truncreg: Truncated Gaussian Response Models
In truncreg: Truncated Gaussian Regression Models

Description Usage Arguments Details Value References See Also Examples

Estimation of models for truncated Gaussian variables by maximum likelihood.

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truncreg(formula, data, subset, weights, na.action,
  point = 0, direction = "left",
  model = TRUE, y = FALSE, x = FALSE, scaled = FALSE, ...)

`formula`	a symbolic description of the model to be estimated,
`data`	the data,
`subset`	an optional vector specifying a subset of observations,
`weights`	an optional vector of weights,
`na.action`	a function which indicates what should happen when the data contains '`NA`'s,
`point`	the value of truncation (the default is 0),
`direction`	the direction of the truncation, either `"left"` (the default) or `"right"`,
`model, y, x`	logicals. If `TRUE` the corresponding components of the fit (model frame, response, model matrix) are returned,
`scaled`	if `TRUE`, scaled parameters (beta / sigma) are estimated,
`...`	further arguments.

The model is estimated with the maxLik package and the Newton-Raphson method, using analytic gradient and Hessian.

A set of standard extractor functions for fitted model objects is available for objects of class "truncreg", including methods to the generic functions print, summary, coef, vcov, logLik, residuals, predict, fitted, model.frame, and model.matrix.

An object of class "truncreg", a list with elements:

`coefficients`	the named vector of coefficients,
`vcov`	the variance matrix of the coefficients,
`fitted.values`	the fitted values,
`logLik`	the value of the log-likelihood,
`gradient`	the gradient of the log-likelihood at convergence,
`nobs`	the number of observations,
`call`	the matched call,
`terms`	the model terms,
`model`	the model frame used (if `model = TRUE`),
`y`	the response vector (if `y = TRUE`),
`x`	the model matrix (if `x = TRUE`),
`point`	the truncation point used,
`direction`	the truncation direction used,
`est.stat`	some information about the estimation (time used, optimization method),

Cragg JG (1971). Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods. Econometrica, 39, 829–844.

Hausman JA, Wise DA (1976). The Evaluation of Results from Truncated Samples: The New-Jersey Negative Income Tax Experiment. Annals of Economic ans Social Measurment, 5, 421–445.

Hausman JA, Wise DA (1976). Social Experimentation, Truncated Distributions and Efficient Estimation. Econometrica, 45, 421–425.

Tobin J (1958). Estimation of Relationships for Limited Dependent Variables. Econometrica, 26, 24–36.

maxLik, mhurdle

########################
## Artificial example ##
########################

## simulate a data.frame
set.seed(1071)
n <- 10000
sigma <- 4
alpha <- 2
beta <- 1
x <- rnorm(n, mean = 0, sd = 2)
eps <- rnorm(n, sd = sigma)
y <- alpha + beta * x + eps
d <- data.frame(y = y, x = x)

## truncated response
d$yt <- ifelse(d$y > 1, d$y, NA)

## binary threshold response
d$yb <- factor(d$y > 0)

## censored response
d$yc <- pmax(1, d$y)

## compare estimates for full/truncated/censored/threshold response
fm_full <- lm(y ~ x, data = d)
fm_trunc <- truncreg(yt ~ x, data = d, point = 1, direction = "left")
fm_thresh <- glm(yb ~ x, data = d, family = binomial(link = "probit"))
library("survival")
fm_cens <- survreg(Surv(yc, yc > 1, type = "left") ~ x, data = d, dist = "gaussian")

## compare scaled regression coefficients
cbind(
  "True"      = c(alpha, beta) / sigma,
  "Full"      = coef(fm_full) / summary(fm_full)$sigma,
  "Truncated" = coef(fm_trunc)[1:2] / coef(fm_trunc)[3],
  "Censored"  = coef(fm_cens) / fm_cens$scale,
  "Threshold" = coef(fm_thresh)
)


################################
## Tobin's durable goods data ##
################################

## Tobit model (Tobin 1958)
data("tobin", package = "survival")
tobit <- survreg(Surv(durable, durable > 0, type = "left") ~ age + quant,
  data = tobin, dist = "gaussian")

## Two-part model (Cragg 1971)
## (see "mhurdle" package for a combined solution)
cragg_probit <- glm(factor(durable > 0) ~ age + quant,
  data = tobin, family = binomial(link = "logit"))
cragg_trunc <- truncreg(durable ~ age + quant, data = tobin, subset = durable > 0)

## Scaled coefficients
cbind(
  "Tobit"     = coef(tobit) / tobit$scale,
  "Binary"    = coef(cragg_probit),
  "Truncated" = coef(cragg_trunc)[1:3] / coef(cragg_trunc)[4])

## likelihood ratio test and BIC
ll <- c("Tobit" = tobit$loglik[1],
        "Two-Part" = as.vector(logLik(cragg_probit) + logLik(cragg_trunc)))
df <- c(4, 3 + 4)
pchisq(2 * diff(ll), diff(df), lower.tail = FALSE)
-2 * ll + log(nrow(tobin)) * df

Loading required package: maxLik
Loading required package: miscTools

Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.

If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
            True      Full Truncated  Censored Threshold
(Intercept) 0.50 0.5071291 0.4932690 0.5218093 0.5242008
x           0.25 0.2465092 0.2466027 0.2481720 0.2514478
                   Tobit      Binary   Truncated
(Intercept)  2.717767303  2.07631898  7.62674516
age         -0.023159868 -0.05284724  0.26094645
quant       -0.008172515 -0.00078640 -0.07190702
  Two-Part 
0.01600393 
   Tobit Two-Part 
70.96733 69.63057