Economics of Management Zone Delineation in Cotton Precision Agriculture. Corresponding Author: Roderick M. Rejesus

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1 Economcs of Management Zone Delneaton n Cotton Precson Agrculture Margarta Velanda 1, Roderck M. Rejesus *, Eduardo Segarra *, and Kevn Bronson 2 Correspondng Author: Roderck M. Rejesus Department of Agrcultural and Appled Economcs Texas Tech Unversty Box Lubbock, TX Phone Number: (806) ext. 253 FAX Number: (806) E-Mal: roderck.rejesus@ttu.edu January, 2006 Selected Paper prepared for presentaton at the Southern Agrculture Economcs Assocaton Annual Meetng, Orlando, Florda, February4-8, Margarta Velanda, Roderck M. Rejesus, and Eduardo Segarra are Graduate Research Assstant, Assstant Professor, and Professor, respectvely, at the Department of Agrcultural and Appled Economcs, Texas Tech Unversty, Lubbock, TX Kevn Bronson s Assocate Professor at the Texas Agrcultural Experment Staton, Texas A&M Unversty, Route 3, Box 219, Lubbock, TX

2 Economcs of Management Zone Delneaton n Cotton Precson Agrculture ABSTRACT Ths paper develops a management zone delneaton procedure based on a spatal statstcs approach and evaluates ts economc mpact for the case of Texas cotton producton. Wth the use of an optmzaton model that utlzes a yeld response functon estmated through spatal econometrc methods, we found that applyng varable N rates based on the management zones delneated would result n hgher cotton yelds and hgher net returns, above Ntrogen cost, relatve to unformly applyng a sngle N rate for the whole feld. In addton, a varable rate N applcaton usng the delneated management zones produced hgher net returns, above Ntrogen cost, relatve to a varable N rate system where the zones are based solely on landscape poston. Ths s ndcatve of the potental economc value of usng a spatal statstcs approach to management zone delneaton. Keywords: Management Zones, Exploratory Spatal Data Analyss, Ste-Specfc Ntrogen Management, Cotton Precson Agrculture. JEL Classfcaton: Q1, Q16. 2

3 Economcs of Management Zone Delneaton n Cotton Precson Agrculture Introducton Optmally confgurng management zones for better management of farm nputs s one of the most fundamental ssues n precson farmng and varable rate applcaton. Management zones are geographcal areas that can be treated as homogenous, so that nput applcaton and decson-makng can be treated separately for each zone, whch wll then lead to more precse management of the farm. The objectve of ths paper s two-fold: (1) to develop a unvarate management zone delneaton procedure based on a specfc ESDA (Exploratory Spatal Data Analyss) technque, and (2) to evaluate the potental economc mpact of ths management zone delneaton procedure for the case of cotton producton n the Texas Hgh Plans. Ths paper mplements spatal econometrc technques and shows ts mportance n economcally evaluatng a partcular management zone delneaton procedure. Emprcal Methodology Data and the ESDA Approach to Management Zone Delneaton The data used to establsh management zones s based on a 2002 agronomc cotton experment desgned to study ntrogen (N) use for cotton producton n the Southern Hgh Plans of Texas. The experment s a randomzed complete block desgn wth three replcates and each replcate was wthn a center pvot rrgaton span. There were three N treatments varable-rate N, blanket-rate N and zero N and there were three defned landscape postons south-facng sde slope, bottom slope, and north-facng sde slope. The data was orgnally collected as pont data (135 data ponts). But we spatally averaged the data nto 443 grds n order to obtan a balanced desgn and reduce measurement errors (Anseln, Bongovann, and Lowenberg-DeBoer, 3

4 2004). The orgnal expermental desgn and the spatal structure of the yeld data used n the analyss are presented n Fgures 1 and 2, respectvely. As mentoned n the ntroductory secton, we use an ESDA approach as the man procedure for establshng management zones. ESDA can be defned as a method that combnes dfferent technques to vsualze spatal dstrbutons, dentfy patterns of dfferent locatons, and dentfy patterns of assocaton between these locatons (Anseln, 1998). Ths method s based on the concept of spatal autocorrelaton, whch s the relatonshp between spatal unts, and makes use of the concept of dstance between locatons. Postve spatal autocorrelaton s the dea that grds wth smlar values of a specfc characterstc are near n space. Ths means that, n the presence of postve spatal autocorrelaton, certan grds located close to each other share smlar characterstcs (Messner & Anseln, 2002, p. 10). The step-by-step procedure for establshng the ESDA approach to management zone delneaton can be descrbed as follows: (1) Defne the neghborhood structure of each grd; (2) Establsh a weght matrx ; (3) Test for the presence of spatal autocorrelaton; (4) Graphcally vsualze the spatal correlaton structure (f step (3) ndcates there s spatal autocorrelaton); and (5) Establsh the management zones. The frst step s to defne the neghbors of each grd. Ths allows us to assess f there are any spatal relatonshps between these ponts, whch can then serve as the bass for management zones. Accordng to Bvand (1998), the neghborhood for each grd can be set by any number of alternatve methods. One approach s to set the neghbors by defnng locatons that share boundares wth each grd. Another possble approach s to draw bands at dfferent dstances of the grds. Snce we have a grd-based data structure, we used a rook structure (four neghbors to each cell, north, south, east and west) to defne the 4

5 neghborhood n our management zone delneaton procedure (Anseln, Bongovann, and Lowenberg-Deboer, 2004). 1 Once, we defned the neghborhood structure, the contguty relatons of each grd wthn a neghborhood must be formally characterzed usng a spatal weghts matrx (Bvand, 1998). A spatal weghts matrx (W) s an NxN (where N regards to the number of observatons), postve defnte matrx wth elements w j, where wj correspond to a par of observatons at locatons and j. By conventon the dagonal elements of the weght matrx are set to be zero, mplyng that each locaton s not a neghbor of tself. Non-zero elements ( w =1) means that locatons and j are neghbors. Typcally, the spatal weghts matrces are also row-standardzed to facltate comparson of spatal characterstcs across rows. The spatal weghts matrx s then used to test for the presence of spatal autocorrelaton n the data. The Moran s I statstc s used to test for the presence of spatal autocorrelaton (Anseln, 1988). Specfcally, we use the global Moran s I calculated as follows: j (1) I = N n n w 1 1 jzz = j= j n 2 S0 z = 1, n n = j j ; j where N s the number of observatons; S0 w = 1 = 1 w s the weght element from the spatal weghts matrx; z and z j are the devatons from the mean (.e. z = x µ, where x s the value of the varable of nterest n locaton and µ s the mean of that varable for all locatons). The null hypothess of the test s that there s no assocaton between the value observed at a locaton and the values observed at the neghborng stes. The alternatve s that the values of the neghborng stes are statstcally smlar. 5

6 Yeld data s often used n prevous studes to delneate management zones due to the dea that yeld captures all the varatons n clmate, sol and nput nteractons (as n Velanda et al., 2004 and Basnet et al., 2003). Therefore, yeld may be a good varable to delneate management zones f these zones are meant to be the bass for the overall management of the feld (.e. for mplementaton of dfferent management practces such N, P, K fertlzaton, water applcaton, etc). However, f the management zones are meant specfcally to mprove N fertlzaton (whch s the case here), then we beleve that usng sol ntrate as the bass for delneatng management zones may be more approprate. Hence, a pror, we chose ntrate n the sol (lbs/acre) as the man varable to serve as the bass for establshng management zones for more precse management of N fertlzer. Ths study also makes a contrbuton to the lterature n ths regard because ths study s the frst (as far as we know) to economcally evaluate a management zone delneaton procedure based on a spatal autocorrelaton statstc for sol ntrate levels. Usng sol ntrate as the varable of nterest, the computed global Moran s I statstc, based on the rook neghborhood structure and weghts matrx defned above, s and ths has a p-value of < Ths ndcates that null hypothess s rejected and that there s spatal autocorrelaton n the data. Based on ths result, a Moran scatterplot s created and management zones based on ths scatterplot s then determned (Fgure 3). There are three management zones establshed based on our procedure. Management zone 1 (MZ1) represents hgh ntrate areas (.e. grds wth hgh ntrate levels have neghbors also wth hgh ntrate levels). Management zone 2 (MZ2) represents low ntrate areas (.e. grds wth low ntrate levels have neghbors wth low ntrate levels). Lastly, management zone 3 (MZ3) represents the area wth a mx of hgh and low ntrate levels (.e. grds wth low ntrate levels have neghbors wth hgh ntrate levels, and vce-versa). 6

7 Economc Model and Estmaton Procedures The economc model to assess the mpact of the management zone delneaton procedure s based on a mathematcal programmng model for spatal proft (or net return) maxmzaton. Ths procedure s consstent wth economc (or proftablty) analyss of precson technologes conducted n the past (See, among others, Lowenberg- Deboer and Boehlje, 1996; Bongovann and Lowenberg- Deboer, 1998; Anseln, Bongovann, and Lowenberg-Deboer, 2001; Bullock, Lowenberg-DeBoer, and Swnton, 2002). In ths framework, we compute the expected net returns from: (1) a unform N rate applcaton based on an agronomc optmum (URA), (2) a unform N rate applcaton based on an economc optmum (URE), and (3) a varable rate N applcaton based on the economc optmum for each of the management zones establshed through our spatal procedure above (VRN). Hence, our economc analyss evaluates the economc mpact of our management zone delneaton procedure relatve to the unform N rate applcaton based on the agronomcally recommended rate and the economcally optmum rate calculated from the model. In addton, we also compare the expected net returns from a varable rate N applcaton that used landscape poston (VRL) as the bass for the management zones versus a varable rate N system that s based on the management zone delneaton procedure usng our spatal approach. For the unform N applcaton, we frst use the agronomcally recommended N rate and then calculate the correspondng net return based on the parameters of the proft maxmzaton model below. Ths s the net returns calculated by smply pluggng-n the agronomc N recommendaton of 52 lbs/acre (See Bronson et al., 2003 for the agronomc bass of ths recommendaton). An economcally optmal unform N rate applcaton s computed usng the optmzaton framework below. We then compare the net return fgures for the unform rate case 7

8 (for both the agronomc and economc optma) to the net returns, above Ntrogen cost, for the case of the varable rate N applcaton usng the delneated management zones (based on landscape poston and the spatal approach). Ths calculaton utlzes the optmzaton model, where the man component s a spatal cotton yeld response functon (for each management unt). As mentoned above, there are studes that have examned the approprate spatal econometrc technques for estmatng spatal yeld response functons usng precson agrculture data (See, among others, Anseln, Bongovann, and Lowenberg-Deboer, 2001; Lambert, Lowenberg-Deboer, and Bongovann, 2004). For the case of varable rate N applcaton, the typcal procedure s to frst start wth standard ordnary least squares (OLS) regresson of a response functon specfed wth varyng coeffcents based on the management zones. Consstent wth prevous studes, we use the quadratc specfcaton by management zone: (2) Yeld = α + β N + γ N + ε 2 j j j j where Yeld j s the cotton yeld, Nj s the N rate, ndexes management zone, and j s the locaton (n ths case, the grds) wthn each management zone. Ths specfcaton allows for the estmaton of management zone effects on the levels α, as well as nteracton effects between the management zones and the varables N and term coeffcents are estmated usng dummy varables. 2 N. These management zone and nteracton Ordnarly, perfect collnearty due to a set of dummy varables s resolved by droppng a sngle dummy. However, because ths analyss ams n part to nvestgate the dfference n cotton yeld response under unform rate applcaton (.e. usng an average N rate for the whole feld) versus varable rate applcaton, we wsh to estmate management zone devatons from the mean yeld, rather than devatons from the yeld of an omtted management zone. The economc 8

9 restrcton requred to do ths s that the dummy varables for all the zones sum to zero. Ths condton s mplemented by subtractng the management zone one dummy from the others, and then droppng management zone one from the data set. As a result, the constant coeffcent estmate can be nterpreted as the mean overall yeld wth zero appled N and the management zone dummes are the dfferences wth respect to ths overall mean. The coeffcent for the dropped varable s then calculated n a supplementary regresson, droppng another dummy varable. Consequently, the management zone dummes and nteracton terms allows us to calculate the zone-specfc response functons. Thus, the parameters of the yeld response functon that excludes the management zone dummes and the nteracton terms represents our estmate of the unform rate response functon whch reflects the average yeld response for the whole feld. Ths procedure allows us to estmate a sngle regresson equaton to generate the yeld response functon for both the unform rate case and a partcular varable rate case (ether based on landscape or the spatal approach). From the estmated regresson of the yeld response functon, the presence of spatal autocorrelaton n the resduals s then evaluated. If t s present, then approprate spatal econometrc technques need to be mplemented to account for the spatal autocorrelaton n the resduals. As s well-known, gnorng such autocorrelaton wll yeld OLS estmates that are neffcent and wll bas the standard errors, t-test statstcs and measures of ft, renderng statstcal nference unrelable (Anseln, 1988). Once the parameters of the cotton yeld response functons are estmated, these estmates are used to formulate an optmzaton model to maxmze proft for a representatve farm. In ths model, we maxmze net returns over fertlzer cost usng the yeld response parameters and estmated prces/costs. 9

10 The net return for the farm s defned as the weghted sum of the net returns n each management zone (for the case of varable rate applcaton), where the weghts are the proporton of the area n the management zone. For the case of fndng the economcally optmum unform N rate applcaton, ths weght s set to one and there s no management zone delneaton. More formally, the mathematcal programmng model can be expressed as: m 2 (3) Max E[ π ] = ( Aω E[ Pc ( α + β N + γ N ) rn N ] where: = 1 E = Expectaton operator π = Total net returns over N fertlzer and fxed cost ($) A = Total land area (22,000 acres) ω = Proporton of total land area allocated to management unt (.e. for the management zones based on the spatal approach, zone 1= 37%, zone 2= 48%, zone 3= 15%) = Management unt (ether the whole feld or the management zones) m = Total number of management unts (m = 1 for unform rate applcaton and m = 3 for varable rate based on the management zones delneated usng the spatal approach). P c = Prce of cotton ( $0.47 per lb, see Bronson et. al, 2005) N = Quantty of N appled n management unt (n lbs/acre) r = Prce of N fertlzer appled ($0.21/lb, see Bronson et. al, 2005 ) N Results and Dscusson 10

11 Response Functon Estmaton Results The results of both the OLS and spatal error estmaton procedures are presented n Table 1. 2 All the coeffcents follow our a pror expectatons and are all statstcally sgnfcant (at the 10% level). Therefore, there are dfferences n the yeld response for each management zone. Further, the magntudes of the coeffcents and standard errors are dfferent n the spatal error model as compared to the tradtonal OLS. Ths suggests that economc nferences from these two models would be dfferent and that ncorrect decsons could be made when only tradtonal OLS technques, rather than spatal econometrc methods, are used n the yeld response estmaton. Addtonally, when the spatal error structure s modeled, the ft of the model mproves as shown by the ncrease of the log lkelhood and a decrease n Akake Informaton Crtera (AIC). The mprovement of the model was also to be expected because of the hghly sgnfcant spatal error (lambda) coeffcent. Mathematcal Programmng Results: Yeld, Ntrogen, and Proftablty Based on the estmated response functon(s) and the optmzaton model descrbed above, we estmated the yeld, the N applcaton levels, and the net returns over fertlzer cost for each of the dfferent applcaton technques consdered: URA, URE, VRN, and VRL. Each of these applcaton scenaros was examned by usng a yeld response functon estmated both by OLS and by usng the spatal error model (SEM) estmated through a maxmum lkelhood technque (ML). Ths allows us to see the potental magntude of nference or recommendaton errors that could be commtted when spatal autocorrelaton s not properly accounted for n the yeld response estmaton. 11

12 A comparson of the returns for the dfferent N rate applcaton technques s presented n Table 2. The OLS technque tends to overestmate the benefts from varable rate applcaton relatve to the unform rate base on the agronomc recommendatons, and OLS tends to underestmate the benefts from varable rate applcaton relatve to the unform rate based on the economc optmzaton model. Note, that wth the use of the spatal error model the varable rate applcaton of N based on the management zones delneated tend to have a hgher net return relatve to the unform rate base on the agronomc recommendatons applcaton, albet smaller than f OLS was used. The spatal error model for the varable rate applcaton of N based on the management zones delneated tend to have a hgher net return relatve to the unform rate base on the economc optmzaton model, albet hgher than f OLS was used. Another notable comparson s the hgher net return of VRN relatve to VRL, once we correct the model for spatal autocorrelaton. Ths shows that our spatal approach to management zone delneaton has added value (n terms of varable rate applcaton of N) relatve to a management zone delneaton technque based solely on landscape poston. The average N levels for the dfferent applcaton technques are presented n Table 3. Our results show that, on average, the varable rate system usng the delneated management zones based on the spatal approach tend to have hgher yelds than the unform rate applcaton technques (Table 3). The VRN scenaro also generated a hgher average yeld than the VRL scenaro. Wth regards to N applcaton levels, the varable rate scenaro (VRN) tends to utlze more N (on average) than the URE technque (Table 4). But the varable rate scenaro tends to have lower N levels relatve to the URA scenaro. Note, however, that the varable rate scenaros (VRN) tend to more effcently utlze N because t apples less N n zones wth hgh sol ntrate levels and more N n zones wth low sol ntrate levels. 3 Therefore, even f N applcaton s 12

13 hgher (on average) for the varable rate technques, the more effcent use of the N fertlzer may possbly reduce ntrate run-off n the sol and, consequently, reduce non-pont source polluton. The results that regards to the net returns are based on an approach that does not take nto account a fxed cost, gven that ths s a short run analyss. However, there s a fxed cost that regards to the ntrate sol test, whch needs to be takng nto account for the mplementaton of the varable rate technology. For the experment consder n ths analyss, the estmated cost for the ntrate sol analyss s $9.60/acre (Bronson et. al, 2005). If we consder ths cost, then VRN s not more proftable than URA and URE anymore. The breakeven analyss, where the breakeven fee s smply calculate as the dfference between net returns under VRN and net return under URA, shows that for VRN to be more proftable than URA and URE, the cost of the sol analyss needs to be less than $2.21/acre. Senstvty Analyss The two mportant components that underle the results presented above are the choce of neghborhood structure and the yeld response estmaton technque. The rook neghborhood structure s used as the bass for the spatal weghts matrx n the delneaton of the management zones and n modelng the error structure of the SEM yeld response functon. Standard OLS technques and a ML approach to estmatng the SEM yeld response functon are the estmaton technques used to produce the economc results above. In order to check for the senstvty of the economc results, we also examne the economc effect of usng an alternatve neghborhood structure and/or alternatve estmaton technques (Table 4). 13

14 In general, we fnd that regardless of neghborhood structure or estmaton technque VRN stll tend to have hgher net returns relatve to the unform rate approaches (URA and URE). Conclusons Based on an ESDA approach that utlzes a spatal autocorrelaton statstc, we are able to develop a procedure for delneatng management zones usng precson agrculture data from cotton producton n the Texas hgh plans. The results of the optmzaton model suggest that applyng varable N rates based on the management zones delneated (usng the spatal approach developed), would result n hgher yelds and hgher net returns over fertlzer cost relatve to the tradtonal unform rate applcaton and relatve to the varable rate applcaton based on landscape poston. Furthermore, the hgher net returns and yelds for the VRN applcaton technque were acheved by more effcently utlzng N for the whole feld. Thus, more precse management of N based on the management zones delneated may have potental mplcatons for fertlzer runoff and non-pont source polluton n the sol. Furthermore, the results of our analyss also renforce the observaton n past studes that ncorrectly estmatng yeld response functons wthout correctng for spatal dependence may lead to msleadng nferences about the economc mpact of varable rate technologes. 14

15 Footnotes: 1 There are other contguty-based neghborhood structures lke the queen (eght neghbors to each cell) or the bshop (four neghbors wth common vertex) structure. We also used these structures for defnng management zones and found very smlar results to the rook structure. The management zone delneaton results for the alternatve neghborhood structures are not reported here, but are avalable from the authors upon request. 2 Note that the yeld response functon estmated n Table 1 s based on the management zones delneated usng our spatal approach. Although not reported here, we also estmate the yeld response functon when the management zones are based on landscape poston. Ths yeld response functon s used to evaluate the net returns from a varable rate N applcaton based on management zones delneated by landscape poston. Ths wll allow us to see whether a varable rate applcaton based on the management zone delneaton procedure we develop generates hgher net returns relatve to a delneaton procedure that s smply based on landscape poston. 3 In the nterest of space, the exact fgures for the appled N n each management zone are not explctly reported here, but are avalable from the authors upon request. 15

16 References: Anseln, L. Spatal Econometrcs: Methods and Models. Dordrecht, Netherlands: Kluwer Academc, Anseln, L. The Moran Scatterplot as an ESDA Tool to Assess Local Instablty n Spatal Assocaton. In: Fscher M., H. Scholten, and D. Unwn (eds), Spatal Analyss Perspectves on GIS. Taylor & Francs, Brstol, PA. pp , 1998 Anseln L., Bongovann R. and Lowenberg-DeBoer J. A spatal econometrc approach to the economcs of ste-specfc ntrogen management n corn producton. Amercan Journal of Agrcultural Economcs. 86 (August 2004): Basnet, B., Kelly, R., Jensen, T., Strong, W., Apan, A. and Butler, D. Delneaton of Management Zones usng Multple Crop Yeld Data. In Internatonal Sol Tllage Research Organzaton Conference, 16th Trennal Conference, pp July Brsbane, Australa: The Unversty of Queensland. Bvand, R. A revew of spatal statstcal technques for locaton studes. Onlne. Avalable at Retreved October Bongovann, R. and J. Lowenberg-DeBoer. Economcs of Varable Rate Lme n Indana, pp Proceedngs of the 4th Internatonal Conference on Precson Agrculture. St. Paul, MN, July 19-22, Bronson, K.F., J. D. Booker, J.P. Bordovsky, J. W. Keelng, T.A. Wheeler, R.K. Boman, M.N. Parajulee, E. Segarra, M. Velanda-Parra, and R.L. Nchols. Ste-Specfc Irrgaton and Ntrogen Management for Cotton Producton n the Southern Hgh Plans, Workng paper. Aprl,

17 Bronson, K., Keelng, W., Booker, J.D., Chua, T., Wheeler, T., Boman, R., and Lascano, R. Influence of Landscape Poston, Sol Seres, and Phosphorus Fertlzer on Cotton Lnt Yeld. Agronomy Journal 95(July-August 2003): Bullock, D. S., Lowenberg-Deboer, J. Swnton, S. Addng value to spatally managed nputs by understandng ste-specfc yeld response. Agrcultural Economcs 27( November 2002): Lambert, D.M., Lowenberg-DeBoer, J., and Bongovann, R.M. Spatal Regresson Models for Yeld Montor Data: A Case Study for Argentna. Precson Agrculture. 5(December 2004): Lowenberg-DeBoer, J. and M. Boehlje. Revoluton, Evaluaton or Deadend:Economc Perspectves on Precson Agrculture. In: Robert, P.; Rust, H. and R. Larson, eds. Proceedngs of the 3rd Internatonal Conference on Precson Agrculture. Mnneapols, MN, June 23-26, Messner, S. and Anseln, L. Spatal analyses of homcde wth aeral data. Workng Paper, Unversty of Illnos at Urbana-Champagn, Velanda, M., R.M. Rejesus, E.Segarra, K. Bronson, and R. Kulkarn. An Economc Analyss of a Spatal Statstcs Approach to Management Zone Delneaton n Precson Agrculture: The Case of Texas Cotton. In R.H. Rust and W.E. Larson (eds.), Proceedngs of the Seventh Internatonal Conference on Precson Agrculture, Bloomngton, MN (July 25-28, 2004). 17

18 Table 1. Parameter estmates of the cotton yeld response functon for the management zones delneated usng the spatal approach Varables OLS (Ordnary Least Squares) COEFF P-value (lbs ac -1 ) SEM (Spatal Error Model) COEFF P-value (lbs ac -1 ) Constant N N MZ MZ MZ N x MZ N x MZ N x MZ N 2 x MZ N 2 x MZ N 2 x MZ Lambda NA NA Measures of ft OLS SEM Log Lkelhood AIC Dagnostc tests d.f. Value Value P-value Lagrange multpler(error) 1 NA Robust LM(error) 1 NA Lagrange multpler (lag) 1 NA Robust LM (lag) 1 NA Table 2. Net returns under dfferent applcaton methods and estmaton procedures OLS SEM Dfference (OLS-SEM) --- Net Returns ($ acre -1 ) --- Unform rate, agronomc optmum (URA) Unform rate, economc optmum (URE)

19 Varable rate, spatal approach (VRN) Varable rate, landscape poston (VRL) Dfferences across applcaton technques URE vs. URA (URE URA) VRN vs. URA (VRN URA) VRN vs. URE (VRN URE) VRL vs. URA (VRL URA) VRL vs. URE (VRL URE) VRN vs. VRL (VRN VRL) Table 3. Ntrogen levels under dfferent applcaton methods and estmaton procedures OLS SEM Dfference (OLS-SEM) --- N level (lbs acre -1 ) --- Unform rate, agronomc optmum (URA) Unform rate, economc optmum (URE) Varable rate, spatal approach (VRN) Varable rate, landscape poston (VRL) Dfferences across applcaton technques URE vs. URA (URE URA) VRN vs. URA (VRN URA) VRN vs. URE (VRN URE) VRL vs. URA (VRL URA) VRL vs. URE (VRL URE) VRN vs. VRL (VRN VRL)

20 Table 4. Senstvty of the dfferences n net returns under alternatve neghborhood structure and estmaton method assumptons Neghborhood structure 1 Dfference n net returns ($ acre -1 ) across applcaton technques 3 Estmaton Method 2 URE-URA VRN-URA VRN-URE VRL-URA VRL-URE VRN-VRL Rook Structure OLS SEM (ML) SEM (GM-Two step) SEM (GM-Iterated) SEM (GM-GHET) Queen Structure OLS SEM (ML) SEM (GM-Two step) SEM (GM-Iterated) SEM (GM-GHET) Note: (1) The neghborhood structures consdered are rook and queen. Note that these structures are assumed both n the delneaton of the management zones for the spatal approach and n specfyng the error structure n the SEM model. (2) The alternatve estmaton methods consdered (asde from the tradtonal OLS and SEM (ML)) are: SEM usng two stage general method of moments (GM-Two step), SEM usng terated general method of moments (GM-Iterated), and SEM usng general method of moments that corrects for groupwse heteroskedastcty (GM-GHET). (3) Applcaton technques are: unform rate based on agronomc optmum (URA), unform rate based on economc optmum (URE), varable rate based on the spatal approach (VRN), and varable rate based on landscape poston (VRL). 20

21 Fgure 1. Dgtzed Grds for Cotton Yeld (lbs/acre) 21

22 Management zone 1 Management zone 2 Management zone 3 Fgure 2. Delneated Management Zones from the ESDA Procedure 22