# Truncated Gaussian Response Models

### Description

Estimation of models for truncated Gaussian variables by maximum likelihood.

### Usage

1 2 3 |

### 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 ' |

`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. |

### 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 |

`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), |

### 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | ```
########################
## 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
``` |