snreg: Linear Regression with Skew-Normal Errors
In snreg: Regression with Skew-Normally Distributed Error Term

snreg

R Documentation

Linear Regression with Skew-Normal Errors

Description

snreg fits a linear regression model where the disturbance term follows a skew-normal distribution. The function supports multiplicative heteroskedasticity of the noise variance via a log-linear specification (ln.var.v) and allows the skewness parameter to vary linearly with exogenous variables (skew.v).

Usage

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

Arguments

`formula`	an object of class `formula` specifying the regression: typically `y ~ x1 + ...`, where `y` is the dependent variable and `x`'s are regressors.
`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 the subset of observations to be used in estimation.
`init.sk`	numeric. Initial value for the (global) skewness parameter of the noise; can be `NULL` if `skew.v` is supplied with its own coefficients to initialize.
`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.
`skew.v`	optional one-sided formula; e.g. `skew.v ~ z3 + z4`. Specifies exogenous variables determining the skewness of the noise via a linear index; if `NULL`, the skewness is constant (scalar).
`start.val`	optional numeric vector of starting values for all free parameters (regression coefficients, variance/heteroskedasticity parameters, skewness parameters).
`technique`	character vector giving the preferred maximization routine(s) in order of preference. Currently recognized keywords include `"nr"` (Newton–Raphson), `"bhhh"`, `"nm"` (Nelder–Mead), `"bfgs"`, `"cg"`. This scaffold does not implement them yet, but records the choice.
`vcetype`	character specifying the variance-covariance estimator type: `"aim"` for the approximated information matrix or `"opg"` for the outer product of gradients. Default is `"aim"`.
`lmtol`	numeric. Convergence tolerance based on the scaled gradient (if applicable). Default is `1e-5`.
`reltol`	numeric. Relative convergence tolerance for likelihood maximization. Default is `1e-12`.
`maxit`	integer. Maximum number of iterations for the optimizer. Default is `199`.
`optim.report`	integer. Verbosity for reporting progress (if implemented). Default is `1`.
`optim.trace`	integer. If positive, tracing information is printed (if implemented). Default is `1`.
`print.level`	integer. Printing level for summaries: `1`—print estimation results; `2`—print optimization details; `3`—print compact summary. Default `3`.
`digits`	integer. Number of digits for printing. Default `4`.
`only.data`	logical. If `TRUE`, the function returns only the constructed model matrices and design sets (no estimation). Default `FALSE`.
`...`	additional arguments reserved for future methods (e.g., box constraints).

Details

Linear Regression with Skew-Normal Errors

The model is

y_i = x_i^\top \beta + \varepsilon_i,\quad \varepsilon_i \sim SN(0, \sigma_i^2, \alpha_i),

where SN denotes the skew-normal distribution (Azzalini).

Heteroskedasticity in the noise variance (if specified via ln.var.v) is modeled as

\log(\sigma_i^2) = w_i^\top \gamma_v,

and the (optional) covariate-driven skewness (if specified via skew.v) as

\alpha_i = s_i^\top \delta.

This function constructs the model frame and design matrices for \beta, \gamma_v, and \delta, and is designed to be paired with a maximum likelihood routine to estimate parameters and (optionally) their asymptotic covariance via either AIM or OPG.

Value

An object of class "snreg" containing the maximum-likelihood results and, depending on the optimization routine, additional diagnostics:

par: Numeric vector of parameter estimates at the optimum.
coef: Named numeric vector equal to par.
vcov: Variance–covariance matrix of the estimates.
sds: Standard errors, computed as sqrt(diag(vcov)).
ctab: Coefficient table with columns: Estimate, Std.Err, Z value, Pr(>z).
RSS: Residual sum of squares.
esample: Logical vector indicating which observations were used in estimation.
n: Number of observations used in the estimation sample.
skewness: Vector of the fitted skewness index.
hessian: (BFGS only) Observed Hessian at the optimum. If vcetype == "opg", this is set to the negative outer product of the individual gradients; otherwise a numerical Hessian is computed.
value: (BFGS only) Objective value returned by optim. With control$fnscale = -1, this equals the maximized log-likelihood.
counts: (BFGS only) Number of iterations / function evaluations returned by optim.
convergence: (BFGS only) Convergence code from optim.
message: (BFGS only) Additional optim message, if any.
ll: Maximized log-likelihood value.
gradient: (NR only) Gradient at the solution.
gg: (NR only) Optional gradient-related diagnostic.
gHg: (NR only) Optional Newton-step diagnostic.
theta_rel_ch: (NR only) Relative parameter change metric across iterations.

The returned object has class "snreg".

References

Azzalini, A. (1985). A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 12(2), 171–178.

Azzalini, A., & Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press.

Examples

library(snreg)

data("banks07")
head(banks07)

# Translog cost function specification

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 and skewness

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

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

summary(m1)

# 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)

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

summary(m2)

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