Stochastic Frontier Model with a Skew-Normally Distributed Error Term

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.report = 1,
  optim.trace = 1,
  reltol = 1e-12,
  lmtol = 1e-05,
  maxit = 199,
  print.level = 3,
  report = 1,
  trace = 1,
  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.

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 with optim. The algorithm stops if it cannot reduce the objective by a factor of reltol * (abs(val) + reltol) at a step. Default is sqrt(.Machine$double.eps).

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 3.

digits

integer. Number of digits for displaying estimates and efficiencies. Default is 4.

...

additional arguments passed to internal methods or to optim, as relevant (e.g., cost.eff.less.one = TRUE for cost-frontier conventions).

optim

logical. If TRUE, estimation proceeds via stats::optim; if FALSE, an internal routine (if provided) would be used. Default is FALSE.

optim.method

character. Method passed to stats::optim when optim = TRUE. Default is "bfgs".

optim.reltol

numeric. Relative tolerance specifically for optim; default 1e-8.

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".

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).

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

Author

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

Examples

if (FALSE) { # \dontrun{

library(snreg)

data("banks07")
head(banks07)

myprod <- FALSE

# Translog cost function
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 & symmetric
# -------------------------------------------------------------
formSV <- NULL   # variance equation
formSK <- NULL   # skewness equation
formSU <- NULL   # inefficiency equation (unused here)

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

coef(m1)


# -------------------------------------------------------------
# Specification 2: heteroskedastic + skewed noise
# -------------------------------------------------------------
formSV <- ~ log(TA)      # heteroskedastic variance
formSK <- ~ ER           # skewness driver
formSU <- ~ LA + ER      # inefficiency

m2 <- snsf(
  formula  = spe.tl,
  data     = banks07,
  prod     = myprod,
  ln.var.v = formSV,
  skew.v   = formSK
)

coef(m2)

} # }