R Under development (unstable) (2026-01-18 r89306 ucrt) -- "Unsuffered Consequences"
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> library( "sampleSelection" )
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/
> suppressPackageStartupMessages( library( "mvtnorm" ) )
> options( digits = 3 )
> set.seed( 123 )
> 
> 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!
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (542 censored and 458 observed)
8 free parameters (df = 993)
Probit selection equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -0.2036     0.0601   -3.39  0.00073 ***
x1            1.9961     0.1229   16.24  < 2e-16 ***
x2            1.0642     0.0821   12.97  < 2e-16 ***
Outcome equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.1380     0.0909    12.5   <2e-16 ***
w             1.4317     0.0367    39.0   <2e-16 ***
Multiple R-Squared:0.785,	Adjusted R-Squared:0.784
   Error terms:
              Estimate Std. Error t value Pr(>|t|)
invMillsRatio   0.1034     0.0965    1.07     0.28
sigma           0.8747         NA      NA       NA
rho             0.1182         NA      NA       NA
--------------------------------------------
> myHeckitIv <- heckit( s ~ x1 + x2, y ~ w, data = myData, inst = ~ x1 )
> summary( myHeckitIv ) # unbiased
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (542 censored and 458 observed)
8 free parameters (df = 993)
Probit selection equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -0.2036     0.0601   -3.39  0.00073 ***
x1            1.9961     0.1229   16.24  < 2e-16 ***
x2            1.0642     0.0821   12.97  < 2e-16 ***
Outcome equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   2.0381     0.1046    19.5   <2e-16 ***
w             0.9892     0.0422    23.4   <2e-16 ***
Multiple R-Squared:0.717,	Adjusted R-Squared:0.716
   Error terms:
              Estimate Std. Error t value Pr(>|t|)  
invMillsRatio   -0.278      0.110   -2.52    0.012 *
sigma            1.013         NA      NA       NA  
rho             -0.274         NA      NA       NA  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
> 
> myHeckitIvWe <- heckit( s ~ x1 + x2, y ~ w, data = myData, inst = ~ x1,
+    weights = rep( 0.5, nObs ) )
Warning message:
In heckit2fit(selection, outcome, data = data, weights = weights,  :
  weights are ignored in instrumental variable estimations
> summary( myHeckitIvWe )
--------------------------------------------
Tobit 2 model (sample selection model)
2-step Heckman / heckit estimation
1000 observations (542 censored and 458 observed)
8 free parameters (df = 993)
Probit selection equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -0.2036     0.0849   -2.40    0.017 *  
x1            1.9961     0.1738   11.49   <2e-16 ***
x2            1.0642     0.1161    9.17   <2e-16 ***
Outcome equation:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   2.0381     0.1049    19.4   <2e-16 ***
w             0.9892     0.0423    23.4   <2e-16 ***
Multiple R-Squared:0.717,	Adjusted R-Squared:0.716
   Error terms:
              Estimate Std. Error t value Pr(>|t|)  
invMillsRatio   -0.278      0.111    -2.5    0.012 *
sigma            1.013         NA      NA       NA  
rho             -0.274         NA      NA       NA  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------
> 
> 
> proc.time()
   user  system elapsed 
   1.18    0.17    1.34