tobit: Tobit Model

Description

Fits a Tobit model.

Usage

tobit(Lower = 0, Upper = Inf, lmu = "identitylink", lsd = "loglink",
      imu = NULL, isd = NULL,
      type.fitted = c("uncensored", "censored", "mean.obs"),
      byrow.arg = FALSE, imethod = 1, zero = "sd")

Arguments

Lower

Numeric. It is the value $L$ described below. Any value of the linear model $x_{i}^{T} β$ that is less than this lowerbound is assigned this value. Hence this should be the smallest possible value in the response variable. May be a vector (see below for more information).

Upper

Numeric. It is the value $U$ described below. Any value of the linear model $x_{i}^{T} β$ that is greater than this upperbound is assigned this value. Hence this should be the largest possible value in the response variable. May be a vector (see below for more information).

lmu, lsd

Parameter link functions 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, byrow.arg

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). See CommonVGAMffArguments for more information.

imethod

Initialization method. Either 1 or 2 or 3, this specifies some methods for obtaining initial values for the parameters. See CommonVGAMffArguments for information.

zero

A vector, e.g., 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. See CommonVGAMffArguments for more information.

Value

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

Warning

If values of the response and Lower and/or Upper are not integers then there is the danger that the value is wrongly interpreted as uncensored. For example, if the first 10 values of the response were runif(10) and Lower was assigned these value then testing y[1:10] == Lower[1:10] is numerically fraught. Currently, if any y < Lower or y > Upper then a warning is issued.

Details

The Tobit model can be written $y_{i}^{*} = x_{i}^{T} β + ε_{i}$ where the $e_{i} \sim N (0, σ^{2})$ independently and $i = 1, \dots, n$ . 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. Fisher scoring is used for the standard and nonstandard models. By default, the mean $x_{i}^{T} β$ 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 $x_{i}^{T} β$ .

References

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

Examples

Run this code

# NOT RUN {
# Here, fit1 is a standard Tobit model and fit2 is a nonstandard Tobit model
tdata <- data.frame(x2 = seq(-1, 1, length = (nn <- 100)))
set.seed(1)
Lower <- 1; Upper <- 4  # For the nonstandard Tobit model
tdata <- transform(tdata,
                   Lower.vec = rnorm(nn, Lower, 0.5),
                   Upper.vec = rnorm(nn, Upper, 0.5))
meanfun1 <- function(x) 0 + 2*x
meanfun2 <- function(x) 2 + 2*x
meanfun3 <- function(x) 2 + 2*x
meanfun4 <- function(x) 3 + 2*x
tdata <- transform(tdata,
  y1 = rtobit(nn, mean = meanfun1(x2)),  # Standard Tobit model
  y2 = rtobit(nn, mean = meanfun2(x2), Lower = Lower, Upper = Upper),
  y3 = rtobit(nn, mean = meanfun3(x2), Lower = Lower.vec, Upper = Upper.vec),
  y4 = rtobit(nn, mean = meanfun3(x2), Lower = Lower.vec, Upper = Upper.vec))
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, data = tdata, trace = TRUE)
coef(fit1, matrix = TRUE)
summary(fit1)

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

fit3 <- vglm(y3 ~ x2, tobit(Lower = with(tdata, Lower.vec),
                            Upper = with(tdata, Upper.vec), type.f = "cens"),
             data = tdata, trace = TRUE)
table(fit3@extra$censoredL)
table(fit3@extra$censoredU)
coef(fit3, matrix = TRUE)

# fit4 is fit3 but with type.fitted = "uncen".
fit4 <- vglm(cbind(y3, y4) ~ x2,
             tobit(Lower = rep(with(tdata, Lower.vec), each = 2),
                   Upper = rep(with(tdata, Upper.vec), each = 2),
                   byrow.arg = TRUE),
             data = tdata, crit = "coeff", trace = TRUE)
head(fit4@extra$censoredL)  # A matrix
head(fit4@extra$censoredU)  # A matrix
head(fit4@misc$Lower)       # A matrix
head(fit4@misc$Upper)       # A matrix
coef(fit4, matrix = TRUE)

# }
# NOT RUN {
 # Plot fit1--fit4
par(mfrow = c(2, 2))

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, data = 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!

plot(y3 ~ x2, data = tdata, las = 1,
     main = "Tobit model with nonconstant censor levels",
     col = as.numeric(attr(y3, "cenL")) + 2 +
           as.numeric(attr(y3, "cenU") * 2),
     pch = as.numeric(attr(y3, "cenL")) + 1 +
           as.numeric(attr(y3, "cenU") * 2))
legend(x = "topleft", leg = c("censoredL", "censoredU", "uncensored"),
       pch = c(2, 3, 1), col = c(3, 4, 2))
legend(-1.0, 3.5, c("Truth", "Estimate", "Naive"),
       col = c("purple", "orange", "black"), lwd = 2, lty = c(1, 2, 2))
lines(meanfun3(x2) ~ x2, tdata, col = "purple", lwd = 2)
lines(fitted(fit3) ~ x2, tdata, col = "orange", lwd = 2, lty = 2)
lines(fitted(lm(y3 ~ x2, tdata)) ~ x2, tdata, col = "black",
      lty = 2, lwd = 2)  # This is simplest but wrong!

plot(y3 ~ x2, data = tdata, las = 1,
     main = "Tobit model with nonconstant censor levels",
     col = as.numeric(attr(y3, "cenL")) + 2 +
           as.numeric(attr(y3, "cenU") * 2),
     pch = as.numeric(attr(y3, "cenL")) + 1 +
           as.numeric(attr(y3, "cenU") * 2))
legend(x = "topleft", leg = c("censoredL", "censoredU", "uncensored"),
       pch = c(2, 3, 1), col = c(3, 4, 2))
legend(-1.0, 3.5, c("Truth", "Estimate", "Naive"),
       col = c("purple", "orange", "black"), lwd = 2, lty = c(1, 2, 2))
lines(meanfun3(x2) ~ x2, data = tdata, col = "purple", lwd = 2)
lines(fitted(fit4)[, 1] ~ x2, tdata, col = "orange", lwd = 2, lty = 2)
lines(fitted(lm(y3 ~ x2, tdata)) ~ x2, data = tdata, col = "black",
      lty = 2, lwd = 2)  # This is simplest but wrong!
# }

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