Truncated Gaussian Response Models
Estimation of models for truncated Gaussian variables by maximum likelihood.
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 ' |
point |
the value of truncation (the default is 0), |
direction |
the direction of the truncation, either |
model, y, x |
logicals. If |
scaled |
if |
... |
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 |
y |
the response vector (if |
x |
the model matrix (if |
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)) * dfPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.