tobit {VGAM}R Documentation

Tobit Model

Description

Fits a Tobit model to a univariate response.

Usage

tobit(Lower = 0, Upper = Inf, lmu="identity", lsd="loge",
      emu=list(), esd=list(), imethod=1, zero=2)

Arguments

Lower Numeric of length 1, it is the value L described below. Any value of the linear model x_i^T beta that is less than this value is assigned this value. Hence this should be the smallest possible value in the response variable.
Upper Numeric of length 1, it is the value U described below. Any value of the linear model x_i^T beta that is greater than this value is assigned this value. Hence this should be the largest possible value in the response variable.
lmu, lsd Parameter link functions applied to the mean and standard deviation parameters. See Links for more choices. The standard deviation is a positive quantity, therefore a log link is its default.
emu, esd List. Extra argument for each of the links. See earg in Links for general information.
imethod Initialization method. Either 1 or 2, this specifies two methods for obtaining initial values for the parameters.
zero An integer vector, containing the value 1 or 2. If so, the mean or standard deviation respectively are modelled as an intercept only. Setting zero=NULL means both linear/additive predictors are modelled as functions of the explanatory variables.

Details

The Tobit model can be written

y_i^* = x_i^T beta + e_i

where the e_i ~ N(0,sigma^2) independently and i=1,...,n. However, we measure y_i = y_i^* only if y_i^* > L and y_i^* < U for some cutpoints L and U. Otherwise we let y_i=L or y_i=U, whatever is closer. The Tobit model is thus a multiple linear regression but with censored responses if it is below or above certain cutpoints.

The defaults for Lower and Upper correspond to the classical Tobit model. By default, the mean x_i^T beta is the first linear/additive predictor, and the log of the standard deviation is the second linear/additive predictor. The Fisher information matrix for uncensored data is diagonal. The fitted values are the estimates of x_i^T beta.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

The response must be univariate. If there is no censoring then normal1 is recommended instead. Any value of the response less than Lower or greater than Upper will be assigned the value Lower and Upper respectively, and a warning will be issued. The fitted object has components censoredL and censoredU in the extra slot which specifies whether observations are censored in that direction.

Author(s)

Thomas W. Yee

References

Tobin, J. (1958) Estimation of relationships for limited dependent variables. Econometrica 26, 24–36.

See Also

normal1, cnormal1. dcnormal1.

Examples

n = 1000
x = seq(-1, 1, len=n)
f = function(x) 1 + 4*x
ystar = f(x) + rnorm(n)
Lower = 1
Upper = 4
y  = pmax(ystar, Lower)
y  = pmin(y, Upper)
table(y==Lower | y==Upper)   # How many censored values?
fit = vglm(y ~ x, tobit(Lower=Lower, Upper=Upper), trace=TRUE)
table(fit@extra$censoredL)
table(fit@extra$censoredU)
coef(fit, matrix=TRUE)
summary(fit)
## Not run: 
plot(x, y, main="Tobit model", las=1)
legend(-0.9, 3, c("Truth", "Estimate"), col=c("Blue", "Red"), lwd=2)
lines(x, f(x), col="blue", lwd=2)  # The truth
lines(x, fitted(fit), col="red", lwd=2, lty="dashed")  # The estimate
## End(Not run)

[Package VGAM version 0.7-7 Index]