VGAM (version 1.1-14)

tobit: Tobit Regression

Description

Fits a Tobit regression 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")

Value

An object of class "vglmff"

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

Details

The Tobit model can be written $$y_i^* = x_i^T \beta + \varepsilon_i$$ where the \(e_i \sim N(0,\sigma^2)\) independently and \(i=1,\ldots,n\). 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 \(x_i^T \beta\) 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 \(x_i^T \beta\).

References

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

See Also

rtobit, cens.normal, uninormal, double.cens.normal, posnormal, CommonVGAMffArguments, round2, mills.ratio, margeff, rnorm.

Examples

# Here, fit1 is a standard Tobit model and fit2 is nonstandard
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) 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))  # Ditto
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)

if (FALSE)  # 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"), lwd = 2,
       col = c("purple", "orange", "black"), 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"), lwd = 2,
       col = c("purple", "orange", "black"), 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", pch = c(2, 3, 1), col = c(3, 4, 2),
       leg = c("censoredL", "censoredU", "uncensored"))
legend(-1.0, 3.5, c("Truth", "Estimate", "Naive"), lwd = 2,
       col = c("purple", "orange", "black"), 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", pch = c(2, 3, 1), col = c(3, 4, 2),
       leg = c("censoredL", "censoredU", "uncensored"))
legend(-1.0, 3.5, c("Truth", "Estimate", "Naive"), lwd = 2, 
       col = c("purple", "orange", "black"), 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!