| Title: | Some EM-Type Estimation Methods for the Heckman Selection Model |
|---|---|
| Description: | Some EM-type algorithms to estimate parameters for the well-known Heckman selection model are provided in the package. Such algorithms are as follow: ECM(Expectation/Conditional Maximization), ECM(NR)(the Newton-Raphson method is adapted to the ECM) and ECME(Expectation/Conditional Maximization Either). Since the algorithms are based on the EM algorithm, they also have EM’s main advantages, namely, stability and ease of implementation. Further details and explanations of the algorithms can be found in Zhao et al. (2020) <doi: 10.1016/j.csda.2020.106930>. |
| Authors: | Kexuan Yang <[email protected]>, Sang Kyu Lee <[email protected]>, Jun Zhao <[email protected]>, and Hyoung-Moon Kim <[email protected] > |
| Maintainer: | Sang Kyu Lee <[email protected]> |
| License: | GPL-2 |
| Version: | 1.1.1 |
| Built: | 2025-02-22 04:25:33 UTC |
| Source: | https://github.com/sangkyustat/emss |
coef method for a class "EMSS".
## S3 method for class 'EMSS' coef(object, only = NULL, ...)## S3 method for class 'EMSS' coef(object, only = NULL, ...)
object |
an object of class "EMSS" made by the function |
only |
a character value for choosing specific variable's coefficients. Initial value is |
... |
not used, but exists because of the compatibility. |
a numeric vector or a list, containing one set or two sets, is given.
# 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)# 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)
confint method for a class "EMSS".
## S3 method for class 'EMSS' confint(object, parm, level = 0.95, ...)## S3 method for class 'EMSS' confint(object, parm, level = 0.95, ...)
object |
an object of class "EMSS" made by the function |
parm |
not used, but exists because of the compatibility. |
level |
a numeric value between 0 and 1 for controlling the significance level of confidence interval; default value is 0.95. |
... |
not used, but exists because of the compatibility. |
# 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)# 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)
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.
EMSS( response, selection, data, method = "ECM", initial.param = NULL, eps = 10^(-10) )EMSS( response, selection, data, method = "ECM", initial.param = NULL, eps = 10^(-10) )
response |
a formula for the response equation. |
selection |
a formula for the selection equation. |
data |
a data frame and data has to be included with the form of |
method |
a character indicating which method to be used. |
initial.param |
a vector, initial parameter values for the estimation. The length of the initial parameters has to be same as the length of parameters, which are to be estimated. |
eps |
a numerical error value for the end of the loop. A minimum value that can be arbitrarily set to terminate the iteration of the function, in order to find the optimal parameter estimation. |
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.
ECM returns an object of class "ECM".
The object class "ECM" is a list
containing the following components.
call |
a matched call. |
estimate_response |
estimated regression coefficients for the response formula. |
estimate_selection |
estimated regression coefficients for the sample selection formula. |
estimate_sigma |
an estimated scale paramter for the bivariate normal distribution. |
estimate_rho |
an estimated correlation coefficient for the bivariate normal distribution. |
hessian_mat |
hessian matrix for parameters. |
resp_leng |
the numbers of coefficients for the response formula |
select_leng |
the numbers of coefficients for the selection formula |
Q_value |
the vallue of the Q function for EM type algorithms |
names_response |
names of regression coefficients for the reponse formula. |
names_selection |
names of regression coefficients for the selection formula. |
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.
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.
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)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)
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.
data(Smoke, package = "EMSS")data(Smoke, package = "EMSS")
a dataframe with 807 observations and 8 variables as below:
years of schooling (numeric)
age of respondents (numeric)
cigarette price(state), cents per pack (numeric)
annual income in us dollar (numeric)
state smoking restrictions for restaurants exist or not (categorical)
smoked at least once or not (categorical)
number of cigarettes smoked per day, with interval boundaries: 0,5,10,20,50 (numeric)
number of cigarettes smoked per day (numeric)
Wooldridge's dataset is available on https://ideas.repec.org/p/boc/bocins/smoke.html#biblio.
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.
summary method for a class "EMSS".
## 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), ...)## 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), ...)
object |
an object of class "EMSS" made by the function |
tidy |
a logical value stands for whether the summary format is in tidy format or not, if |
conf.int |
a logical value stands for whether the confidence interval is included in the tiny format or not. If |
conf.level |
a numeric value between 0 and 1 for controlling the significance level of confidence interval; default value is 0.95. |
... |
not used, but exists because of the compatibility. |
x |
an object of class "summary.EMSS". |
digits |
a numeric number of significant digits. |
# 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)# 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)
vcov method for a class "EMSS".
## S3 method for class 'EMSS' vcov(object, ...)## S3 method for class 'EMSS' vcov(object, ...)
object |
an object of class "EMSS" made by the function |
... |
not used, but exists because of the compatibility. |
# 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)# 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)