A SIMPLE APPROACH TO STOCHASTIC TECHNOLOGY ESTIMATION USING FARM-LEVEL 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 Urbana-Champaign 1301 W. Gregory Dr. Urbana, IL 61801 217-333-1822 (voice) 217-333-5538(fax)
[email protected]
First Draft April 16, 2003
∗
corresponding author
A SIMPLE APPROACH TO STOCHAS TIC TECHNOLOGY ESTIMATION USING FARM-LEVEL 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 farm- level stochastic technology estimation (Antle(1983a), Griffiths and Anderson, Nelson and Preckel, Love and Buccola, Khumbhakar). The development of estimation methods for use with farmlevel 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 farm- level data, both cross-sectional and more recently, panel.
The various methods developed for farm- level 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 JustPope production function restricts the effects of inputs on the third and higher moments of output, he developed a nonparametric moment-based approach that regressed each (estimated) moment of output on the input vector in a multi-stage 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 six-step sequence that incorporates an error-components 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 two- moment approach akin to the Just-Pope function, the moments are jointly estimated, thereby avoiding a multi- stage approach. The two-moment approach significantly eases convergence and computational difficulty in the non-linear setting, while also providing more moment conditions for model identification than does a single-moment approach. Other important advantages of this approach are: (i) the problem of contemporaneous input-error 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 pre-determined/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, risk-averse 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(., θ) : Utility function. θ 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 wx
: Vectors of variable inputs applied to the production of C and S, respectively. :
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 multiple-output production setting can be represented by the joint conditional density function, f(qc, qs / xc, xs , zc, zs ; α ) (Antle (1988) ). α 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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(1)
Max
∫ ∫
U[ W0 + ( p q - w /x x c ) + ( p q - w /x xs ) ; θ] c
c
s
s
q s∈Ss q c∈Sc
f( q , q / x c , x s , zc , zs ; α) dq dq c
s
c
s
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 :
(2)
Max
∫ ∫
U[ W0 + ( p q - w /x x c) + ( p q - w /x xs ) ; θ] c
c
s
s
q s∈Ss q c∈Sc
f( q , q / xc , xs , zc* , z s* ; α ) dq dq c
s
c
5
s
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:
(3)
c
c
s
s
q = q ( x c , z*c , εc , α c) q = q ( x s , z*s , εs , α s), (ε c , ε s) ~ F( εc , εs)
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 ε c. 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 multi-output 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 corn-soybean 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 output-specific basis. However, it is best avoided where possible, since it entails assumptions that may be quite undesirable. In technology estimation in a corn-soybeans 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 multi-output 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
(4)
∫ ∫
Max
c
c
U[ W 0 + ( p q ( xc , z*c , εc , αc ) - w /x x c) +
εs ∈S s εc∈Sc
( p q ( x s , z*s , εs , α s) - w /x x s) ; θ ] f( εc , εs ) d εc d εs s
s
In this formulation, α c and α s are technology parameters attached exclusively to the production of C and S, respectively. The structural model composed of first-order conditions for variable inputs applied to C and S is then given by:
∂ qc - w j) ] = 0 for variable inputs attached to C, and ∂ xcj
(5)
E[ U′( W0j + πcj + πsj) (P
(6)
∂ qs E[ U′( W0j + π + π ) (P s - w j) ] = 0 for variable inputs attached to S, and of ∂ xj c j
s j
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 moment-based 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 decision- making 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, U (W ) = −e − λW . Here W is final wealth, and λ is the (constant) Arrow-Pratt 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 B S. 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
∫e
tε
dε are the moment- generating 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 Cobb-Douglas Just-Pope representation of stochastic (corn) technology1 , given by2 : (6)
c
c
c
c
q j = A c ( x 1c j ) α1 ( x c2 j ) α 2 + B c ( x 1cj ) β1 ( x c2 j ) β2 ε j , c
c
ε j iid N(0, 1)
Joint estimation preserves estimation consistency, and the cross-equation 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 estimation-friendly 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 cross-sectional context involve the use of prices, which can be problematic. Very often, farm- level 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, cross-sectional variation in farm- level 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 Just-Pope production representation (6) for the two outputs, c and s, as: (7)
k
k
k
k
q jt = A k ( x1kjt ) α1 ( x k2 jt ) α 2 + B k ( x 1kjt ) β1 ( x k2 jt ) β2 ε jt , k = c, s; ε jt ~ (0,1) for all j, t k
k
k
Note that a normal distribution is no longer assumed for the error term. One way to proceed is to write that: E(,k jt * St ) = 0, where St consists of the information set at the beginning of the growing year, t. In other words,
(q jt k − A k ( x 1kjt ) α1 ( x k2 jt ) α 2 ) E | Ω k k t = 0 Bk ( x1kjt ) β1 ( x k2 jt ) β 2 k
(8)
k
St includes all past input values, and this implies the following set of orthogonality conditions:
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(9)
E[,k jt xis ] = 0 t = 2...T; s=1...t-1
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) (10)
k
k
E[(q jt − A k ( x 1kjt ) α1 ( x k2 jt ) α2 ) * St ] = 0, and k
k
k
k
k
E [ { ( q jt k − A k ( x 1kjt ) α1 ( x k2 jt ) α 2 ) 2 − ( Bk ) 2 ( x 1kjt ) 2β1 ( x k2 jt ) 2β 2 } *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 conditions 3 . 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 .
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The entire set of moment conditions implied by (9) and (10) can be concisely summarized as (11)
E[ Ui (Β) ⊗ z i ] = 0
where % is the vector of parameters, Ui(%) = [{ui1 (%), vi1 (%)}, {ui2 (%), vi2 (%)},...,{uiT (%), viT (%)}], k
k
uit (%) = q jt k − A k ( x1kjt ) α1 ( x k2 jt ) α 2 , k
k
k
k
vit (%) = ( q jt k − A k ( x 1kjt ) α1 ( x k2 jt ) α 2 ) 2 − ( Bk ) 2 ( x 1kjt ) 2β1 ( x k2 jt ) 2β 2 , zi = [zi1 , ..., zim ], and m = number of instruments available per year. These population moment conditions have sample equivalents given by (12)
F(#) =
1 N ∑ U i (Β) ⊗ z i N i=1
The GMM estimator is then obtained by minimizing, with respect to #, (13)
M(#) = F/(#) W F(#)
where W (the weighting matrix) is a symmetric positive definite matrix.
Where the number of moment conditions exceeds the number of parameters to be estimated, the GMM estimator depends upon the choice of the weighting matrix. Suppose the covariance matrix of the orthogonality conditions is denoted by Γ . Then choosing a consistent estimate of the inverse of Γ for W results in the GMM estimator (13) being consistent and asymptotically efficient, given the instrument set (Hansen (1982)). This ‘optimal’ GMM estimator is arrived at by starting with an arbitrary positive semidefinite weighting matrix (such as the identity matrix), calculating the ‘first step’ GMM estimator from this, computing the covariance matrix of the orthogonality
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conditions for the first step estimator, and using it as the weighting matrix in a second step. Iterating this process until convergence, or until gains from further iterations are small, produces a GMM estimator with better small sample properties. Note that in our context of non- linear instrumental variables estimation, the assumption of serially uncorrelated disturbances implies that the optimal estimator of Γ is White’s hetroskedasticity-consistent estimator. Where the disturbance is assumed serially correlated, the Newey and West estimator is available. GMM estimates are thus robust to heteroskedasticity and autocorrelation.
Note that although we have used Kronecker product notation in (11) and (12) to summarize the moment conditions, in practice the use of panel data and a particular maintained assumption regarding exogeneity implies that different years have different numbers of instruments available to them. For example, if there are four years of data and the assumption is that only instruments up to (t-1) are valid, then year 3 will have more moment conditions than year 2, and year 4 will have more conditions than year 3. A straightforward specification test is the overidentifying restrictions test, or Hansen’s Jtest (Hansen). If the number of moment conditions exceeds the number of parameters to be estimated, under the null hypothesis that the overidentifying restrictions are valid, N times the value of the criterion function, ∧
(14)
JN(%) = N F/(#) Γ F(#)
is asymptotically chi-squared distributed with degrees of freedom equal to the number of overidentifying restrictions. Another class of specification tests that can be used profitably in our scenario is the likelihood ratio type test developed by Eichenbaum,
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Hansen and Singleton. Given an unrestricted GMM (baseline) estimator (JN(%u)), and a GMM estimator incorporating restrictions, (JN(%r) ), under the null hypothesis that the restrictions are valid, the statistic (15)
(JN(%r) - JN(%u) )
is asymptotically chi-square distributed with degrees of freedom equal to the number of restrictions imposed 5 . This test has sometimes been used in prior research to verify instrument exogeneity in GMM models, with the restricted model containing extra overidentifying instruments compared to the baseline model, and the degrees of freedom equal to the number of extra instruments (e.g. Ziliak and Kniesner). In our context, under the maintained hypothesis of no producer heterogeneity, this translates to a handy test of whether inputs are applied in a pre-determined or a sequential fashion. As discussed before, sequential application of a particular input would result in contemporaneous correlation between errors and input values at time t. Thus if the test did not reject the null hypothesis of exogeneity of a time t input instrument, it would strengthen the case for predetermined application of the input and the direct estimation of the stochastic production function.
Data Since agricultural production the world over is predominantly multi-output, production function estimation from farm- level data has often proceeded by aggregating outputs either simply by using revenue, or by constructing output indices. However, the risk effects of inputs are intuitively best understood on a crop-specific basis, where the parallels with biological response functions can be clearly made. For example, the risk
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effects of fertilizer input upon one crop may be substantially different from the risk effects on another crop in the output- mix – a distinction that would be entirely lost in aggregation. As discussed earlier, however, crop-specific input data are seldom available, and the only recourse usually is to use aggregate outputs (e.g. Carpentier and Weaver) or to seek out a single-output production scenario to apply the method to (e.g. Saha, Shumway, and Talpaz).
In our instance, we were fortunate to obtain a dataset containing output-specific input information. Our dataset is a panel of 50 Illinois grain farms over the period 1989-92 6 . Several thousand farmers maintain accounting records with the Illinois Farm Business Farm Management (FBFM) association. However, the expenditure records maintained in this databank are generally not output-specific. A smaller subset of farmers do maintain output-specific input expenditure records, aside from information on most other variables essential for our analysis, such as corn and soybean outputs, acreage, etc. The use of accounting data for production analysis is quite widespread in agricultural economics. A common approach to inferring quantity and price information from such data involves the use of weighted prices from state or national- level data (e.g. Saha, Shumway and Talpaz). Price aggregates for the fertilizer and pesticide input categories were computed using Illinois state level price and quantity data on commonly used pesticides and fertilizers, with quantity shares as weights. The pesticide and fertilizer expenditures of individual farms in our dataset were divided by these constructed price aggregates to obtain measures of pesticide and fertilizer input quantities. Other variable input expenditures such as seed, hired labor and fuel were aggregated into a composite input category called
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‘Other variable inputs’. A price aggregate for this category was computed using expenditure shares. All nominal variables were deflated to 1989 dollars using the Consumer Price Index for the North Central United States. Some summary statistics are presented in Table 1.
Results The two- moment GMM approach outlined previously was applied to the dataset. Although the specification outlined in (3) allows the error terms of the stochastic production functions for C and S to be correlated, the functions were estimated separately in the application. An initial specification with the input vector consisting of land, pesticides, fertilizers and other variable inputs proved problematic because of collinearity between land and other inputs. Hence a per-acre specification was employed in all subsequent analysis. It can be confirmed that an original Cobb-Douglas CRS specification for both mean and variance portions of a Just-Pope function results in a Cobb-Douglas form for both portions under the per-acre specification. The instruments used in the baseline model were a constant and lagged values (1,...,t-1) of each of the three inputs. Since the dataset extends from 1989 to 1992, this implies that the data for 1989 were used only to provide instruments. The results from the baseline models for corn and soybeans are presented in table 2.
Since the number of orthogonality conditions exceed the number of parameters estimated, the overidentifying restrictions test, often interpreted as a test of overall model specification, can be performed using the J-statistic. The J-statistic is distributed as P2
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with degrees of freedom equal to the number of orthogonality conditions minus the number of parameters, under the null hypothesis that the overidentifying restrictions hold. The critical P2 values for 34 degrees of freedom at 5% and 1% are 48.60 and 56.06 respectively. The J-stat for the corn model is 44.05, and for the soybeans model is 36.35, and thus we can conclude that the overidentifying restrictions do hold for our twomoment corn and soybeans models.
Most previous studies that have estimated the effects of individual inputs on yields of specific crops have used experimental data. Where farm level data have been used, the output measures have predominantly been aggregates from multiple crops. Thus the use of farm- level crop- level data in this study enables a fresh basis for comparison with previous studies. As can be seen from Table 2, the majority of parameters are significantly different from zero at the 1 % leve l. The effects of all inputs on the mean of both corn and soybeans are positive, and significantly different from zero. Previous studies have occasionally estimated a mean-decreasing effect for some inputs (Anderson and Griffiths, Nelson and Preckel). This is theoretically possible if the input is riskdecreasing and producers are significantly risk-averse, but appears implausible in practice. As Just and Pope point out, ‘Although it is conceivable that some input may be used at a point where its expected marginal product is negative if it were also leading to substantial reduction in variance, no such examples seem apparent ’ (Just and Pope, p.69, footnote 4).
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The signs of the ‘Yield: Standard Deviation’ parameter estimates attached to each input in Table 2 directly provide information on whether the input is risk- increasing or decreasing. The fertilizer parameters for both corn and soybeans are positive, indicating a variance- increasing effect. The parameter is significant at the 1% level for the corn case, but only at the 10% level for the soybeans case. This evidence on corn yield variance is broadly in agreement with previous work using experimental or plot- level data. For example, the primal system used by Love and Buccola found a varianceincreasing effect of Nitrogen, Phosphorous, as well as Potassium on corn yields. Just and Pope also found that nitrogen fertilizer increases corn yield variance. As for pesticides, our results show significant risk-reducing effects on the outputs of both corn and soybeans. This is in line with a conventional wisdom (Horowitz and Lichtenberg; Carlson; Robison and Barry) which holds that pesticides are applied in anticipation of the risk of pest infestation, and are thus naturally to be regarded as risk-decreasing, ‘insurance’ inputs. This notion is by no means unchallenged, however. For instance, Horowitz and Lichtenberg argue that pesticides could well be risk- increasing if sources of randomness other than pest infestation are also important. They point to Pannell’s survey finding that there seem to be no actual empirical studies showing pesticides as reducing risk in cases where pesticides are applied ex-ante. Our estimate here that pesticides reduce risk then naturally raises the question: is there any indication here that pesticides are applied ex-ante?
One way to answer this question is, of course, to test whether ‘time t’ pesticide inputs are valid instruments in our models or not. As noted before, GMM provides a convenient chi-
21
square test under the null hypothesis that the instruments are valid (exogenous). This test was performed for all 3 input categories for both crops, and the results are presented in Table 3. Interestingly, in not one case can we reject the hypothesis that time t inputs are valid instruments. Thus we are able to advance some evidence, noted as deficient in the literature by Pannell, that pesticides are risk-reducing and applied ex-ante.
Summary and Discussion Knowledge of the ways in which particular inputs affect output variability is acknowledged as important and policy-relevant in our profession. Initial methods advanced for the estimation of such effects used experimental data and therefore did not run into issues of explanatory variable endogeneity. However, it is equally recognized that the most useful and reliable estimates are those derived from farm- level information on actual choices and realizations. Here the endogeneity problem must be confronted.
Several methods have been advanced for the estimation of such effects using farm- level data. Some have simply ignored the endogeneity issue. Others that have laid stress upon it have put forth structural estimation procedures that are complex and rely on strong behavioral assumptions to overcome endogeneity bias. Where the behavioral assump tions (such as CARA) are weakened, data requirements (initial wealth) are correspondingly more severe. Nor is it particularly convenient for applied researchers who are primarily interested in the risk effects of inputs to set about estimating large systems of equations involving complete behavioral specifications (Shankar and Nelson).
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In this paper, we have suggested that the literature on production functions estimation using GMM techniques and past input levels as instruments may be profitably employed in this situation. Such instrumentation ameliorates endogeneity without relying on strong behavioral assumptions or requiring wealth data, and appears a better solution than relying on price-based instrumentation. Price information, especially with sufficient variation, is difficult to obtain. On the other hand, the sort of basic farm panel data on input and output quantities required for the estimation procedure we have suggested is collected by many farm data organizations around the world. In the US, the Illinois and Kansas ‘Farm Business & Farm Management’ surveys are notable examples, while in Europe, the Farm Accountancy Data Network is collecting such information for every EU member and candidate country.
Our estimation strategy has been based on the use of two conditional moments instead of one, thereby enabling adequate moment conditions even when the time dimension of the available dataset is very short. Specification tests indicate reasonable performance of this model, and the parameter estimates generated appear plausible and quite significant for both. Multi-stage estimation is obviated, distributional assumptions based on normality of outputs are not needed, and convenient tests of the endogeneity of inputs are available. The approach thus appears worthy of inclusion in the portfolio of methods available to estimate farm- level stochastic technology7 .
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Endnotes 1
In the rest of the paper, and in our empirical application, we continue to use the Cobb-
Douglas version of the Just-Pope function. This version has not only proven to be the most popular in empirical applications, but is also the most convenient in that the ∂E ( y ) directions (signs) of the mean effects as well as the variance effects ∂x ∂Var ( y ) of an input x are directly given by individual estimated parameters. This is in ∂x
contrast to some alternate forms of the Just-Pope function, for example the one used in Saha, Shumway, and Talpaz, where input effects at one of the moments are not independent of the sample values. 2
Equation (6) has a two- input representation for convenience, and uses the superscript c
to refer to a parameter or variable pertaining to the production of output c. 3
Indeed, in an article by Carpentier and Weaver that deals more generally with
information use by French cereal farmers, just such an approach is taken. 4
Jointly using the conditional mean along with the conditional variance and higher order
moments in GMM estimation has been discussed in other contexts by various authors. Breitung and Lechner discuss this for limited dependent variable models in general, and Woolridge analyzes the case of multiplicative unobserved effect models. Lee and Wirjanto demonstrate in the context of conditionally heteroskedastic time series models that using two moments jointly results in efficiency gains. 5
The same estimator for Γ has to be used in both the restricted and unrestricted models
for this test to be valid.
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6
Although this is admittedly a small cross-section, especially for a method whose most
desirable properties are asymptotic, the sample size is comparable with those in studies of a similar nature. For example, Saha, Shumway, and Talpaz use data on 15 farms over 4 years, Antle (1987) uses data on 30 Indian farms, and the sample sizes for the county-bycounty analysis of Love and Buccola range from 55 to 106. 7
Of course, this is not to deny that this approach has its weaknesses, as well. For
instance, we have included no farm or time-specific effects (although neither does much of the literature in this area, such as Antle(1983a), and Love and Buccola.
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Table 1. Summary Statistics for 50 Illinois Grain Farms, 1989-92 1
Mean
Coefficient of Variation (%)
Corn area
274
55
Soybeans area
202
59
Corn output per acre
160
35
Soybeans output per acre
58
38
Corn fertilizer expenditure per acre
44
33
Soybeans fertilizer exp. Per acre
11
106
Corn pesticide expenditure per acre
23
37
Soybeans pesticide exp. Per acre
24
37
Corn other input exp. Per acre
39
42
Soybeans other input exp. Per acre
31
54
Note: All areas are in hectares. Corn and Soybeans yields are in bushels. All input expenditures are in constant (1989) dollars.
1
The table presents input expenditure data instead of the constructed measures of input quantities used in estimation since the former are more ‘natural’ and comparable across studies. For similar reasons, output quantities and input expenditures are presented in per-acre terms. More extensive descriptions of the data, including year-by-year breakdowns are available from the authors upon request. 26
Table 2. Estimates for Baseline Models Estimate a
Input
Estimate a
Input
Corn: Mean
Soybeans: Mean
Ac
52.44*** (7.21)
As
24.20*** (4.20)
Pesticide
0.16*** (0.02)
Pesticide
0.25*** (0.03)
0.17***
Fertilizer
Fertilizer (0.02)
Other
0.13*** (0.03)
0.05*** (0.01)
Other
Corn: Standard Deviation
0.09*** (0.01)
Soybeans: Standard Deviation
Bc
0.07 (0.05)
Bs
0.38 (0.31)
Pesticide
-0.26*** (0.04)
Pesticide
-0.14 (0.09)
Fertilizer
0.51*** (0.09)
Fertilizer
0.25* (0.13)
Other
0.27*** (0.03)
Other
-0.14** (0.06)
J-Statistic: Degrees of Freedom
36.35 34
J-Statistic: 44.05 Degrees of Freedom: 34 a Asymptotic Standard Errors in Parantheses
b *** indicates significance at the 1% level, ** at the 5% level, and * at the 10% level.
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Table 3.
Input
‘Time t’ exogeneity tests* i 2 Statistic
Upper Tail Area
Input
Corn
i 2 Statistic
Upper Tail Area
Soybeans
Pesticide
3.16
0.78
Pesticide
5.75
0.45
Fertilizer
3.13
0.79
Fertilizer
5.49
0.48
Other
3.81
0.70
Other
8.54
0.20
*In all cases, the null hypothesis is that the time t input is a valid instrument, i.e., exogenous
28
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