Tobit Regression

Fits a Tobit regression model.

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 $x_i^T \beta$ 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 (see below for more information).

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 (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.

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 $x_i^T \beta$.

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. The function round2 may be useful.

Value

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

References

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

Author

Thomas W. Yee

Note

The response can be a matrix. If so, then Lower and Upper are recycled into a matrix with the number of columns equal to the number of responses, and the recycling is done row-wise if byrow.arg = TRUE. The default order is as matrix, which is byrow.arg = FALSE. For example, these are returned in fit4@misc$Lower and fit4@misc$Upper below.

If there is no censoring then uninormal 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 cens.normal is an alternative to tobit().

When obtaining initial values, if the algorithm would otherwise want to fit an underdetermined system of equations, then it uses the entire data set instead. This might result in rather poor quality initial values, and consequently, monitoring convergence is advised.

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?
#> 
#> FALSE  TRUE 
#>    52    48 
with(tdata, table(y2 == Lower | y2 == Upper))  # Ditto
#> 
#> FALSE  TRUE 
#>    66    34 
with(tdata, table(attr(y2, "cenL")))
#> 
#> FALSE  TRUE 
#>    75    25 
with(tdata, table(attr(y2, "cenU")))
#> 
#> FALSE  TRUE 
#>    91     9 

fit1 <- vglm(y1 ~ x2, tobit, data = tdata, trace = TRUE)
#> Iteration 1: loglikelihood = -94.952489
#> Iteration 2: loglikelihood = -90.601839
#> Iteration 3: loglikelihood = -90.335693
#> Iteration 4: loglikelihood = -90.334971
#> Iteration 5: loglikelihood = -90.334971
coef(fit1, matrix = TRUE)
#>                     mu loglink(sd)
#> (Intercept) 0.06794616 -0.06234739
#> x2          2.02595920  0.00000000
summary(fit1)
#> 
#> Call:
#> vglm(formula = y1 ~ x2, family = tobit, data = tdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  0.06795    0.13614   0.499    0.618    
#> (Intercept):2 -0.06235    0.10273  -0.607    0.544    
#> x2             2.02596    0.23949   8.459   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: mu, loglink(sd)
#> 
#> Log-likelihood: -90.335 on 197 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 5 
#> 
#> No Hauck-Donner effect found in any of the estimates
#> 

fit2 <- vglm(y2 ~ x2,
             tobit(Lower = Lower, Upper = Upper, type.f = "cens"),
             data = tdata, trace = TRUE)
#> Iteration 1: loglikelihood = -117.40947
#> Iteration 2: loglikelihood = -115.71547
#> Iteration 3: loglikelihood = -115.66898
#> Iteration 4: loglikelihood = -115.66819
#> Iteration 5: loglikelihood = -115.66818
#> Iteration 6: loglikelihood = -115.66818
table(fit2@extra$censoredL)
#> 
#> FALSE  TRUE 
#>    75    25 
table(fit2@extra$censoredU)
#> 
#> FALSE  TRUE 
#>    91     9 
coef(fit2, matrix = TRUE)
#>                   mu loglink(sd)
#> (Intercept) 2.053261 -0.04893087
#> x2          1.880404  0.00000000

fit3 <- vglm(y3 ~ x2, tobit(Lower = with(tdata, Lower.vec),
                            Upper = with(tdata, Upper.vec),
                            type.f = "cens"),
             data = tdata, trace = TRUE)
#> Iteration 1: loglikelihood = -118.69013
#> Iteration 2: loglikelihood = -117.11926
#> Iteration 3: loglikelihood = -117.07583
#> Iteration 4: loglikelihood = -117.07438
#> Iteration 5: loglikelihood = -117.07433
#> Iteration 6: loglikelihood = -117.07433
#> Iteration 7: loglikelihood = -117.07433
table(fit3@extra$censoredL)
#> 
#> FALSE  TRUE 
#>    84    16 
table(fit3@extra$censoredU)
#> 
#> FALSE  TRUE 
#>    77    23 
coef(fit3, matrix = TRUE)
#>                   mu loglink(sd)
#> (Intercept) 2.904743  0.09173307
#> x2          2.178193  0.00000000

# 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)
#> Iteration 1: coefficients = 
#> 2.89014257, 0.25304478, 2.96031053, 0.20508806, 2.12538698, 2.02573207
#> Iteration 2: coefficients = 
#> 2.885765205, 0.093423424, 2.942093302, 0.043317927, 2.232760342, 2.239483502
#> Iteration 3: coefficients = 
#> 2.905736069, 0.096906039, 2.951169180, 0.023582998, 2.181484799, 2.175403035
#> Iteration 4: coefficients = 
#> 2.904410822, 0.091524289, 2.950632485, 0.016355913, 2.179996601, 2.164449979
#> Iteration 5: coefficients = 
#> 2.904781633, 0.091888984, 2.950534264, 0.014991137, 2.178221623, 2.160988668
#> Iteration 6: coefficients = 
#> 2.904730135, 0.091715330, 2.950515287, 0.014622934, 2.178246473, 2.160270160
#> Iteration 7: coefficients = 
#> 2.904742983, 0.091733074, 2.950510499, 0.014541499, 2.178193053, 2.160085708
#> Iteration 8: coefficients = 
#> 2.904740949, 0.091727259, 2.950509419, 0.014521425, 2.178195929, 2.160043739
#> Iteration 9: coefficients = 
#> 2.904741408, 0.091728014, 2.950509157, 0.014516762, 2.178194244, 2.160033552
#> Iteration 10: coefficients = 
#> 2.904741329, 0.091727813, 2.950509096, 0.014515643, 2.178194396, 2.160031165
#> Iteration 11: coefficients = 
#> 2.904741346, 0.091727843, 2.950509081, 0.014515379, 2.178194341, 2.160030595
#> Iteration 12: coefficients = 
#> 2.904741343, 0.091727836, 2.950509078, 0.014515317, 2.178194347, 2.160030461
#> Iteration 13: coefficients = 
#> 2.904741344, 0.091727837, 2.950509077, 0.014515302, 2.178194345, 2.160030429
#> Iteration 14: coefficients = 
#> 2.904741344, 0.091727837, 2.950509077, 0.014515298, 2.178194346, 2.160030421
#> Iteration 15: coefficients = 
#> 2.904741344, 0.091727837, 2.950509077, 0.014515297, 2.178194346, 2.160030419
head(fit4@extra$censoredL)  # A matrix
#>      y3    y4
#> 1 FALSE FALSE
#> 2 FALSE  TRUE
#> 3  TRUE  TRUE
#> 4  TRUE  TRUE
#> 5  TRUE FALSE
#> 6 FALSE FALSE
head(fit4@extra$censoredU)  # A matrix
#>      y3    y4
#> 1 FALSE FALSE
#> 2 FALSE FALSE
#> 3 FALSE FALSE
#> 4 FALSE FALSE
#> 5 FALSE FALSE
#> 6 FALSE FALSE
head(fit4@misc$Lower)       # A matrix
#>           [,1]      [,2]
#> [1,] 0.6867731 0.6867731
#> [2,] 1.0918217 1.0918217
#> [3,] 0.5821857 0.5821857
#> [4,] 1.7976404 1.7976404
#> [5,] 1.1647539 1.1647539
#> [6,] 0.5897658 0.5897658
head(fit4@misc$Upper)       # A matrix
#>          [,1]     [,2]
#> [1,] 3.689817 3.689817
#> [2,] 4.021058 4.021058
#> [3,] 3.544539 3.544539
#> [4,] 4.079014 4.079014
#> [5,] 3.672708 3.672708
#> [6,] 4.883644 4.883644
coef(fit4, matrix = TRUE)
#>                  mu1 loglink(sd1)      mu2 loglink(sd2)
#> (Intercept) 2.904741   0.09172784 2.950509    0.0145153
#> x2          2.178194   0.00000000 2.160030    0.0000000

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!

 # \dontrun{}