1. Tobit and Selection Models HISHAM ABOU-TALEB R3 basics course - 2014.06.02

2. Tobit Model a censored regression The TOBIT model is a statistical model proposed by James Tobin (1958). It also called a censored regression model. It is designed to estimate linear relationships between variables when there is either left- or right- censoring in the dependent variable (also known as censoring from below and above, respectively). http://www.ats.ucla.edu/stat/stata/dae/tobit.htm

3. Tobit Model Censoring from aboveCensoring from above takes place when cases with a value at or above some threshold, all take on the value of the threshold. The true value might be equal to the threshold, but it might also be higher. In the case of censoring from below, values those that fall at or below some threshold are censored.

4. Type 1 Tobit Simple Tobit model where the outcome variable is observed only if it is above or below a given threshold. Censoring can be at a value yL, different from zero Example: Restricting speedometer readings to 85 mph. So, if we want to predict a vehicle's top- speed from a combination of horse-power and engine size, you would get a reading no higher than 85, regardless of how fast the vehicle was really traveling.

5. Type 1 Tobit N <- 1000 u <- rnorm(N) x <- - 1 + rnorm(N) ystar <- 1 + x + u y 0) hist(y) ols <- lm(y ~ x) summary(ols) library(AER) tobit <- tobit(y ~ x,left=0,right=Inf,dist = "gaussian")

6. Type 2 Tobit Heckit modelIt is also known as Heckit model The outcome y2 is observed only if y1 is equal to 0. y1 a binary variable which may be correlated with the error term of y2, resulting in endogenous selection process.

7. Type 2 Tobit N <- 1000 u <- rnorm(N) v <- rnorm(N) x <- - 1 + rnorm(N) z <- 1 + rnorm(N) d 0) ystar <- 1 + x + u y <- ystar*(d == 1) hist(y) ols <- lm(y ~ x) summary(ols)

8. Type 2 Tobit Also we can use heckit() and selection() in sampleSelection package library(sampleSelection) heckit.ml <- heckit(selection = d ~ x + z, outcome = y ~ x, method = "ml") summary(heckit.ml) heckit.2step <- heckit(selection = d ~ x + z, outcome = y ~ x, method = "2step") summary(heckit.2step)

9. Truncation Truncated regression is used to model dependent variables for which some of the observations are not included in the analysis because of the value of the dependent variable. We can use the truncreg package in R for this analysis Example: students are required to have a minimum score of 40 to pass the exam. Thus, the sample is truncated at an achievement score of 40. Doubly truncated data analysis (DTDA package)