micEcon (version 0.4-0)

selection: Heckman-style selection models

Description

This is the frontend for estimating Heckman-style selection models either with one or two outcomes (also known as generalized tobit models).

Usage

selection(selection, outcome, data = sys.frame(sys.parent()),
   subset, method = "ml", start = NULL, 
   ys = FALSE, xs = FALSE, yo = FALSE, xo = FALSE,
   mfs = FALSE, mfo = FALSE, print.level = 0, ...)
heckit( selection, outcome, data = sys.frame(sys.parent()),
   method = "2step", ... )

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 env

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, or "2step" for 2-step estimation.

start

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

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.

print.level

integer. Various debugging information, higher value gives more information.

...

additional parameters for the corresponding fitting functions tobit2fit, tobit5fit, heckit2fit

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 'maxLik', and in addition the elements 'twoStep', 'init', 'param', termS, termO, 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 list with components 'probit', 'coefficients', 'param', 'vcov', 'lm', 'sigma', 'rho', 'invMillsRatio', and 'imrDelta'. If a tobit-2 (sample selection) model is estimated by the two-step method (argument method = "2step"), the returned object is 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
• init
  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.
• 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.
• 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 $\delta$s calculated from the inverse Mills Ratios and the results of the 1st step probit estimation.

Details

The endogenous variable of the 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').

For tobit-2 (sample selection) models, only those observations are included in the second step estimation (argument 'outcome'), where this variable 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 this endogenous variable of the selections 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').

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.

The (generic) function 'coef' ('coef.selection') can be used to extract the estimated coefficients. The (generic) function 'vcov' ('vcov.selection') can be used to extract the estimated variance covariance matrix of the coefficients. The (generic) function 'print' ('print.selection') can be used to print a few results. The (generic) function 'summary' ('summary.selection') can be used to obtain and print detailed results.

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.

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

See Also

lm, glm, binomial

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 ) )

## Wooldridge( 2003 ): example 17.5, page 590
data( Mroz87 )
# Two-step estimation
summary( heckit( lfp ~ nwifeinc + educ + exper + I( exper^2 ) + age +
   kids5 + kids618, log( wage ) ~ educ + exper + I( exper^2 ), Mroz87,
   method = "2step" ) )

## Cameron and Trivedi (2005): Section 16.6, page 553ff
data( RandHIE )
subsample <- RandHIE$year == 2 & !is.na( RandHIE$educdec )
selectEq <- binexp ~ logc + idp + lpi + fmde + physlm + disea +
   hlthg + hlthf + hlthp + linc + lfam + educdec + xage + female +
   child + fchild + black
outcomeEq <- lnmeddol ~ logc + idp + lpi + fmde + physlm + disea +
   hlthg + hlthf + hlthp + linc + lfam + educdec + xage + female +
   child + fchild + black
# ML estimation
cameron <- selection( selectEq, outcomeEq, data = RandHIE[ subsample, ] )
summary( cameron )

## example using random numbers
library( MASS )
nObs <- 1000
sigma <- matrix( c( 1, -0.7, -0.7, 1 ), ncol = 2 )
errorTerms <- mvrnorm( 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( MASS )
nObs <- 1000
sigma <- matrix( c( 1, 0.5, 0.1, 0.5, 1, -0.3, 0.1, -0.3, 1 ), ncol = 3 )
errorTerms <- mvrnorm( 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)