R Programming/Tobit And Selection Models

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Tobit (type 1 Tobit)[edit]

In this section, we look at simple tobit model where the outcome variable is observed only if it is above or below a given threshold.

  • tobit() in the AER package[1]. This is a wrapper for survreg().
N <- 1000
u <- rnorm(N)
x <- - 1 + rnorm(N)
ystar <- 1 + x + u 
y <- ystar*(ystar > 0)
hist(y)

ols <- lm(y ~ x)
summary(ols)

library(AER)
tobit <- tobit(y ~ x,left=0,right=Inf,dist = "gaussian")


Selection models (type 2 tobit or heckit)[edit]

In this section we look at endogenous selection process. The outcome y is observe only if d is equal to one with d a binary variable which is correlated with the error term of y.

  • heckit() and selection() in sampleSelection [2]. The command is called heckit() in honor of James Heckman[3].
N <- 1000
u <- rnorm(N)
v <- rnorm(N)
x <- - 1 + rnorm(N)
z <- 1 + rnorm(N)
d <- (1 + x + z + u + v> 0)
ystar <- 1 + x + u 
y <- ystar*(d == 1)
hist(y)

ols <- lm(y ~ x)
summary(ols)

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)

Multi-index selection models[edit]

In this section we look at endogenous selection processes in matching markets. Matching is concerned with who transacts with whom, and how. For example, which students attend which college. The outcome y is observed only for equilibrium student-college pairs (or matches). These matches are indicated with d equal to one with d a binary variable which is correlated with the error term of y.

  • stabit() and stabit2() in matchingMarkets.[4][5] The command is called stabit() in reference to the application in stable matching markets.

Simulate two-sided matching data for 20 markets (m=20) with 100 students (nStudents=100) per market and 20 colleges with quotas of 5 students, each (nSlots=rep(5,20)). True parameters in selection and outcome equations are all equal to 1.

library(matchingMarkets)
xdata <- stabsim2(m=20, nStudents=100, nSlots=rep(5,20), 
  colleges = "c1",
  students = "s1",
  outcome = ~ c1:s1 + eta + nu,
  selection = ~ -1 + c1:s1 + eta
)

Observe the bias from sorting between students and colleges.

lm1 <- lm(y ~ c1:s1, data=xdata$OUT)
summary(lm1)

Correct for sorting bias by running the Gibbs sampler in Sorensen (2007).[6]

fit2 <- stabit2(OUT = xdata$OUT, 
           colleges = "c1",
           students = "s1",
           outcome = y ~ c1:s1, 
           selection = ~ -1 + c1:s1,
           niter=1000
)
summary(fit2)

Truncation[edit]

  • truncreg package
  • DTDA "An R package for analyzing truncated data" pdf.

References[edit]

  1. Christian Kleiber and Achim Zeileis (2008). Applied Econometrics with R. New York: Springer-Verlag. ISBN 978-0-387-77316-2. URL http://CRAN.R-project.org/package=AER
  2. Sample Selection Models in R: Package sampleSelection http://www.jstatsoft.org/v27/i07
  3. James Heckman "Sample selection bias as a specification error", Econometrica: Journal of the econometric society, 1979
  4. Klein, T. (2015). "Analysis of Stable Matchings in R: Package matchingMarkets". Vignette to R Package matchingMarkets. http://cran.at.r-project.org/web/packages/matchingMarkets/vignettes/matching.pdf. 
  5. "matchingMarkets: Analysis of Stable Matchings". R Project. http://cran.at.r-project.org/web/packages/matchingMarkets/index.html. 
  6. Sorensen, M. (2007). "How Smart is Smart Money? A Two-Sided Matching Model of Venture Capital". Journal of Finance 62 (6): 2725-2762.