1. The Basic Procedure

In this section, we focus on single-group mean and structure analysis and will extend to multigroup analysis later. Let the $p\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vector $x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x} $$\end{document} denote the manifest variables, $\mu (·)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\cdot )$$\end{document} and $\Sigma (·)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\cdot )$$\end{document} the model of interest, and the $q\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times 1$$\end{document} vector ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} the population model parameter values specified by the researcher. The goal is to find some noise $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} and $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} to imitate the lack of fit in reality and construct ${\mu }^{\ast }=\mu \left({\theta }_0\right)+t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^* = \varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0) + \varvec{t} $$\end{document} and ${\Sigma }^{\ast }=\Sigma \left({\theta }_0\right)+E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^* = \varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0) + \varvec{E} $$\end{document} . Then, ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} are considered the population moments of $x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x} $$\end{document} and used to generate random data for simulation studies. We hope when fitting the model to ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} , the discrepancy function achieves its minimum at the specified parameter values ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} , and the amount of misfit equals some desired values. Here, we select the normal theory maximum likelihood (ML) fit function as the measure of misfit:

(1) $\begin{array}{c}{F}_{\mathrm{ML}}={[{\mu }^{\ast }-\mu \left(\theta \right)]}^{\prime }{\Sigma }^{-1}\left(\theta \right)[{\mu }^{\ast }-\mu \left(\theta \right)]+ln\left|\Sigma \left(\theta \right)\right|-ln|{\Sigma }^{\ast }|+\mathrm{tr}\left[{\Sigma }^{\ast }{\Sigma }^{-1}\left(\theta \right)\right]-p.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_\mathrm{ML} = [\varvec{{{\upmu }}} ^* - \varvec{{{\upmu }}} (\varvec{{{\uptheta }}})]' \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}}) [\varvec{{{\upmu }}} ^* - \varvec{{{\upmu }}} (\varvec{{{\uptheta }}})] + \ln |\varvec{{\Sigma }} (\varvec{{{\uptheta }}})| - \ln |\varvec{{\Sigma }} ^*| + \mathrm{tr}[\varvec{{\Sigma }} ^* \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}})] - p. \end{aligned}$$\end{document}

In constructing ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} (or equivalently $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} and $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} ), we seek to satisfy the following three requirements simultaneously:

(2) $\begin{array}{cc}& {\theta }_0=argminF[{\mu }^{\ast },{\Sigma }^{\ast };\mu \left(\theta \right),\Sigma \left(\theta \right)];\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\varvec{{{\uptheta }}} _0 = \arg \min F[\varvec{{{\upmu }}} ^*, \varvec{{\Sigma }} ^*; \varvec{{{\upmu }}} (\varvec{{{\uptheta }}}), \varvec{{\Sigma }} (\varvec{{{\uptheta }}})]; \end{aligned}$$\end{document}

(3) $\begin{array}{cc}& [{\mu }^{\ast }-\mu \left({\theta }_0\right){]}^{\prime }{\Sigma }^{-1}\left({\theta }_0\right)[{\mu }^{\ast }-\mu \left({\theta }_0\right)]=F_{\mathrm{mean}};\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{[}\varvec{{{\upmu }}} ^* - \varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0)]' \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}} _0) [\varvec{{{\upmu }}} ^* - \varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0)] = F_\mathrm{mean}; \end{aligned}$$\end{document}

(4) $\begin{array}{cc}& ln|\Sigma \left({\theta }_0\right)|-ln|{\Sigma }^{\ast }|+\mathrm{tr}\left[{\Sigma }^{\ast }{\Sigma }^{-1}\left({\theta }_0\right)\right]-p=F_{\mathrm{cov}};\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\ln |\varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0)| - \ln |\varvec{{\Sigma }} ^*| + \mathrm{tr}[\varvec{{\Sigma }} ^* \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}} _0)] - p = F_\mathrm{cov}; \end{aligned}$$\end{document}

where ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} , $F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean}$$\end{document} , and $F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov}$$\end{document} are constants specified by the researcher. The desired $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} value for the whole model is then $F_{\mathrm{mean}}+F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean} + F_\mathrm{cov}$$\end{document} . The necessary condition for Eq. (2) is (see “Appendix”):

(5) $\begin{array}{c}\dot{F}\left({\theta }_0\right)=2\dot{\mu }\left({\theta }_0\right){\Sigma }^{-1}\left({\theta }_0\right)·t+\dot{\Sigma }\left({\theta }_0\right)[{\Sigma }^{-1}\left({\theta }_0\right)\otimes {\Sigma }^{-1}\left({\theta }_0\right)]·\mathrm{vec}(t{t}^{\prime }+E)=0,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\dot{F}}(\varvec{{{\uptheta }}} _0) = 2\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0) \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}} _0) \cdot \varvec{t} + \dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0) [\varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}} _0) \otimes \varvec{{\Sigma }} ^{-1}(\varvec{{{\uptheta }}} _0)] \cdot \mathrm{vec}(\varvec{t} \varvec{t} ' + \varvec{E}) = \varvec{0}, \end{aligned}$$\end{document}

where $\dot{F}\left({\theta }_0\right)=\partial F/\partial \theta {\mid }_{\theta ={\theta }_0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{F}}(\varvec{{{\uptheta }}} _0) = \partial F / \partial \varvec{{{\uptheta }}} \mid _{\varvec{{{\uptheta }}} = \varvec{{{\uptheta }}} _0}$$\end{document} (a $q\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times 1$$\end{document} vector), $\dot{\mu }\left({\theta }_0\right)=\partial \mu {\left(\theta \right)}^{\prime }/\partial \theta {\mid }_{\theta ={\theta }_0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0) = \partial \varvec{{{\upmu }}} (\varvec{{{\uptheta }}})' / \partial \varvec{{{\uptheta }}} \mid _{\varvec{{{\uptheta }}} = \varvec{{{\uptheta }}} _0}$$\end{document} (a $q\times p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times p$$\end{document} matrix), $\dot{\Sigma }\left({\theta }_0\right)=\partial {\mathrm{vec}}^{\prime }\left[\Sigma \left(\theta \right)\right]/\partial \theta {\mid }_{\theta ={\theta }_0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0) = \partial {\mathrm{vec}'}[\varvec{{\Sigma }} (\varvec{{{\uptheta }}})] / \partial \varvec{{{\uptheta }}} \mid _{\varvec{{{\uptheta }}} = \varvec{{{\uptheta }}} _0}$$\end{document} (a $q\times p^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times p^2$$\end{document} matrix), $\mathrm{vec}(·)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{vec}(\cdot )$$\end{document} stack the columns of a matrix into a vector, and $\otimes$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\otimes $$\end{document} is the Kronecker product. We define the derivatives in this way so that the j-th row in $\dot{F}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{F}}(\varvec{{{\uptheta }}})$$\end{document} is the derivative of $F\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(\varvec{{{\uptheta }}})$$\end{document} with respect to ${\theta }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\uptheta }} _j$$\end{document} . That is, we define $\dot{F}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{F}}(\varvec{{{\uptheta }}})$$\end{document} as the transpose of the Jacobian matrix of $F\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(\varvec{{{\uptheta }}})$$\end{document} with respect to $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} $$\end{document} , because doing so facilitates the following exposition of our method. We define $\dot{\mu }\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}})$$\end{document} and $\dot{\Sigma }\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}})$$\end{document} in the same manner.

Note some rows in $\dot{\mu }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)$$\end{document} and $\dot{\Sigma }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)$$\end{document} are zeros because some model parameters do not play a role in the mean or the covariance structure. More specifically, parameters for the mean or intercept of a variable (i.e., coefficients originating from the triangle “1” in a path diagram) affect $\mu \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}})$$\end{document} but not $\Sigma \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}})$$\end{document} , and we refer to them as ${\theta }_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _a$$\end{document} or Type-a parameters. Regression coefficients from one variable to another (e.g., factor loadings, structural coefficients) affect both $\mu \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}})$$\end{document} and $\Sigma \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}})$$\end{document} , and we refer to them as ${\theta }_b$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _b$$\end{document} or Type-b parameters. Variances and covariances of variables affect $\Sigma \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}})$$\end{document} only but not $\mu \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}})$$\end{document} , and we refer to them as ${\theta }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _c$$\end{document} or Type-c parameters. An example for such notations is in Fig. 1. Throughout the paper, we use the subscripts “a,” “b,” and “c” to denote the subsets corresponding to the Type-a, b, and c parameters. For example, $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} , $q_b$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_b$$\end{document} , and $q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c$$\end{document} denote the number of elements in ${\theta }_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _a$$\end{document} , ${\theta }_b$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _b$$\end{document} , and ${\theta }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _c$$\end{document} . Next, we study the specific forms of $\dot{\mu }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)$$\end{document} and $\dot{\Sigma }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)$$\end{document} . Without loss of generality, let us assume the model parameters are arranged in such an order that $\theta ={({\theta }_a^{\prime },{\theta }_b^{\prime },{\theta }_c^{\prime })}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} = (\varvec{{{\uptheta }}} _a', \varvec{{{\uptheta }}} _b', \varvec{{{\uptheta }}} _c')'$$\end{document} . For $\dot{\mu }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)$$\end{document} , the first $(q_a+q_b)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q_a + q_b)$$\end{document} rows are nonzero, whereas the last $q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c$$\end{document} rows are all zeros. For $\dot{\Sigma }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)$$\end{document} , the first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} rows are all zeros, but the last $(q_b+q_c)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q_b + q_c)$$\end{document} rows are nonzero. In this paper, we focus on the case where $\mathrm{rank}\left[\dot{\mu }{\left({\theta }_0\right)}_a\right]=q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}[\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)_a] = q_a$$\end{document} and $\mathrm{rank}\left[\dot{\Sigma }{\left({\theta }_0\right)}_{b,c}\right]=q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}[\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)_{b,c}] = q_b + q_c$$\end{document} , meaning the first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} rows in $\dot{\mu }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)$$\end{document} are linearly independent, and so are the last $(q_b+q_c)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q_b + q_c)$$\end{document} rows in $\dot{\Sigma }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)$$\end{document} . This is generally true for SEM analyses in practice.

Using the shorthand $\Omega =2\dot{\mu }\left({\theta }_0\right){\Sigma }_0^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Omega }} = 2\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0) \varvec{{\Sigma }} ^{-1}_0$$\end{document} and $\Delta =\dot{\Sigma }\left({\theta }_0\right)({\Sigma }_0^{-1}\otimes {\Sigma }_0^{-1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Delta }} = \dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0) (\varvec{{\Sigma }} ^{-1}_0 \otimes \varvec{{\Sigma }} ^{-1}_0)$$\end{document} (recall ${\Sigma }_0=\Sigma \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} _0 = \varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0)$$\end{document} ), we rewrite Eq. (5) as

(6) $\begin{array}{c}\Omega ·t+\Delta ·\mathrm{vec}\left(t{t}^{\prime }\right)+\Delta ·\mathrm{vec}E=0.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{{\Omega }} \cdot \varvec{t} + \varvec{{\Delta }} \cdot \mathrm{vec}(\varvec{t} \varvec{t} ') + \varvec{{\Delta }} \cdot \mathrm{vec}\varvec{E} = \varvec{0}. \end{aligned}$$\end{document}

Because the last $q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c$$\end{document} rows in $\dot{\mu }\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)$$\end{document} are zeros, the last $q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_c$$\end{document} rows in $\Omega$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Omega }} $$\end{document} are also zeros. Similarly, the first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} rows in $\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Delta }} $$\end{document} are also zeros. Accordingly, Eq. (6) has the form

(7) $\begin{array}{c}\left(\begin{array}{c}{\Omega }_a\\ {\Omega }_b\\ O\end{array}\right)·t+\left(\begin{array}{c}O\\ {\Delta }_b\\ {\Delta }_c\end{array}\right)·\mathrm{vec}\left(t{t}^{\prime }\right)+\left(\begin{array}{c}O\\ {\Delta }_b\\ {\Delta }_c\end{array}\right)·\mathrm{vec}E=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{bmatrix} \varvec{{\Omega }} _a \\ \varvec{{\Omega }} _b \\ \varvec{O} \end{bmatrix} \cdot \varvec{t} + \begin{bmatrix} \varvec{O} \\ \varvec{{\Delta }} _b \\ \varvec{{\Delta }} _c \end{bmatrix} \cdot \mathrm{vec}(\varvec{t} \varvec{t} ') + \begin{bmatrix} \varvec{O} \\ \varvec{{\Delta }} _b \\ \varvec{{\Delta }} _c \end{bmatrix} \cdot \mathrm{vec}\varvec{E} = \begin{bmatrix} \varvec{0} \\ \varvec{0} \\ \varvec{0} \end{bmatrix}, \end{aligned}$$\end{document}

where $O$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{O} $$\end{document} is a matrix of zeros. The strategy for finding $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} and $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} is as follows. First, we find an initial solution to

(8) $\begin{array}{c}{\Omega }_a·t=0,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{{\Omega }} _a\cdot \varvec{t} = \varvec{0}, \end{aligned}$$\end{document}

and denote this solution as $\tilde{t}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{t}}$$\end{document} . Second, we find a scalar $\pi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\uppi }} $$\end{document} for $\tilde{t}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{t}}$$\end{document} such that ${\left(\pi \tilde{t}\right)}^{\prime }{\Sigma }_0^{-1}\left(\pi \tilde{t}\right)=F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\uppi }} \tilde{\varvec{t}})' \varvec{{\Sigma }} ^{-1}_0 ({{\uppi }} \tilde{\varvec{t}}) = F_\mathrm{mean}$$\end{document} , and thus, $t=\pi \tilde{t}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} = {{\uppi }} \tilde{\varvec{t}}$$\end{document} satisfies Eqs. () and (8). Third, given $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} , an identity about $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} can be obtained based on Eqs. (6) and (7):

(9) $\begin{array}{c}\Delta ·\mathrm{vec}E=-\Omega ·t-\Delta ·\mathrm{vec}\left(t{t}^{\prime }\right)\equiv \eta .\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{{\Delta }} \cdot \mathrm{vec}\varvec{E} = -\varvec{{\Omega }} \cdot \varvec{t} - \varvec{{\Delta }} \cdot \mathrm{vec}(\varvec{t} \varvec{t} ') \equiv \varvec{{{\upeta }}}. \end{aligned}$$\end{document}

Fourth, we substitute Eq. (9) into Eq. (4) and solve for $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} .

Now let us consider how to implement these four steps. We do not consider the trivial case where the intercept of an endogenous manifest variable is freely estimated. This is because, say if $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_j$$\end{document} is such a variable and its intercept $a_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j$$\end{document} is free, then the model can always fit the mean of $X_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_j$$\end{document} perfectly. Thus, there is no nonzero solution for $t_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_j$$\end{document} in $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} . In a later section, when we extend the procedure to multigroup analyses, due to between-group constraints, it becomes possible to create misfit on the intercepts of manifest variables. Returning to Eq. (6), given that $\mathrm{rank}\left[\dot{\mu }{\left({\theta }_0\right)}_a\right]=q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}[\dot{\varvec{{{\upmu }}}}(\varvec{{{\uptheta }}} _0)_a] = q_a$$\end{document} and ${\Sigma }_0^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^{-1}_0$$\end{document} has full rank p, it follows $\mathrm{rank}\left({\Omega }_a\right)=q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{{\Omega }} _a) = q_a$$\end{document} . Because $\mathrm{rank}\left([{\Omega }_a\mid 0]\right)=\mathrm{rank}\left({\Omega }_a\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}([\varvec{{\Omega }} _a \mid \varvec{0} ]) = \mathrm{rank}(\varvec{{\Omega }} _a)$$\end{document} , ${\Omega }_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Omega }} _a$$\end{document} is a $q_a\times p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a \times p$$\end{document} matrix, and $\mathrm{rank}\left({\Omega }_a\right)<p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{{\Omega }} _a) < p$$\end{document} , it follows that Eq. (8) has infinitely many solutions. The solutions are of the form

(10) $\begin{array}{c}t=(I-{\Omega }_a^{-}{\Omega }_a)y_t,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{t} = (\varvec{I} - \varvec{{\Omega }} ^{-}_a \varvec{{\Omega }} _a) \varvec{y} _t, \end{aligned}$$\end{document}

where ${\Omega }_a^{-}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Omega }} ^{-}_a$$\end{document} is a generalized inverse of ${\Omega }_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Omega }} _a$$\end{document} , and $y_t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _t$$\end{document} can be any $p\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times 1$$\end{document} vector. For convenience, one can just use the Moore–Penrose inverse for the generalized inverse. Given a randomly generated $\widetilde{y_t}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{y} _t}$$\end{document} , an initial solution to Eq. (8) is thus $\tilde{t}=(I-{\Omega }_a^{-}{\Omega }_a)\widetilde{y_t}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{t}} = (\varvec{I} - \varvec{{\Omega }} ^{-}_a \varvec{{\Omega }} _a) \tilde{\varvec{y} _t}$$\end{document} . Next, we calculate ${\tilde{t}}^{\prime }{\Sigma }_0^{-1}\tilde{t}\equiv {\tilde{F}}_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{t}}' \varvec{{\Sigma }} ^{-1}_0 \tilde{\varvec{t}} \equiv {\tilde{F}}_\mathrm{mean}$$\end{document} . Defining $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} as $t=\tilde{t}\sqrt{F_{\mathrm{mean}}/{\tilde{F}}_{\mathrm{mean}}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} = \tilde{\varvec{t}} \sqrt{F_\mathrm{mean}/ {\tilde{F}}_\mathrm{mean}}$$\end{document} will satisfy Eqs. (8) and () simultaneously. Given $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} , the value of $\mathrm{vec}\left(t{t}^{\prime }\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{vec}(\varvec{t} \varvec{t} ')$$\end{document} is also determined, and thus, the only unknown in Eq. (7) is $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} .

Next, we construct $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} . The first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} rows in Eq. (7) always hold regardless of the choice of $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} , and thus, we just need to focus on the nonzero rows concerning ${\theta }_b$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _b$$\end{document} and ${\theta }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _c$$\end{document} . To ensure the solution $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} is symmetric, we rewrite Eq. (9) as

(11) $\begin{array}{c}\Delta D·\mathrm{vec}h\left(E\right)=B·\mathrm{vec}h\left(E\right)=\eta ,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{{\Delta }} \varvec{D} \cdot \mathrm{vec}h(\varvec{E}) = \varvec{B} \cdot \mathrm{vec}h(\varvec{E}) = \varvec{{{\upeta }}}, \end{aligned}$$\end{document}

where $\mathrm{vec}h(·)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{vec}h(\cdot )$$\end{document} stacks the lower triangular elements of a matrix into a vector, $B=\Delta D$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B} =\varvec{{\Delta }} \varvec{D} $$\end{document} , and $D$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{D} $$\end{document} is the duplication matrix such that $\mathrm{vec}\left(E\right)=D·\mathrm{vec}h\left(E\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{vec}(\varvec{E}) = \varvec{D} \cdot \mathrm{vec}h(\varvec{E})$$\end{document} . Given that $\mathrm{rank}\left[\dot{\Sigma }\left({\theta }_0\right)\right]=q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}[\dot{\varvec{{\Sigma }}}(\varvec{{{\uptheta }}} _0)] = q_b + q_c$$\end{document} , $({\Sigma }_0^{-1}\otimes {\Sigma }_0^{-1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varvec{{\Sigma }} ^{-1}_0 \otimes \varvec{{\Sigma }} ^{-1}_0)$$\end{document} has full rank $p^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^2$$\end{document} , and $D$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{D} $$\end{document} has full column rank $p^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*$$\end{document} (where $p^{\ast }=p(p+1)/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^* = p(p+1)/2$$\end{document} ), we have $\mathrm{rank}\left(\Delta \right)=q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{{\Delta }}) = q_b+q_c$$\end{document} and $\mathrm{rank}\left(B\right)\le q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{B}) \le q_b+q_c$$\end{document} . The first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} rows of $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B} $$\end{document} are all zeros, and if $\mathrm{rank}\left(B\right)=q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{B}) = q_b+q_c$$\end{document} , then the rest $(q_b+q_c)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q_b+q_c)$$\end{document} rows are linearly independent. Because the first $q_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a$$\end{document} elements of $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upeta }}} $$\end{document} are all zeros, it follows that $\mathrm{rank}\left(\right[B\mid \eta \left]\right)=\mathrm{rank}\left(B\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}([\varvec{B} \mid \varvec{{{\upeta }}} ]) = \mathrm{rank}(\varvec{B})$$\end{document} and Eq. (11) is a consistent linear system. Given that $B$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B} $$\end{document} is a $q\times p^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times p^2$$\end{document} matrix and $\mathrm{rank}\left(B\right)<p^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{B}) < p^2$$\end{document} , Eq. (11) has infinitely many solutions. If $\mathrm{rank}\left(B\right)<q_b+q_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{B}) < q_b+q_c$$\end{document} , $\mathrm{rank}\left(\right[B\mid \eta \left]\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}([\varvec{B} \mid \varvec{{{\upeta }}} ]) $$\end{document} may not equal $\mathrm{rank}\left(B\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}(\varvec{B}) $$\end{document} and thus sometimes $B·\mathrm{vec}h\left(E\right)=\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{B} \cdot \mathrm{vec}h(\varvec{E}) = \varvec{{{\upeta }}} $$\end{document} is not a consistent system. In that case, one can simply find a different $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upeta }}} $$\end{document} to satisfy $\mathrm{rank}\left(\right[B\mid \eta \left]\right)=\mathrm{rank}\left(B\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{rank}([\varvec{B} \mid \varvec{{{\upeta }}} ]) = \mathrm{rank}(\varvec{B})$$\end{document} by generating a new $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} from Eq. (10). Therefore, Eq. (11) can be guaranteed to have infinitely many solutions, and their general form is

(12) $\begin{array}{c}\mathrm{vec}h\left(E\right)=B^{-}\eta +(I-B^{-}B)y_E,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathrm{vec}h(\varvec{E}) = \varvec{B} ^{-} \varvec{{{\upeta }}} + (\varvec{I} - \varvec{B} ^{-}\varvec{B}) \varvec{y} _E, \end{aligned}$$\end{document}

where $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} can be any $p^{\ast }\times 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^* \times 1$$\end{document} vector. Next, we rewrite $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} in Eq. (4) in terms of $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} and solve the equation for $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} . More specifically, we define a new function:

(13) $\begin{array}{c}\varphi =ln|{\Sigma }_0|-ln|{\Sigma }_0+E|+\mathrm{tr}(I+E{\Sigma }_0^{-1})-p-F_{\mathrm{cov}},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\upphi }} = \ln |\varvec{{\Sigma }} _0| - \ln |\varvec{{\Sigma }} _0 + \varvec{E} | + \mathrm{tr}(\varvec{I} + \varvec{E} \varvec{{\Sigma }} ^{-1}_0) - p - F_\mathrm{cov}, \end{aligned}$$\end{document}

and then find a root for $\varphi =0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upphi }} = 0$$\end{document} in terms of $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} using the Newton method. In particular, the Jacobian of $\varphi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upphi }} $$\end{document} with respect to $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} is $J_{\varphi }\left(y_E\right)=\partial \varphi /\partial y_E^{\prime }=(\partial \varphi /\partial E)(\partial E/\partial y_E^{\prime })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{J} _{{{\upphi }}}(\varvec{y} _E) = \partial {{\upphi }}/ \partial \varvec{y} _E' = (\partial {{\upphi }}/ \partial \varvec{E}) (\partial \varvec{E} / \partial \varvec{y} _E')$$\end{document} , where

(14) $\begin{array}{c}\frac{\partial \varphi }{\partial E}={\mathrm{vec}}^{\prime }[{\Sigma }_0^{-1}-{({\Sigma }_0+E)}^{-1}]·\frac{\partial E}{\partial E};\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial {{\upphi }}}{\partial \varvec{E}} = {\mathrm{vec}'}[\varvec{{\Sigma }} ^{-1}_0 - (\varvec{{\Sigma }} _0 + \varvec{E})^{-1}] \cdot \frac{\partial \varvec{E}}{\partial \varvec{E}}; \end{aligned}$$\end{document}

(15) $\begin{array}{c}\frac{\partial E}{\partial y_E}=\frac{\partial \left[D\mathrm{vec}h\right(E\left)\right]}{\partial y_E}=D(I-B^{-}B).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \varvec{E}}{\partial \varvec{y} _E} = \frac{\partial [\varvec{D} \mathrm{vec}h(\varvec{E})]}{\partial \varvec{y} _E} = \varvec{D} (\varvec{I}-\varvec{B} ^{-}\varvec{B}). \end{aligned}$$\end{document}

Note $\partial E/\partial E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \varvec{E}/\partial \varvec{E} $$\end{document} does not equal $I$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{I} $$\end{document} but another constant matrix instead, because each off-diagonal element of $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} appears twice in $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} , leading to $\partial e_{\mathrm{ij}}/\partial e_{\mathrm{ji}}=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial e_{ij} / \partial e_{ji} = 1$$\end{document} even if $i\ne j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \ne j$$\end{document} . Then we can find a root for $\varphi \left(y_E\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upphi }} (\varvec{y} _E) = 0$$\end{document} using an iterative process. The update from step k to step $(k+1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1)$$\end{document} is:

(16) $\begin{array}{c}{y}_E^{(k+1)}=y_E^{\left(k\right)}-J_{\varphi }^{-1}\left[y_E^{\left(k\right)}\right]·\varphi \left[y_E^{\left(k\right)}\right].\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{y} ^{(k+1)}_E = \varvec{y} ^{(k)}_E - \varvec{J} ^{-1}_{{{\upphi }}}[\varvec{y} ^{(k)}_E] \cdot {{\upphi }} [\varvec{y} ^{(k)}_E]. \end{aligned}$$\end{document}

At convergence, the $y_E^{(k+1)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} ^{(k+1)}_E$$\end{document} value is a root to $\varphi \left(y_E\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upphi }} (\varvec{y} _E) = 0$$\end{document} . Then, we can construct $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} based on Eq. (12). Now that $t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{t} $$\end{document} and $E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{E} $$\end{document} are available, it is straightforward to compute ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} .

The ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} obtained with the above procedure will satisfy Eqs. () to (5), but any $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} $$\end{document} satisfying Eq. (5) is only a stationary point of $F_{\mathrm{ML}}[{\mu }^{\ast },{\Sigma }^{\ast };\mu \left(\theta \right),\Sigma \left(\theta \right)]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}[\varvec{{{\upmu }}} ^*, \varvec{{\Sigma }} ^*; \varvec{{{\upmu }}} (\varvec{{{\uptheta }}}), \varvec{{\Sigma }} (\varvec{{{\uptheta }}})]$$\end{document} . Equation (2) requires ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} to be the global minimizer of $F_{\mathrm{ML}}[{\mu }^{\ast },{\Sigma }^{\ast };\mu \left(\theta \right),\Sigma \left(\theta \right)]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}[\varvec{{{\upmu }}} ^*, \varvec{{\Sigma }} ^*; \varvec{{{\upmu }}} (\varvec{{{\uptheta }}}), \varvec{{\Sigma }} (\varvec{{{\uptheta }}})]$$\end{document} , not just a stationary point. Therefore, it needs to continue to show that ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} is indeed the global minimizer. To prove this, we borrow from the arguments that established the consistency of the ML point estimator (Kano, Reference Kano1986; Shapiro, Reference Shapiro1984; see also Chun & Shapiro, Reference Chun and Shapiro2010). More specifically, let $m\left(\theta \right)={[\mu {\left(\theta \right)}^{\prime },{\mathrm{vec}}^{\prime }\left[\Sigma \left(\theta \right)\right]]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{m} (\varvec{{{\uptheta }}}) = [\varvec{{{\upmu }}} (\varvec{{{\uptheta }}})', {\mathrm{vec}'}[\varvec{{\Sigma }} (\varvec{{{\uptheta }}})]]'$$\end{document} denote the model-implied moments, $\tilde{m}={[{\tilde{\mu }}^{\prime },{\mathrm{vec}}^{\prime }\left(\tilde{\Sigma }\right)]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{m}} = [\tilde{\varvec{{{\upmu }}}}', {\mathrm{vec}'}(\tilde{\varvec{{\Sigma }}})]'$$\end{document} the moments of the data, and $m_0={[{\mu }_0^{\prime },{\mathrm{vec}}^{\prime }\left({\Sigma }_0\right)]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{m} _0 = [\varvec{{{\upmu }}} _0', {\mathrm{vec}'}(\varvec{{\Sigma }} _0)]'$$\end{document} . Because $m_0=m\left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{m} _0 = \varvec{m} (\varvec{{{\uptheta }}} _0) $$\end{document} , ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} is always a global minimizer of $F_{\mathrm{ML}}[m_0,m\left(\theta \right)]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}[\varvec{m} _0, \varvec{m} (\varvec{{{\uptheta }}})]$$\end{document} . Moreover, if the model is identified at ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} , then ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} is the unique minimizer (Kano, Reference Kano1986; Shapiro, Reference Shapiro1984). Let $\tilde{\theta }=\underset{\theta \in \Theta }{\mathrm{argmin}}{F}_{\mathrm{ML}}[\tilde{m},m\left(\theta \right)]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{{{\uptheta }}}} = \underset{\varvec{{{\uptheta }}} \in \varvec{{\Theta }}}{\mathrm {argmin}} F_\mathrm{ML}[\tilde{\varvec{m}}, \varvec{m} (\varvec{{{\uptheta }}})]$$\end{document} , and $\tilde{\theta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tilde{\varvec{{{\uptheta }}}} $$\end{document} is consistent if, for all $\tilde{m}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{m}}$$\end{document} sufficiently close to $m_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{m} _0$$\end{document} , $\tilde{\theta }\to {\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{{{\uptheta }}}} \rightarrow \varvec{{{\uptheta }}} _0$$\end{document} as $\tilde{m}\to m_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{m}} \rightarrow \varvec{m} _0$$\end{document} . Under mild regularity conditions, the consistency of $\tilde{m}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\varvec{m}}$$\end{document} holds if the model is identified at ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} and the set $\Theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Theta }} $$\end{document} is compact (Kano, Reference Kano1986; Shapiro, Reference Shapiro1984).

2. Demonstration 1

In this section we use the proposed method to create several sets of ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} for Model 1 in Fig. 1. The model form and ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} values are presented in Fig. 1. For the amount of misfit, we choose values that render the misfit quite serious and greater than values of interest to a researcher when designing simulation studies or analyzing real data. We use huge $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} values because, based on the above discussion of Proposition 1, for the proposed method to work well, it needs the misfit being not too large. Therefore, if our method works well given huge $F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean}$$\end{document} and $F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov}$$\end{document} in this demonstration, it should perform even better given smaller $F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean}$$\end{document} and $F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov}$$\end{document} values in practice. In choosing $F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean}$$\end{document} and $F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov}$$\end{document} for this demonstration, we draw analogy with the definition of RMSEA $\epsilon =\sqrt{F/df}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upvarepsilon }} = \sqrt{F/df}$$\end{document} , and set $F_{\mathrm{mean}}={(.15)}^{2}d{f}_{\mathrm{mean}}=.135$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean} = (.15)^2df_\mathrm{mean} = .135$$\end{document} and $F_{\mathrm{cov}}={(.20)}^{2}d{f}_{\mathrm{cov}}=.96$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov} = (.20)^2df_\mathrm{cov} = .96$$\end{document} . Accordingly, $F_{\mathrm{all}}=1.095$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{all} = 1.095$$\end{document} and the model’s RMSEA is .191, indicating serious misfit. Next, we used the proposed method to create four sets of ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} randomly (as $y_t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _t$$\end{document} and $y_E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y} _E$$\end{document} can be chosen randomly, see Eqs. (10) and (12)). We then fitted Model 1 to ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} using the “lavaan” package (Rosseel, Reference Rosseel2012) in R (R Core Team, 2017) and recorded ${\theta }_{\mathrm{fit}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{fit}$$\end{document} , $F_{\mathrm{mean}}^{\left(\mathrm{fit}\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^\mathrm{(fit)}_\mathrm{mean}$$\end{document} , and $F_{\mathrm{cov}}^{\left(\mathrm{fit}\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^\mathrm{(fit)}_\mathrm{cov}$$\end{document} , where ${\theta }_{\mathrm{fit}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{fit}$$\end{document} is the fitted model parameter.

Table 1 presents the four sets of ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} and model estimation results. Comparing $\mu \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0)$$\end{document} and $\Sigma \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0)$$\end{document} to ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} , we see the model can largely reproduce the data and the misfit spreads somewhat evenly over all the elements of $\mu \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0)$$\end{document} and $\Sigma \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0)$$\end{document} , representing a realistic modeling scenario. For all the four sets of ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} , the ${\theta }_{\mathrm{fit}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{fit}$$\end{document} obtained is exactly equal to the specified value ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} , indicating the goal in Eq. (2) is satisfied. Regarding the amount of misfit, the difference in $F_{\mathrm{cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{cov}$$\end{document} between fitted and desired values is around ${10}^{-9}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-9}$$\end{document} , and the difference in $F_{\mathrm{mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{mean}$$\end{document} between fitted and desired values is around ${10}^{-17}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-17}$$\end{document} (see Table 1). Such a small difference between the desired and fitted $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} values can be well explained by rounding, and the goals in Eqs. () and (4) are also satisfied.Footnote ¹ Therefore, even when such a large amount of model misfit is specified, the proposed method still performs extremely well and is able to yield ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} as desired.

Table 1. Population model-implied moments and four generated moments with specified misfit and parameter values in Demonstration 1.

The lower triangle contains covariances and upper triangle correlations. $F_{\mathrm{Cov}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{Cov}$$\end{document} Diff & $F_{\mathrm{Mean}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{Mean}$$\end{document} Diff = Fitted $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} value for the covariance (or mean) part − Desired $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} value for the covariance (or mean) part.

2.1. Contrast to the Type I Approach

For Model 1, if a researcher wants to create a misspecified model from the Type I perspective, there are only six model parameters possible for removal, namely the three latent means and the three latent covariances. To better illustrate the limitations of the Type I perspective, we fit the six misspecified models (only one parameter is missing in each model) to $\mu \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}} _0)$$\end{document} and $\Sigma \left({\theta }_0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}} _0)$$\end{document} and record the fit indices and $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} values in Table 2. First, it is almost impossible for the Type I approach to control the amount of model misfit. Among the six wrong models, the best-fitting model is the one without $c_{F1F3}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{F1F3}$$\end{document} , which yields RMSEA = .073; CFI = .950; SRMR = .122. Unless changing the ${\theta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _0$$\end{document} values, it is impossible to have a wrong model with, say an RMSEA of .05 or .06. It is also impossible to achieve a larger RMSEA values, say .08 or .10 in the current example, and the only available RMSEA values are those six values in Table 2. However, the method we proposed allows the researcher to specify the desired $F_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\mathrm{ML}$$\end{document} values on a continuous scale. Second, if one removes a latent covariance from Model 1, the model is wrong in the covariance part only and its $F_{\mathrm{mean}}^{\left(\mathrm{fit}\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^\mathrm{(fit)}_\mathrm{mean}$$\end{document} is always 0 (see Table 2). To introduce misfit on the mean part, one has to remove a latent mean, but in that case the model will be too wrong to be useful in a simulation study (see Table 2 for the fit indices). By contrast, our method is able to control both $F_{\mathrm{mean}}^{\left(\mathrm{fit}\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^\mathrm{(fit)}_\mathrm{mean}$$\end{document} and $F_{\mathrm{cov}}^{\left(\mathrm{fit}\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^\mathrm{(fit)}_\mathrm{cov}$$\end{document} at a reasonable level. Third, the Type I approach is unable to control the location of model misfit. If one removes a latent covariance, say $c_{F1F2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{F1F2}$$\end{document} , then only some elements in $\Sigma \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}})$$\end{document} will have misfit and the other elements will remain in perfect fit, showing an unrealistic pattern in the residual matrix (see Table ). In addition, if one removes a latent mean (e.g., $a_{F1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{F1}$$\end{document} ), the model-implied means will all be zeros for the indicators that load on this latent factor (e.g., $X_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1$$\end{document} to $X_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_3$$\end{document} ; see Table ). Again this is not a problem for our method, as we saw earlier in Table 1 that for the four sets of ${\mu }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} ^*$$\end{document} and ${\Sigma }^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} ^*$$\end{document} , all the elements in $\mu \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}})$$\end{document} and $\Sigma \left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}})$$\end{document} have some lack of fit, depicting a more realistic scenario.

Table 2. Population fit indices and ${\text{F}}_{\mathrm{ML}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {F}_\mathrm{ML}$$\end{document} values after removing a parameter from the model in Demonstration 1.

Table 3. Residuals or model-implied moments after removing a parameter from Model 1 in Demonstration 1.

3. Multiple Group Analysis

In this section, we first introduce some new notations for the multigroup context. We illustrate the notations with Model 2 in Fig. 2. We focus on the cases where the model form is the same across all the G groups. We use “(g)” flexibly in the subscript or superscript (e.g., ${\theta }_a^{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(g)}_a$$\end{document} , ${\theta }_{\left(g\right)}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}'$$\end{document} ) to denote the elements within group g (where $g=1,2,\cdots ,G$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g= 1,2,\cdots ,G$$\end{document} ) and use vectors or matrices without “(g)” to denote those that contain the corresponding elements of all the G groups. For example, for the model in Fig. 2, we have $\theta ={[{\theta }_{\left(1\right)}^{\prime },{\theta }_{\left(2\right)}^{\prime }]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} = [\varvec{{{\uptheta }}} _{(1)}', \varvec{{{\uptheta }}} _{(2)}']'$$\end{document} and $\Sigma =Diag[{\Sigma }_{\left(1\right)},{\Sigma }_{\left(2\right)}]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} = Diag[\varvec{{\Sigma }} _{(1)}, \varvec{{\Sigma }} _{(2)}]$$\end{document} , where $Diag[·]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Diag[\cdot ]$$\end{document} denotes a block diagonal matrix. Suppose Model 2 has the constraints $a_F^{\left(1\right)}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{(1)}_F = 0$$\end{document} , $a_x^{\left(1\right)}=a_x^{\left(2\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{a} ^{(1)}_x = \varvec{a} ^{(2)}_x$$\end{document} , and $b^{\left(1\right)}=b^{\left(2\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{b} ^{(1)} = \varvec{b} ^{(2)}$$\end{document} , where $a_x={[a_1,a_2,a_3]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{a} _x = [a_1, a_2, a_3]'$$\end{document} and $b={[b_2,b_3]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{b} = [b_2, b_3]'$$\end{document} . Then, the Type-a parameters in the two groups are written as ${\theta }_a^{\left(1\right)}={[0,a_1^{\left(1\right)},a_2^{\left(1\right)},a_3^{\left(1\right)}]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(1)}_a = [0, a^{(1)}_1, a^{(1)}_2, a^{(1)}_3]'$$\end{document} and ${\theta }_a^{\left(2\right)}={[a_F^{\left(2\right)},a_1^{\left(2\right)},a_2^{\left(2\right)},a_3^{\left(2\right)}]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(2)}_a = [a^{(2)}_F, a^{(2)}_1, a^{(2)}_2, a^{(2)}_3]'$$\end{document} . That is, the number of elements in ${\theta }_a^{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(g)}_a$$\end{document} remains the same across all groups, and we define ${\theta }_b^{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(g)}_b$$\end{document} and ${\theta }_c^{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(g)}_c$$\end{document} in the same manner. If a parameter is set to a special value in a certain group g (e.g., $a_F^{\left(1\right)}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{(1)}_F = 0$$\end{document} ), we still include the fixed value in ${\theta }_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}$$\end{document} . (Thus, the first element of ${\theta }_a^{\left(1\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(1)}_a$$\end{document} is 0.) In addition, for the parameters constrained to be equal over some groups, we consider them as the same parameter but give them different labels in ${\theta }_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}$$\end{document} (e.g., ${\theta }_b^{\left(1\right)}={[b_2^{\left(1\right)},b_3^{\left(1\right)}]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(1)}_b = [b^{(1)}_2, b^{(1)}_3]'$$\end{document} , ${\theta }_b^{\left(2\right)}={[b_2^{\left(2\right)},b_3^{\left(2\right)}]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} ^{(2)}_b = [b^{(2)}_2, b^{(2)}_3]'$$\end{document} ). Regarding the type of constraints, we focus on between-group equality constraints only, because other constraint types seldom appear in practice. Let q denote the number of elements in ${\theta }_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}$$\end{document} , $q_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{(g)}$$\end{document} the number of free parameters in ${\theta }_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}$$\end{document} , and $q_{\mathrm{all}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{all}$$\end{document} the total number of free parameters in $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} $$\end{document} . The subsets corresponding to the Type-a, b, and c parameters are defined in the same way. For example, in Fig. 2, $q_a^{\left(1\right)}=3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{(1)}_a = 3$$\end{document} and $q_a=q_a^{\left(2\right)}=4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_a = q^{(2)}_a = 4$$\end{document} . Vector ${\theta }_a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _a$$\end{document} has $q_{a}G=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_aG = 8$$\end{document} elements, but $q_a^{\left(\mathrm{all}\right)}=4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^\mathrm{(all)}_a = 4$$\end{document} . We consider the model-implied moments in group g as functions of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} $$\end{document} (not ${\theta }_{\left(g\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\uptheta }}} _{(g)}$$\end{document} ) and denote them as ${\mu }_{\left(g\right)}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} _{(g)}(\varvec{{{\uptheta }}})$$\end{document} and ${\Sigma }_{\left(g\right)}\left(\theta \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} _{(g)}(\varvec{{{\uptheta }}})$$\end{document} . The model-implied moments of the whole multigroup analysis are $\mu \left(\theta \right)={[{\mu }_{\left(1\right)}{\left(\theta \right)}^{\prime },{\mu }_{\left(2\right)}{\left(\theta \right)}^{\prime },\cdots ,{\mu }_{\left(G\right)}{\left(\theta \right)}^{\prime }]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{{\upmu }}} (\varvec{{{\uptheta }}}) = [\varvec{{{\upmu }}} _{(1)}(\varvec{{{\uptheta }}})', \varvec{{{\upmu }}} _{(2)}(\varvec{{{\uptheta }}})', \cdots , \varvec{{{\upmu }}} _{(G)}(\varvec{{{\uptheta }}})']'$$\end{document} and $\Sigma \left(\theta \right)=Diag[{\Sigma }_{\left(1\right)}\left(\theta \right),{\Sigma }_{\left(2\right)}\left(\theta \right),\cdots ,{\Sigma }_{\left(G\right)}\left(\theta \right)]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\Sigma }} (\varvec{{{\uptheta }}}) = Diag[\varvec{{\Sigma }} _{(1)}(\varvec{{{\uptheta }}}), \varvec{{\Sigma }} _{(2)}(\varvec{{{\uptheta }}}), \cdots , \varvec{{\Sigma }} _{(G)}(\varvec{{{\uptheta }}})]$$\end{document} .