crch – 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())

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 `NA`s.
`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.

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.

References

Messner JW, Mayr GJ, Zeileis A (2016). Heteroscedastic Censored and Truncated Regression with crch. The R Journal, 3(1), 173–181. 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. doi:10.1002/we.1666

Examples

library("crch")

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)

[1] TRUE

## print
CRCH


Call:
crch(formula = sqrt(rain) ~ sqrtensmean, data = RainIbk, dist = "gaussian")

Coefficients (location model):
(Intercept)  sqrtensmean  
     0.1468       0.5817  

Coefficients (scale model with log link):
(Intercept)  
     0.4945  

Distribution: gaussian

## summary
summary(CRCH)


Call:
crch(formula = sqrt(rain) ~ sqrtensmean, data = RainIbk, dist = "gaussian")

Standardized residuals:
    Min      1Q  Median      3Q     Max 
-2.4256 -0.7120 -0.1562  0.5786  4.8408 

Coefficients (location model):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  0.14683    0.05456   2.691  0.00713 ** 
sqrtensmean  0.58173    0.01540  37.781  < 2e-16 ***

Coefficients (scale model with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  0.49454    0.01003   49.31   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Distribution: gaussian
Log-likelihood: -9512 on 3 Df
Number of iterations in BFGS optimization: 3

## 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)

      df      AIC
CRCH   3 19029.75
CRCH2  3 17961.76
CRCH3  4 17914.41
CRCH4  4 17867.35