EM type Estimation Methods for the Heckman's Sample Selection Model

Description

Some algorithms: ECM, ECMnr and ECME can be used to estimate parameters in Heckman selection model and contain the advantages of the EM algorithm: easy implementation and numerical stability. "ECMnr" stands for Expectation/Conditioncal Maximization with Newton-Raphson, and "ECME" for Expectation/Conditional Maximization Either.

Usage

EMSS(
  response,
  selection,
  data,
  method = "ECM",
  initial.param = NULL,
  eps = 10^(-10)
)

Arguments

Details

The dependent variable of the selection equation (specified by argument selection) must have exactly two levels (e.g., 'FALSE' and 'TRUE', or '0' and '1'). The default argument method is "ECM" and the default start values ("NULL") are obtained by two-step estimation of this model through the command selection from the package sampleSelection. NA's are allowed in the data. These are ignored if the corresponding outcome is unobserved, otherwise observations which contain NA (either in selection or outcome) are changed to 0.

Value

ECM returns an object of class "ECM". The object class "ECM" is a list containing the following components.

Background

Heckman selection model is classic to deal with the data where the outcome is partially observed and the missing part is not at random. Heckman (1979) developed 2-step and maximum likelihood estimation (MLE) to do the estimation for this selection model. And these two method are described in R package sampleSelection by Toomet and Henningsen (2008). Zhelonkin et al. (2016) developed robust 2-stage method which performs more robustly than the 2-step method to deal with the data where outlying observations exist and ssmrob package is available. Zhao et al. (2020) extended EM algorithm to more general cases resulting in three algorithms: ECM, ECM(NR), and ECME. They also own EM algorithm's main advantages, namely, stability and ease of implementation.

References

Heckman, J. (1979) Sample selection bias as a specication error. Econometrica, 47, 153-161.

Toomet, O. and Henningsen, A. (2008) Sample selection models in R:Package sampleSelection. Journal of Statistical Software, 27, 1-23.

Zhao,J., Kim, H.-J. and Kim, H.-M. (2020) New EM-type algorithms for the Heckman selection model. Computational Statistics and Data Analysis, 146, https://doi.org/10.1016/j.csda.2020.106930.

Zhelonkin, M., Genton, M.G. and Ronchetti, E. (2016) Robust inference in sample selection models. Journal of the Royal Statistical Society Series B, 78, 805-827.

Examples

data(Smoke, package = "EMSS")
ex1 <- EMSS(response = cigs_intervals ~ educ,
           selection = smoker ~ educ + age,
           data = Smoke)
print(ex1)

data(Smoke, package = "EMSS")
ex2 <- EMSS(response = cigs_intervals ~ educ,
           selection =  smoker ~ educ + age,
           data = Smoke, method="ECMnr")
print(ex2)

## example using random numbers with exclusion restriction

N <- 1000
errps <- mvtnorm::rmvnorm(N,c(0,0),matrix(c(1,0.5,0.5,1),2,2) )
xs <- runif(N)
ys <- xs+errps[,1]>0
xo <- runif(N)
yo <- (xo+errps[,2])*(ys>0)
ex3 <- EMSS(response = yo ~ xo,
           selection = ys ~ xs,
           initial.param = c(rep(0,4), 0.3, 0.6), method="ECMnr")
print(ex3)

Survey Data on Smoking Behaviour

Description

The Data is the subset of the original data from Mullahy (1985) and Mullahy (1997). The dataset is from Wooldridge (2009) used for researches on cross sectinal data studies. The dataset is also available from Smoke from the package sampleSelection.

Usage

data(Smoke, package = "EMSS")

Format

a dataframe with 807 observations and 8 variables as below:

educ

years of schooling (numeric)

age

age of respondents (numeric)

cigpric

cigarette price(state), cents per pack (numeric)

income

annual income in us dollar (numeric)

restaurn

state smoking restrictions for restaurants exist or not (categorical)

smoker

smoked at least once or not (categorical)

cigs_intervals

number of cigarettes smoked per day, with interval boundaries: 0,5,10,20,50 (numeric)

cigs

number of cigarettes smoked per day (numeric)

Source

Wooldridge's dataset is available on https://ideas.repec.org/p/boc/bocins/smoke.html#biblio.

References

Jeffrey, M. Wooldridge (2009) Introductory Econometrics: A modern approach, Canada: South-Western Cengage Learning.

Mullahy, John (1985) Cigarette Smoking: Habits, Health Concerns, and Heterogeneous Unobservables in a Microeconometric Analysis of Consumer Demand, Ph.D. dissertation, University of Virginia.

Mullahy, John (1997), Instrumental-Variable Estimation of Count Data Models: Applications to Models of Cigarette Smoking Behavior, Review of Economics and Statistics, 79, 596-593.

Getting Coefficients of EM type Sample Selection Model Fits

Description

coef method for a class "EMSS".

Usage

## S3 method for class 'EMSS'
coef(object, only = NULL, ...)

Arguments

Value

a numeric vector or a list, containing one set or two sets, is given.

Examples

# examples continued from EMSS
data(Smoke, package = "EMSS")
ex1 <- EMSS(response = cigs_intervals ~ educ,
           selection = smoker ~ educ + age,
           data = Smoke)
coef(ex1)

data(Smoke, package = "EMSS")
ex2 <- EMSS(response = cigs_intervals ~ educ,
           selection =  smoker ~ educ + age,
           data = Smoke, method="ECMnr")
coef(ex2)

## example using random numbers with exclusion restriction

N <- 1000
errps <- mvtnorm::rmvnorm(N,c(0,0),matrix(c(1,0.5,0.5,1),2,2) )
xs <- runif(N)
ys <- xs+errps[,1]>0
xo <- runif(N)
yo <- (xo+errps[,2])*(ys>0)

ex3 <- EMSS(response = yo ~ xo,
           selection = ys ~ xs,
           initial.param = c(rep(0,4), 0.3, 0.6), method="ECMnr")
coef(ex3)

Getting Confidence Intervals for Parameters of EM type Sample Selection Model Fits

Description

confint method for a class "EMSS".

Usage

## S3 method for class 'EMSS'
confint(object, parm, level = 0.95, ...)

Arguments

Examples

# examples continued from EMSS
data(Smoke, package = "EMSS")
ex1 <- EMSS(response = cigs_intervals ~ educ,
           selection = smoker ~ educ + age,
           data = Smoke)
confint(ex1)

data(Smoke, package = "EMSS")
ex2 <- EMSS(response = cigs_intervals ~ educ,
           selection =  smoker ~ educ + age,
           data = Smoke, method="ECMnr")
confint(ex2)

## example using random numbers with exclusion restriction

N <- 1000
errps <- mvtnorm::rmvnorm(N,c(0,0),matrix(c(1,0.5,0.5,1),2,2) )
xs <- runif(N)
ys <- xs+errps[,1]>0
xo <- runif(N)
yo <- (xo+errps[,2])*(ys>0)

ex3 <- EMSS(response = yo ~ xo,
           selection = ys ~ xs,
           initial.param = c(rep(0,4), 0.3, 0.6), method="ECMnr")
confint(ex3)

Summarizing EM type Sample Selection Model Fits

Description

summary method for a class "EMSS".

Usage

## S3 method for class 'EMSS'
summary(object, tidy = FALSE, conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'summary.EMSS'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Examples

# examples continued from EMSS
data(Smoke, package = "EMSS")
ex1 <- EMSS(response = cigs_intervals ~ educ,
           selection = smoker ~ educ + age,
           data = Smoke)
summary(ex1)

data(Smoke, package = "EMSS")
ex2 <- EMSS(response = cigs_intervals ~ educ,
           selection =  smoker ~ educ + age,
           data = Smoke, method="ECMnr")
summary(ex2)

## example using random numbers with exclusion restriction

N <- 1000
errps <- mvtnorm::rmvnorm(N,c(0,0),matrix(c(1,0.5,0.5,1),2,2) )
xs <- runif(N)
ys <- xs+errps[,1]>0
xo <- runif(N)
yo <- (xo+errps[,2])*(ys>0)

ex3 <- EMSS(response = yo ~ xo,
           selection = ys ~ xs,
           initial.param = c(rep(0,4), 0.3, 0.6), method="ECMnr")
summary(ex3)

Getting Variance-Covariance Matrix for Parameters of EM type Sample Selection Model Fits

Description

vcov method for a class "EMSS".

Usage

## S3 method for class 'EMSS'
vcov(object, ...)

Arguments

Examples

# examples continued from EMSS
data(Smoke, package = "EMSS")
ex1 <- EMSS(response = cigs_intervals ~ educ,
           selection = smoker ~ educ + age,
           data = Smoke)
vcov(ex1)

data(Smoke, package = "EMSS")
ex2 <- EMSS(response = cigs_intervals ~ educ,
           selection =  smoker ~ educ + age,
           data = Smoke, method="ECMnr")
vcov(ex2)

## example using random numbers with exclusion restriction

N <- 1000
errps <- mvtnorm::rmvnorm(N,c(0,0),matrix(c(1,0.5,0.5,1),2,2) )
xs <- runif(N)
ys <- xs+errps[,1]>0
xo <- runif(N)
yo <- (xo+errps[,2])*(ys>0)

ex3 <- EMSS(response = yo ~ xo,
           selection = ys ~ xs,
           initial.param = c(rep(0,4), 0.3, 0.6), method="ECMnr")
vcov(ex3)