A Luenberger Soil Quality Indicator
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1 A Luenberger Soil Quality Indicator Atakelty Hailu Univerity of Wetern Autralia and Robert G. Chamber Univerity of Maryland and Univerity of Wetern Autralia EWEPA X preentation 29 June 2007
2 Outline Soil quality index contruction Role for economit Definition of a Luenberger oil quality indicator Provide an empirical example uing crop rotation trial data from Wetern Autralia Compare reult from three frontier: Bayeian etimated tochatic frontier Determinitic frontier Parametric (etimated mathematical programming DEA Tet tranlation homotheticity property (Bayeian Obtain aggregation weight implied by Luenberger indicator (pot etimation exercie
3 Soil quality reearch: role for economit Depite coniderable reearch effort over the lat decade and a half current procedure in the oil cience literature rely on expert opinion and ad hoc coring and weighting rule: Example: Recale quality variable to [01] Ue equal weight to aggregate core However there i a wealth of tool in economic that would help reolve the oil quality index contruction problem. Economit are good at: Etimating complex technologie/production relationhip Defining indexe
4 Soil quality index ue Potential area of application for oil quality indexe include: imple way of decribing oil cot management through preciion agriculture monitoring of agricultural and mine ite rehabilitation; and conervation contract deign
5 Previou ditance function baed approach Indexe baed on DEA efficiency meaure (Jaenicke and Lengnick 1999: Soil quality indicator defined a a ratio of ditance function value with and without oil quality indicator The oil quality index (SQI i then regreed on oil quality attribute to derive weight But that approach had problem
6 Directional Ditance Function The directional ditance function meaure the maximum amount by which the input (output vector can be contracted (inflated in the direction of a vector (g. For a directional ditance function on oil quality indicator ( we have: D( y x ; g = up { θ : ( y x θg T θ R+ } θ
7 Propertie of the directional ditance function The function i: homogeneou of degree -1 in the directional vector (tranlation property concave in deflated/contracted vector non-decreaing in input (and bad output but non-increaing in output non-negative for all technically feaible (xy combination
8 Soil quality indicator The quality of a vector i relative to a vector 0 can be defined a follow: Technology-0 indicator: ; ( ; ( i g x y D g x y D L = Technology-1 indicator: Soil quality indicator a average of the two indicator: L = (L0 + L1/2 ; ( ; ( 0 g x y D g x y D L = ; ( ; ( i g x y D g x y D L =
9 Luenberger indicator g i 0 I(y i x i I(y 0 x 0
10 Soil quality indicator: homotheticity The idea of contructing a quality indicator without reference to (yx i baed on a hypothei of a impler tructure for the technology Tetable hypothei within our framework If the directional ditance function atifie the following homotheticity property then the oil quality indicator i independent of the point of evaluation: ( ( ( ( ( ; ( 0 0 A A L x y D A g x y D i i i i = =
11 Etimation
12 Determinitic frontier 1 DEA/VRS 2 Parametric (MP Aigner and Chu (1968 method Quadratic functional form Parameter etimation by linear programming or goal programming Minimize um of deviation of etimated ditance function value ubject to the following condition: Feaibility or incluion (poitivity of ditance function value Monotonicity (derivative ign Tranlation property etc.
13 Stochatic Frontier Bayeian etimated Ue tranlation property and manipulate function to define a likelihood/poterior function (thi preentation (O Donnell and Coelli 2005 for output ditance function Literature: Ue latent variable (unoberved weight to define aggregate output and reolve endogeneity problem: GMM method Atkinon; Fare et al (2005 Fernandez et al. 2000/2005 pecial tructure (CES O Donnell (2007 quadratic form Clarification needed on conitency between montonicity requirement and the requirement of the latent variable definition
14 Bayeian poterior imulation Set prior u i ~ f G (1 λ ~ f (1 g f G 1 λ p( β = I ( β B p( σ ~ f n 2 c 2 2 o o G
15 Bayeian poterior imulation Likelihood traight forward Poterior: Prior. Likelihood MCMC imulation: Gibb with Metropoli-Hating Parameter block: h lambda efficiency term and the directional ditance function parameter (KOS 1997
16 Bayeian poterior imulation In um there i till work to be done on the econometric front for multiple output technologie But thank God that the production function till exit! Thi i a ingle output technology: Therefore we could etimate a tochatic production function and then compute the directional ditance function value from it Frontier etimated but calculation not done
17 Data Long term crop rotation trial from three experiment tation in Wetern Autralia: Salmon Gum Newdegate and Eperance aroona wheat variety data 224 obervation Output: grain yield Input: 3 oil quality variable: organic carbon PH and Nitrogen (NH4 3 other input variable: rainfall N P
18 Reult
19 Reult Bayeian Baed on MCMC tep with tep dicarded a burn-in The mean of the Luenberger indicator value range from to 1.02 with a mean= but a median value cloe to zero. MP: A imilar range from the MP etimated frontier: to 1.11 with a mean = DEA: Smaller range to 0.54 and with a mean of High correlation among the three erie
20 Luenberger indicator: Bayeian Denity plot of SQIL 0.6 mean 2.5% quantile 97.5% quantile Denity SQIL
21 SQIL/Baye 0.5 SQIL/Baye SQIL/MP (a SQIL/DEA (b
22 Tranlation homotheticity: I it rejected by the data?
23 Tet: tranlation homotheticity Poterior odd ratio: Baye factor: Prior odd ratio:? DIC: General model: Dbar = pot.mean of -2logL: Dbar = pot.mean of -2logL: Dhat = -2LogL at pot.mean: pd: DIC: Homothetic model: Dbar Dhat pd DIC
24 Bayeian etimated tochatic frontier: Homothetic v. Non-homothetic (.97 LSQI: determinitic and tochatic frontier LSQI: Baye. tocha atic dit LSQI: Baye. tochatic dit. ret
25 Table 1: Correlation among oil quality indicator obtained from different repreentation of the technology Baye BayeHT MP DEA Baye BayeHT MP DEA
26 Derivation of aggregation weight
27 Table 1: Soil quality index and it relationhip to oil quality attribute: OLS regreion reult Etimated Coefficient (β q Standardized Coefficient (t value in parenthee (β q ( σ SQIL σ q Station R 2 Intercept ORGC PH NH4 ORGC PH NH4 Baye ( ( ( BayeHT ( ( ( MP ( (14.65 ( DEA ( ( (
28 Summary Method: conitency of reult acro method Acidity (PH mot important attribute Tranlation homotheticity not rejected uggeting we might be able to build good indicator baed on oil attribute only But thi i jut one data et: Further exploration i needed Next tep: Tet method with different data et (different crop and varietie and ae how variable/table reult are
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