sfalcmcross is a symbolic formula based function for the
estimation of the latent class stochastic frontier model (LCM) in the case
of cross-sectional or pooled cross-sectional data. The model is estimated
using maximum likelihood (ML). See Orea and Kumbhakar (2004), Parmeter and
Kumbhakar (2014, p282).
Only the half-normal distribution is possible for the one-sided error term. Eleven optimization algorithms are available.
The function also accounts for heteroscedasticity in both one-sided and two-sided error terms, as in Reifschneider and Stevenson (1991), Caudill and Ford (1993), Caudill et al. (1995) and Hadri (1999).
The model can estimate up to five classes.
sfalcmcross(
formula,
uhet,
vhet,
thet,
logDepVar = TRUE,
data,
subset,
weights,
wscale = TRUE,
S = 1L,
udist = "hnormal",
start = NULL,
whichStart = 2L,
initAlg = "nm",
initIter = 100,
lcmClasses = 2,
method = "bfgs",
hessianType = 1,
itermax = 2000L,
printInfo = FALSE,
tol = 1e-12,
gradtol = 1e-06,
stepmax = 0.1,
qac = "marquardt"
)# S3 method for sfalcmcross
print(x, ...)
# S3 method for sfalcmcross
bread(x, ...)
# S3 method for sfalcmcross
estfun(x, ...)
sfalcmcross returns a list of class 'sfalcmcross'
containing the following elements:
The matched call.
Multi parts formula describing the estimated model.
The argument 'S'. See the section ‘Arguments’.
Character string. 'Latent Class Production/Profit Frontier, e
= v - u' when S = 1 and 'Latent Class Cost Frontier, e = v + u' when
S = -1.
Number of observations used for optimization.
Number of main explanatory variables.
Number of variables in the logit specification of the finite mixture model (i.e. number of covariates).
The argument 'logDepVar'. See the section
‘Arguments’.
Number of variables explaining heteroscedasticity in the one-sided error term.
Number of variables explaining heteroscedasticity in the two-sided error term.
Total number of parameters estimated.
The argument 'udist'. See the section
‘Arguments’.
Numeric vector. Starting value for ML estimation.
A data frame (tibble format) containing information on data
used for optimization along with residuals and fitted values of the OLS and
ML estimations, and the individual observation log-likelihood. When
weights is specified an additional variable is also provided in
dataTable.
When start = NULL and whichStart == 2L.
Initial ML estimation with half normal distribution for the one-sided error
term. Model to construct the starting values for
the latent class estimation. Object of class 'maxLik' and
'maxim' returned.
Logical. If TRUE weighted log-likelihood is
maximized.
The optimization algorithm used.
Number of iterations of the ML estimation.
An optimization algorithm termination message.
Log-likelihood at the starting values.
The number of classes estimated.
Log-likelihood value of the ML estimation.
Numeric vector. Parameters obtained from ML estimation.
Double. Matrix of ML parameters by class.
Numeric vector. Each variable gradient of the ML estimation.
Matrix. Each variable individual observation gradient of the ML estimation.
Numeric. Gradient norm of the ML estimation.
The covariance matrix of the parameters obtained from the ML estimation.
The argument 'hessianType'. See the section
‘Arguments’.
Date and time of the estimated model.
A symbolic description of the model to be estimated based on
the generic function formula (see section ‘Details’).
A one-part formula to account for heteroscedasticity in the one-sided error variance (see section ‘Details’).
A one-part formula to account for heteroscedasticity in the two-sided error variance (see section ‘Details’).
A one-part formula to account for technological heterogeneity in the construction of the classes.
Logical. Informs whether the dependent variable is logged
(TRUE) or not (FALSE). Default = TRUE.
The data frame containing the data.
An optional vector specifying a subset of observations to be used in the optimization process.
An optional vector of weights to be used for weighted
log-likelihood. Should be NULL or numeric vector with positive values.
When NULL, a numeric vector of 1 is used.
Logical. When weights is not NULL, a scaling
transformation is used such that the weights sums to the sample
size. Default TRUE. When FALSE no scaling is used.
If S = 1 (default), a production (profit) frontier is
estimated: \(\epsilon_i = v_i-u_i\). If S = -1, a cost frontier is
estimated: \(\epsilon_i = v_i+u_i\).
Character string. Distribution specification for the one-sided
error term. Only the half normal distribution 'hnormal' (Aigner
et al., 1977, Meeusen and Vandenbroeck, 1977) is currently
implemented.
Numeric vector. Optional starting values for the maximum likelihood (ML) estimation.
Integer. If 'whichStart = 1', the starting values
are obtained from the method of moments. When 'whichStart = 2'
(Default), the model is initialized by solving the homoscedastic pooled
cross section SFA model.
Character string specifying the algorithm used for
initialization and obtain the starting values (when 'whichStart = 2').
Only maxLik package algorithms are available:
'bfgs', for Broyden-Fletcher-Goldfarb-Shanno
(see maxBFGS)
'bhhh', for Berndt-Hall-Hall-Hausman
(see maxBHHH)
'nr', for Newton-Raphson (see maxNR)
'nm', for Nelder-Mead - Default -
(see maxNM)
'cg', for Conjugate Gradient
(see maxCG)
'sann', for Simulated
Annealing (see maxSANN)
Maximum number of iterations for initialization algorithm.
Default 100.
Number of classes to be estimated (default = 2). A
maximum of five classes can be estimated.
Optimization algorithm used for the estimation. Default =
'bfgs'. 11 algorithms are available:
'bfgs',
for Broyden-Fletcher-Goldfarb-Shanno (see
maxBFGS)
'bhhh', for
Berndt-Hall-Hall-Hausman (see maxBHHH)
'nr', for Newton-Raphson (see maxNR)
'nm', for Nelder-Mead (see maxNM)
'cg', for Conjugate Gradient
(see maxCG)
'sann', for Simulated
Annealing (see maxSANN)
'ucminf', for a quasi-Newton type optimization with BFGS updating of
the inverse Hessian and soft line search with a trust region type monitoring
of the input to the line search algorithm
(see ucminf)
'mla', for general-purpose optimization based on
Marquardt-Levenberg algorithm (see mla)
'sr1', for Symmetric Rank 1 (see
trust.optim)
'sparse',
for trust regions and sparse Hessian
(see trust.optim)
'nlminb', for optimization using PORT routines (see
nlminb)
Integer. If 1 (default), analytic Hessian is
returned. If 2, bhhh Hessian is estimated (\(g'g\)).
Maximum number of iterations allowed for optimization.
Default = 2000.
Logical. Print information during optimization. Default =
FALSE.
Numeric. Convergence tolerance. Default = 1e-12.
Numeric. Convergence tolerance for gradient. Default =
1e-06.
Numeric. Step max for ucminf algorithm. Default =
0.1.
Character. Quadratic Approximation Correction for 'bhhh'
and 'nr' algorithms. If 'qac = stephalving', the step length
is decreased but the direction is kept. If 'qac = marquardt'
(default), the step length is decreased while also moving closer to the pure
gradient direction. See maxBHHH and
maxNR.
an object of class sfalcmcross (returned by the function
sfalcmcross).
additional arguments of frontier are passed to sfalcmcross; additional arguments of the print, bread, estfun, nobs methods are currently ignored.
LCM is an estimation of a finite mixture of production functions:
$$y_i = \alpha_j + \mathbf{x_i^{\prime}} \bm{\beta_j} + v_{i|j} - Su_{i|j}$$
$$\epsilon_{i|j} = v_{i|j} - Su_{i|j}$$
where \(i\) is the observation, \(j\) is the class, \(y\) is the output (cost, revenue, profit), \(x\) is the vector of main explanatory variables (inputs and other control variables), \(u\) is the one-sided error term with variance \(\sigma_{u}^2\), and \(v\) is the two-sided error term with variance \(\sigma_{v}^2\).
S = 1 in the case of production (profit) frontier function and
S = -1 in the case of cost frontier function.
The contribution of observation \(i\) to the likelihood conditional on class \(j\) is defined as:
$$P(i|j) = \frac{2}{\sqrt{\sigma_{u|j}^2 + \sigma_{v|j}^2}}\phi\left(\frac{S\epsilon_{i|j}}{\sqrt{ \sigma_{u|j}^2 +\sigma_{v|j}^2}}\right)\Phi\left(\frac{ \mu_{i*|j}}{\sigma_{*|j}}\right)$$
where
$$\mu_{i*|j}=\frac{- S\epsilon_{i|j} \sigma_{u|j}^2}{\sigma_{u|j}^2 + \sigma_{v|j}^2}$$
and
$$\sigma_*^2 = \frac{\sigma_{u|j}^2 \sigma_{v|j}^2}{\sigma_{u|j}^2 + \sigma_{v|j}^2}$$
The prior probability of using a particular technology can depend on some covariates (namely the variables separating the observations into classes) using a logit specification:
$$\pi(i,j) = \frac{\exp{(\bm{\theta}_j'\mathbf{Z}_{hi})}}{ \sum_{m=1}^{J}\exp{(\bm{\theta}_m'\mathbf{Z}_{hi})}}$$
with \(\mathbf{Z}_h\) the covariates, \(\bm{\theta}\) the coefficients estimated for the covariates, and \(\exp(\bm{\theta}_J'\mathbf{Z}_h)=1\).
The unconditional likelihood of observation \(i\) is simply the average over the \(J\) classes:
$$P(i) = \sum_{m=1}^{J}\pi(i,m)P(i|m)$$
The number of classes to retain can be based on information criterion (see
for instance ic).
Class assignment is based on the largest posterior probability. This probability is obtained using Bayes' rule, as follows for class \(j\):
$$w\left(j|i\right)=\frac{P\left(i|j\right) \pi\left(i,j\right)}{\sum_{m=1}^JP\left(i|m\right) \pi\left(i, m\right)}$$
To accommodate heteroscedasticity in the variance parameters of the error
terms, a single part (right) formula can also be specified. To impose the
positivity on these parameters, the variances are modelled respectively as:
\(\sigma^2_{u|j} = \exp{(\bm{\delta}_j'\mathbf{Z}_u)}\) and \(\sigma^2_{v|j} =
\exp{(\bm{\phi}_j'\mathbf{Z}_v)}\), where \(Z_u\) and \(Z_v\) are the
heteroscedasticity variables (inefficiency drivers in the case of \(\mathbf{Z}_u\))
and \(\bm{\delta}\) and \(\bm{\phi}\) the coefficients. 'sfalcmcross' only
supports the half-normal distribution for the one-sided error term.
sfalcmcross allows for the maximization of weighted log-likelihood.
When option weights is specified and wscale = TRUE, the weights
are scaled as:
$$new_{weights} = sample_{size} \times \frac{old_{weights}}{\sum(old_{weights})}$$
For complex problems, non-gradient methods (e.g. nm or
sann) can be used to warm start the optimization and zoom in the
neighborhood of the solution. Then a gradient-based methods is recommended
in the second step. In the case of sann, we recommend to significantly
increase the iteration limit (e.g. itermax = 20000). The Conjugate
Gradient (cg) can also be used in the first stage.
A set of extractor functions for fitted model objects is available for
objects of class 'sfalcmcross' including methods to the generic functions
print,
summary,
coef,
fitted,
logLik,
residuals,
vcov,
efficiencies,
ic,
marginal,
estfun and
bread (from the sandwich package),
lmtest::coeftest() (from the lmtest package).
Aigner, D., Lovell, C. A. K., and P. Schmidt. 1977. Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1), 21--37.
Caudill, S. B., and J. M. Ford. 1993. Biases in frontier estimation due to heteroscedasticity. Economics Letters, 41(1), 17--20.
Caudill, S. B., Ford, J. M., and D. M. Gropper. 1995. Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. Journal of Business & Economic Statistics, 13(1), 105--111.
Hadri, K. 1999. Estimation of a doubly heteroscedastic stochastic frontier cost function. Journal of Business & Economic Statistics, 17(3), 359--363.
Meeusen, W., and J. Vandenbroeck. 1977. Efficiency estimation from Cobb-Douglas production functions with composed error. International Economic Review, 18(2), 435--445.
Orea, L., and S.C. Kumbhakar. 2004. Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29, 169--183.
Parmeter, C.F., and S.C. Kumbhakar. 2014. Efficiency analysis: A primer on recent advances. Foundations and Trends in Econometrics, 7, 191--385.
Reifschneider, D., and R. Stevenson. 1991. Systematic departures from the frontier: A framework for the analysis of firm inefficiency. International Economic Review, 32(3), 715--723.
print for printing sfalcmcross
object.
summary for creating and printing
summary results.
coef for extracting coefficients of the
estimation.
efficiencies for computing
(in-)efficiency estimates.
fitted for extracting the fitted frontier
values.
ic for extracting information criteria.
logLik for extracting log-likelihood
value(s) of the estimation.
marginal for computing marginal effects of
inefficiency drivers.
residuals for extracting residuals of the
estimation.
vcov for computing the variance-covariance
matrix of the coefficients.
bread for bread for sandwich estimator.
estfun for gradient extraction for each
observation.
## Using data on eighty-two countries production (GDP)
# LCM Cobb Douglas (production function) half normal distribution
# Intercept and initStat used as separating variables
cb_2c_h1 <- sfalcmcross(formula = ly ~ lk + ll + yr, thet = ~initStat,
data = worldprod)
summary(cb_2c_h1)
# summary of the initial ML model
summary(cb_2c_h1$InitHalf)
# Only the intercept is used as the separating variable
# and only variable initStat is used as inefficiency driver
cb_2c_h3 <- sfalcmcross(formula = ly ~ lk + ll + yr, uhet = ~initStat,
data = worldprod)
summary(cb_2c_h3)
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