lcmcross: Latent class stochastic frontier using cross-section data
In sfaR: Stochastic Frontier Analysis using R
lcmcross R Documentation
Latent class stochastic frontier using cross-section data
Description
lcmcross 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-section 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.
Nine 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.
Usage
lcmcross(formula, uhet, vhet, thet, logDepVar = TRUE, data, subset, S = 1,
udist = "hnormal", start = NULL, lcmClasses = 2, method = "bfgs", hessianType = 1,
itermax = 2000, printInfo = FALSE, tol = 1e-12, gradtol = 1e-06, stepmax = 0.1,
qac = "marquardt", initStart = FALSE, initAlg = "nlminb", initIter = 100,
initFactorLB = 0.5, initFactorUB = 1.5)
Arguments
formula
A symbolic description of the model to be estimated based on
the generic function formula (see section ‘Details’).
uhet
A one-part formula to account for heteroscedasticity in the
one-sided error variance (see section ‘Details’).
vhet
A one-part formula to account for heteroscedasticity in the
two-sided error variance (see section ‘Details’).
thet
A one-part formula to account for technological heterogeneity in the
construction of the classes.
logDepVar
Logical. Informs whether the dependent variable is logged
(TRUE) or not (FALSE). Default = TRUE.
data
The data frame containing the data.
subset
An optional vector specifying a subset of observations to be
used in the optimization process.
S
If S = 1 (default), a production (profit) frontier is
estimated: ε_i = v_i-u_i. If S = -1, a cost frontier is
estimated: ε_i = v_i+u_i.
udist
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.
start
Numeric vector. Optional starting values for the maximum
likelihood (ML) estimation.
lcmClasses
Number of classes to be estimated (default = 2).
A maximum of five classes can be estimated.
method
Optimization algorithm used for the estimation.
Default = "bfgs". 9 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)
-
"ucminf", implements a quasi-Newton type 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)
hessianType
Integer. If 1 (default), analytic Hessian is
returned for all the distributions except "gamma",
"lognormal" and "weibull" for which the numeric Hessian is
returned. If 2, bhhh Hessian is estimated (g'g). If 3,
robust Hessian is computed (H^{-1}GH^{-1}).
itermax
Maximum number of iterations allowed for optimization.
Default = 2000.
printInfo
Logical. Print information during optimization. Default =
FALSE.
tol
Numeric. Convergence tolerance. Default = 1e-12.
gradtol
Numeric. Convergence tolerance for gradient. Default = 1e-06.
stepmax
Numeric. Step max for ucminf algorithm. Default = 0.1.
qac
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.
initStart
Logical. If TRUE, the model is jump-started using an
alternative algorithm ("nlminb") within certain bounds. Default =
FALSE.
initAlg
Character. Algorithm used to jump-start the latent class
model. Only "nlminb" is currently available.
initIter
Maximum number of iterations for the algorihtm when
initStart = TRUE. Default = 100.
initFactorLB
A numeric value indicating by which factor the
starting value should be multiplied to define the lower bounds
for the jump-start algorithm. Default = 0.5.
initFactorUB
A numeric value indicating by which factor the
starting value should be multiplied to define the upper bounds
for the jump-start algorithm. Default = 1.5.
Details
LCM is an estimation of a finite mixture of production functions:
y_i = α_j + x'_iβ_j + v_{i|j} - Su_{i|j}
ε_{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 σ_{u}^2, and v is the two-sided error term with
variance σ_{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}{√{σ_{u|j}^2 + σ_{v|j}^2}}
φ≤ft(\frac{Sε_{i|j}}{√{σ_{u|j}^2 +
σ_{v|j}^2}}\right) Φ≤ft(\frac{μ_{i*|j}}{σ_{*|j}}\right)
where
μ_{i*|j}=\frac{- Sε_{i|j}σ_{u|j}^2}{σ_{u|j}^2 +
σ_{v|j}^2}
and
σ_*^2 = \frac{σ_{u|j}^2 σ_{v|j}^2}{σ_{u|j}^2 +
σ_{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:
π(i,j) = \frac{\exp{(θ_j'Z_h)}}{∑_{m=1}^{J}\exp{(θ_m'Z_h)}}
with Z_h the covariates, θ the coefficients estimated for
the covariates, and \exp(θ_J'Z_h)=1.
The unconditional likelihood of observation i is simply the average over
the J classes:
P(i) = ∑_{m=1}^{J}π(i,m)P(i|m)
The number of classes can be retained 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≤ft(j|i\right)=\frac{P≤ft(i|j\right)π≤ft(i,
j\right)}{∑_{m=1}^JP≤ft(i|m\right)π≤ft(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:
σ^2_{u|j} = \exp{(δ_j'Z_u)} and σ^2_{v|j} = \exp{(φ_j'Z_v)},
where Z_u and Z_v are the heteroscedasticity variables (inefficiency
drivers in the case of Z_u) and δ and φ the coefficients.
In the case of heterogeneity in the truncated mean μ, it is modelled
as μ=ω'Z_{μ}.
Value
lcmcross returns a list of class 'lcmcross' containing
the following elements:
call
The matched call.
formula
Multi parts formula describing the estimated model.
S
The argument 'S'. See the section ‘Arguments’.
typeSfa
Character string. "Latent Class Production/Profit Frontier,
e = v - u" when S = 1 and "Latent Class Cost Frontier, e = v + u" when S = -1.
Nobs
Number of observations used for optimization.
nXvar
Number of main explanatory variables.
nZHvar
Number of variables in the logit specification of the
finite mixture model (i.e. number of covariates).
logDepVar
The argument 'logDepVar'. See the section ‘Arguments’.
nuZUvar
Number of variables explaining heteroscedasticity in
the one-sided error term.
nvZVvar
Number of variables explaining heteroscedasticity in
the two-sided error term.
nParm
Total number of parameters estimated.
udist
The argument 'udist'. See the section ‘Arguments’.
startVal
Numeric vector. Starting value for ML estimation.
dataTable
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.
InitHalf
When start = NULL. 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.
optType
The optimization algorithm used.
nIter
Number of iterations of the ML estimation.
optStatus
An optimization algorithm termination message.
startLoglik
Log-likelihood at the starting values.
nClasses
The number of classes estimated.
mlLoglik
Log-likelihood value of the ML estimation.
mlParam
Numeric vector. Parameters obtained from ML estimation.
gradient
Numeric vector. Each variable gradient of the ML
estimation.
gradL_OBS
Matrix. Each variable individual observation gradient of
the ML estimation.
gradientNorm
Numeric. Gradient norm of the ML estimation.
invHessian
The covariance matrix of the parameters obtained from
the ML estimation.
hessianType
The argument 'hessianType'. See the section ‘Arguments’.
mlDate
Date and time of the estimated model.
Note
In the case of panel data, lcmcross estimates a pooled cross-section
where the probability of belonging to a class a priori is not permanent (not fixed over time).
Author(s)
K Hervé Dakpo, Yann Desjeux and Laure Latruffe
References
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.
See Also
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.
Examples
## Using data on eighty-two countries production (DGP)
# LCM Cobb Douglas (production function) half normal distribution
# Intercept and initStat used as separating variables
cb_2c_h1 <- lcmcross(formula = ly ~ lk + ll + yr, thet = ~initStat, data = worldprod)
summary(cb_2c_h1)
# summary of the initial ML model
summary(cb_2c_h1$InitHalf)
# same result by jump-starting the estimation
cb_2c_h2 <- lcmcross(formula = ly ~ lk + ll + yr, data = worldprod, initStart = TRUE)
summary(cb_2c_h2)
# Only the intercept is used as the separating variable and only variable
# initStat is used as inefficiency driver
cb_2c_h3 <- lcmcross(formula = ly ~ lk + ll + yr, uhet = ~initStat, data = worldprod)
summary(cb_2c_h3)
sfaR documentation built on May 3, 2022, 3 p.m.
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| lcmcross | R Documentation |
Latent class stochastic frontier using cross-section data
Description
lcmcross 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-section 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. Nine 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.
Usage
lcmcross(formula, uhet, vhet, thet, logDepVar = TRUE, data, subset, S = 1, udist = "hnormal", start = NULL, lcmClasses = 2, method = "bfgs", hessianType = 1, itermax = 2000, printInfo = FALSE, tol = 1e-12, gradtol = 1e-06, stepmax = 0.1, qac = "marquardt", initStart = FALSE, initAlg = "nlminb", initIter = 100, initFactorLB = 0.5, initFactorUB = 1.5)
Arguments
formula |
A symbolic description of the model to be estimated based on
the generic function |
uhet |
A one-part formula to account for heteroscedasticity in the one-sided error variance (see section ‘Details’). |
vhet |
A one-part formula to account for heteroscedasticity in the two-sided error variance (see section ‘Details’). |
thet |
A one-part formula to account for technological heterogeneity in the construction of the classes. |
logDepVar |
Logical. Informs whether the dependent variable is logged
( |
data |
The data frame containing the data. |
subset |
An optional vector specifying a subset of observations to be used in the optimization process. |
S |
If |
udist |
Character string. Distribution specification for the one-sided
error term. Only the half normal distribution |
start |
Numeric vector. Optional starting values for the maximum likelihood (ML) estimation. |
lcmClasses |
Number of classes to be estimated (default = |
method |
Optimization algorithm used for the estimation.
Default =
|
hessianType |
Integer. If |
itermax |
Maximum number of iterations allowed for optimization.
Default = |
printInfo |
Logical. Print information during optimization. Default =
|
tol |
Numeric. Convergence tolerance. Default = |
gradtol |
Numeric. Convergence tolerance for gradient. Default = |
stepmax |
Numeric. Step max for |
qac |
Character. Quadratic Approximation Correction for |
initStart |
Logical. If |
initAlg |
Character. Algorithm used to jump-start the latent class
model. Only |
initIter |
Maximum number of iterations for the algorihtm when
|
initFactorLB |
A numeric value indicating by which factor the
starting value should be multiplied to define the lower bounds
for the jump-start algorithm. Default = |
initFactorUB |
A numeric value indicating by which factor the
starting value should be multiplied to define the upper bounds
for the jump-start algorithm. Default = |
Details
LCM is an estimation of a finite mixture of production functions:
y_i = α_j + x'_iβ_j + v_{i|j} - Su_{i|j}
ε_{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 σ_{u}^2, and v is the two-sided error term with variance σ_{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}{√{σ_{u|j}^2 + σ_{v|j}^2}} φ≤ft(\frac{Sε_{i|j}}{√{σ_{u|j}^2 + σ_{v|j}^2}}\right) Φ≤ft(\frac{μ_{i*|j}}{σ_{*|j}}\right)
where
μ_{i*|j}=\frac{- Sε_{i|j}σ_{u|j}^2}{σ_{u|j}^2 + σ_{v|j}^2}
and
σ_*^2 = \frac{σ_{u|j}^2 σ_{v|j}^2}{σ_{u|j}^2 + σ_{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:
π(i,j) = \frac{\exp{(θ_j'Z_h)}}{∑_{m=1}^{J}\exp{(θ_m'Z_h)}}
with Z_h the covariates, θ the coefficients estimated for the covariates, and \exp(θ_J'Z_h)=1.
The unconditional likelihood of observation i is simply the average over the J classes:
P(i) = ∑_{m=1}^{J}π(i,m)P(i|m)
The number of classes can be retained 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≤ft(j|i\right)=\frac{P≤ft(i|j\right)π≤ft(i, j\right)}{∑_{m=1}^JP≤ft(i|m\right)π≤ft(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: σ^2_{u|j} = \exp{(δ_j'Z_u)} and σ^2_{v|j} = \exp{(φ_j'Z_v)}, where Z_u and Z_v are the heteroscedasticity variables (inefficiency drivers in the case of Z_u) and δ and φ the coefficients. In the case of heterogeneity in the truncated mean μ, it is modelled as μ=ω'Z_{μ}.
Value
lcmcross returns a list of class 'lcmcross' containing
the following elements:
call |
The matched call. |
formula |
Multi parts formula describing the estimated model. |
S |
The argument |
typeSfa |
Character string. "Latent Class Production/Profit Frontier,
e = v - u" when |
Nobs |
Number of observations used for optimization. |
nXvar |
Number of main explanatory variables. |
nZHvar |
Number of variables in the logit specification of the finite mixture model (i.e. number of covariates). |
logDepVar |
The argument |
nuZUvar |
Number of variables explaining heteroscedasticity in the one-sided error term. |
nvZVvar |
Number of variables explaining heteroscedasticity in the two-sided error term. |
nParm |
Total number of parameters estimated. |
udist |
The argument |
startVal |
Numeric vector. Starting value for ML estimation. |
dataTable |
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. |
InitHalf |
When |
optType |
The optimization algorithm used. |
nIter |
Number of iterations of the ML estimation. |
optStatus |
An optimization algorithm termination message. |
startLoglik |
Log-likelihood at the starting values. |
nClasses |
The number of classes estimated. |
mlLoglik |
Log-likelihood value of the ML estimation. |
mlParam |
Numeric vector. Parameters obtained from ML estimation. |
gradient |
Numeric vector. Each variable gradient of the ML estimation. |
gradL_OBS |
Matrix. Each variable individual observation gradient of the ML estimation. |
gradientNorm |
Numeric. Gradient norm of the ML estimation. |
invHessian |
The covariance matrix of the parameters obtained from the ML estimation. |
hessianType |
The argument |
mlDate |
Date and time of the estimated model. |
Note
In the case of panel data, lcmcross estimates a pooled cross-section
where the probability of belonging to a class a priori is not permanent (not fixed over time).
Author(s)
K Hervé Dakpo, Yann Desjeux and Laure Latruffe
References
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.
See Also
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.
Examples
## Using data on eighty-two countries production (DGP) # LCM Cobb Douglas (production function) half normal distribution # Intercept and initStat used as separating variables cb_2c_h1 <- lcmcross(formula = ly ~ lk + ll + yr, thet = ~initStat, data = worldprod) summary(cb_2c_h1) # summary of the initial ML model summary(cb_2c_h1$InitHalf) # same result by jump-starting the estimation cb_2c_h2 <- lcmcross(formula = ly ~ lk + ll + yr, data = worldprod, initStart = TRUE) summary(cb_2c_h2) # Only the intercept is used as the separating variable and only variable # initStat is used as inefficiency driver cb_2c_h3 <- lcmcross(formula = ly ~ lk + ll + yr, uhet = ~initStat, data = worldprod) summary(cb_2c_h3)
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