Du, Xiaodong

Hennessy, David A.

Yu, Cindy L.

English

Research Papers in Economics

Are Crop Yield Distributions Negatively Skewed? 
A Bayesian Examination 
 
Xiaodong Du 
Postdoc Research Associate 
Center for Agricultural and Rural Development 
Iowa State University 
xdu@iastate.edu 
 
David A. Hennessy 
Professor 
Department of Economics, and 
Center for Agricultural and Rural Development 
Iowa State University 
hennessy@iastate.edu 
 
Cindy Yu 
Assistant Professor 
Department of Statistics 
Iowa State University 
Cindyyu@iastate.edu 
 
 
Poster prepared for presentation at the Agricultural & Applied Economics Association 2010 
AAEA, CAES, & WAEA Joint Annual Meeting, Denver, CO, July 25-27, 2010. 
 
 
 
Copyright 2010 by Fatma Xiaodong Du, David A. Hennessy, and Cindy Yu. All rights 
reserved. Readers may make verbatim copies of this document for non-commercial purposes 
by any means, provided that this copyright notice appears on all such copies. 
  RISKS ORIGINATED FROM the random nature of 
yield have significant impacts on farmers’ production 
and marketing decisions. A better understanding of 
crop yield distributions is important both for crop pro-
ducers and for the crop insurance industry, where 
contract payout patterns are sensitive to distribution 
tails. There is a long-established literature on how 
inputs affect mean and variance of crop yield distri-
butions while controversy surrounds skewness attri-
butes of typical yield distributions. In this study, we 
extend the stochastic production model in Just and 
Pope (1978) to accommodate skewness. The model is 
applied to several experimental yield datasets for corn 
and cotton at the small plot-level. We conduct infer-
ence within a Bayesian Monte Carlo Markov Chain 
(MCMC) framework (Gelman et al., 2004).
Xiaodong Du
David A. Hennessy
Cindy Yu
Department of Economics Are Crop Yield Distributions Negatively Skewed? 
    A Bayesian Examination
Center for Agricultural and Rural Development
Introduction
LET EXPERIMENTAL PLOT yield be given by
    y=f(N)+g(N)ε
h(N)
    (1)
where N is an input, e.g., nitrogen. Here ε is random 
while f, g, and h are functions defined to be consistent 
with the Just-Pope method. Let
f(N) = α0 Nα1;       g(N) = β0 Nβ1;       h(N) = γ0 N  γ1 (2)
The moments for crop yield are:
 (3)
The stochastic production function in eqn. (1) is mean, 
variance, and skewness flexible.
GIVEN THAT FIELD experiments are conducted on 
discrete and limited nitrogen application levels, we esti-
mate the yield distribution for each nitrogen level indi-
vidually. Following (1), for nitrogen level i (i =  1,2..., I ), 
yields are modeled as
y   i = a0
i + X′ β + b  iε
1/c 
i, ε ~ D · J(α)  
                                                     J(α) = Beta (α, α). (4)
where y  i = [y0
i   ,     y 2
i ,     y k
i ]′ denotes k yield observation and a0
i  
denotes the constant term which varies with nitrogen 
input, f     (N   i  ). The matrix X = [x1, x2, ... xL]′ denotes the L 
controlled variables including, e.g., rotation effect, and 
location/time dummies with the corresponding coeffi-
cient vector β = [β1, β2, ... βL]′. Scale of crop yield is rep-
resented by b  i = g(N   i   ).
We specify the yield random variations as ε1/c  i, where ε is 
assumed to follow a symmetric beta distribution Beta (α, 
α) on the range (0, D), and skewness is represented by 
the parameter c i. The beta distribution has long been 
popular in yield distribution models (Nelson and Preckel, 
1989). Skewness is introduced into an originally symmet-
ric distribution on ε through c i. In doing so, it is possible 
to retain some well known properties of symmetric dis-
tributions and compare different classes of skewed distri-
butions in a common framework. The corresponding 
yield skewness Skew (y  i), is
   (5)
In this study, inference is conducted within a Bayesian 
MCMC method, which has the advantage of easily in-
corporating inequality constraints on parameters into 
the estimation procedure. The inequality constraint is to 
ensure that parameter estimates are consistent with rela-
tionships implied by the underlying distribution asump-
tion, which is
                  0 < [  
  1 — – –D
   (y  i - a0
i + X′ β)  /b  i  ]
  c  i 
< 1.       
Furthermore, the Bayesian procedure is particularly 
suitable as the model specified in (4) is highly nonlinear 
in the parameters. We employ the random-walk Metrop-
olis-Hasting algorithms for updating posterior draws. 
Normal priors are adopted for all parameters. After con-
vergence, draws by the Gibbs sampler are used to com-
pute the mean and standard error of each parameter.
Data
THREE DATASETS ARE employed: (i) corn yield 
dataset A: experiments in Floyd County, Iowa from 
1979 to 2003 with four nitrogen levels (lb./ac.), 0, 80, 
160, and 240. Control variables include time trend and 
corn-after-corn (CC) rotation; (2) corn yield dataset B: 
experiments on four Iowa farms in 1986-1991 with ten 
nitrogen levels (lb./ac.): 0, 25, 50, 75, 100, 125, 150, 
200, 250, and 300. Variables include dummies for four 
years and CC rotation; (3) cotton yield dataset C: ex-
periments in three Texas counties, 1998-2002, with four 
nitrogen levels (lb./ac.), 0, 50, 100, and 150. Control 
variables are dummies for four years and two locations.
Results 
THE GIBBS SAMPLER is coded in Matlab and ran for 
10,000 cycles with the first 5000 as burn-in. The mean 
derived and standard error of the skewness in (5) for 
each nitrogen level are presented in the table below. The 
results indicate that the skewness parameter ci generally 
increases with the levels of nitrogen inputs, while the 
corresponding derived distribution skewness decreases. 
In other words, the impact of the nitrogen input on the 
crop yield distribution is in general skewness-decreasing.
The Figure plots the posterior predictive   yield distribu-
tion at mean parameter values of Dataset A.
Stochastic Technology
Empirical Methodology
Posterior Predictive Yield Distributions for nitrogen 
levels (N) of 0, 80, 160, 240 lb./ac.
Gelman, A., J.B. Carlin, H.S. Stern, & D.B. Rubin. 2004. Bayesian Data 
Analysis. 2nd edition. Chapman & Hall/CRC.
Just, R. & R.D. Pope. 1978. Stochastic Specification of Production Func-
tion and Economic Implication. Journal of Econometrics 7:67-86.
Nelson, C.H. and P.V. Preckel. 1989. The Conditional Beta Distribution 
As a Stochastic Production Function. American Journal of Agricul-
tural Economics 71: 370-378.
References
J
u
l
y
 
2
0
1
0
Estimated skewness and standard error 
(in parentheses)
0 50 100 150 200 250
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Yield(bu/ac)
 
 
N=0
N=80
N=160
N=240

Are Crop Yield Distributions Negatively Skewed? A Bayesian Examination

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