Assumption 1

Suppose Eq. (2) holds and that $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} and $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} has at least two finite moments. Further suppose

1. (1) $E\epsilon =0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {E} \varepsilon = 0$$\end{document} , and the cross-covariance matrix $\text{Cov}(f,\epsilon )=E\left[(f-Ef){\epsilon }^{\prime }\right]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Cov}} \, (f,\varepsilon ) = \mathbb {E} [(f - \mathbb {E} f) \varepsilon ']$$\end{document} is zero.

2. (2) $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} has full column rank.

3. (3) $\Phi =\text{Cov}\left(f\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi = {\text {Cov}} \, (f)$$\end{document} is positive definite.

4. (4) $\Psi =\text{Cov}\left(\epsilon \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi = {\text {Cov}} \, (\varepsilon )$$\end{document} is positive definite.

Let $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} be the set of all left-inverses of $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} . That is, $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\in \mathcal {G}(\Lambda )$$\end{document} means $A\Lambda =I_{d_f}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \Lambda = I_{d_f}$$\end{document} .

Lemma 1

Suppose given Assumption 1.

1. (1) Let A be a deterministic matrix, and let $r_A=A\tilde{z}-f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_A = A \tilde{z} - f$$\end{document} . Then $\text{Cov}(f,r_A)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Cov}} \, (f,r_A) = 0$$\end{document} if and only if $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal {G}(\Lambda )$$\end{document} . This holds also if $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} is singular.

2. (2) Let A be a deterministic matrix. If $E\left[\epsilon \right|f]=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {E} [\varepsilon |f] = 0$$\end{document} , then $E\left[A\tilde{z}\right|f]=f$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {E} [A \tilde{z} | f] = f$$\end{document} if and only if $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal {G}(\Lambda )$$\end{document} .

3. (3) The transformation matrix $T=\Phi {\Lambda }^{\prime }{\Sigma }^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = \Phi \Lambda ' \Sigma ^ {-1} $$\end{document} used in the Thurstone factor score exists, but is not in $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} .

4. (4) The Bartlett matrix $\Delta ={\left({\Lambda }^{\prime }{\Psi }^{-1}\Lambda \right)}^{-1}{\Lambda }^{\prime }{\Psi }^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta = (\Lambda ' \Psi ^ {-1} \Lambda )^ {-1} \Lambda ' \Psi ^ {-1} $$\end{document} exists and is in $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} , and is such that for all $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal {G}(\Lambda )$$\end{document} we have that $\text{Cov}\left(r_{\Delta }\right)-\text{Cov}\left(r_A\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {Cov}} \, (r_\Delta ) - {\text {Cov}} \, (r_A)$$\end{document} is non-positive definite.

Proof

See Section E.4.1. $\square$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The set of left inverses of $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is non-empty if and only if $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} has full column rank (Harville, 1997, Lemma 8.1.1). Therefore, Assumption 1 (2) is foundational. Assumption 1 (4) can be avoided, see, e.g., Eq. (7) in Wall and Amemiya (Reference Wall and Amemiya2000) and Fuller (Reference Fuller1987) for a Bartlett formula that avoids inverting $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} . We will not consider singular $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} matrices in this paper. Further, since $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is assumed to have full column rank, the set of left inverses $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} equals the set of generalized inverses of $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} (Harville, 1997, Lemma 9.2.8). This set can be described constructively, see Theorem 9.2.7 in Harville (1997). Due to Lemma 1 (4), we single out the Bartlett factor score out of the elements from $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} in most of our study.

In applications, the transformation matrix A has to be estimated. This introduces estimation error, as discussed in the upcoming Sect. 3. Taking this estimation error into account is outside the scope of this paper.

Let us now consider the regression representation in Eq. (3). For a given $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal {G}(\Lambda )$$\end{document} , such as the Bartlett score $A=\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = \Delta $$\end{document} , we write $r=r_A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r = r_A$$\end{document} and

(4) $\begin{array}{c}\ddot{f}={({\ddot{\xi }}^{\prime },{\ddot{\eta }}^{\prime })}^{\prime }=A\tilde{z}=A(\Lambda f+\epsilon )=f+r={({\xi }^{\prime },{\eta }^{\prime })}^{\prime }+(r_{\xi }^{\prime },r_{\eta }^{\prime })\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ddot{f} = (\ddot{\xi }', \ddot{\eta }')' = A \tilde{z} = A (\Lambda f + \varepsilon ) = f + r = (\xi ', \eta ')' + (r_\xi ', r_\eta ') \end{aligned}$$\end{document}

where $r_{\xi },r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi , r_\eta $$\end{document} are, respectively, the first $d_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\xi $$\end{document} and last $d_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\eta $$\end{document} coordinates of r, and $\ddot{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\xi }$$\end{document} , $\ddot{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\eta }$$\end{document} are respectively the first $d_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\xi $$\end{document} and last $d_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\eta $$\end{document} coordinates of $\ddot{f}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{f}$$\end{document} . From Eq. (4), we reach

(5) $\begin{array}{c}\ddot{\xi }=\xi +r_{\xi },\ddot{\eta }=\eta +r_{\eta }.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ddot{\xi }= \xi + r_\xi , \qquad \ddot{\eta }= \eta + r_\eta . \end{aligned}$$\end{document}

Since $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\in \mathcal {G}(\Lambda )$$\end{document} , we have that $r=A\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=A\varepsilon $$\end{document} . Now, since $r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta $$\end{document} is a linear transformation of r, we get $E\left[\ddot{\eta }\right|\xi ]=E\left[\eta \right|\xi ]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb {E} [\ddot{\eta }| \xi ] = \mathbb {E} [\eta | \xi ]$$\end{document} as long as $\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is independent to f. We therefore make the following assumption.

Assumption 2

Suppose $\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is independent to f.

In the classical literature on covariance models (see, e.g., the survey paper Shapiro, 2007), the strong Assumption 2 is not made. We need this assumption, and not merely the covariance Assumption 1 (1) to identify H. With only covariance restrictions the distribution of $f,\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,\varepsilon $$\end{document} is not identified (see Mardia et al., Reference Mardia, Kent and Bibby1979, Exercise 9.2.2). Also Kelava et al. (Reference Kelava, Kohler, Krzyżak and Schaffland2017) made this assumption to identify H.

Lemma 2

Suppose given Assumption 1 and 2. For a given $A\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \in \mathcal {G}(\Lambda )$$\end{document} , we have that $H\left(x\right)=E\left[\eta \right|\xi =x]=E\left[\ddot{\eta }\right|\xi =x]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H(x) = \mathbb {E} [ \eta | \xi = x] = \mathbb {E} [ \ddot{\eta }| \xi = x ] $$\end{document} for $\ddot{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\eta }$$\end{document} given by Eq. (4) and Eq. (5).

Proof

See Section E.4.2. $\square$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Under the following assumption, $\ddot{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\xi }$$\end{document} and $\ddot{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\eta }$$\end{document} are computable based on identifiable quantities. Therefore, we may suppose that we observe $\ddot{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\xi }$$\end{document} and $\ddot{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\eta }$$\end{document} directly when analyzing identification of H.

Assumption 3

Suppose $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} , $\mu$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} , $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} are identified from the distribution of z.

The identification of these matrices is a classical problem, and the measurement model is usually only considered valid when they are identified up to scaling of the latent variables through the covariance matrix of z (Anderson, 2003; Bollen, Reference Bollen1989; Mardia et al., Reference Mardia, Kent and Bibby1979). However, conditional expectations are well-behaved with regard to changes of the units of measurement (see Appendix H in the online supplementary material).

We are interested in identifying H(x) based on $\ddot{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\xi }$$\end{document} and $\ddot{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\eta }$$\end{document} . Using Lemma 2, Eq. (5) has the same structure as a non-parametric regression problem with measurement noise, see, e.g., Delaigle et al. (Reference Delaigle, Fan and Carroll2009); Delaigle (Reference Delaigle2014); Apanasovich and Liang (2021), or Huang and Zhou (Reference Huang and Zhou2017). This appears to be noticed also by Wall & Amemiya (Reference Wall and Amemiya2000, see the discussion immediately following their Eq. (9)), but at the time that paper was written, no generally applicable non-parametric regression methods with measurement error were available. These approaches generally need a known distribution and independence conditions to hold for the measurement error $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} , which has to be independent noise. Therefore, we make the following additional assumptions.

Assumption 4

Suppose

1. (1) $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} has a known distribution.

2. (2) $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} and $r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta $$\end{document} are independent.

We will later consider approximating the measurement error $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} by zero, meaning we ignore the measurement error, and will show that this approximation works as $d_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_x$$\end{document} increases. In these arguments, we also use Assumption 4. When ignoring measurement errors, we conjecture that exact independence can be weakened to appropriate dependence bounds. We do not investigate this in the present paper.

The following result is the starting point of the literature on non-parametric regression with measurement error with some papers cited above. We state the result with our notation and provide its short proof for completeness.

Proposition 1

Suppose given Assumption 1, 2, 3, and 4. Then H is identified.

Proof

See Section E.4.3. $\square$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now consider when Assumption 4 (2) can be justified. Assumption 4 (1) will be considered in the next subsection.

Since ${(r_{\xi }^{\prime },r_{\eta }^{\prime })}^{\prime }=r=A\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r_\xi ', r_\eta ')' = r = A \varepsilon $$\end{document} we have $r_{\eta }=(0_{d_{\eta },d_{\xi }},I_{d_{\eta }})A\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta = ( \varvec{0}_{d_\eta ,d_\xi }, I_{d_\eta } ) A \varepsilon $$\end{document} and $r_{\xi }=(I_{d_{\xi }},0_{d_{\xi },d_{\eta }})A\epsilon$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi = ( I_{d_\xi }, \varvec{0}_{d_\xi , d_\eta }) A \varepsilon $$\end{document} , where $0_{a,b}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{0}_{a,b}$$\end{document} is the $a\times b$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\times b$$\end{document} zero matrix. Unless strong distributional assumptions are made, $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} and $r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta $$\end{document} will not be independent unless firstly A is partitioned diagonal (thereby avoiding cross terms from ${\epsilon }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _x$$\end{document} and ${\epsilon }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _y$$\end{document} ), and secondly ${\epsilon }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _x$$\end{document} is independent of ${\epsilon }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _y$$\end{document} . If this is the case, i.e., if

$\begin{array}{c}A=\left(\begin{array}{cc}{A}_x& 0_{d_{\xi },d_y}\\ 0_{d_{\eta },d_x}& A_y\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A = \begin{pmatrix} A_x &{} \varvec{0}_{d_\xi , d_y}\\ \varvec{0}_{d_\eta , d_x} &{} A_y \end{pmatrix}, \end{aligned}$$\end{document}

then $r_{\xi }=A_x{\epsilon }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi = A_x \varepsilon _x$$\end{document} and $r_{\eta }=A_y{\epsilon }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta = A_y \varepsilon _y$$\end{document} , and $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} will be independent to $r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta $$\end{document} , as long as ${\epsilon }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _x$$\end{document} and ${\epsilon }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _y$$\end{document} are independent.

In general, we may write $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} as a partitioned matrix

$\begin{array}{c}\Lambda =\left(\begin{array}{cc}{\Lambda }_x& {\Lambda }_{x,y}\\ {\Lambda }_{y,x}& {\Lambda }_y\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Lambda = \begin{pmatrix} \Lambda _{x} &{} \Lambda _{x,y} \\ \Lambda _{y,x} &{} \Lambda _y \end{pmatrix} \end{aligned}$$\end{document}

where ${\Lambda }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _x$$\end{document} is a $d_x\times d_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_x \times d_\xi $$\end{document} matrix, ${\Lambda }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _y$$\end{document} is a $d_y\times d_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_y \times d_\eta $$\end{document} matrix, ${\Lambda }_{x,y}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{x,y}$$\end{document} is a $d_x\times d_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_x \times d_\eta $$\end{document} matrix, and ${\Lambda }_{y,x}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{y,x}$$\end{document} is a $d_y\times d_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_y \times d_\xi $$\end{document} matrix. When A is a partition diagonal matrix with diagonal matrix entries $A_x,A_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_x, A_y$$\end{document} , we have

(6) $\begin{array}{c}{({\ddot{\xi }}^{\prime },{\ddot{\eta }}^{\prime })}^{\prime }=A\tilde{z}=A(\Lambda f+\epsilon )=\left(\begin{array}{c}{A}_x{\Lambda }_x\xi +A_x{\Lambda }_{x,y}\eta +A_x{\epsilon }_x\\ A_y{\Lambda }_y\eta +A_y{\Lambda }_{y,x}\xi +A_y{\epsilon }_y\end{array}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\ddot{\xi }', \ddot{\eta }')' = A \tilde{z} = A (\Lambda f + \varepsilon ) = \begin{pmatrix} A_x \Lambda _x \xi + A_x \Lambda _{x,y} \eta + A_x \varepsilon _x \\ A_y \Lambda _y \eta + A_y \Lambda _{y,x} \xi + A_y \varepsilon _y \end{pmatrix}. \end{aligned}$$\end{document}

The matrix $A_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_x$$\end{document} will not in general be a generalized inverse of both ${\Lambda }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _x$$\end{document} and ${\Lambda }_{x,y}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{x,y}$$\end{document} . Therefore, the factor scores will contain residual dependency between $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} and $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} which distort the identification of H as then Assumption 4 (2) no longer holds, if used directly as input to non-parametric regression methods.

In order to fulfill Assumption 4 (2), we, therefore, do not allow cross-loadings or error correlations between endogenous and exogenous parts of the model, that is between $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} and $\eta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} . If such are part of the model, one would have to delete corresponding observed variables in order to directly apply our analysis. A less wasteful method might hopefully be derived as an extension of this work, though such an extension is outside the scope of the present paper. Within the measurement part of the endogenous and exogenous variables, cross-loadings or error correlations are allowed. Hence, we make the following assumptions.

Assumption 5

Suppose ${\epsilon }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _x$$\end{document} and ${\epsilon }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _y$$\end{document} are independent, and that

$\begin{array}{c}\Psi =\left(\begin{array}{cc}{\Psi }_x& 0_{d_x,d_y}\\ 0_{d_y,d_x}& {\Psi }_y\end{array}\right),\Lambda =\left(\begin{array}{cc}{\Lambda }_x& 0_{d_x,d_{\eta }}\\ 0_{d_y,d_{\xi }}& {\Lambda }_y\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Psi = \begin{pmatrix} \Psi _{x} &{} \varvec{0}_{d_x, d_y}\\ \varvec{0}_{d_y, d_x} &{} \Psi _y \end{pmatrix}, \quad \Lambda = \begin{pmatrix} \Lambda _{x} &{} \varvec{0}_{d_x, d_\eta }\\ \varvec{0}_{d_y, d_\xi } &{} \Lambda _y \end{pmatrix}, \end{aligned}$$\end{document}

where ${\Lambda }_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{x}$$\end{document} and ${\Lambda }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{y}$$\end{document} have full column ranks.

Under Assumption 1 (4), ${\Psi }_x,{\Psi }_y$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi _x, \Psi _y$$\end{document} are positive definite, as they are principle sub-matrices of a positive-definite matrix $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi $$\end{document} (Horn & Johnson, Reference Horn and Johnson2013, Observation 7.1.2). Under Assumption 5, a direct calculation shows that if $A_x\in G\left({\Lambda }_x\right),A_y\in G\left({\Lambda }_y\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_x \in \mathcal {G}(\Lambda _x), A_y \in \mathcal {G}(\Lambda _y)$$\end{document} , then $A=\left(\begin{array}{cc}{A}_x& 0_{d_{\xi },d_y}\\ 0_{d_{\eta },d_x}& A_y\end{array}\right)\in G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = \begin{pmatrix} A_x &{} \varvec{0}_{d_\xi , d_y}\\ \varvec{0}_{d_\eta , d_x} &{} A_y \end{pmatrix} \in \mathcal {G}(\Lambda )$$\end{document} . While there are also elements in $G\left(\Lambda \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(\Lambda )$$\end{document} of different forms (Harville, 1997, Exercise 9.7), partitioned diagonal generalized inverses of $\Lambda$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} imply that Assumption 4 (2) holds.

Lemma 3

Suppose given Assumption 1 and 5. Suppose that $A=\left(\begin{array}{cc}{A}_x& 0_{d_{\xi },d_y}\\ 0_{d_{\eta },d_x}& A_y\end{array}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = \begin{pmatrix} A_x &{} \varvec{0}_{d_\xi , d_y}\\ \varvec{0}_{d_\eta , d_x} &{} A_y \end{pmatrix}$$\end{document} where $A_x\in G\left({\Lambda }_x\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_x \in \mathcal {G}(\Lambda _x)$$\end{document} and $A_y\in G\left({\Lambda }_y\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_y \in \mathcal {G}(\Lambda _y)$$\end{document} . Then $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} and $r_{\eta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\eta $$\end{document} are independent, i.e., Assumption 4 (2) holds.

Proof

See Section E.4.4. $\square$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Under Assumption 5, Bartlett (Reference Bartlett1937) factor scores are partitioned diagonal as shown in the following result, and hence using Bartlett scores under Assumption 5 leads to Assumption 4 (2).

Proposition 2

Suppose given Assumption 1 and 5. Then,

$\begin{array}{c}\Delta =\left(\begin{array}{cc}{\Delta }_x& 0_{d_{\xi },d_y}\\ 0_{d_{\eta },d_x}& {\Delta }_y\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta = \begin{pmatrix} \Delta _x &{} \varvec{0}_{d_\xi , d_y}\\ \varvec{0}_{d_\eta , d_x} &{} \Delta _y \end{pmatrix}, \end{aligned}$$\end{document}

with ${\Delta }_x:={\left({\Lambda }_x^{\prime },{\Psi }_x^{-1},{\Lambda }_x\right)}^{-1}{\Lambda }_x^{\prime }{\Psi }_x^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _x:= \left( \Lambda _x'\Psi _{x}^ {-1} \Lambda _x\right) ^ {-1} \Lambda _x'\Psi _x^ {-1} $$\end{document} and ${\Delta }_y:={\left({\Lambda }_y^{\prime },{\Psi }_y^{-1},{\Lambda }_y\right)}^{-1}{\Lambda }_y^{\prime }{\Psi }_y^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _y:= \left( \Lambda _y'\Psi _y^ {-1} \Lambda _y\right) ^ {-1} \Lambda _y'\Psi _y^ {-1} $$\end{document} both existing. Additionally,

$\begin{array}{c}\text{Cov}r=\left(\begin{array}{cc}{\left({\Lambda }_x^{\prime },{\Psi }_x^{-1},{\Lambda }_x\right)}^{-1}& 0_{d_{\xi },d_{\eta }}\\ 0_{d_{\eta },d_{\xi }}& {\left({\Lambda }_y^{\prime },{\Psi }_y^{-1},{\Lambda }_y\right)}^{-1}\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\text {Cov}} \, r=\begin{pmatrix} \left( \Lambda _x'\Psi _{x}^ {-1} \Lambda _x\right) ^ {-1} &{} \varvec{0}_{d_\xi , d_\eta }\\ \varvec{0}_{d_\eta , d_\xi } &{} \left( \Lambda _y'\Psi _y^ {-1} \Lambda _y\right) ^ {-1} \end{pmatrix}, \end{aligned}$$\end{document}

which is positive definite, and whose diagonal partitions are positive-definite matrices.

Proof

See Section E.4.5. $\square$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now consider Assumption 4 (1). We examine two approximations for sufficiently large $d_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_x$$\end{document} : Firstly, that $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} is approximately zero, and secondly that $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} is approximately normal. Since $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} will go to zero as $d_x$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_x$$\end{document} increases under weak assumptions, asymptotic normality is closely connected to techniques that treat $r_{\xi }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_\xi $$\end{document} as zero, and a normality approximation can potentially improve approximations of H. This issue will be further discussed though not resolved at the end of Sect. 2.2.