tobit: Tobit Model

Description

Fits a Tobit model.

Usage

tobit(Lower = 0, Upper = Inf, lmu = "identity", lsd = "loge",
      nsimEIM = 250, imu = NULL, isd = NULL, 
      type.fitted = c("uncensored", "censored", "mean.obs"),
      imethod = 1, zero = -2)

Arguments

Lower

Numeric. 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. May be a vector (se

Upper

Numeric. 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. May be a vector

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

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 observ

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 explanato

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.

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. 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 $x_i^T \beta$.

References

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

Examples

Run this code

# 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, 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)

fit3 <- vglm(y3 ~ x2,
            tobit(Lower = with(tdata, Lower.vec),
                  Upper = with(tdata, Upper.vec), type.f = "cens"),
            tdata, crit = "coeff", trace = TRUE) # ditto
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)),
            tdata, crit = "coeff", trace = TRUE) # ditto
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)

# Plot the results
par(mfrow = c(2, 2))
# Plot fit1
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 fit2
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!

# Plot fit3
plot(y3 ~ x2, tdata, las = 1,
     main = "Tobit model with nonconstant censor levels",
     col = as.numeric(attr(y3, "cenL")) + 3 +
           as.numeric(attr(y3, "cenU")),
     pch = as.numeric(attr(y3, "cenL")) + 1 +
           as.numeric(attr(y3, "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(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 fit4
plot(y3 ~ x2, tdata, las = 1,
     main = "Tobit model with nonconstant censor levels",
     col = as.numeric(attr(y3, "cenL")) + 3 +
           as.numeric(attr(y3, "cenU")),
     pch = as.numeric(attr(y3, "cenL")) + 1 +
           as.numeric(attr(y3, "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(meanfun3(x2) ~ x2, tdata, col = "purple", lwd = 2)
lines(fitted(fit4)[, 1] ~ 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!

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