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 0.7-9 Index]