It implements a Method of Moments (MM) estimation of stochastic production frontier with explanatory variables influencing technical inefficiency (i.e. the technical inefficiency effects) and accounting for one single endogenous input.
sfaendog(y, x.exo, x.endo, c.var, ineff, inst, data, nls.algo = c("GN", "LM"),
gmm.kernel = c("Bartlett", "Quadratic Spectral", "Truncated", "Parzen",
"Tukey-Hanning"), gmm.optim = c("BFGS", "Nelder-Mead", "CG", "SANN"),
maxiter = 100)A character string specifying the name of the dependent variable of the
production function (output). Values of the y variable will be internally
coerced to their logged values.
A vector of strings containing the names of the exogenous inputs
of the production function. Values of the x.exo variables will be internally
coerced to their logged values.
A character string specifying the name of the input of the
production function considered as endogenous. Values of the the x.endo
variable will be internally coerced to their logged values.
A vector of strings containing the names of non-input variables influencing the output (production environment conditions, time trend, etc.).
A vector of strings containing the names of the variables influencing technical inefficiency.
A vector of strings containing the names of the external instrumental variables.
A data frame containing the variables called in y, x.exo,
x.endo, c.var, ineff and inst.
A character string specifying the algorithm used in the Non-linear Least
Squares (NLS) estimation (see Estimation 'recipe'). The default algorithm is the Gauss-Newton
algorithm ("GN"). The other possible value is "LM" for the Levenberg-Marquardt
algorithm.
Levenberg-Marquardt being a more robust variation of Gauss-Newton algorithm,
specifiying nls.algo = "LM" may help in case of non convergence.
A character string specifying the type of kernel used to compute
the covariance matrix of the vector of sample moment conditions in the Generalised Method of
Moments (GMM) estimations (see Estimation 'recipe'). The default value is "Bartlett"
kernel. Other possible options are "Quadratic Spectral", "Truncated",
"Parzen", or "Tukey-Hanning" (see gmm for more details).
A character string specifying the optimisation method to be applied
in the GMM estimations (see Estimation 'recipe'). The default value ("BFGS")
is the quasi-Newton method. Other possibles methods are "Nelder-Mead", "CG",
"SANN" (see optim for more details).
A positive integer specifying the maximum number of iterations
allowed for the NLS and GMM estimations. The default value is 100.
sfaendog returns a list of class 'sfaendog'.
The object of class 'sfaendog' is a list containing at least the following
components:
Results of Step1 (see Estimation 'recipe'), inheriting from
class 'lm' for which details on components are given in lm.
The list of external instrumental variables (inst) used in
the specification, as well as Fisher test results showing the strength of these
external instrumental variables.
Results of Step2b (see Estimation 'recipe'), inheriting from
class 'nls' for which details on components are given in nls.
Results of Step3 (see Estimation 'recipe'), inheriting from
class 'gmm' for which details on components are given in gmm.
Results of Step4 (see Estimation 'recipe'), inheriting from
class 'gmm' for which details on components are given in gmm.
The function summary is used to obtain and print a summary of the results.
The function sfaendog() implements the 4-step ‘recipe’ detailed
in Latruffe et al. (2017, p.788).
The use of basic formula operators generally involved in model formulae, such as
":", "*", or "^", are allowed in x.exo, x.endo,
c.var, ineff and inst.
As in function formula, the function I()
can also be used to inhibit the interpretation of operators such as "+",
"-", "*" and "^" as formula operators, so that they are used as
arithmetical operators.
sfaendog() assumes a Cobb-Douglas functional form for the production frontier.
In this case, the stochastic frontier production model is written as: $$ \ln\textrm{y}=\boldsymbol{\alpha'}_0\ln\textbf{x}-\eta\exp(\boldsymbol{\theta'}_0 \textbf{z})+\textrm{v}$$ where \(\textrm{ln}\) is the log; \(\textrm{y}\) is the observed output; \(\boldsymbol{\alpha}_0\) and \(\boldsymbol{\theta}_0\) are vectors of parameters to be estimated; \(\textbf{x}\) is a vector containing the inputs as well as a constant term one; \(\textrm{v}\) is a random term which accounts for the effects of unobserved heterogeneity across observations and for stochastic events affecting the production process; \(\eta\exp(\boldsymbol{\theta'}_0\textbf{z})\) is a non-negative term accounting for the presence of technical inefficiency; \(\textbf{z}\) is a vector of variables influencing technical inefficiency (the inefficiency effects), including a constant term one; and \(\eta\) is a positive random term with mean one.
If all inputs are exogenous, the above equation can be estimated by NLS.
sfaendog() accounts for the endogeneity of one input, with a MM estimator based
on Chamberlain's (1987) ‘efficient instruments’.
In the case of endogeneity of one input, the stochastic frontier production model
defined above can be rewritten as:
$$
\ln\textrm{y}=\boldsymbol{\alpha}_{\textup{x},0}\ln\mathbf{x}_x+\alpha_{e,0}\ln{x_e}-
\eta\exp(\boldsymbol{\theta'}_0\textbf{z})+\textrm{v}$$
where \(\mathbf{x}_x\) is the vector containing the exogenous inputs and the
constant term one; \(x_e\) is the endogenous input; and the subscript \(0\)
denotes the ‘true’ parameters value. The vector of exogenous variables is
denoted by \(\textbf{w}=(\ln\textbf{x}_x,\textbf{q},\textbf{z})\)
where \(\textbf{q}\) is the vector of external instrumental variables.
Assuming that \(E[\textrm{v}|\textbf{w}]=0\), and that \(\eta\) and \((\textrm{v},\textbf{w})\) are independent, the stochastic frontier production model can be rewritten as: $$\ln\textrm{y}=\boldsymbol{\alpha}_{\textup{x},0}\ln\mathbf{x}_x+\alpha_{e,0}\ln{x_e}- \exp(\boldsymbol{\theta'}_0\textbf{z})+e\;\;\textrm{with}\;\;E[e|\textbf{w}]=0$$ where the error term \(e\) is defined as \(e\equiv e(\boldsymbol{\delta}_0)=\ln\textrm{y}-\boldsymbol{\alpha}_{\textup{x},0}\ln\textbf{x}_x -\alpha_{e,0}\ln{x_e}+\exp(\boldsymbol{\theta'}_0\textbf{z})\)
The estimation 'recipe' detailed in Latruffe et al. (2017, p.788) and implemented
through sfaendog() consists in the following four steps:
Step1: Ordinary Least Squares (OLS) estimation of \(\ln{x_e}\)
on a set of external instrumental variables (\(\textbf{q}\)) and all exogenous
variables included in the stochastic frontier production model. This generates the
predicted values of \(\ln{x_e}\) to be used as instruments in Step3.
The strength of the external instrumental variables (\(\textbf{q}\)) is measured by
testing, with a Fisher test, the nullity of the parameters related to these external
instrumental variables.
This Step1 corresponds to Step 1 of the estimation 'recipe' presented in
Latruffe et al. (2017, p.788).
Step2: NLS estimation of the stochastic frontier production model,
to compute a non consistent and non efficient estimator to be used in Step3.
Step2 consists in three sub-steps (not detailed as such in
Latruffe et al.'s (2017, p.788) recipe):
Step2a.i: OLS estimation of a production model with the output
as the dependent variable, and the explanatory variables being the inputs and the
non-input variables influencing the output.
This provides predicted residuals to be used in Step2a.ii, and parameters to be
used as starting values for the variables in the production part of the stochastic frontier
production model in Step2b.
Step2a.ii: OLS estimation of the predicted residuals of Step2a.i
on the variables influencing technical inefficiency.
This provides parameters to be used as starting values for the variables in the inefficiency
effect part of the stochastic frontier production model in Step2b.
Step2b: NLS estimation of the stochastic frontier production model,
using starting values obtained from Step2a.i and Step2a.ii.
This sub-step corresponds to Step 2 of the estimation 'recipe' presented in
Latruffe et al. (2017, p.788).
Step3: Estimation of the stochastic frontier production model, using the
predicted values of the endogenous input obtained from Step1 and using the parameters
obtained from Step2b as starting values. The estimation is done with GMM.
Step3 computes a consistent but non efficient estimator to be used in Step4.
This Step3 corresponds to Step 3 of the estimation 'recipe' presented in
Latruffe et al. (2017, p.788).
Step4: Estimation of the stochastic frontier production model, using the
predicted values of the endogenous input obtained from Step1 the parameters
obtained from Step3 as starting values. This estimation is done with GMM.
Step4 computes a consistent and efficient estimator, and returns
the final results of the estimation of the stochastic frontier production model.
This Step4 corresponds to Step 4 of the estimation 'recipe' presented in
Latruffe et al. (2017, p.788).
Please note that, the applicability of default options in nls.algo, gmm.kernel, and gmm.optim
is highly data-dependent and the user may have to play around with different options.
Chamberlain G. (1987). Asymptotic Efficiency in Estimation with Conditional Moment Restrictions. Journal of Econometrics, 34(3), 305--334. https://doi.org/10.1016/0304-4076(87)90015-7
Latruffe L., Bravo-Ureta B.E., Carpentier A., Desjeux Y., and Moreira V.H. (2017). Subsidies and Technical Efficiency in Agriculture: Evidence from European Dairy Farms. American Journal of Agricultural Economics, 99(3), 783--799. https://doi.org/10.1093/ajae/aaw077
summary for creating and printing summary results.
y <- "farm_output"
x.exo <- c("agri_land", "tot_lab", "tot_asset")
x.endo <- "costs"
c.var <- c("LFA", "T", "I(T^2)")
ineff <- c("hired_lab", "rented_land", "debt_asset", "subs", "region", "region:T")
inst <- c("milkprice", "I(milkprice^2)", "price_ind", "milkprice:region")
RES <- sfaendog(y, x.exo, x.endo, c.var, ineff, inst, data=Farms)
summary(RES)
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