snsf: Stochastic Frontier Model with a Skew-Normally Distributed...

In snreg: Regression with Skew-Normally Distributed Error Term

Stochastic Frontier Model with a Skew-Normally Distributed Error Term

Description

snsf performs maximum likelihood estimation of the parameters and technical or cost efficiencies in a Stochastic Frontier Model with a skew-normally distributed error term.

Usage

snsf(
  formula,
  data,
  subset,
  distribution = "e",
  prod = TRUE,
  start.val = NULL,
  init.sk = NULL,
  ln.var.u = NULL,
  ln.var.v = NULL,
  skew.v = NULL,
  mean.u = NULL,
  technique = c("nr"),
  vcetype = c("aim"),
  optim.method = "bfgs",
  optim.report = 1,
  optim.trace = 1,
  reltol = 1e-12,
  optim.reltol = 1e-12,
  lmtol = 1e-05,
  maxit = 199,
  print.level = 0,
  threads = 1,
  only.data = FALSE,
  digits = 4,
  ...
)

Arguments

formula	an object of class formula specifying the frontier: a typical model is y ~ x1 + ..., where y is the log of output (or total cost), and x's are inputs (or outputs and input prices, in logs). See Details.
data	an optional data.frame containing the variables in formula. If not found in data, variables are taken from environment(formula).
subset	an optional logical or numeric vector specifying a subset of observations for which the model is estimated and efficiencies are computed.
distribution	character scalar specifying the distribution of the inefficiency term: default "e" (exponential). "h" (half-normal) and "t" (truncated normal) to be implemented.
prod	logical. If TRUE, estimates correspond to a stochastic production frontier and technical efficiencies are returned; if FALSE, estimates correspond to a stochastic cost frontier and cost efficiencies are returned. Default is TRUE.
start.val	optional numeric vector of starting values for the optimizer.
init.sk	numeric. Initial value for the skewness parameter of the noise component; default is 0.5.
ln.var.u	optional one-sided formula; e.g. ln.var.u = ~ z3 + z4. Specifies exogenous variables entering the (log) variance of the inefficiency component. If NULL, the inefficiency variance is homoskedastic, i.e., \sigma_{u0}^2 = \exp(\gamma_{u0}[0]).
ln.var.v	optional one-sided formula; e.g. ln.var.v = ~ z1 + z2. Specifies exogenous variables entering the (log) variance of the random noise component. If NULL, the noise variance is homoskedastic, i.e., \sigma_{v0}^2 = \exp(\gamma_{v0}[0]).
skew.v	optional one-sided formula; e.g. skew.v = ~ z5 + z6. Allows the skewness of the noise to depend linearly on exogenous variables. If NULL, the skewness is constant across units.
mean.u	optional one-sided formula; e.g. mean.u = ~ z7 + z8. Specifies whether the mean of the pre-truncated normal distribution of the inefficiency term is a linear function of exogenous variables. In cross-sectional models, used only when distribution = "t". If NULL, the mean is constant across units. To be implemented.
technique	Optimization technique to use.
vcetype	Type of variance-covariance matrix estimation.
optim.method	character. Method passed to stats::optim when optim = TRUE. Default is "bfgs".
optim.report	integer. Verbosity level for reporting during optimization (if implemented). Default is 1.
optim.trace	integer. Trace level for optimization (if implemented). Default is 1.
reltol	numeric. Relative convergence tolerance used when maximizing the log-likelihood.
optim.reltol	numeric. Relative tolerance specifically for optim; default 1e-8.
lmtol	numeric. Convergence tolerance based on the scaled gradient (when applicable). Default is 1e-5.
maxit	numeric. Maximum number of iterations for the optimizer. Default is 199.
print.level	integer. Printing level: 1—estimation results; 2—optimization details; 3—summary of (cost/technical) efficiencies; 4—unit-specific point and interval estimates of efficiencies. Default is 0.
threads	Number of threads for parallel computation.
only.data	Logical; if TRUE, return only processed data.
digits	integer. Number of digits for displaying estimates and efficiencies. Default is 4.
...	Additional arguments (currently unused).
optim	logical. If TRUE, estimation proceeds via stats::optim; if FALSE, an internal routine (if provided) would be used. Default is FALSE.
report	Reporting level for optimization progress.
trace	Logical; if TRUE, trace optimization progress.

Details

Stochastic Frontier Model with a Skew-Normally Distributed Error Term

Models for snsf are specified symbolically. A typical model has the form y ~ x1 + ..., where y represents the logarithm of outputs or total costs and {x1, ...} is a set of inputs (for production) or outputs and input prices (for cost), all typically in logs.

Options ln.var.u and ln.var.v allow for multiplicative heteroskedasticity in the inefficiency and/or noise components; i.e., their variances can be modeled as exponential functions of exogenous variables (including an intercept), as in Caudill et al. (1995).

Value

An object of class "snreg" with maximum-likelihood estimates and diagnostics:

par

Numeric vector of ML parameter estimates at the optimum.

coef

Named numeric vector equal to par.

vcov

Variance–covariance matrix of the estimates.

sds

Standard errors, sqrt(diag(vcov)).

ctab

Coefficient table with columns Coef., SE, z, P>|z|.

ll

Maximized log-likelihood value.

hessian

(When computed) Observed Hessian or OPG used to form vcov.

value

(Optim-only, before aliasing) Objective value from optim.

counts

(Optim-only) Iteration and evaluation counts from optim.

convergence

Convergence code).

message

(Optim-only) Message returned by optim, if any.

gradient

(NR-only) Gradient at the solution.

gg

(NR-only) Gradient-related diagnostic.

gHg

(NR-only) Newton-step diagnostic.

theta_rel_ch

(NR-only) Relative parameter change metric across iterations.

resid

Regression residuals.

RSS

Residual sum of squares crossprod(resid).

shat2

Residual variance estimate var(resid).

shat

Residual standard deviation sqrt(shat2).

aic

Akaike Information Criterion.

bic

Bayesian Information Criterion.

Mallows

Mallows' C_p-like statistic.

u

Estimated inefficiency term (vector). Returned for models with an inefficiency component (e.g., exponential).

eff

Efficiency scores exp(-u) (technical or cost, depending on prod).

sv

Estimated (possibly unit-specific) standard deviation of the noise term.

su

Estimated (possibly unit-specific) standard deviation or scale of the inefficiency term. For exponential models.

skewness

Estimated skewness index (e.g., from the skewness equation).

esample

Logical vector marking observations used in estimation.

n

Number of observations used.

The returned object has class "snreg".

Author(s)

Oleg Badunenko <Oleg.Badunenko.brunel.ac.uk>

References

Badunenko, O., & Henderson, D. J. (2023). Production analysis with asymmetric noise. Journal of Productivity Analysis, 61(1), 1–18. https://doi.org/10.1007/s11123-023-00680-5

Caudill, S. B., Ford, J. M., & Gropper, D. M. (1995). Frontier estimation and firm-specific inefficiency measures in the presence of heteroskedasticity. Journal of Business & Economic Statistics, 13(1), 105–111.

See Also

sf, snreg, lm.mle

Examples

library(snreg)

data("banks07")

# Translog cost function specification

myprod <- FALSE

spe.tl <- log(TC) ~ (log(Y1) + log(Y2) + log(W1) + log(W2))^2 +
  I(0.5 * log(Y1)^2) + I(0.5 * log(Y2)^2) +
  I(0.5 * log(W1)^2) + I(0.5 * log(W2)^2)

# Specification 1: homoskedastic noise, skewness, inefficiency

formSV <- NULL   # variance equation; constant variance
formSK <- NULL   # skewness equation; constant skewness
formSU <- NULL   # inefficiency variance equation; constant variance

m1 <- snsf(
  formula  = spe.tl,
  data     = banks07,
  prod     = myprod,
  ln.var.v = formSV,
  skew.v   = formSK,
  ln.var.u = formSU
)

# Specification 2: heteroskedastic

formSV <- ~ log(TA)      # variance equation; heteroskedastic noise (variance depends on TA)
formSK <- ~ ER           # skewness equation; with determinants (skewness is determined by ER)
formSU <- ~ LA + ER      # inefficiency variance equation; heteroskedastic noise of inefficiency 
# ([variance of] inefficiency depends on LA and ER)#' 
m2 <- snsf(
  formula  = spe.tl,
  data     = banks07,
  prod     = myprod,
  ln.var.v = formSV,
  skew.v   = formSK,
  ln.var.u = formSU
)

snreg documentation built on Feb. 6, 2026, 5:08 p.m.