selection {sampleSelection}

Heckman-style selection and treatment effect models

Description

This is the frontend for estimating Heckman-style selection models either with one or two outcomes (also known as generalized tobit models). It supports binary outcomes and interval outcomes in the single-outcome case. It also supports normal-distribution based treatment effect models.

For model specification and more details, see Toomet and Henningsen (2008) and the included vignettes “Sample Selection Models”, “Interval Regression with Sample Selection”, and “All-Normal Treatment Effects”.

Usage

selection(selection, outcome, data = sys.frame(sys.parent()),
   weights = NULL, subset, method = "ml",
   type = NULL,
   start = NULL,
   boundaries = NULL,
   ys = FALSE, xs = FALSE, yo = FALSE, xo = FALSE,
   mfs = FALSE, mfo = FALSE,
   printLevel = print.level, print.level=0,
   ...)

heckit( selection, outcome, data = sys.frame(sys.parent()),
   method = "2step", ... )

treatReg(selection, outcome,
   data=sys.frame(sys.parent()),
   mfs=TRUE, mfo=TRUE,
   ...)

Arguments

selection

formula, the selection equation.

outcome

the outcome equation(s). Either a single equation (for tobit 2 models), or a list of two equations (tobit 5 models).

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which selection is called.

weights

an optional vector of ‘prior weights’ to be used in the fitting process. Should be NULL or a numeric vector. Weights are currently only supported in type-2 models.

subset

an optional index vector specifying a subset of observations to be used in the fitting process.

method

how to estimate the model. Either "ml" for Maximum Likelihood, "2step" for 2-step estimation, or "model.frame" for returning the model frame (only).

type

model type. NULL will automatically detect the type, but one may also manually determine it. Numeric ‘2’ and ‘5’ are tobit-2 and tobit-5 models respectively, and character “treatment” is treatment effect model.

start

vector, initial values for the ML estimation. If start does not have names, names are constructed based on the model frame.

boundaries

an optional vector of boundaries of the intervals of the dependent variable of the outcome equation for sample selection models with interval regression of the outcome equation.

ys, yo, xs, xo, mfs, mfo

logicals. If true, the response (y), model matrix (x) or the model frame (mf) of the selection (s) or outcome (o) equation(s) are returned.

printLevel, print.level

integer. Various debugging information, higher value gives more information. The preferred option is ‘printLevel’.

...

additional parameters for the corresponding fitting functions tobit2fit, tobit5fit, heckit2fit, heckit5fit, and tobit2Bfit.

Details

The dependent variable of of the selection equation (specified by argument selection) must have exactly two levels (e.g., 'FALSE' and 'TRUE', or '0' and '1'). By default the levels are sorted in increasing order ('FALSE' is before 'TRUE', and '0' is before '1'). If the dependent variable of the outcome equation (specified by argument outcome) has exactly two levels, this variable is modelled as a binary variable. If argument boundaries is specified, the outcome equation is estimated as interval regression model and the dependent variable of the outcome equation must be a categorical (factor) variable or a variable of strictly positive integer values, whereas the vector specified by argument boundaries must have one more element than the number of levels or the largest integer, respectively. In all other cases, it is assumed that the dependent variable of the outcome equation is continuous and an ordinary sample selection model is estimated.

For tobit-2 (sample selection) models, only those observations are included in the second step estimation (argument 'outcome'), where the dependent variable variable of the selection equation equals the second element of its levels (e.g., 'TRUE' or '1').

For tobit-5 (switching regression) models, in the second step the first outcome equation (first element of argument 'outcome') is estimated only for those observations, where the dependent variable of the selection equation equals the first element of its levels (e.g., 'FALSE' or '0'). The second outcome equation is estimated only for those observations, where this variable equals the second element of its levels (e.g., 'TRUE' or '1').

Treatment effect models are a version of tobit-5 models where the two outcomes are “participation” and “non-participation”. treatReg takes an equal set of explanatory variables for both of these choices and assumes that the corresponding parameters are equal. In typical treatment effect model the selection outcome variable (participation decision) is included as an explanatory variable for the outcome. If this is not done, treatReg amounts to estimating two equations with correlated error structure.

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 removed.

These selection models assume a known (multivariate normal) distribution of error terms. Because of this, the instruments (exclusion restrictions) are not necessary. However, if no instruments are supplied, the results are based solely on the assumption on multivariate normality. This may or may not be an appropriate assumption for a particular problem. Note also that standard errors tend to be large without a strong exclusion restriction.

If argument method is equal to "ml" (the default), the estimation is done by the maximum likelihood method, where the Newton-Raphson algorithm is used by default. Argument maxMethod (see tobit2fit) can be used to chose other algorithms for the maximisation of the (log) likelihood function.

If argument method is equal to "ml" (the default) and argument start is NULL (the default), the starting values for the maximum-likelihood (ML) estimation of a tobit-2 or tobit-5 model are obtained by an initial two-step estimation of this model.

The two-step estimation of interval-regression models with sample-selection has not yet been implemented. If no starting values for a maximum-likelihood (ML) estimation of an interval-regression model with sample-selection are specified (i.e., argument start is NULL, the default), starting values are obtained by an initial estimation of a tobit-2 model, where the dependent variable of the outcome equation is set to the mid points of the boudaries of the intervals. By default, the starting values are obtained by a maximum-likelihood (ML) estimation of the tobit-2 model, whereas the starting values for the maximum-likelihood (ML) estimation of the tobit-2 model are obtained by a 2-step estimation of the tobit-2 model. If argument start is set to "2step", the starting values for the maximum-likelihood (ML) estimation of an interval-regression model with sample-selection are directly obtained by a 2-step estimation of the tobit-2 model (i.e., without a subsequent ML estimation of the tobit-2 model).

Methods that can be applied to objects returned by selection() are described on the help page selection-methods.

Value

'selection' returns an object of class "selection". If the model estimated by Maximum Likelihood (argument method = "ml"), this object is a list, which has all the components of a 'maxLik' object, and in addition the elements 'twoStep', 'start, 'param', 'termS', 'termO', 'outcomeVar', and if requested 'ys', 'xs', 'yo', 'xo', 'mfs', and 'mfo'. If a tobit-2 (sample selection) model is estimated by the two-step method (argument method = "2step"), the returned object is a list with components 'probit', 'coefficients', 'param', 'vcov', 'lm', 'sigma', 'rho', 'invMillsRatio', and 'imrDelta'. If a tobit-5 (switching regression) model is estimated by the two-step method (argument method = "2step"), the returned object is a list with components 'coefficients', 'vcov', 'probit', 'lm1', 'lm2', 'rho1', 'rho2', 'sigma1', 'sigma2', 'termsS', 'termsO', 'param', and if requested 'ys', 'xs', 'yo', 'xo', 'mfs', and 'mfo'.

probit

object of class 'probit' that contains the results of the 1st step (probit estimation) (only for two-step estimations).

twoStep

(only if initial values not given) results of the 2-step estimation, used for initial values

start

initial values for ML estimation

termsS, termsO

terms for the selection and outcome equation

ys, xs, yo, xo, mfs, mfo

response, matrix and frame of the selection- and outcome equations (as a list of two for the latter). NULL, if not requested. The response is represented internally as 0/1 integer vector with 0 denoting either the unobservable outcome (tobit 2) or the first selection (tobit 5).

coefficients

estimated coefficients, the complete model. coefficient for the Inverse Mills ratio is treated as a parameter (=\varrho \sigma).

vcov

variance covariance matrix of the estimated coefficients.

param

a list with following components: index, a list of numeric vectors: where in the coef the component are located; oIntercept, a logical: whether the outcome equation includes intercept; N0, N1, integer, number of observations with unobserved and observed outcomes; nObs, integer, number of valid observations; nParam, integer, number of the parameters in the model (not all are independent); df, integer, degrees of freedom. Note this is not equal to nObs - nParam because of the parameters are not independent in all the cases; levels levels for the response of the selection equation. levels[1] corresponds to the outcome 1, levels[2] to the outcome 2.

lm, lm1, lm2

objects of class 'lm' that contain the results of the 2nd step estimation(s) of the outcome equation(s). Note: the standard errors of this estimation are biased, because they do not account for the estimation of \gamma in the 1st step estimation (the correct standard errors are returned by summary and they are contained in vcov component).

sigma, sigma1, sigma2

the standard error(s) of the error terms of the outcome equation(s).

rho, rho1, rho2

the estimated correlation coefficient(s) between the error term of the selection equation and the outcome equation(s).

invMillsRatio

the inverse Mills Ratios calculated from the results of the 1st step probit estimation.

imrDelta

the \deltas calculated from the inverse Mills Ratios and the results of the 1st step probit estimation.

outcomeVar

character string indicating whether the dependent variable of the outcome equation is "continuous" or "binary".

Note

The 2-step estimate of 'rho' may be outside of the [-1,1] interval. In that case the standard errors of invMillsRatio may be meaningless.

Author(s)

Arne Henningsen, Ott Toomet [e-mail]

References

Cameron, A. C. and Trivedi, P. K. (2005) Microeconometrics: Methods and Applications, Cambridge University Press.

Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.

Heckman, J. (1976) The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models, Annals of Economic and Social Measurement, 5(4), p. 475-492.

Johnston, J. and J. DiNardo (1997) Econometric Methods, Fourth Edition, McGraw-Hill.

Lee, L., G. Maddala and R. Trost (1980) Asymetric covariance matrices of two-stage probit and two-stage tobit methods for simultaneous equations models with selectivity. Econometrica, 48, p. 491-503.

Petersen, S., G. Henningsen and A. Henningsen (2017) Which Households Invest in Energy-Saving Home Improvements? Evidence From a Danish Policy Intervention. Unpublished Manuscript. Department of Management Engineering, Technical University of Denmark.

Toomet, O. and A. Henningsen, (2008) Sample Selection Models in R: Package sampleSelection. Journal of Statistical Software 27(7), https://www.jstatsoft.org/v27/i07/

Wooldridge, J. M. (2003) Introductory Econometrics: A Modern Approach, 2e, Thomson South-Western.

See Also

summary.selection, selection-methods, probit, lm, and Mroz87 and RandHIE for further examples.

Examples

## Greene( 2003 ): example 22.8, page 786
data( Mroz87 )
Mroz87$kids  <- ( Mroz87$kids5 + Mroz87$kids618 > 0 )
# Two-step estimation
summary( heckit( lfp ~ age + I( age^2 ) + faminc + kids + educ,
   wage ~ exper + I( exper^2 ) + educ + city, Mroz87 ) )
# ML estimation
summary( selection( lfp ~ age + I( age^2 ) + faminc + kids + educ,
   wage ~ exper + I( exper^2 ) + educ + city, Mroz87 ) )

## Example using binary outcome for selection model.
## We estimate the probability of womens' education on their
## chances to get high wage (> $5/hr in 1975 USD), using PSID data
## We use education as explanatory variable
## and add age, kids, and non-work income as exclusion restrictions.
data(Mroz87)
m <- selection(lfp ~ educ + age + kids5 + kids618 + nwifeinc,
   wage >= 5 ~ educ, data = Mroz87 )
summary(m)


## example using random numbers
library( "mvtnorm" )
nObs <- 1000
sigma <- matrix( c( 1, -0.7, -0.7, 1 ), ncol = 2 )
errorTerms <- rmvnorm( nObs, c( 0, 0 ), sigma )
myData <- data.frame( no = c( 1:nObs ), x1 = rnorm( nObs ), x2 = rnorm( nObs ),
   u1 = errorTerms[ , 1 ], u2 =  errorTerms[ , 2 ] )
myData$y <- 2 + myData$x1 + myData$u1
myData$s <- ( 2 * myData$x1 + myData$x2 + myData$u2 - 0.2 ) > 0
myData$y[ !myData$s ] <- NA
myOls <- lm( y ~ x1, data = myData)
summary( myOls )
myHeckit <- heckit( s ~ x1 + x2, y ~ x1, myData, print.level = 1 )
summary( myHeckit )

## example using random numbers with IV/2SLS estimation
library( "mvtnorm" )
nObs <- 1000
sigma <- matrix( c( 1, 0.5, 0.1, 0.5, 1, -0.3, 0.1, -0.3, 1 ), ncol = 3 )
errorTerms <- rmvnorm( nObs, c( 0, 0, 0 ), sigma )
myData <- data.frame( no = c( 1:nObs ), x1 = rnorm( nObs ), x2 = rnorm( nObs ),
   u1 = errorTerms[ , 1 ], u2 = errorTerms[ , 2 ], u3 = errorTerms[ , 3 ] )
myData$w <- 1 + myData$x1 + myData$u1
myData$y <- 2 + myData$w + myData$u2
myData$s <- ( 2 * myData$x1 + myData$x2 + myData$u3 - 0.2 ) > 0
myData$y[ !myData$s ] <- NA
myHeckit <- heckit( s ~ x1 + x2, y ~ w, data = myData )
summary( myHeckit )  # biased!
myHeckitIv <- heckit( s ~ x1 + x2, y ~ w, data = myData, inst = ~ x1 )
summary( myHeckitIv ) # unbiased

## tobit-5 example
N <- 500
   library(mvtnorm)
   vc <- diag(3)
   vc[lower.tri(vc)] <- c(0.9, 0.5, 0.6)
   vc[upper.tri(vc)] <- vc[lower.tri(vc)]
   eps <- rmvnorm(N, rep(0, 3), vc)
   xs <- runif(N)
   ys <- xs + eps[,1] > 0
   xo1 <- runif(N)
   yo1 <- xo1 + eps[,2]
   xo2 <- runif(N)
   yo2 <- xo2 + eps[,3]
   a <- selection(ys~xs, list(yo1 ~ xo1, yo2 ~ xo2))
   summary(a)

## tobit2 example
   vc <- diag(2)
   vc[2,1] <- vc[1,2] <- -0.7
   eps <- rmvnorm(N, rep(0, 2), vc)
   xs <- runif(N)
   ys <- xs + eps[,1] > 0
   xo <- runif(N)
   yo <- (xo + eps[,2])*(ys > 0)
   a <- selection(ys~xs, yo ~xo)
   summary(a)

## Example for treatment regressions
## Estimate the effect of treatment on income
## selection outcome: treatment participation, logical (treatment)
## selection explanatory variables: age, education (years)
##   unemployment in 1974, 1975, race
## outcome: log real income 1978
## outcome explanatory variables: treatment, age, education, race.
## unemployment variables are treated as exclusion restriction
data(Treatment, package="Ecdat")
a <- treatReg(treat~poly(age,2) + educ + u74 + u75 + ethn,
              log(re78)~treat + poly(age,2) + educ + ethn,
              data=Treatment)
print(summary(a))