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A SIMPLE APPROACH TO STOCHASTIC TECHNOLOGY
ESTIMATION USING FARMLEVEL DATA
Bhavani Shankar
Department of Agricultural & Food Economics
University of Reading
PO Box 237
Reading RG6 6AR
UK
Carl H. Nelson*
Department of Agricultural and Consumer Economics
University of Illinois at UrbanaChampaign
1301 W. Gregory Dr.
Urbana, IL 61801
2173331822 (voice)
2173335538(fax)
chnelson@uiuc.edu
First Draft
April 16, 2003
∗ corresponding author
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A SIMPLE APPROACH TO STOCHASTIC TECHNOLOGY
ESTIMATION USING FARMLEVEL DATA
Introduction
Methods for estimating production risk that is endogenous to inputs (‘stochastic
technology’) have received significant attention in agricultural economics. While initial
attention was focused on estimation from experimental data varying a single input (Day,
Just and Pope(1979)), more recent attention has centered on farmlevel stochastic
technology estimation (Antle(1983a), Griffiths and Anderson, Nelson and Preckel, Love
and Buccola, Khumbhakar). The development of estimation methods for use with farm
level data has enabled the examination of the complex situation where farmers use
several inputs, each of which may have a distinct effect upon the mean, variance and
higher moments of output. It has also mirrored the increasing availability of farmlevel
data, both crosssectional and more recently, panel.
The various methods developed for farmlevel stochastic technology estimation have
embodied alternate assumptions, and accordingly, alternate levels of estimation
complexity. Antle (1983a) pioneered research in this area. Having noted that the Just
Pope production function restricts the effects of inputs on the third and higher moments
of output, he developed a nonparametric momentbased approach that regressed each
(estimated) moment of output on the input vector in a multistage approach using feasible
GLS. In Antle (1987), estimates of the distribution of risk attitudes were also obtained,
conditional upon the estimated stochastic technology. Love and Buccola pointed out that
Antle’s approach of separate technology estimation raises the specter of estimation
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inconsistency, since production error terms and input variables are likely to be
contemporaneously correlated. Assuming Constant Absolute Risk Aversion (CARA) and
a normally distributed production error term, they demonstrated a method for the joint
estimation of risk preference and stochastic technology parameters. Technology
parameters derived from this method are consistent conditional upon the maintained
assumptions. This literature has also branched off in another direction – providing
procedures for stochastic technology estimation with error components and panel data.
Griffiths and Anderson presented a method that extends the three step Just and Pope
approach into a sixstep sequence that incorporates an errorcomponents specification.
Khumbhakar extended this to enable the estimation of individual technical efficiency
parameters under stochastic technology.
The continued development of such alternative methods is important, since it enables the
applied researcher to choose an approach based upon (i) the assumptions that he or she is
comfortable making, (ii) the estimation complexity that he or she is willing to take on,
and (iii) the nature of the available data. The primary objective of this paper is to present
a relatively simple approach to stochastic technology estimation from farm level data,
when the applied researcher is willing to assume an error structure without heterogeneity
effects. The approach is employable whenever a panel data set is available, even when
the time dimension of the panel is very short and only basic production data on input and
output quantities are available. In other words, the approach is for typically available
farm accounting data sets. This chief advantage of this approach, which relies on
Generalized Method of Moments (GMM) estimation using past inputs as instruments, is
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its relative computational simplicity. While employing a twomoment approach akin to
the JustPope function, the moments are jointly estimated, thereby avoiding a multistage
approach. The twomoment approach significantly eases convergence and computational
difficulty in the nonlinear setting, while also providing more moment conditions for
model identification than does a singlemoment approach. Other important advantages of
this approach are: (i) the problem of contemporaneous inputerror correlation and
resulting inconsistency of separate technology estimates is ameliorated by
instrumentation, (ii) there is no need to make an explicit assumption regarding the
parametric distribution of the error term, and thereby of output itself, and (iii) relatively
simple tests can be devised that test whether individual inputs such as pesticides and
fertilizers are applied in a predetermined/prophylactic fashion (i.e., as ‘insurance inputs’,
Antle(1987) ) or on the basis of sequential decision making within a season.
We proceed by laying out the theoretical model and assumptions underlying the approach
in the next section. Several assumptions and model features described here are common
to Antle(1987) and Love and Buccola as well. These commonalties are described
explicitly so that the debate over the legitimacy of separating technology estimation from
risk preference estimation, and the contrast between the empirical approaches can be
better understood, given the common context. The subsequent sections describe our
empirical strategy and apply it to a dataset of Illinois grain producers. Although
presentation of an alternate method is at the heart of the paper, the application is of
interest in itself as well, since it provides estimates of the risk effects of fertilizers and
pesticides upon Midwestern corn and soybeans enterprises separately.
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Assumptions and Theory
Farm operators are modeled as static, riskaverse expected utility maximizers facing the
same price vectors. We present the model in terms of two outputs, denoted C and S, in
order to simplify the notation. Extrapolation to more outputs is straightforward.
Notation :
U(., q) : Utility function. q is a vector of utility function parameters.
W0 : Endowed initial wealth.
qc,qs : Quantities of outputs C and S, respectively.
pc, ps : Prices of C and S, respectively.
xc, xs : Vectors of variable inputs applied to the production of C and S, respectively.
wx : Vector of prices of variable inputs x.
zc, zs : Vectors of fixed inputs (such as land) used in the production of C and S,
respectively.
Sc, Ss : Supports over which random variables qc and qs are defined, respectively.
Stochastic technology in a multipleoutput production setting can be represented by the
joint conditional density function, f(qc, qs / xc, xs, zc, zs; a ) (Antle (1988) ). a here
denotes the vector of technology parameters. The optimization problem for a producer is
to choose input vectors (xs, xs, zc, zs ) to solve :
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Subject to zc + zs # z
It is important to note that even though z represents fixed factors (land, in our
application), the problem is specified such that an optimizing decision also has to be
taken regarding the division of land between C and S production. However, such a
specification , where even in the very short run agents allocate land exclusively on the
basis of optimization under risk, is a problematic one in many application settings. The
division of land between outputs may not be very flexible in the short run owing to
rotational and government program considerations. Crop rotations cycles are conducted
for soil health and fertility reasons, and therefore land allocation in individual years is not
completely responsive to economic considerations. Government program clauses such as
the former base acres requirement for deficiency payments may have the effect of locking
land into certain crops in the short run. Hence, we make another assumption: that fixed
land input allocation is taken as given in the short run. Thus zc = zc* and zs = zs*, where
zc* and zs* are constants. With the fixed land input allocation assumption, problem (1)
can be rewritten as follows:
Choose (xc, xs) to solve :
dq
dq
) ;
z ,z ,x ,x
/
q ,q f(
] ; )
xw
 qp ( + )
xw
 qp ( +
0
W
U[ Max (2)
sc
s* c*sc
sc
s/
x
ss
c/
x
cc
S
q
S
q
c
c
s
s
α
θ
∫
∈
∫
∈
dq
dq
) ;
z ,z ,x ,x
/
q ,q f(
] ; )
xw
 qp ( + )
xw
 qp ( +
0
W
U[ Max (1)
sc
scsc
sc
s/
x
ss
c/
x
cc
S
q
S
q
c
c
s
s
α
θ
∫
∈
∫
∈
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Another significant aspect of the specifications in (1) and (2) is that all inputs have been
denoted as ‘allocable’ to the production of either C or S, for convenience in exposition.
However, the allocable representation of all inputs has been only notional in that inputs
applicable to any one output have also been allowed to affect the production of the other
output. Thus, stochastic technology has been represented by a joint density function of
the two outputs conditional on all inputs.
At this stage, another restriction placed upon the theoretical structure can help set up
optimality conditions more tractable for estimation purposes. Stochastic nonjointness
(Antle (1987,1988) ) is assumed, which implies that the marginal distribution of any
output is only affected by inputs previously denoted as ‘allocable’ to it. This assumption
is most easily explained by rewriting the technological specification in production
function form. Under stochastic nonjointness, we can write:
) , F(~) ,(
), ,
ε
,
ε
z ,x (
ε
q =
ε
q
) , ,
z , x (
q = q
(3)
scsc
sss*s
ss
cc*cc
cc
αε
αε
The production function representation (3) implies that the marginal output distribution
of C is determined by inputs allocated to C, and random shocks to C, given by εc. The
marginal output distribution of S is defined in a parallel fashion. The random shocks are
jointly distributed with a distribution function F. Thus, neither the random shocks to C
and S, nor the outputs of C and S need necessarily be statistically independent. The
assumption of stochastic nonjointness ensures that the covariances and other product
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moments of the two output distributions are not affected by inputs. These covariances
need not be zero, however. This assumption is first explicitly stated and explained in
Antle (1987), where the assumption becomes necessary to estimate technology and
preference parameters in a multioutput setting using data from a single crop. It is also
implicitly assumed in other studies where estimation is based on a single output. For
instance, Love and Buccola estimated risk preference and corn technology parameters for
a set of cornsoybean farms in Iowa. The omission of soybeans production from the
analysis implies the assumption of stochastic nonjointness.
While this is doubtless a strong assumption in most application settings, it is often
essential in order to make a potentially very complex estimation scenario more
manageable. One alternative is to seek out production scenarios where only a single
output is produced in order to illustrate the method. For example, Saha, Shumway and
Talpaz demonstrated their estimation procedure with data from Kansas wheat farms.
Such scenarios are, however, rarely found in agricultural sectors the world over. Another
alternative that is often utilized is to aggregate multiple outputs into a single output,
either total revenue or more elaborately constructed output indices. This may be the only
feasible approach in contexts where there are multiple outputs but input data is available
only for the farm as a whole, and not on an outputspecific basis. However, it is best
avoided where possible, since it entails assumptions that may be quite undesirable. In
technology estimation in a cornsoybeans farming context, for instance, the use of total
revenue as the dependent variable and whole farm inputs as the independent variables
implies a one to one correspondence between total revenues and the total inputs applied
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on the multioutput farm, which is clearly quite unreasonable. We now proceed by
retaining the convenient production function representation of stochastic technology.
These additional assumptions enable the expected utility maximization problem (2) to be
rewritten as follows. Producers choose input vectors (xc, xs) to solve
εεεε
θ
αε
αε
∫
∈
z ,
∫
∈
x (
εε
s
sc
d
s
d )
cs/
x
sss*s
s
c
/
x
cc
*cc
cc
0
p ( +
SS
, f( ] ; )
x
w
 ) , ,
q p (
+ )
xw
 ) , ,
z ,x (
q
W
U[ Max (4)
ccss
In this formulation, ac and as are technology parameters attached exclusively to the
production of C and S, respectively. The structural model composed of firstorder
conditions for variable inputs applied to C and S is then given by:
(5)
0 = ] )
j
w

c
j
x
q
(P ) +
j
π
+
0j
W
(U E[
c
s
j
c
∂
∂
π
′
for variable inputs attached to C, and
(6)
0 = ] )
w

s
j
x
q
(P ) +
j
π
+
0j
W
(U E[
j
s
s
j
c
∂
∂
π
′
for variable inputs attached to S, and of
course, the production functions in (3).
Estimation Issues and Previous Strategies
Although the stochastic technology parameters are embedded in (5) and (6), these are a
set of simultaneous nonlinear equations that, in general, have no closed form solution that
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makes them amenable to straightforward estimation. Antle’s approach is to separately
estimate stochastic technology using a momentbased approach. Having noted that most
distribution functions are well approximated by their first three moments, he pursues a
sequential estimation strategy where output is first regressed on the contemporaneous
input variables to provide an estimate of the ‘mean’ effect. The estimated errors from the
mean effect regression are then squared cubed and regressed in turn on the inputs,
providing second and third moment effects. The inherent heteroskedasticity is accounted
for by using feasible GLS methods.
Love and Buccola point out that Antle’s estimation strategy is problematic given that
contemporaneous correlation between inputs and production error terms is likely. This
argument, with its roots in work by Marschak and Andrews, is now well recognized in
the production function estimation literature. Where such correlation exists, separate least
squares technology estimates are inconsistent. For the most part, it is the possibility of
unobserved heterogeneity (often interpreted as technical efficiency in the production
literature) existing as part of the error term that is thought to be the origin of the
correlation problem. In agricultural production situations, however, such
contemporaneous correlation problems are possible even where heterogeneity is assumed
away, as in the works of Antle and Love and Buccola, and indeed, in this study. This is
because of the possibility of decisionmaking in agricultural production being sequential
within a season, a feature first modeled in Antle (1983b). While it is possible that some
inputs are applied in a predetermined fashion (‘insurance inputs’), other inputs may be
applied in a sequence of steps in response to the unraveling uncertainty, creating a
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correlation between those inputs and production error. For example, pesticide
applications may proceed in response to sequentially updated information on infestations.
In such situations, Antle (1983b) shows that consistent separate technology estimation is
possible only if data on each stage within the season is available, a requirement that is
almost never realized in practice. The alternative is simultaneous estimation along with
behavioral equation sets (5) and (6).
Love and Buccola present a joint estimation framework that relies on the assumption of
CARA preferences embodied in the use of the negative exponential utility function,
()
W
U We
l
−
= −
. Here W is final wealth, and l is the (constant) ArrowPratt coefficient of
absolute risk aversion. The CARA assumption enables two major simplifications. First, it
enables the simultaneous estimation procedure to proceed on the basis of data on a single
output alone. Where simultaneous estimation of technology and preferences is concerned,
stochastic nonjointness is a necessary, but not sufficient assumption to allow estimation
based on data for only one output. The CARA assumption is also needed, and this can be
seen by examining the set of equations for output C, (5). Even though the C production
function is free of inputs and parameters relating to S, the equations in (5) contain BS.
The CARA assumption, however, implies that U/ [W0 + Bc + Bs] = U/ [Bc]. Thus not only
can the estimation proceed without consideration of the crop S, but additionally without
information on initial wealth levels of farmers, on which reliable data is hard to obtain.
The second function of CARA, or more specifically, the negative exponential utility
function is that, in tandem with the assumption of a normally distributed production error,
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it produces convenient closed form expressions for equations in (5). This is because
integral expressions of the form
ε
∫
εdet
are the momentgenerating functions (MGF) of
random variables ,, and a number of statistical distributions possess convenient
analytical expressions for MGF integrals. This feature, termed the Expected Utility
Moment Generating Function (EUMGF) approach, was initially noted by Hammond, and
first applied to agricultural risk problems in Yassour, Zilberman and Rausser. Love and
Buccola exploit this feature to derive closed form versions of (5). These are estimated
along with the popular CobbDouglas JustPope representation of stochastic (corn)
technology1, given by2:
(6)
c
j
c
2j
c
j 1
cc
2j
c
1j
c
c
j
c
2
c
1
c
2
c
1
) x () x (B) x ()x (Aq
ε+=
ββαα
,
jε iid N(0, 1)
Joint estimation preserves estimation consistency, and the crossequation constraints on
parameters also likely improve efficiency.
The point raised by Love and Buccola regarding the inconsistency of separate technology
estimation is an important one. However, the alternative that is offered might not always
be attractive to applied researchers. Firstly, for researchers who are primarily interested in
learning of the risk properties of individual inputs in a production situation, estimation of
an entire structural system would be excessive and inconvenient. Secondly, the
assumption of normality of production error, and hence of output and profits, may not be
desirable given the conventional wisdom that agricultural outputs (yields) are not well
represented by the normal distribution. Replacing the normal with other parametric
distributions in the EUMGF approach does not produce estimationfriendly results. For
example, it can be confirmed that if output is assumed to be Gamma distributed (with the
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parameters of the distribution expressed as functions of inputs), the EUMGF approach
results in first order conditions from which it is impossible to identify technology
parameters separately. Thirdly, and most importantly, the technology estimates obtained
from a system of first order conditions are conditional upon the behavioral assumptions
under which the system is derived. For instance, the CARA assumption employed by
Love and Buccola is a strong one, and the weight of empirical evidence is not in its favor
(Saha, Shumway and Talpaz (1993)). Where CARA and the negative exponential
functional form are abandoned, convenient closed form expressions are not available.
Furthermore, consideration of all outputs and data on initial wealth become necessary.
Wealth data are typically not available in most datasets, and additionally, there is no
consensus on what an appropriate measure of initial wealth might be (Net worth? Some
measure of permanent income?).
There is value, then, in looking for a strategy that allows separate stochastic technology
estimation while leaving the estimation less exposed to inconsistency. Some form of
instrumentation is an obvious answer. However, the most obvious forms of
instrumentation within the crosssectional context involve the use of prices, which can be
problematic. Very often, farmlevel data (such as those collected by farm accounting
services) come in the form of input expenditures rather than in the form of physical
quantities and prices. The researcher is constrained to using such expenditure in place of
physical quantities in technology estimation, with division by regional price indices being
the only possible way to get closer to actual quantities. At any rate, crosssectional
variation in farmlevel prices tends to be minimal unless the geographical spread of the
sample is vast.
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A different solution is suggested by the recent literature on production function
estimation using panel data. This literature that has developed over the last decade
exploits the orthogonality of the error term with predetermined but not strictly exogenous
explanatory variables that are in the information set at the time of decision making
(Mairesse and Hall, Griliches and Mairesse, Blundell and Bond). Past values of input
variables can therefore be used in building orthogonality conditions, thus making
available different numbers of valid orthogonality conditions for observations at different
times. GMM is the standard econometric approach in these cases. In the next section, we
develop such a GMM estimation strategy for stochastic technology. Particular attention is
paid to developing a computationally tractable and robust estimation approach.
The Estimation Approach
Since the approach is based upon panel data, we rewrite the JustPope production
representation (6) for the two outputs, c and s, as:
(7)
k
jt
k
2 jt
k
1 jt
kk
2 jt
k
1 jt
k
k
jt
k
2
k
1
k
2
k
1
) x () x (B) x () x (Aq
ε+=
ββαα
, k = c, s;
) 1 , 0 (~
k
jt
ε
for all j, t
Note that a normal distribution is no longer assumed for the error term. One way to
proceed is to write that: E(,kjt* St) = 0, where St consists of the information set at the
beginning of the growing year, t. In other words,
(8)
0
) x () x (B
)) x () x (A q (
E
t
k
2 jt
k
1 jt
k
k
2
β
jt
k
1
β
jt
k
k
jt
k
2
k
1
k
2
k
1
=
Ω
−
αα
St includes all past input values, and this implies the following set of orthogonality
conditions:
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(9) E[,kjt xis ] = 0 t = 2...T; s=1...t1
Although (9) could be used as the basis for panel GMM estimation, identification is
problematic in practice due to the extent and type of nonlinearity in the term inside the
brackets in (8). GMM involves a minimization algorithm whose success in convergence
and identification of parameters depends upon the amount and type of nonlinearity. The
expression in (8) involves nonlinearity of a form that obviates iterative convergence and
identification.
The alternative we employ is to utilize the two moments each for k = c and s, implied by
(7) as the basis for generating orthogonality conditions, i.e.,
(9)
)) x () x (Aq [(E
k
2
k
1
k
2jt
k
1jt
k
k
jt
αα
−
* St] = 0, and
(10)
E [ {
k
2
k
1
k
2
k
1
2k
2jt
2k
1 jt
2k2k
2 jt
k
1 jt
k
k
jt
) x () x ()B()) x () x (Aq (
ββαα
−−
} *St ] = 0
Three points are worth noting about this specification. First, all the parameters of interest
are contained in (10) itself, and it is possible to use (10) alone as the basis for generating
orthogonality conditions3. However, this would ignore potentially valuable additional
information on the technology/distribution that is contained in (9)4. Second, having two
conditional moments rather than one is helpful when only limited data are available.
Given data on a set of variables contained in information set St, two conditional moments
provide twice as many orthogonality conditions as one. This may be useful when there is
no possibility of getting more data on variables contained in St, either by obtaining more
years of data, or by finding data on other variables (some of which may make for poor
instruments) that may be in St.