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Bootstrapping for Model Comparison
implied moments obtained from fitting Model 1 to the b-th bootstrap sample. Thus,
is a measure of how much the population moments differ from the
moments estimated from the b-th bootstrap sample using Model 1.
The average of over 1,000 bootstrap samples was 64.162 with a standard
error of 0.292. Similar histograms, along with means and standard errors, are displayed
for the other four models but are not reproduced here. The average discrepancies for
the five competing models are shown in the table below, along with values of BCC,
AIC, and CAIC. The table provides fit measures for five competing models (standard
errors in parentheses).
The Failures column in the table indicates that the likelihood function of Model 2 could
not be maximized for 19 of the 1,000 bootstrap samples, at least not with the iteration
limit of 40. Nineteen additional bootstrap samples were generated for Model 2 in order
to bring the total number of bootstrap samples to the target of 1,000. The 19 samples
where Model 2 could not be fitted successfully caused no problem with the other four
models. Consequently, 981 bootstrap samples were common to all five models.
No attempt was made to find out why Model 2 estimates could not be computed for
19 bootstrap samples. As a rule, algorithms for analysis of moment structures tend to
Model Failures
Mean
Discrepancy
BCC AIC CAIC
1 0 64.16 (0.29) 68.17 66.94 114.66
2 19 29.14 (0.35 36.81 35.07 102.68
2R 0 26.57 (0.30) 30.97 29.64 81.34
Sat. 0 32.05 (0.37) 44.15 42.00 125.51
Indep. 0 334.32 (0.24) 333.93 333.32 357.18
C
ML
α
ˆ
b
a,()
ML discrepancy (implied vs pop) (Default model)
|--------------------
48.268 |**
52.091 |*********
55.913 |*************
59.735 |*******************
63.557 |*****************
67.379 |************
71.202 |********
N = 1000 75.024 |******
Mean = 64.162 78.846 |***
S. e. = .292 82.668 |*
86.490 |**
90.313 |**
94.135 |*
97.957 |*
101.779 |*
|--------------------
C
ML
α
ˆ
b
a,()