Linear Regression via MLE — lm.mle • snreg

lm.mle fits a linear regression model by maximum likelihood, allowing for optional multiplicative heteroskedasticity in the disturbance variance via a log-linear specification provided through ln.var.v.

Usage

lm.mle(
  formula,
  data,
  subset,
  ln.var.v = NULL,
  technique = c("bfgs"),
  lmtol = 1e-05,
  reltol = 1e-12,
  maxit = 199,
  optim.report = 1,
  optim.trace = 10,
  print.level = 3,
  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 the x's are regressors.
data: an optional data.frame containing the variables referenced 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.
ln.var.v: optional one-sided formula; e.g. ln.var.v ~ z1 + z2. When provided, the error variance is modeled as $\log(\sigma_i^2) = w_i^\top \gamma_v$. If NULL, the variance is homoskedastic.
technique: character vector specifying the preferred optimization routine(s) in order of preference. Recognized keywords (for future implementation) include "bfgs" "bhhh", "nm" (Nelder–Mead), "bfgs", and "cg". Default is "bfgs". This scaffold records but does not execute the chosen routine.
lmtol: numeric. Convergence tolerance based on scaled gradient (when applicable). Default 1e-5.
reltol: numeric. Relative convergence tolerance for likelihood maximization. Default 1e-12.
maxit: integer. Maximum number of iterations for the optimizer. Default 199.
optim.report: integer. Verbosity level for reporting progress (if implemented). Default 1.
optim.trace: integer. Trace level for optimization (if implemented). Default 1.
print.level: integer. Printing level for summaries. Default 3.
digits: integer. Number of digits for printing. Default 4.
only.data: logical. If TRUE, returns only constructed data/matrices without estimation. Default FALSE.
...: additional arguments reserved for future methods (e.g., bounds, penalties).

Value

A list of class "snreg" containing (and extending) the fields returned by optim:

par — numeric vector of the MLE parameter estimates.
value — numeric scalar: maximized log-likelihood value.
ll — numeric scalar: maximized log-likelihood value.
counts, convergence, message — standard optim outputs.
hessian — the observed Hessian at the solution (as returned by optim(hessian=TRUE)).
coef — named numeric vector equal to par (estimates).
vcov — variance–covariance matrix, computed as solve(-hessian).
sds — standard errors, sqrt(diag(vcov)).
ctab — coefficient table with columns: Estimate, Std.Err, Z value, Pr(>z).
esample — logical vector indicating the observations used in estimation.
n — scalar, number of observations in the estimation sample.

The object inherits the default optim components and is assigned class "snreg".

Details

Linear Model by Maximum Likelihood (with optional heteroskedasticity)

The model is $$y_i = x_i^\top \beta + \varepsilon_i,\quad \varepsilon_i \sim \mathcal{N}(0, \sigma_i^2).$$ When ln.var.v is supplied, the variance follows $$\log(\sigma_i^2) = w_i^\top \gamma_v,$$ otherwise $\sigma_i^2 = \sigma^2$ is constant (homoskedastic).

This function:

Builds the model frame and X, y.
Builds Zv for the log-variance index when ln.var.v is provided.
Returns a structured object with placeholders for coef, vcov, loglik.

Insert your MLE engine to estimate $\beta$, and (optionally) $\sigma^2$ or $\gamma_v$; compute standard errors via AIM/OPG as required by vcetype.

Examples

if (FALSE) { # \dontrun{

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 (ln.var.v = NULL)
# -------------------------------------------------------------
formSV <- NULL

m1 <- lm.mle(
  formula   = spe.tl,
  data      = banks07,
  ln.var.v  = formSV
)

coef(m1)


# -------------------------------------------------------------
# Specification 2: heteroskedastic noise (variance depends on TA)
# -------------------------------------------------------------
formSV <- ~ log(TA)

m2 <- lm.mle(
  formula   = spe.tl,
  data      = banks07,
  ln.var.v  = formSV
)

coef(m2)

} # }