tobit {VGAM}R Documentation

Tobit Model

Description

Fits a Tobit model.

Usage

tobit(Lower = 0, Upper = Inf, lmu = "identity", lsd = "loge",
      emu = list(), esd = list(), nsimEIM = 250,
      imu = NULL, isd = NULL, 
      type.fitted = c("uncensored", "censored", "mean.obs"),
      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 lowerbound 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 upperbound is assigned this value. Hence this should be the largest possible value in the response variable.

lmu, lsd, emu, esd

Parameter link functions and extra arguments for 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.

imu, isd

See CommonVGAMffArguments for information.

type.fitted

Type of fitted value returned. The first choice is default and is the ordinary uncensored or unbounded linear model. If "censored" then the fitted values in the interval [L, U]. If "mean.obs" then the mean of the observations is returned; this is a doubly truncated normal distribution augmented by point masses at the truncation points (see dtobit).

imethod

Initialization method. Either 1 or 2, this specifies two methods for obtaining initial values for the parameters.

nsimEIM

Used if nonstandard Tobit model. See CommonVGAMffArguments for information.

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 and lmu correspond to the standard Tobit model. Then Fisher scoring is used, else simulated Fisher scoring. 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.

Warning

Convergence is often slow. Setting crit = "coeff" is recommended since premature convergence of the log-likelihood is common. Simulated Fisher scoring is implemented for the nonstandard Tobit model. For this, the working weight matrices for some observations are prone to not being positive-definite; if so then some checking of the final model is recommended and/or try inputting some initial values.

Note

The response can be a matrix. Then Lower and Upper are recycled to the number of columns. 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. The function cennormal1 is an alternative to tobit().

Author(s)

Thomas W. Yee

References

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

See Also

rtobit, cennormal1, normal1, dcennormal1, posnormal1, rnorm.

Examples

# Here, fit1 is a standard Tobit model and fit2 is a nonstandard Tobit model
Lower = 1; Upper = 4; set.seed(1)  # For the nonstandard Tobit model
tdata = data.frame(x2 = seq(-1, 1, len = (nn <- 100)))
meanfun1 = function(x) 0 + 2*x
meanfun2 = function(x) 2 + 2*x
tdata = transform(tdata,
  y1 = rtobit(nn, mean = meanfun1(x2)),  # Standard Tobit model 
  y2 = rtobit(nn, mean = meanfun2(x2), Lower = Lower, Upper = Upper))
with(tdata, table(y1 == 0)) # How many censored values?
with(tdata, table(y2 == Lower | y2 == Upper)) # How many censored values?
with(tdata, table(attr(y2, "cenL")))
with(tdata, table(attr(y2, "cenU")))

fit1 = vglm(y1 ~ x2, tobit, tdata, trace = TRUE,
            crit = "coeff")  # crit = "coeff" is recommended
coef(fit1, matrix = TRUE)
summary(fit1)

fit2 = vglm(y2 ~ x2, tobit(Lower = Lower, Upper = Upper, type.f = "cens"),
            tdata, crit = "coeff", trace = TRUE) # ditto
table(fit2@extra$censoredL)
table(fit2@extra$censoredU)
coef(fit2, matrix = TRUE)

## Not run:  # Plot the results
par(mfrow = c(2, 1))
plot(y1 ~ x2, tdata, las = 1, main = "Standard Tobit model",
     col = as.numeric(attr(y1, "cenL")) + 3,
     pch = as.numeric(attr(y1, "cenL")) + 1)
legend(x = "topleft", leg = c("censored", "uncensored"),
       pch = c(2, 1), col = c("blue", "green"))
legend(-1.0, 2.5, c("Truth", "Estimate", "Naive"),
       col = c("purple", "orange", "black"), lwd = 2, lty = c(1, 2, 2))
lines(meanfun1(x2) ~ x2, tdata, col = "purple", lwd = 2)
lines(fitted(fit1) ~ x2, tdata, col = "orange", lwd = 2, lty = 2)
lines(fitted(lm(y1 ~ x2, tdata)) ~ x2, tdata, col = "black",
      lty = 2, lwd = 2) # This is simplest but wrong!

plot(y2 ~ x2, tdata, las = 1, main = "Tobit model",
     col = as.numeric(attr(y2, "cenL")) + 3 +
           as.numeric(attr(y2, "cenU")),
     pch = as.numeric(attr(y2, "cenL")) + 1 +
           as.numeric(attr(y2, "cenU")))
legend(x = "topleft", leg = c("censored", "uncensored"),
       pch = c(2, 1), col = c("blue", "green"))
legend(-1.0, 3.5, c("Truth", "Estimate", "Naive"),
       col = c("purple", "orange", "black"), lwd = 2, lty = c(1, 2, 2))
lines(meanfun2(x2) ~ x2, tdata, col = "purple", lwd = 2)
lines(fitted(fit2) ~ x2, tdata, col = "orange", lwd = 2, lty = 2)
lines(fitted(lm(y2 ~ x2, tdata)) ~ x2, tdata, col = "black",
      lty = 2, lwd = 2) # This is simplest but wrong!

## End(Not run)

[Package VGAM version 0.8-4 Index]