files include data-point information about yields, latitude, longitude, elevation, and

grain moisture, which is used to generate a geo-positioned database and the site-specific

yield maps.

Some manipulation of the original data was required, since the spatial layout of the

raw data (8288 point observations) was such that it included points located closer together

within the same row than between rows. In order to obtain a balanced design suitable for

conversion to a regular grid layout, the original data yield points were spatially averaged.

This was executed in the geographic information system (GIS) software SSToolboxTM,

creating 13.6 by 13.6 meter grids over the observations, and rotating them by 10.5 degrees

(Figure 2). Data points at the extreme west and extreme east side of the plot were deleted,

because they reflect an empty combine entering the row. Finally, and after averaging the

data within each grid, a layout of 1738 regular polygons was obtained (Figure 3).

<INSERT FIGURES 2 AND 3 ABOUT HERE>

RESULTS

The coefficient estimates that form the basis for the economic analysis are obtained

for a series of models that incorporate increasing complexity in terms of spatial variability

(spatial regimes and/or heteroskedasticity) and spatial autocorrelation (autoregression).

This complexity was warranted by the outcome of specification tests at each stage of the

analysis. The progression starts with a standard regression model, quadratic in N and with

constant coefficients across all four landscape categories, estimated by ordinary least

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squares (OLS). Next is the same specification with varying coefficients across the

landscape categories, again estimated by OLS. Third are spatial autoregressive error

models with spatial regimes according to the topographic categories. These models are

estimated by both a maximum likelihood procedure (ML) as well as by a generalized

moments techniques (GM). Finally, a spatial autoregressive error model is considered that

incorporates both coefficient variation across topographic regimes as well as groupwise

heteroskedasticity (non-constant error variance) corresponding to these regimes. Again,

these models are estimated by ML (ML-GHET, for groupwise heteroskedasticity) as well

as GM (GM-GHET). All spatial models (and all specification tests for spatial effects) were

estimated for a rook (4 neighbors) as well as for a queen (8 neighbors) measure of

contiguity. This allows for a careful analysis of the sensitivity of the economic results to

the specification of the spatial model, the estimation method used and the choice of the

spatial weights matrix.

In the interest of conserving space, only the most salient results are reported here

(the complete set is available from the authors). Specifically, the focus is on the difference

between a standard regimes model that incorporated landscape position and is estimated by

OLS (referred to as REGIMES in the Tables) and a spatial regimes model with a spatial

autoregressive error term as well as groupwise heteroskedasticity, estimated by ML-GHET

(referred to as AUTO in the Tables).

Coefficient Estimates and Model Diagnostics. The estimation results for the

REGIMES and AUTO specifications are summarized in Table 1. These final specifications

are selected on the basis of the outcome of a series of diagnostics carried out on

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increasingly complex models. Starting with the estimates for REGIMES in Table 1, the

coefficients of both N and N2 are highly significant and with the expected sign. Moreover,

it is clear that the landscape position dummies have coefficients that are significantly

different from the mean value at the 1% significance level, providing the motivation for the

use of the “regime” specification in terms of the “level”. However, this finding is not

sustained for the linear and quadratic terms of the nitrogen variable, which do not seem to

vary significantly with landscape position.

Diagnostics for spatial autocorrelation in this model suggest a spatial error model as

the proper alternative. While both LM-Error (1993 as χ2 with 1 degree of freedom) and

LM-Lag tests (295 as χ2 with 1 degree of freedom) strongly reject the null of no spatial

autocorrelation at very high signifcance levels (p < 0.001), the decision rules outlined in

Anselin and Florax (1995) suggest the error model as the alternative. In addition, standard

diagnostics for heteroskedasticity suggest the presence of this form of misspecification as

well. Even after implementing a spatial error model for the regimes, there remains

heteroskedasticity. Both spatial error autoregression as well as heteroskedasticity are

therefore incorporated in the AUTO model. The coefficient estimates vary slightly relative

to the values obtained for REGIMES, with the exception of the landscape dummies, where

the estimates for Low E, Hilltop and Slope W are quite different (even yielding an opposite

sign for the latter). In addition, there now is evidence of coefficient instability across

landscape position for the interaction terms with N (for three of the four categories) and

with N2 (for low E).

<INSERT TABLE 1 ABOUT HERE>

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Spatial Variability. In order to visualize the variability in the coefficient estimates

for the marginal response to N (the coefficient for N in the model) by topographic location

in a manner that incorporates the efficiency of the estimates (i.e., their standard errors) the

distribution of the estimates was simulated. This was based on the results in Table 1 for the

coefficient estimates and their standard errors, and assuming normality. Figure 4 illustrates

the differences and highlights the higher degree of precision obtained in the AUTO model.

Clearly, the estimates for Low E and Hilltop are different from the mean response (across

all landscape categories) and this is even more pronounced for the AUTO model.

A second aspect of the variability of the model estimates across landscape position

is illustrated in Figure 5. The expected yield is computed by varying N, based on the

landscape-specific coefficients and interaction terms estimated for the AUTO model (from

Table 1). The graphs illustrate both how the nonlinear (quadratic) nature of the response is

different for each landscape position as well as how the yield level varies by location.

Yields are highest in the Low E area, but the response to N is greatest in the Hilltop.

In Figure 6, the difference between the marginal effect of N by landscape position is

highlighted (the additional yield for the first Kg. of N), computed by combining the mean

effect and the landscape interaction terms (based on the estimates for the AUTO model in

Table 1). Similarly to the findings in Figure 5, the highest response is for Hilltop and the

lowest in Low E, the latter yielding less than half the return of the former.

The effect of coefficient variability on the computation of optimal N rates is

illustrated in Figure 7. The highest optimal rate is found for Slope W (147.8 kg per ha),

20

followed by Hilltop (92.9 kg per ha), whereas the optimal rate for Low E was zero. This

may in part be explained by the fact that Slope W corresponds with a lower quality soil.

Low E, Slope E and Hilltop are type IIIes soils, while Slope W is type IVes. Soils type

IIIes present excessive drainage, and are developed from sandy-loam materials. They have

low water holding capacity, low structural stability, low organic matter content, important

weather limitations, and moderate susceptibility to wind erosion. However, soils type IVes

have even higher susceptibility to wind erosion, lower water holding capacity, lower

organic matter content, and very low structural stability (Jarsun et al., 1993), characteristics

that explain the high optimal N rate.

<INSERT FIGURES 5, 6 AND 7 ABOUT HERE>

Finally, in Figure 8, the variability is illustrated of economic returns by landscape

position as a result of optimizing the application of N. This is relative to a strategy of not

applying any fertilizer. The profit maximizing or economic response to N was obtained

using a net price of corn of $6.85 per quintal, which is a three-year average of corn prices

in Argentina, a cost of elemental N of $0.4348 per Kg ($0.4674 per Kg with a 15% annual

interest rate), and a VRA application fee of $6 per hectare (this fee was not deducted in

Figure 8). As in Figure 7, the highest economic return is obtained for Slope W and Hilltop,

with much lower returns for Slope E.


Returns by N Rate Application. A comparison of the returns from different N

rate applications is given in Table 2. The returns were estimated for two uniform

application rates and for a variable rate application following the four landscape positions

21

in our study. The two uniform rates were used to represent the range of N rates currently

practiced in the Río Cuarto area. The lower uniform N rate (36.8 Kg ha-1) was

recommended by Castillo et al. (1998). The higher uniform N rate (83.49 Kg ha-1) was the

profit-maximizing rate for the whole field using the response function estimated with the

AUTO model (Table 1). The estimated VRA assumed that N varies by landscape position

according to the profit maximizing levels identified in Table 3 for that part of the

topography. All three estimates use the response curves by landscape to estimate yields

(Table 1), which are then weighted by the area in the corresponding topographical zone

(Lowland East 2.12 ha or 26.52%; Slope East 1.69 ha or 21.17%; Hilltop 1.63 ha or:

20.37%; Slope West 2.53 ha or 31.93%).

Returns above fertilizer cost for a uniform rate of N, applied to the whole field

(traditional fertilizer application), using the N fertilizer rate recommended by Castillo et al

(1998) were estimated as $415.35 ha-1. Returns above fertilizer cost for a uniform rate of

N, applied to the whole field (traditional fertilizer application), using the whole field profit

maximizing N rate from Table 4, were estimated as $419.56 ha-1. Returns above fertilizer

cost for variable rate (VRA) of N, which included the VRT-N fee of $6 ha-1, were

estimated as $417.00 ha-1. The breakeven fee obtained from the econometric estimates for

the REGIME model is $3.28 ha-1, whereas the estimates from the AUTO model yielded

$7.65 ha-1 (Table 2).


The difference between the physical and economic results by topography area that

follow from using different econometric model estimates is highlighted in Tables 3 and 4.

22

Table 3 is calculated using the OLS estimates in the REGIMES model, even though there

is evidence that it is not efficient as the spatial AUTO model, estimated by ML-GHET. The

AUTO model provides a higher confidence in the difference between the regions, and

therefore more accurate estimates. This results in a significant shift in the breakeven point

for the variable rate fee charged by the service provider. The estimates from the AUTO

model result in more than a doubling of this breakeven point, suggesting that VRT may be

profitable for farmers, because the return is $1.65 higher than the estimated market VRT

fee of $6.00.



The explicit incorporation of a spatial component in the yield model specification

reveals patterns of interaction among yield points that are not accounted for in conventional

models. The spatial model also shows how OLS estimates may be significantly imprecise

when this interaction is not taken into account in the estimation process. The spatial

autoregressive error model provides a better fit, as well as a higher accuracy for the

coefficient estimates that are at the basis for the economic calculations. Interestingly, the

two models considered also lead to different economic conclusions, where one would

discourage the adoption of VRT N fertilization (REGIMES), while the other demonstrates

the economic feasibility of a $6 per ha VRT fee (AUTO).

Sensitivity Analysis. In order to further assess the sensitivity of the computed

physical and economic returns to econometric modeling strategy, the twelve models

considered are compared in Table 5 and Figure 9. Specifically, the characteristics that vary

23

between the models pertain to the specification of the yield response model (regimes or

not), the incorporation of spatial autoregressive error terms, the estimation method used

(ML or GM) and the spatial weights matrix employed (rook or queen). The most striking

feature of these results is that all spatial autoregressive models yield a break even fee that

exceeds $6 per ha, irrespective of estimation method or spatial weights matrix. While the

individual results vary slightly, they consistently suggest the profitability of the VRT

practice, which the non-autoregressive models do not. In other words, the spatial

econometric methodology yields qualitatively different economic recommendations, as a

result of the explicit consideration of spatial variability in the field, which results in more

precise estimates.



The Cost of a Wrong Decision. The added value provided by an explicit spatial

econometric approach can be measured by the concept of the “cost of a wrong decision”

outlined by Havlicek and Seagraves (1962). This is accomplished by comparing the

economic results for optimal N rates from the REGIMES model (the “wrong” model) to

those obtained with the “true” AUTO model. More precisely, the AUTO model is

considered to be the best available information and thus treated as the true response. If the

optimal N rate estimated with the coefficients for the REGIMES model (fourth column of

Table 2) were to be used instead of the one estimated with the proper spatial AUTO model

in the yield response function for AUTO, the difference would indicate the cost of a wrong

decision. The results here suggest that the returns above fertilizer cost for variable rate

24

application (VRA) of N would drop from $417.00 ha-1 to $414.60 ha-1 (Table 6). Thus, the

cost of a wrong decision from using the non-spatial estimates would be $2.40 ha-1, or

$2,400 per year for a 1,000-hectare farm.


CONCLUSION

The key benefit of an explicit spatial econometric methodology, such as the one

employed in this paper is that any spatial structure in the data is exploited to yield more

precise (and, in some cases, less biased) estimates for the parameters in a yield response

function. Since these parameters form the basis for all ensuing economic computations,

increased precision affects the precision of the estimates for yield, return and profitability.

In the case study considered in this paper, the qualitative economic results were

significantly different between a spatial and a traditional approach. The spatial

autoregressive models, irrespective of details in their implementation (estimation method,

spatial weights matrix) consistently pointed to the profitability of a VRT N application,

whereas the traditional models did not. Assuming that the estimates from the spatial

models are therefore superior, there is a clear economic payoff to adopting them.

While this is an interesting result, and encouraging for the potential of spatial

modeling techniques to provide cost effective tools to assess VRT profitability, it should be

kept into perspective. The case study pertained only to one farm at one point in time. New

data have been obtained to extend the analysis to multiple farms as well as to multiple

years. It remains to be seen whether the spatial econometric methodology can be

25

successfully adopted to these designs and confirm the preliminary results in the current

paper.

ACKNOWLEDGEMENTS.

The authors would like to thank Mario Bragachini, and the team of the Precision

Agriculture Project of INTA, Manfredi Experimental Station, for conducting the field trials

in Argentina. Authors also appreciate the Site-Specific Technology Development Group

(SST) for providing the GIS software SSToolboxTM for this analysis, as well as the

necessary technical support. The research was made possible in part by an assistantship

funded by the National Institute for Agricultural Technology (INTA) of Argentina.

Anselin’s research was supported in part by Grant BCS-9978058 from the National

Science Foundation to the Center for Spatially Integrated Social Science (CSISS).

26

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Figure 1. Experimental design for the “Las Rosas farm.” Landscape positions and N rates.

Figure 2. Reference grid clipped to the field boundaries of “Las Rosas”, 1999.

Figure 3. Digitized grids reflecting average yields within each grid.

132 kg/ha66 kg/ha106 kg/ha


Hilltop (3)Slope W (4)

Slope E (2)Low E (1)





Hilltop (3)Slope W (4)

Slope E (2)Low E (1)

33

Figure 4. Comparison of the expected crop response functions to N by landscape

position for the REGIMES and the AUTO model.

Figure 5. Expected Crop Response Functions to N by Landscape Position, AUTO Model.

Yield=a1*N (OLS model)

0

0.5

1

1.5

2

2.5

3

3.5

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24Marginal yield response to N

Dis

tribu

tion

Low E

Slope E

Hilltop

Slope W

OLS

Yield=a1*N (spatial model)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24

Marginal yield response to N

Dis

tribu

tion

Low E

Slope E

Hilltop

Slope W

Spatial

Corn response to N

55

60

65

70

75

0 20 40 60 80 100 120 140 160 180 200N, Kg/ha

Yiel

d, q

uint

als/

ha

Low E

Slope E

Hilltop

Slope W

34

Figure 6: Expected Corn Response to the First Kg of N, AUTO Model.

Figure 7: Optimal N Rates by Topography, AUTO Model.

Figure 8. Expected Net Returns from N Above Fertilizer Cost by Landscape Position,

AUTO Model.

Corn response to the 1st Kg of N

6.699.65

14.04 13.16

0

5

10

15

20

Low E Slope E Hilltop Slope W

Cor

n Yi

eld,

Kg/

ha

Optimal N rates by topography

147.84

92.87

41.210.00

050

100150200


Kg/

ha

Net returns to N

0.003.99

22.9432.07

05

101520253035


$/ha

35

Figure 9. Breakeven VRT fees for the different models. An application fee of $6 per hectare is the assumed as the market extra cost for using VRT in Argentina. Models correspond to the list in Table 5.

$2.40

$3.28

$6.58

$7.57

$6.58

$7.67

$2.40

$3.28

$6.72 $6.93 $6.96 $7.11

$0.00

$1.00

$2.00

$3.00

$4.00

$5.00

$6.00

$7.00

$8.00

$9.00

1 2 3 4 5 6 7 8 9 10 11 12Model

Brea

keve

n VR

T fe

e ($

/ha)

36

Table 1. Coefficient Estimates.

REGIMES AUTO VARIABLE COEFF S.D. t-value Prob COEFF S.D. z-value Prob Constant 5863.68 0.3072 190.8795 0.0000 5942.87 0.6946 85.5550 0.0000 N 11.5415 0.0108 10.7182 0.0000 10.8791 0.0062 17.4180 0.0000 N2 -0.0358 0.0001 -4.6416 0.0000 -0.0243 0.0000 -5.4190 0.0000 Low E 851.1340 0.5176 16.4431 0.0000 418.8830 0.8347 5.0185 0.0000 Slope E 199.9670 0.5573 3.5884 0.0003 205.0530 0.6664 3.0769 0.0021 Hilltop -1206.1200 0.5608 -21.5087 0.0000 -406.5670 0.8070 -5.0383 0.0000 Slope W 155.1700 0.4917 3.1560 0.0016 -217.6550 0.8673 -2.5097 0.0121 N x Low E -2.8057 0.0181 -1.5512 0.1210 -4.1845 0.0101 -4.1278 0.0000 N x Slope E -1.0599 0.0196 -0.5399 0.5893 -1.2280 0.0098 -1.2526 0.2103 N x Hilltop 3.3528 0.0198 1.6897 0.0913 3.1562 0.0130 2.4356 0.0149 N x Slope W 0.5143 0.0172 0.2991 0.7649 2.2770 0.0103 2.2153 0.0267 N² x Low E 0.0100 0.0001 0.7768 0.4374 0.0215 0.0001 2.9737 0.0029 N² x Slope E -0.0056 0.0001 -0.3948 0.6930 -0.0100 0.0001 -1.4173 0.1564 N² x Hilltop -0.0071 0.0001 -0.4957 0.6202 -0.0145 0.0001 -1.5507 0.1210 N² x Slope W -0.0332 0.0001 0.2087 0.8347 -0.0214 0.0001 0.3900 0.6965 Coefficient estimates, standard errors, significant test and probability. REGIMES pertains to a model with spatial regimes estimated by OLS; AUTO pertains to a model with spatial regimes, a spatial autoregressive error and groupwise heteroskedasticity estimated by GM. Dummy variables are specified as differences from the mean. Units are Kg per hectare.

Table 2. Net Returns.

Net returns ($ ha-1) REGIMES AUTO Difference Uniform rate agronomic (36.8) $414.94 $415.35 $0.41 Uniform rate economic (83.49) $416.95 $419.56 $2.62 Variable Rate $418.22 $423.00 $4.78 Variable Rate - $6 ha-1 fee $412.22 $417.00 $4.78 Breakeven VR fee $3.28 $7.65 $4.37 Returns computation based on coefficients obtained from the REGIMES specification and the AUTO model.

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Table 3. Physical and Economic Results for the REGIMES model.

N rate for Maximum Optimal Optimal Net max. yield yield N rate yield returns (Kg ha-1) (Kg ha-1) (Kg ha-1) (Kg ha-1) ($ha-1) Low E 169.49 7455.11 37.11 7003.48 $456.40 Slope E 126.68 6727.53 44.21 6446.20 $414.90 Hilltop 173.65 5950.77 94.10 5679.37 $339.06 Slope W 181.43 7112.50 78.75 6762.18 $420.40 Weighted average 165.08 6885.24 63.52 6538.73 $412.22 Trial area total (8 ha) 1315.53 54875.39 506.07 52113.82 $3,285.44

Table 4. Physical and Economic Results for the AUTO model.

N rate for Maximum Optimal Optimal Net max. yield yield N rate yield returns (Kg ha-1) (Kg ha-1) (Kg ha-1) (Kg ha-1) ($ ha-1) Low E 1183.33 10322.68 0.00 6361.75 $429.78 Slope E 140.64 6826.57 41.21 6487.36 $419.12 Hilltop 180.73 6804.58 92.87 6504.83 $396.18 Slope W 307.13 7745.55 147.84 7202.11 $418.24 Weighted average 478.54 8042.88 74.85 6685.85 $417.00 Trial area total (8 ha) 3815.41 64107.32 595.45 53283.24 $3,323.77

Table 5. Physical and Economic Results by Model and Estimation Technique.

N rate for Maximum Optimal Optimal Net Breakeven max. yield yield N rate yield returns VRT fee Queen criterion (8) (Kg ha-1) (Kg ha-1) (Kg ha-1) (Kg ha-1) ($ ha-1) ($ha-1)

1 OLS 705.38 27686.97 279.48 26233.96 $ 1,642.40 $2.40 2 OLS w/interactions 651.24 27245.91 254.17 25891.23 $ 1,630.75 $3.28 3 SAR (GM-iterated) 1198.15 28951.58 282.01 25826.04 $ 1,613.28 $6.58 4 SAR (ML) 1673.59 31055.99 270.53 26269.30 $ 1,649.00 $7.57 5 SAR (GM-GHET) 1213.40 29054.73 279.62 25869.05 $ 1,617.34 $6.58 6 SAR (ML-GHET) 1811.82 31699.38 259.17 26402.33 $ 1,663.42 $7.67

Rook criterion (4) 7 OLS 705.38 27686.97 279.48 26233.96 $ 1,642.40 $2.40 8 OLS w/interactions 651.24 27245.91 254.17 25891.23 $ 1,630.75 $3.28 9 SAR (GM-iterated) 1212.67 29200.34 307.42 26111.97 $ 1,620.98 $6.72

10 SAR (ML) 1373.95 29946.15 279.83 26213.44 $ 1,640.83 $6.93 11 SAR (GM-GHET) 1368.63 29676.25 297.08 26020.51 $ 1,619.55 $6.96 12 SAR (ML-GHET) 1485.11 30371.01 267.17 26215.86 $ 1,646.91 $7.11

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Table 6. Physical and Economic Results from Using the N Rates from the REGIMES

Model, and the Coefficients from the AUTO Model. N rate for Maximum Optimal Optimal Net max. yield Yield N rate yield returns (Kg ha-1) (Kg ha-1) (Kg ha-1) (Kg ha-1) ($ha-1) Low E 1183.33 10322.68 37.11 6606.26 $429.19 Slope E 140.64 6826.57 44.21 6507.56 $419.10 Hilltop 180.73 6804.58 94.10 6513.19 $396.17 Slope W 307.13 7745.55 78.75 6628.40 $411.24 Weighted average 478.54 8042.88 63.52 6573.48 $414.60 Trial area total (8ha) 3815.41 64107.32 506.07 52395.35 $3,304.72

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