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. Author manuscript; available in PMC: 2024 Jun 1.
Published in final edited form as: Can J Stat. 2022 Apr 15;51(2):355–374. doi: 10.1002/cjs.11701

Table 3:

Simulation results for different choices of d(j)(X(j)) and comparisons to Gu et al. (2019) based on 1000 replications. The proposed method here is Method 2 without using working models.

n = 400 n = 1000
βc β X 1 β X 2 βZ β X 2 Z βc β X 1 β X 2 βZ β X 2 Z
MLE bias 0.010 −0.005 −0.013 0.006 0.016 0.006 −0.006 −0.003 0.006 0.002
emp 0.122 0.136 0.144 0.140 0.139 0.076 0.084 0.087 0.083 0.085
GTCM-1 bias 0.036 0.015 0.008 0.044 −0.087 0.040 0.009 0.013 0.047 −0.093
emp 0.059 0.081 0.138 0.134 0.096 0.037 0.049 0.084 0.082 0.060
EL2-1-1 bias −0.000 0.018 −0.013 0.006 0.016 0.007 0.009 −0.003 0.006 0.002
emp 0.052 0.076 0.144 0.140 0.139 0.033 0.045 0.087 0.083 0.085
EL2-1-2 bias −0.000 0.018 −0.013 0.006 0.016 0.007 0.009 −0.002 0.006 0.003
emp 0.050 0.075 0.144 0.141 0.130 0.031 0.044 0.087 0.084 0.078
GTCM-2 bias 0.028 −0.017 0.087 0.044 −0.089 0.032 −0.016 0.098 0.048 −0.093
emp 0.054 0.071 0.076 0.133 0.095 0.035 0.042 0.047 0.081 0.060
EL2-2-1 bias 0.000 0.019 −0.033 0.006 0.015 0.006 0.018 −0.012 0.006 0.002
emp 0.048 0.065 0.062 0.141 0.138 0.030 0.037 0.035 0.083 0.085
EL2-2-2 bias −0.009 0.011 −0.034 0.008 0.049 −0.002 0.011 −0.013 0.008 0.028
emp 0.042 0.060 0.062 0.141 0.104 0.026 0.034 0.036 0.084 0.061
EL2-2-3 bias −0.010 0.011 −0.034 0.008 0.051 −0.002 0.011 −0.013 0.009 0.029
emp 0.042 0.060 0.062 0.142 0.104 0.026 0.034 0.036 0.084 0.061
GTCM-3 bias 0.031 0.001 0.001 0.044 −0.087 0.034 −0.001 0.018 0.047 −0.093
emp 0.064 0.081 0.081 0.134 0.095 0.041 0.049 0.049 0.082 0.060
EL2-3-1 bias −0.005 0.000 −0.012 0.006 0.014 0.000 −0.003 0.007 0.006 0.001
emp 0.059 0.076 0.075 0.140 0.139 0.039 0.044 0.045 0.083 0.085
EL2-3-2 bias −0.008 −0.001 −0.013 0.006 0.024 −0.002 −0.006 0.005 0.007 0.013
emp 0.057 0.073 0.073 0.140 0.114 0.036 0.043 0.044 0.083 0.069
EL2-3-3 bias −0.010 −0.001 −0.017 0.008 0.032 0.000 −0.008 0.012 0.005 0.005
emp 0.057 0.073 0.070 0.140 0.111 0.036 0.043 0.040 0.084 0.068

emp: empirical standard error. GTCM-1,2 or 3: the method in Gu et al. (2019) using Calculator 1, 2, or 3, respectively. EL2-j-1,2 or 3: the proposed Method 2 without working models using Calculator j and different specifications of d(j)(X(j)).