selection: Heckman-style selection and treatment effect models
In sampleSelection: Sample Selection Models
Description
Usage
Arguments
Details
Value
Note
Author(s)
References
See Also
Examples
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
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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
(= rho * 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 γ 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 δs 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 otoomet@ut.ee
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
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## 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))
Example output
Loading required package: maxLik
Loading required package: miscTools
Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.
If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
753 observations (325 censored and 428 observed)
14 free parameters (df = 740)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.157e+00 1.402e+00 -2.965 0.003127 **
age 1.854e-01 6.597e-02 2.810 0.005078 **
I(age^2) -2.426e-03 7.735e-04 -3.136 0.001780 **
faminc 4.580e-06 4.206e-06 1.089 0.276544
kidsTRUE -4.490e-01 1.309e-01 -3.430 0.000638 ***
educ 9.818e-02 2.298e-02 4.272 2.19e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9712003 2.0593505 -0.472 0.637
exper 0.0210610 0.0624646 0.337 0.736
I(exper^2) 0.0001371 0.0018782 0.073 0.942
educ 0.4170174 0.1002497 4.160 3.56e-05 ***
city 0.4438379 0.3158984 1.405 0.160
Multiple R-Squared:0.1264, Adjusted R-Squared:0.116
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -1.098 1.266 -0.867 0.386
sigma 3.200 NA NA NA
rho -0.343 NA NA NA
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 5 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -1581.258
753 observations (325 censored and 428 observed)
13 free parameters (df = 740)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.120e+00 1.401e+00 -2.942 0.003368 **
age 1.840e-01 6.587e-02 2.794 0.005345 **
I(age^2) -2.409e-03 7.723e-04 -3.119 0.001886 **
faminc 5.680e-06 4.416e-06 1.286 0.198782
kidsTRUE -4.506e-01 1.302e-01 -3.461 0.000568 ***
educ 9.528e-02 2.315e-02 4.115 4.3e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.9630242 1.1982209 -1.638 0.102
exper 0.0278683 0.0615514 0.453 0.651
I(exper^2) -0.0001039 0.0018388 -0.056 0.955
educ 0.4570051 0.0732299 6.241 7.33e-10 ***
city 0.4465290 0.3159209 1.413 0.158
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 3.1084 0.1138 27.307 <2e-16 ***
rho -0.1320 0.1651 -0.799 0.424
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model with binary outcome (sample selection model)
Maximum Likelihood estimation
BHHH maximisation, 8 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -653.2037
753 observations (325 censored and 428 observed)
9 free parameters (df = 744)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.430362 0.475966 0.904 0.366
educ 0.156223 0.023811 6.561 1.00e-10 ***
age -0.034713 0.007649 -4.538 6.61e-06 ***
kids5 -0.890560 0.112663 -7.905 9.69e-15 ***
kids618 -0.038167 0.039320 -0.971 0.332
nwifeinc -0.020948 0.004318 -4.851 1.49e-06 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.5213 0.5611 -8.058 3.08e-15 ***
educ 0.2879 0.0369 7.800 2.09e-14 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
rho 0.1164 0.2706 0.43 0.667
--------------------------------------------
Call:
lm(formula = y ~ x1, data = myData)
Residuals:
Min 1Q Median 3Q Max
-2.17074 -0.64543 -0.03873 0.57432 2.28424
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.59142 0.05825 27.32 <2e-16 ***
x1 1.23876 0.05686 21.79 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8834 on 464 degrees of freedom
(534 observations deleted due to missingness)
Multiple R-squared: 0.5057, Adjusted R-squared: 0.5046
F-statistic: 474.7 on 1 and 464 DF, p-value: < 2.2e-16
Tobit 2 model
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (534 censored and 466 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.23585 0.06271 -3.761 0.000179 ***
x1 2.14721 0.12650 16.974 < 2e-16 ***
x2 1.13917 0.08906 12.792 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.09042 0.09157 22.83 <2e-16 ***
x1 0.91730 0.07365 12.45 <2e-16 ***
Multiple R-Squared:0.56, Adjusted R-Squared:0.5581
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -0.8430 0.1053 -8.004 3.35e-15 ***
sigma 0.9422 NA NA NA
rho -0.8947 NA NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (537 censored and 463 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.18249 0.05951 -3.066 0.00223 **
x1 2.05944 0.12152 16.947 < 2e-16 ***
x2 0.91713 0.07861 11.666 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.03962 0.08915 11.66 <2e-16 ***
w 1.42953 0.03609 39.61 <2e-16 ***
Multiple R-Squared:0.7861, Adjusted R-Squared:0.7852
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio 0.24587 0.09516 2.584 0.00992 **
sigma 0.87190 NA NA NA
rho 0.28200 NA NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (537 censored and 463 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.18249 0.05951 -3.066 0.00223 **
x1 2.05944 0.12152 16.947 < 2e-16 ***
x2 0.91713 0.07861 11.666 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.63086 0.09495 17.18 <2e-16 ***
w 1.13645 0.03849 29.53 <2e-16 ***
Multiple R-Squared:0.7555, Adjusted R-Squared:0.7544
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -0.01126 0.10223 -0.11 0.912
sigma 0.92074 NA NA NA
rho -0.01223 NA NA NA
--------------------------------------------
--------------------------------------------
Tobit 5 model (switching regression model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 8 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -894.8142
500 observations: 158 selection 1 (FALSE) and 342 selection 2 (TRUE)
10 free parameters (df = 490)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.07410 0.08906 0.832 0.406
xs 0.88461 0.14145 6.254 8.74e-10 ***
Outcome equation 1:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03602 0.17938 0.201 0.841
xo1 1.06831 0.16801 6.359 4.66e-10 ***
Outcome equation 2:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.2229 0.1226 -1.818 0.0697 .
xo2 0.9898 0.1654 5.983 4.22e-09 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma1 1.06558 0.10531 10.118 < 2e-16 ***
sigma2 1.08002 0.08407 12.846 < 2e-16 ***
rho1 0.92899 0.03397 27.348 < 2e-16 ***
rho2 0.81995 0.09766 8.396 5.01e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 5 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -735.544
500 observations (163 censored and 337 observed)
6 free parameters (df = 494)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01765 0.11727 0.151 0.88
xs 0.83856 0.20047 4.183 3.41e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0251 0.1618 0.155 0.877
xo 0.8827 0.1612 5.474 6.99e-08 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 0.93966 0.07971 11.79 <2e-16 ***
rho -0.55189 0.22433 -2.46 0.0142 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit treatment model (switching regression model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 4 iterations
Return code 1: gradient close to zero
Log-Likelihood: -2651.502
2344 observations: 2204 non-participants (selection FALSE) and 140 participants (selection TRUE)
17 free parameters (df = 2327)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.94272 0.38051 -5.106 3.57e-07 ***
poly(age, 2)1 -41.64058 7.63374 -5.455 5.42e-08 ***
poly(age, 2)2 2.65968 4.97762 0.534 0.593166
educ -0.13661 0.03207 -4.260 2.13e-05 ***
u74TRUE 0.79452 0.22374 3.551 0.000391 ***
u75TRUE 2.31494 0.21291 10.873 < 2e-16 ***
ethnblack 1.35300 0.18734 7.222 6.89e-13 ***
ethnhispanic 1.31932 0.29465 4.478 7.91e-06 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.983926 0.069341 129.561 < 2e-16 ***
treatTRUE -0.963132 0.075837 -12.700 < 2e-16 ***
poly(age, 2)1 6.512273 0.797670 8.164 5.25e-16 ***
poly(age, 2)2 -4.428831 0.773235 -5.728 1.15e-08 ***
educ 0.080227 0.005231 15.338 < 2e-16 ***
ethnblack -0.256112 0.035865 -7.141 1.23e-12 ***
ethnhispanic -0.007786 0.079273 -0.098 0.922
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 0.69304 0.01014 68.359 < 2e-16 ***
rho 0.17699 0.06502 2.722 0.00654 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
sampleSelection documentation built on Jan. 13, 2021, 7:49 p.m.
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Description Usage Arguments Details Value Note Author(s) References See Also Examples
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | 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 |
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 |
type |
model type. |
start |
vector, initial values for the ML estimation. If
|
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 ( |
printLevel, print.level |
integer. Various debugging information, higher value gives more information. The preferred option is ‘printLevel’. |
... |
additional parameters for the corresponding fitting
functions |
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 (= rho * sigma). |
vcov |
variance covariance matrix of the estimated coefficients. |
param |
a list with following components: |
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 γ in the 1st step estimation
(the correct standard errors are returned by |
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 δs 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 |
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 otoomet@ut.ee
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | ## 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))
|
Example output
Loading required package: maxLik
Loading required package: miscTools
Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.
If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
753 observations (325 censored and 428 observed)
14 free parameters (df = 740)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.157e+00 1.402e+00 -2.965 0.003127 **
age 1.854e-01 6.597e-02 2.810 0.005078 **
I(age^2) -2.426e-03 7.735e-04 -3.136 0.001780 **
faminc 4.580e-06 4.206e-06 1.089 0.276544
kidsTRUE -4.490e-01 1.309e-01 -3.430 0.000638 ***
educ 9.818e-02 2.298e-02 4.272 2.19e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.9712003 2.0593505 -0.472 0.637
exper 0.0210610 0.0624646 0.337 0.736
I(exper^2) 0.0001371 0.0018782 0.073 0.942
educ 0.4170174 0.1002497 4.160 3.56e-05 ***
city 0.4438379 0.3158984 1.405 0.160
Multiple R-Squared:0.1264, Adjusted R-Squared:0.116
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -1.098 1.266 -0.867 0.386
sigma 3.200 NA NA NA
rho -0.343 NA NA NA
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 5 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -1581.258
753 observations (325 censored and 428 observed)
13 free parameters (df = 740)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.120e+00 1.401e+00 -2.942 0.003368 **
age 1.840e-01 6.587e-02 2.794 0.005345 **
I(age^2) -2.409e-03 7.723e-04 -3.119 0.001886 **
faminc 5.680e-06 4.416e-06 1.286 0.198782
kidsTRUE -4.506e-01 1.302e-01 -3.461 0.000568 ***
educ 9.528e-02 2.315e-02 4.115 4.3e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.9630242 1.1982209 -1.638 0.102
exper 0.0278683 0.0615514 0.453 0.651
I(exper^2) -0.0001039 0.0018388 -0.056 0.955
educ 0.4570051 0.0732299 6.241 7.33e-10 ***
city 0.4465290 0.3159209 1.413 0.158
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 3.1084 0.1138 27.307 <2e-16 ***
rho -0.1320 0.1651 -0.799 0.424
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model with binary outcome (sample selection model)
Maximum Likelihood estimation
BHHH maximisation, 8 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -653.2037
753 observations (325 censored and 428 observed)
9 free parameters (df = 744)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.430362 0.475966 0.904 0.366
educ 0.156223 0.023811 6.561 1.00e-10 ***
age -0.034713 0.007649 -4.538 6.61e-06 ***
kids5 -0.890560 0.112663 -7.905 9.69e-15 ***
kids618 -0.038167 0.039320 -0.971 0.332
nwifeinc -0.020948 0.004318 -4.851 1.49e-06 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.5213 0.5611 -8.058 3.08e-15 ***
educ 0.2879 0.0369 7.800 2.09e-14 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
rho 0.1164 0.2706 0.43 0.667
--------------------------------------------
Call:
lm(formula = y ~ x1, data = myData)
Residuals:
Min 1Q Median 3Q Max
-2.17074 -0.64543 -0.03873 0.57432 2.28424
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.59142 0.05825 27.32 <2e-16 ***
x1 1.23876 0.05686 21.79 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8834 on 464 degrees of freedom
(534 observations deleted due to missingness)
Multiple R-squared: 0.5057, Adjusted R-squared: 0.5046
F-statistic: 474.7 on 1 and 464 DF, p-value: < 2.2e-16
Tobit 2 model
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (534 censored and 466 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.23585 0.06271 -3.761 0.000179 ***
x1 2.14721 0.12650 16.974 < 2e-16 ***
x2 1.13917 0.08906 12.792 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.09042 0.09157 22.83 <2e-16 ***
x1 0.91730 0.07365 12.45 <2e-16 ***
Multiple R-Squared:0.56, Adjusted R-Squared:0.5581
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -0.8430 0.1053 -8.004 3.35e-15 ***
sigma 0.9422 NA NA NA
rho -0.8947 NA NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (537 censored and 463 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.18249 0.05951 -3.066 0.00223 **
x1 2.05944 0.12152 16.947 < 2e-16 ***
x2 0.91713 0.07861 11.666 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.03962 0.08915 11.66 <2e-16 ***
w 1.42953 0.03609 39.61 <2e-16 ***
Multiple R-Squared:0.7861, Adjusted R-Squared:0.7852
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio 0.24587 0.09516 2.584 0.00992 **
sigma 0.87190 NA NA NA
rho 0.28200 NA NA NA
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (537 censored and 463 observed)
8 free parameters (df = 993)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.18249 0.05951 -3.066 0.00223 **
x1 2.05944 0.12152 16.947 < 2e-16 ***
x2 0.91713 0.07861 11.666 < 2e-16 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.63086 0.09495 17.18 <2e-16 ***
w 1.13645 0.03849 29.53 <2e-16 ***
Multiple R-Squared:0.7555, Adjusted R-Squared:0.7544
Error terms:
Estimate Std. Error t value Pr(>|t|)
invMillsRatio -0.01126 0.10223 -0.11 0.912
sigma 0.92074 NA NA NA
rho -0.01223 NA NA NA
--------------------------------------------
--------------------------------------------
Tobit 5 model (switching regression model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 8 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -894.8142
500 observations: 158 selection 1 (FALSE) and 342 selection 2 (TRUE)
10 free parameters (df = 490)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.07410 0.08906 0.832 0.406
xs 0.88461 0.14145 6.254 8.74e-10 ***
Outcome equation 1:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03602 0.17938 0.201 0.841
xo1 1.06831 0.16801 6.359 4.66e-10 ***
Outcome equation 2:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.2229 0.1226 -1.818 0.0697 .
xo2 0.9898 0.1654 5.983 4.22e-09 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma1 1.06558 0.10531 10.118 < 2e-16 ***
sigma2 1.08002 0.08407 12.846 < 2e-16 ***
rho1 0.92899 0.03397 27.348 < 2e-16 ***
rho2 0.81995 0.09766 8.396 5.01e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Tobit 2 model (sample selection model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 5 iterations
Return code 2: successive function values within tolerance limit
Log-Likelihood: -735.544
500 observations (163 censored and 337 observed)
6 free parameters (df = 494)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01765 0.11727 0.151 0.88
xs 0.83856 0.20047 4.183 3.41e-05 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0251 0.1618 0.155 0.877
xo 0.8827 0.1612 5.474 6.99e-08 ***
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 0.93966 0.07971 11.79 <2e-16 ***
rho -0.55189 0.22433 -2.46 0.0142 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Tobit treatment model (switching regression model)
Maximum Likelihood estimation
Newton-Raphson maximisation, 4 iterations
Return code 1: gradient close to zero
Log-Likelihood: -2651.502
2344 observations: 2204 non-participants (selection FALSE) and 140 participants (selection TRUE)
17 free parameters (df = 2327)
Probit selection equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.94272 0.38051 -5.106 3.57e-07 ***
poly(age, 2)1 -41.64058 7.63374 -5.455 5.42e-08 ***
poly(age, 2)2 2.65968 4.97762 0.534 0.593166
educ -0.13661 0.03207 -4.260 2.13e-05 ***
u74TRUE 0.79452 0.22374 3.551 0.000391 ***
u75TRUE 2.31494 0.21291 10.873 < 2e-16 ***
ethnblack 1.35300 0.18734 7.222 6.89e-13 ***
ethnhispanic 1.31932 0.29465 4.478 7.91e-06 ***
Outcome equation:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.983926 0.069341 129.561 < 2e-16 ***
treatTRUE -0.963132 0.075837 -12.700 < 2e-16 ***
poly(age, 2)1 6.512273 0.797670 8.164 5.25e-16 ***
poly(age, 2)2 -4.428831 0.773235 -5.728 1.15e-08 ***
educ 0.080227 0.005231 15.338 < 2e-16 ***
ethnblack -0.256112 0.035865 -7.141 1.23e-12 ***
ethnhispanic -0.007786 0.079273 -0.098 0.922
Error terms:
Estimate Std. Error t value Pr(>|t|)
sigma 0.69304 0.01014 68.359 < 2e-16 ***
rho 0.17699 0.06502 2.722 0.00654 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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