crch: Censored and Truncated Regression with Conditional Heteroscedasticy

Description

Fitting censored (tobit) or truncated regression models with conditional heteroscedasticy.

Usage

crch(formula, data, subset, na.action, weights, offset, 
  link.scale = c("log", "identity", "quadratic"),
  dist = c("gaussian", "logistic", "student"), df = NULL, 
  left = -Inf, right = Inf, truncated = FALSE, 
  type = c("ml", "crps"), control = crch.control(...), 
  model = TRUE, x = FALSE, y = FALSE, ...)
trch(formula, data, subset, na.action, weights, offset, 
  link.scale = c("log", "identity", "quadratic"),
  dist = c("gaussian", "logistic", "student"), df = NULL, 
  left = -Inf, right = Inf, truncated = TRUE, 
  type = c("ml", "crps"), control = crch.control(...), 
  model = TRUE, x = FALSE, y = FALSE, ...)
crch.fit(x, z, y, left, right, truncated = FALSE, dist = "gaussian",
  df = NULL, link.scale = "log", type = "ml", weights = NULL, offset = NULL, 
  control = crch.control())

Value

An object of class "crch" or "crch.boost", i.e., a list with the following elements.

coefficients: list of coefficients for location, scale, and df. Scale and df coefficients are in log-scale.
df: if dist = "student": degrees of freedom of student-t distribution. else NULL.
residuals: the residuals, that is response minus fitted values.
fitted.values: list of fitted location and scale parameters.
dist: assumed distribution for the dependent variable y.
cens: list of censoring points.
optim: output from optimization from optim.
method: optimization method used for optim.
type: used loss function (maximum likelihood or minimum CRPS).
control: list of control parameters passed to optim
start: starting values of coefficients used in the optimization.
weights: case weights used for fitting.
offset: list of offsets for location and scale.
n: number of observations.
nobs: number of observations with non-zero weights.
loglik: log-likelihood.
vcov: covariance matrix.
link: a list with element "scale" containing the link objects for the scale model.
truncated: logical indicating wheter a truncated model has been fitted.
converged: logical variable whether optimization has converged or not.
iterations: number of iterations in optimization.
call: function call.
formula: the formula supplied.
terms: the terms objects used.
levels: list of levels of the factors used in fitting for location and scale respectively.
contrasts: (where relevant) the contrasts used.
y: if requested, the response used.
x: if requested, the model matrix used.
model: if requested, the model frame used.
stepsize, mstop, mstopopt, standardize: return values of boosting optimization. See crch.boost for details.

Arguments

formula: a formula expression of the form y ~ x | z where y is the response and x and z are regressor variables for the location and the scale of the fitted distribution respectively.
data: an optional data frame containing the variables occurring in the formulas.
subset: an optional vector specifying a subset of observations to be used for fitting.
na.action: a function which indicates what should happen when the data contain NAs.
weights: optional case weights in fitting.
offset: optional numeric vector with a priori known component to be included in the linear predictor for the location. For crch.fit, offset can also be a list of 2 offsets used for the location and scale respectively.
link.scale: character specification of the link function in the scale model. Currently, "identity", "log", "quadratic" are supported. The default is "log". Alternatively, an object of class "link-glm" can be supplied.
dist: assumed distribution for the dependent variable y.
df: optional degrees of freedom for dist="student". If omitted the degrees of freedom are estimated.
left: left limit for the censored dependent variable y. If set to -Inf, y is assumed not to be left-censored.
right: right limit for the censored dependent variable y. If set to Inf, y is assumed not to be right-censored.
truncated: logical. If TRUE truncated model is fitted with left and right interpreted as truncation points, If FALSE censored model is fitted. Default is FALSE
type: loss function to be optimized. Can be either "ml" for maximum likelihood (default) or "crps" for minimum continuous ranked probability score (CRPS).
control: a list of control parameters passed to optim or to the internal boosting algorithm if control=crch.boost(). Default is crch.control().
model: logical. If TRUE model frame is included as a component of the returned value.
x, y: for crch: logical. If TRUE the model matrix and response vector used for fitting are returned as components of the returned value. for crch.fit: x is a design matrix with regressors for the location and y is a vector of observations.
z: a design matrix with regressors for the scale.
...: arguments to be used to form the default control argument if it is not supplied directly.

Details

crch fits censored (tobit) or truncated regression models with conditional heteroscedasticy with maximum likelihood estimation. Student-t, Gaussian, and logistic distributions can be fitted to left- and/or right censored or truncated responses. Different regressors can be used to model the location and the scale of this distribution. If control=crch.boost() optimization is performed by boosting.

trch is a wrapper function for crch with default truncated = TRUE.

crch.fit is the lower level function where the actual fitting takes place.

References

Messner JW, Mayr GJ, Zeileis A (2016). Heteroscedastic Censored and Truncated Regression with crch. The R Journal, 3(1), 173--181. tools:::Rd_expr_doi("10.32614/RJ-2016-012")

Messner JW, Zeileis A, Broecker J, Mayr GJ (2014). Probabilistic Wind Power Forecasts with an Inverse Power Curve Transformation and Censored Regression. Wind Energy, 17(11), 1753--1766. tools:::Rd_expr_doi("10.1002/we.1666")

Examples

Run this code

data("RainIbk", package = "crch")
## mean and standard deviation of square root transformed ensemble forecasts
RainIbk$sqrtensmean <- 
  apply(sqrt(RainIbk[,grep('^rainfc',names(RainIbk))]), 1, mean)
RainIbk$sqrtenssd <- 
  apply(sqrt(RainIbk[,grep('^rainfc',names(RainIbk))]), 1, sd)

## fit linear regression model with Gaussian distribution 
CRCH <- crch(sqrt(rain) ~ sqrtensmean, data = RainIbk, dist = "gaussian")
## same as lm?
all.equal(
  coef(lm(sqrt(rain) ~ sqrtensmean, data = RainIbk)),
  head(coef(CRCH), -1),
  tol = 1e-6)

## print
CRCH
## summary
summary(CRCH)

## left censored regression model with censoring point 0:
CRCH2 <- crch(sqrt(rain) ~ sqrtensmean, data = RainIbk, 
  dist = "gaussian", left = 0)

## left censored regression model with censoring point 0 and 
## conditional heteroscedasticy:
CRCH3 <- crch(sqrt(rain) ~ sqrtensmean|sqrtenssd, data = RainIbk, 
  dist = "gaussian",  left = 0)

## left censored regression model with censoring point 0 and 
## conditional heteroscedasticy with logistic distribution:
CRCH4 <- crch(sqrt(rain) ~ sqrtensmean|sqrtenssd, data = RainIbk, 
  dist = "logistic", left = 0)

## compare AIC 
AIC(CRCH, CRCH2, CRCH3, CRCH4)

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