Fits a sample selection model based on the Student's t-distribution, extending the classical Heckman model to account for heavy-tailed error terms. The estimation is performed via Maximum Likelihood using the BFGS algorithm.
Usage
HeckmantS(selection, outcome, data = sys.frame(sys.parent()), df, start = NULL)Value
A list containing:
coefficients: Named vector of estimated model parameters.value: Negative of the maximum log-likelihood.loglik: Maximum log-likelihood.counts: Number of gradient evaluations performed.hessian: Hessian matrix at the optimum.fisher_infotS: Approximate Fisher information matrix.prop_sigmatS: Standard errors for the parameter estimates.level: Levels of the selection variable.nObs: Number of observations.nParam: Number of model parameters.N0: Number of censored (unobserved) observations.N1: Number of uncensored (observed) observations.NXS: Number of parameters in the selection equation.NXO: Number of parameters in the outcome equation.df: Degrees of freedom (observations minus parameters).aic: Akaike Information Criterion.bic: Bayesian Information Criterion.initial.value: Initial parameter values used in the optimization.
Details
The function implements the Heckman sample selection model using the Student's t-distribution for the error terms, as proposed by Marchenko and Genton (2012) . This extension allows for robustness against outliers and heavy-tailed distributions. Initial parameter values can be specified by the user or default to standard starting values.
References
Yulia V Marchenko, Marc G Genton (2012). “A Heckman selection-t model.” Journal of the American Statistical Association, 107(497), 304–317.
Examples
data(MEPS2001)
attach(MEPS2001)
#> The following objects are masked from Mroz87:
#>
#> age, educ
#> The following objects are masked from MEPS2001 (pos = 4):
#>
#> age, age2, agefem, ambexp, blhisp, dambexp, dhospexp, educ,
#> fairpoor, female, ffs, good, hospexp, income, ins, instype,
#> instype_s1, lambexp, lnambx, totchr, vgood, year01
#> The following objects are masked from MEPS2001 (pos = 5):
#>
#> age, age2, agefem, ambexp, blhisp, dambexp, dhospexp, educ,
#> fairpoor, female, ffs, good, hospexp, income, ins, instype,
#> instype_s1, lambexp, lnambx, totchr, vgood, year01
#> The following objects are masked from MEPS2001 (pos = 6):
#>
#> age, age2, agefem, ambexp, blhisp, dambexp, dhospexp, educ,
#> fairpoor, female, ffs, good, hospexp, income, ins, instype,
#> instype_s1, lambexp, lnambx, totchr, vgood, year01
selectEq <- dambexp ~ age + female + educ + blhisp + totchr + ins + income
outcomeEq <- lnambx ~ age + female + educ + blhisp + totchr + ins
HeckmantS(selectEq, outcomeEq, data = MEPS2001, df = 12)
#> Start not provided using default start values.
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> Warning: NaNs produced
#> $coefficients
#> (Intercept) age female educ blhisp totchr
#> -0.748019043 0.098533957 0.724842140 0.064846420 -0.393556230 0.890059436
#> ins income (Intercept) age female educ
#> 0.180016364 0.002977867 5.205886164 0.206829239 0.306506438 0.017318512
#> blhisp totchr ins sigma rho df
#> -0.192934870 0.512686542 -0.052509944 1.194892986 -0.322168670 12.934344590
#>
#> $value
#> [1] -5822.076
#>
#> $loglik
#> [1] -5822.076
#>
#> $counts
#> gradient
#> 69
#>
#> $hessian
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -1045.335913 -3975.20743 -359.414455 -13504.5652 -4.300910e+02
#> [2,] -3975.207432 -16388.30432 -1349.322374 -51289.5058 -1.583016e+03
#> [3,] -359.414455 -1349.32237 -359.414455 -4656.7914 -1.695611e+02
#> [4,] -13504.565168 -51289.50582 -4656.791414 -181859.6871 -5.232408e+03
#> [5,] -430.090984 -1583.01619 -169.561150 -5232.4082 -4.300910e+02
#> [6,] -145.860600 -602.85462 -46.163000 -1835.1456 -5.929394e+01
#> [7,] -346.558168 -1411.42865 -103.902184 -4536.3727 -1.329523e+02
#> [8,] -35004.285459 -138431.90013 -10349.321495 -473374.4981 -1.240440e+04
#> [9,] -190.799948 -739.56329 -76.244731 -2507.6750 -7.078001e+01
#> [10,] -739.562821 -3106.34398 -287.796589 -9704.5152 -2.661973e+02
#> [11,] -76.244731 -287.79693 -76.244731 -996.1872 -3.293481e+01
#> [12,] -2507.658742 -9704.45232 -996.176053 -34305.5602 -8.752624e+02
#> [13,] -70.780011 -266.19737 -32.934807 -875.2644 -7.078001e+01
#> [14,] -35.569055 -148.80304 -12.327036 -450.6581 -1.347519e+01
#> [15,] -66.954815 -274.03304 -23.866914 -892.0566 -2.415969e+01
#> [16,] -27.596750 -128.70745 -24.945396 -415.2929 -7.303460e-01
#> [17,] -89.557989 -341.59549 -17.435199 -1094.2383 -4.616415e+01
#> [18,] -3.457874 -14.60579 -2.228612 -47.5399 -7.609129e-01
#> [,6] [,7] [,8] [,9] [,10]
#> [1,] -145.860600 -346.558168 -35004.2855 -190.79995 -739.56282
#> [2,] -602.854624 -1411.428645 -138431.9001 -739.56329 -3106.34398
#> [3,] -46.163000 -103.902184 -10349.3215 -76.24473 -287.79659
#> [4,] -1835.145604 -4536.372685 -473374.4981 -2507.67495 -9704.51523
#> [5,] -59.293944 -132.952337 -12404.3954 -70.78001 -266.19730
#> [6,] -166.289194 -43.198594 -4820.5854 -35.56906 -148.80274
#> [7,] -43.198594 -346.558168 -13210.0409 -66.95482 -274.03283
#> [8,] -4820.585397 -13210.040947 -1752607.6357 -6589.10677 -26630.18862
#> [9,] -35.569058 -66.954815 -6589.1068 -1795.39082 -7408.16204
#> [10,] -148.802745 -274.032834 -26630.1886 -7408.16204 -32811.75078
#> [11,] -12.327037 -23.866914 -2304.4776 -998.52227 -4103.17504
#> [12,] -450.649269 -892.049315 -90457.5089 -24387.44513 -100669.64445
#> [13,] -13.475193 -24.159689 -2088.0579 -498.02689 -1963.48937
#> [14,] -41.544562 -11.173235 -1209.6813 -1005.56255 -4506.03994
#> [15,] -11.173236 -66.954815 -2605.2613 -689.10720 -2935.24072
#> [16,] -18.577696 -13.956406 -1350.3027 34.23691 92.53846
#> [17,] 2.124710 -23.738318 -2961.7796 -537.92964 -2052.81383
#> [18,] -1.593787 -1.356436 -131.7931 -3.26218 -14.02076
#> [,11] [,12] [,13] [,14] [,15]
#> [1,] -76.244731 -2507.65874 -70.7800111 -35.569055 -66.954815
#> [2,] -287.796926 -9704.45232 -266.1973677 -148.803036 -274.033038
#> [3,] -76.244731 -996.17605 -32.9348068 -12.327036 -23.866914
#> [4,] -996.187168 -34305.56020 -875.2643948 -450.658097 -892.056601
#> [5,] -32.934807 -875.26241 -70.7800111 -13.475191 -24.159689
#> [6,] -12.327037 -450.64927 -13.4751928 -41.544562 -11.173236
#> [7,] -23.866914 -892.04931 -24.1596887 -11.173235 -66.954815
#> [8,] -2304.477593 -90457.50893 -2088.0579023 -1209.681264 -2605.261296
#> [9,] -998.522272 -24387.44513 -498.0268913 -1005.562550 -689.107202
#> [10,] -4103.175038 -100669.64445 -1963.4893720 -4506.039945 -2935.240722
#> [11,] -998.522272 -13532.11319 -294.9030200 -620.817336 -353.790925
#> [12,] -13532.113186 -342591.76954 -6339.2923835 -13568.013755 -9447.606329
#> [13,] -294.903020 -6339.29238 -498.0268913 -254.221224 -173.624910
#> [14,] -620.817336 -13568.01375 -254.2212239 -1757.351110 -349.650848
#> [15,] -353.790925 -9447.60633 -173.6249104 -349.650848 -689.107202
#> [16,] -8.626206 343.03363 28.0011939 -31.910629 11.632587
#> [17,] -185.076727 -6938.23634 -221.0280269 -71.886791 -175.804616
#> [18,] -1.939404 -46.04109 -0.8021633 -2.085468 -1.016321
#> [,16] [,17] [,18]
#> [1,] -27.596750 -89.557989 -3.4578743
#> [2,] -128.707453 -341.595494 -14.6057940
#> [3,] -24.945396 -17.435199 -2.2286117
#> [4,] -415.292925 -1094.238253 -47.5399006
#> [5,] -0.730346 -46.164151 -0.7609129
#> [6,] -18.577696 2.124710 -1.5937867
#> [7,] -13.956406 -23.738318 -1.3564356
#> [8,] -1350.302702 -2961.779624 -131.7930574
#> [9,] 34.236914 -537.929642 -3.2621802
#> [10,] 92.538456 -2052.813834 -14.0207611
#> [11,] -8.626206 -185.076727 -1.9394037
#> [12,] 343.033627 -6938.236339 -46.0410879
#> [13,] 28.001194 -221.028027 -0.8021633
#> [14,] -31.910629 -71.886791 -2.0854683
#> [15,] 11.632587 -175.804616 -1.0163213
#> [16,] -3227.190075 -314.286394 18.1021806
#> [17,] -314.286394 -378.037383 1.9028018
#> [18,] 18.102181 1.902802 -0.2655836
#>
#> $fisher_infotS
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 4.313375e-02 -3.500000e-03 -1.214053e-03 -2.126291e-03 -4.654934e-03
#> [2,] -3.500000e-03 8.847170e-04 -1.109301e-05 3.884487e-05 1.471318e-04
#> [3,] -1.214053e-03 -1.109301e-05 4.697741e-03 -5.235787e-05 -4.567688e-04
#> [4,] -2.126291e-03 3.884487e-05 -5.235787e-05 1.639570e-04 1.706921e-04
#> [5,] -4.654934e-03 1.471318e-04 -4.567688e-04 1.706921e-04 4.425352e-03
#> [6,] -9.486884e-04 -2.578003e-04 4.667081e-04 5.822508e-05 -1.436173e-04
#> [7,] -6.955367e-05 -3.081641e-04 2.678786e-04 -9.540082e-06 -9.178129e-06
#> [8,] 2.931381e-05 -7.887625e-06 1.434196e-05 -5.703822e-06 5.655671e-06
#> [9,] -4.189627e-03 5.093448e-04 -3.375106e-04 1.636414e-04 7.990585e-04
#> [10,] 3.199742e-04 -9.669708e-05 3.086669e-05 3.036477e-06 -3.661025e-05
#> [11,] -3.512132e-05 -3.154244e-05 -3.452484e-04 2.062261e-05 -3.509919e-05
#> [12,] 2.093716e-04 -3.432091e-06 2.207445e-05 -1.510017e-05 -3.072496e-05
#> [13,] 6.678148e-04 -2.657410e-07 -4.669232e-05 -3.204896e-05 -5.027497e-04
#> [14,] -1.501501e-04 5.039836e-06 1.050776e-04 1.080070e-05 -4.802927e-05
#> [15,] 3.466940e-05 1.064825e-05 -2.476736e-05 6.229279e-06 -2.605442e-05
#> [16,] 3.389544e-04 -2.199003e-05 -2.833154e-04 -1.840409e-05 1.341094e-04
#> [17,] -5.510202e-04 -1.613765e-04 2.902144e-04 5.860347e-05 -1.458891e-04
#> [18,] 4.323498e-02 -6.392716e-03 -4.021904e-02 -1.870223e-03 1.900890e-02
#> [,6] [,7] [,8] [,9] [,10]
#> [1,] -9.486884e-04 -6.955367e-05 2.931381e-05 -4.189627e-03 3.199742e-04
#> [2,] -2.578003e-04 -3.081641e-04 -7.887625e-06 5.093448e-04 -9.669708e-05
#> [3,] 4.667081e-04 2.678786e-04 1.434196e-05 -3.375106e-04 3.086669e-05
#> [4,] 5.822508e-05 -9.540082e-06 -5.703822e-06 1.636414e-04 3.036477e-06
#> [5,] -1.436173e-04 -9.178129e-06 5.655671e-06 7.990585e-04 -3.661025e-05
#> [6,] 7.604425e-03 3.483078e-04 2.728479e-06 -1.366306e-03 9.106068e-05
#> [7,] 3.483078e-04 4.624768e-03 -8.482870e-06 -2.853336e-04 4.299572e-05
#> [8,] 2.728479e-06 -8.482870e-06 2.092491e-06 2.038862e-06 -1.224739e-07
#> [9,] -1.366306e-03 -2.853336e-04 2.038862e-06 4.358989e-02 -2.539082e-03
#> [10,] 9.106068e-05 4.299572e-05 -1.224739e-07 -2.539082e-03 5.102522e-04
#> [11,] 3.326870e-04 5.141118e-05 -6.502066e-07 -4.908609e-03 1.606443e-04
#> [12,] 4.473854e-05 1.212803e-05 -1.624229e-08 -1.719292e-03 1.716642e-05
#> [13,] -1.864809e-04 -7.574758e-05 3.701026e-07 -6.221111e-04 2.720135e-05
#> [14,] -1.262210e-05 3.458621e-05 -2.631148e-07 -2.382221e-03 -5.714482e-05
#> [15,] 2.700121e-05 -4.434396e-04 -2.784404e-07 -1.619448e-03 -6.834446e-05
#> [16,] -4.730626e-04 -7.624058e-05 -7.863843e-07 1.378454e-03 -6.191654e-05
#> [17,] 9.441059e-04 2.867290e-04 -2.282407e-06 -1.409395e-02 5.316861e-04
#> [18,] -5.929615e-02 -6.893023e-03 -1.538194e-04 -4.128248e-02 6.738710e-05
#> [,11] [,12] [,13] [,14] [,15]
#> [1,] -3.512132e-05 2.093716e-04 6.678148e-04 -1.501501e-04 3.466940e-05
#> [2,] -3.154244e-05 -3.432091e-06 -2.657410e-07 5.039836e-06 1.064825e-05
#> [3,] -3.452484e-04 2.207445e-05 -4.669232e-05 1.050776e-04 -2.476736e-05
#> [4,] 2.062261e-05 -1.510017e-05 -3.204896e-05 1.080070e-05 6.229279e-06
#> [5,] -3.509919e-05 -3.072496e-05 -5.027497e-04 -4.802927e-05 -2.605442e-05
#> [6,] 3.326870e-04 4.473854e-05 -1.864809e-04 -1.262210e-05 2.700121e-05
#> [7,] 5.141118e-05 1.212803e-05 -7.574758e-05 3.458621e-05 -4.434396e-04
#> [8,] -6.502066e-07 -1.624229e-08 3.701026e-07 -2.631148e-07 -2.784404e-07
#> [9,] -4.908609e-03 -1.719292e-03 -6.221111e-04 -2.382221e-03 -1.619448e-03
#> [10,] 1.606443e-04 1.716642e-05 2.720135e-05 -5.714482e-05 -6.834446e-05
#> [11,] 3.162279e-03 9.156810e-05 -6.173213e-04 4.292646e-04 4.166586e-04
#> [12,] 9.156810e-05 1.050257e-04 5.579628e-05 7.022270e-05 1.537829e-05
#> [13,] -6.173213e-04 5.579628e-05 3.327632e-03 -2.969169e-04 -9.295577e-05
#> [14,] 4.292646e-04 7.022270e-05 -2.969169e-04 1.275292e-03 2.862802e-04
#> [15,] 4.166586e-04 1.537829e-05 -9.295577e-05 2.862802e-04 2.546589e-03
#> [16,] -3.081356e-04 -4.241507e-05 1.861276e-04 -2.102128e-04 -6.424215e-05
#> [17,] 3.286068e-03 3.626051e-04 -2.025724e-03 2.211183e-03 1.107330e-03
#> [18,] 1.184755e-02 1.044868e-05 -7.938867e-03 7.300716e-03 1.000511e-02
#> [,16] [,17] [,18]
#> [1,] 3.389544e-04 -5.510202e-04 4.323498e-02
#> [2,] -2.199003e-05 -1.613765e-04 -6.392716e-03
#> [3,] -2.833154e-04 2.902144e-04 -4.021904e-02
#> [4,] -1.840409e-05 5.860347e-05 -1.870223e-03
#> [5,] 1.341094e-04 -1.458891e-04 1.900890e-02
#> [6,] -4.730626e-04 9.441059e-04 -5.929615e-02
#> [7,] -7.624058e-05 2.867290e-04 -6.893023e-03
#> [8,] -7.863843e-07 -2.282407e-06 -1.538194e-04
#> [9,] 1.378454e-03 -1.409395e-02 -4.128248e-02
#> [10,] -6.191654e-05 5.316861e-04 6.738710e-05
#> [11,] -3.081356e-04 3.286068e-03 1.184755e-02
#> [12,] -4.241507e-05 3.626051e-04 1.044868e-05
#> [13,] 1.861276e-04 -2.025724e-03 -7.938867e-03
#> [14,] -2.102128e-04 2.211183e-03 7.300716e-03
#> [15,] -6.424215e-05 1.107330e-03 1.000511e-02
#> [16,] 6.589598e-04 -1.082931e-03 4.013272e-02
#> [17,] -1.082931e-03 1.311913e-02 6.043407e-02
#> [18,] 4.013272e-02 6.043407e-02 8.151115e+00
#>
#> $prop_sigmatS
#> [1] 0.207686662 0.029744192 0.068540069 0.012804568 0.066523321 0.087203352
#> [7] 0.068005646 0.001446544 0.208781917 0.022588764 0.056234142 0.010248202
#> [13] 0.057685628 0.035711227 0.050463743 0.025670212 0.114538789 2.855015718
#>
#> $level
#> [1] "0" "1"
#>
#> $nObs
#> [1] 3328
#>
#> $nParam
#> [1] 18
#>
#> $N0
#> [1] 526
#>
#> $N1
#> [1] 2802
#>
#> $NXS
#> [1] 8
#>
#> $NXO
#> [1] 7
#>
#> $df
#> [1] 3310
#>
#> $aic
#> [1] 11680.15
#>
#> $bic
#> [1] 11790.13
#>
#> $initial.value
#> [1] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
#> [9] 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000
#> [17] 0.000000 2.484907
#>
#> attr(,"class")
#> [1] "HeckmantS" "list"