samples. Yellow (light color) indicates “inadmissible” estimates ($\hat{\psi}_3 < 0$). Top: maximum-likelihood estimates. Bottom: a single Bayesian trace plot produced by `stan`. []{data-label="fig:shapiro"}](shapiro-bayes.pdf "fig:")

These problems occur *only* with the estimates $\hat{\psi_3}$. Standard results, such as convergence to normality, large-sample unbiasedness, and correctness of normal-theory standard errors, do accrue for every other parameter of the model. The reason for this is that the null space of $\Delta$, namely $\text{Null}(\Delta_{\text{Shapiro}}) = [\begin{array}{cccccc}
0&0&0&0&0 &1
\end{array}]^T$, does not involve any of these other parameters. We have noted a similar phenomenon in the CTUM SB-MTMM model, where the correlations among traits are not involved in any rank deficiency.

In short, although @shapiro_investigation_1983 were correct in pointing out there is no *identification* problem with this model, real-life analysis of this model using finite sample will yield plenty of other problems, regardless of the estimation method used. The same phenomenon occurs in more complex models such as the reduced-group split-ballot multitrait-multimethod model, discussed in the following section.

Convergence and “admissibility” of SB-MTMM model
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The previous section illustrated the basic problems that result from a simple rank deficiency in a simple model. I now illustrate how rank deficiencies affect estimation of the reduced-group split-ballot multitrait-multimethod 