Truncated Gaussian Response Models

Description

Estimation of models for truncated Gaussian variables by maximum likelihood.

Usage

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

Arguments

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.

Details

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.

Value

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),

References

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.

See Also

maxLik, mhurdle

Examples

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########################
## 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

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