### Simulation results of coverage rates using different methods of computing rho in the Heckman two-step estimator

Results from heckman simulations to measure coverage rates for the null hypotheses that each estimated model parameter is equal to its true value. Simulations performed with program simrho.do.

Each column in the tables below represents a coeffient from the regression equation of the model. The reported coverage rates are shown for each method of computing rho, and there are separate tables for tests at the 1%, 5%, and 10% level of significance.

                   Simulation 1

Data generating process:  N = 50
rho = 0.8
selection rate = .4
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 338
Number of times rhoforce produced non-PD VCE = 7

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.051    0.030    0.034    0.020
rhotrunc   |    0.030    0.023    0.026    0.015
rholimited |    0.030    0.023    0.026    0.015
rhosigma   |    0.026    0.022    0.026    0.015
MLE        |    0.202    0.233    0.177    0.161

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.107    0.099    0.073    0.079
rhotrunc   |    0.076    0.076    0.067    0.066
rholimited |    0.076    0.076    0.066    0.066
rhosigma   |    0.070    0.072    0.060    0.062
MLE        |    0.268    0.307    0.235    0.237

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.174    0.167    0.130    0.129
rhotrunc   |    0.138    0.141    0.116    0.108
rholimited |    0.138    0.140    0.116    0.104
rhosigma   |    0.127    0.131    0.109    0.096
MLE        |    0.323    0.365    0.299    0.306

Simulation 2

Data generating process:  N = 100
rho = -0.85
selection rate = .4
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 268
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.032    0.022    0.026    0.026
rhotrunc   |    0.014    0.016    0.026    0.020
rholimited |    0.014    0.016    0.026    0.020
rhosigma   |    0.011    0.016    0.026    0.020
MLE        |    0.145    0.176    0.130    0.126

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.091    0.077    0.070    0.080
rhotrunc   |    0.075    0.068    0.064    0.061
rholimited |    0.074    0.068    0.064    0.060
rhosigma   |    0.067    0.065    0.063    0.056
MLE        |    0.215    0.234    0.190    0.203

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.155    0.138    0.122    0.129
rhotrunc   |    0.136    0.117    0.114    0.112
rholimited |    0.136    0.117    0.113    0.110
rhosigma   |    0.130    0.109    0.109    0.102
MLE        |    0.271    0.270    0.237    0.261

Simulation 3

Data generating process:  N = 100
rho = 0.4
selection rate = .8
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 93
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.015    0.016    0.015    0.019
rhotrunc   |    0.012    0.015    0.015    0.016
rholimited |    0.012    0.015    0.015    0.016
rhosigma   |    0.011    0.015    0.015    0.016
MLE        |    0.048    0.053    0.037    0.056

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.060    0.061    0.062    0.066
rhotrunc   |    0.058    0.060    0.060    0.062
rholimited |    0.058    0.060    0.060    0.061
rhosigma   |    0.054    0.059    0.059    0.057
MLE        |    0.100    0.095    0.085    0.105

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.112    0.116    0.114    0.117
rhotrunc   |    0.108    0.116    0.113    0.113
rholimited |    0.108    0.116    0.113    0.110
rhosigma   |    0.106    0.114    0.111    0.106
MLE        |    0.166    0.149    0.141    0.165

Simulation 4

Data generating process:  N = 200
rho = 0.6
selection rate = .6
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 34
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.017    0.017    0.016    0.017
rhotrunc   |    0.017    0.017    0.016    0.017
rholimited |    0.017    0.017    0.016    0.017
rhosigma   |    0.017    0.017    0.015    0.017
MLE        |    0.025    0.015    0.017    0.021

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.055    0.062    0.055    0.066
rhotrunc   |    0.055    0.062    0.054    0.063
rholimited |    0.055    0.062    0.054    0.063
rhosigma   |    0.054    0.062    0.053    0.060
MLE        |    0.064    0.061    0.061    0.079

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.113    0.104    0.107    0.106
rhotrunc   |    0.113    0.103    0.107    0.106
rholimited |    0.113    0.103    0.107    0.104
rhosigma   |    0.113    0.102    0.107    0.104
MLE        |    0.119    0.109    0.113    0.126

Simulation 5

Data generating process:  N = 300
rho = 0.6
selection rate = .6
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 7
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.010    0.013    0.005    0.012
rhotrunc   |    0.010    0.013    0.005    0.012
rholimited |    0.010    0.013    0.005    0.012
rhosigma   |    0.010    0.013    0.005    0.012
MLE        |    0.012    0.014    0.006    0.016

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.060    0.055    0.067    0.070
rhotrunc   |    0.059    0.055    0.067    0.070
rholimited |    0.059    0.055    0.067    0.070
rhosigma   |    0.059    0.055    0.067    0.069
MLE        |    0.058    0.058    0.066    0.067

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.117    0.119    0.123    0.117
rhotrunc   |    0.117    0.119    0.123    0.117
rholimited |    0.117    0.119    0.123    0.117
rhosigma   |    0.116    0.119    0.123    0.117
MLE        |    0.127    0.112    0.125    0.122

Simulation 6

Data generating process:  N = 1000
rho = 0.6
selection rate = .4
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 0
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.007    0.014    0.011    0.012
rhotrunc   |    0.007    0.014    0.011    0.012
rholimited |    0.007    0.014    0.011    0.012
rhosigma   |    0.007    0.014    0.011    0.012
MLE        |    0.006    0.015    0.011    0.011

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.036    0.048    0.039    0.051
rhotrunc   |    0.036    0.048    0.039    0.051
rholimited |    0.036    0.048    0.039    0.051
rhosigma   |    0.036    0.048    0.039    0.051
MLE        |    0.036    0.048    0.039    0.044

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.083    0.100    0.094    0.116
rhotrunc   |    0.083    0.100    0.094    0.116
rholimited |    0.083    0.100    0.094    0.116
rhosigma   |    0.083    0.100    0.094    0.116
MLE        |    0.088    0.096    0.096    0.104

Simulation 7

Data generating process:  N = 1000
rho = 0.4
selection rate = .8
repetitions = 1000

Number of times estimate of rho outside [-1,1] = 0
Number of times rhoforce produced non-PD VCE = 0

. Report .01

Rejection rates for 1% tests against true nulls
(nominal rates are .01)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.009    0.009    0.013    0.008
rhotrunc   |    0.009    0.009    0.013    0.008
rholimited |    0.009    0.009    0.013    0.008
rhosigma   |    0.009    0.009    0.013    0.008
MLE        |    0.011    0.010    0.014    0.009

. Report .05

Rejection rates for 5% tests against true nulls
(nominal rates are .05)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.045    0.045    0.049    0.057
rhotrunc   |    0.045    0.045    0.049    0.057
rholimited |    0.045    0.045    0.049    0.057
rhosigma   |    0.045    0.045    0.049    0.057
MLE        |    0.055    0.046    0.051    0.054

. Report .10

Rejection rates for 10% tests against true nulls
(nominal rates are .10)
Rho method |      B_1      B_2      B_3   B_cons
-----------+------------------------------------
rhoforce   |    0.086    0.094    0.090    0.106
rhotrunc   |    0.086    0.094    0.090    0.106
rholimited |    0.086    0.094    0.090    0.106
rhosigma   |    0.086    0.094    0.090    0.106
MLE        |    0.096    0.098    0.089    0.103