Appendix A: Mathematical Proofs
Proof of Theorem
3.1\((i)\ \)Given that the structural coefficients of \(S_{1}\) are finite and that \(E(U^{2})\) and \( E(U_{x}U_{x}^{\prime })\ \)exist and are finite, the following moments exist and are finite:
$$\begin{aligned} E(XX^{\prime })&=\left[ \begin{array}{cc} \phi _{u}E(U^{2})\phi _{u}^{\prime }+\alpha _{x}E(U_{x}U_{x}^{\prime })\alpha _{x},&0 \\ 0,&1 \end{array}\right] \\ E(ZX^{\prime })&=\alpha _{u}E(UX^{\prime })=[ \alpha _{u}E(U^{2})\phi _{u}^{\prime }, 0 ] \\ E(YX^{\prime })&=\beta _{o}E(XX^{\prime })+\alpha _{u}E(UX^{\prime })=\beta _{o}E(XX^{\prime })+[ \alpha _{u}E(U^{2})\phi _{u}^{\prime },&0 ]. \end{aligned}$$
\((ii)\) Substituting for \(\alpha _{u}U\) in \((3)\) with its expression from \( (1) \), \(\alpha _{u}U=Z-\alpha _{z}U_{z},\) gives
$$\begin{aligned} Y-Z=\beta _{o}X-\alpha _{z}U_{z}+\alpha _{y}U_{y},\text{ and} \text{ thus}\;E[(Y-Z)X^{\prime }]=\beta _{o}E(XX^{\prime }). \end{aligned}$$
It follows from the nonsingularity of \(E(XX^{\prime })\) that \(\beta _{o}\) is point identified as
$$\begin{aligned} \beta _{o}=\pi _{y-z.x}\equiv E[(Y-Z)X^{\prime }]E(XX^{\prime })^{-1}. \square \end{aligned}$$
Proof of Theorem
4.1\((i) \) Given that the structural coefficients of \(\mathcal S _{2}\) are finite and that \(E(U^{2})\) and \(E(U_{x}U_{x}^{\prime })\) exist and are finite, we have that
$$\begin{aligned} E(XX^{\prime })&=\left[ \begin{array}{cc} \eta _{u}E(U^{2})\eta _{u}^{\prime }+\alpha _{x}E(U_{x}U_{x}^{\prime })\alpha _{x}^{\prime },&0 \\ 0,&1 \end{array} \right], \quad \mathrm{and} \\ E(YX^{\prime })&=\beta _{o}E(XX^{\prime })+[ \begin{array}{llll} \alpha _{u}E(UX_{1}^{\prime }),&\alpha _{u}E(UX_{2}^{\prime }),&\alpha _{u}E(UX_{31}^{\prime }),&\alpha _{u}E(U) \end{array}] \\&=\beta _{o}E(XX^{\prime })+[ \begin{array}{llll} \alpha _{u}^{2}E(U^{2}),&\alpha _{u}^{2}E(U^{2}),&\alpha _{u}E(U^{2})\phi _{u}^{\prime },&0 \end{array}] \end{aligned}$$
exist and are finite. \((ii)\) Further, \(\alpha _{u}^{2}E(U^{2})\) is identified by \(\alpha _{u}^{2}E(U^{2})=E(X_{2}X_{1}^{\prime })\) and \(\phi _{u}E(U^{2})\alpha _{u}\) is overidentified by \(\phi _{u}E(U^{2})\alpha _{u}=E(X_{31}X_{1}^{\prime })=E(X_{31}X_{2}^{\prime })\). Given that \( E(XX^{\prime })\) is nonsingular, it follows that \(\beta _{o}\) is fully (over)identified by
$$\begin{aligned} \beta _{o}&=\beta _{JC}^{*}\equiv \pi _{y.x}-[ \begin{array}{lll} E(X_{2}X_{1}^{\prime }),&E(X_{2}X_{1}^{\prime }),&E(X_{1}X_{3}^{\prime }) \end{array} ]E(XX^{\prime })^{-1} \\&=\beta _{JC}^{\dag }\equiv \pi _{y.x}-[ \begin{array}{lll} E(X_{2}X_{1}^{\prime }),&E(X_{2}X_{1}^{\prime }),&E(X_{2}X_{3}^{\prime }) \end{array} ]E(XX^{\prime })^{-1}.\square \end{aligned}$$
Proof of Theorem
5.1\((i)\) Given that the structural coefficients of \(\mathcal S _{3}\) and \(E(U^{2})\) and \( E(U_{x}U_{x}^{\prime })\) exist and are finite we have
$$\begin{aligned} E(XX^{\prime })&=\left[ \begin{array}{cc} \eta _{u}E(U^{2})\eta _{u}^{\prime }+\alpha _{x}E(U_{x}U_{x}^{\prime })\alpha _{x}^{\prime },&0 \\ 0,&1 \end{array} \right],\quad \mathrm{and} \\ E(YX^{\prime })&=\beta _{o}E(XX^{\prime })+\alpha _{u}\iota _{p}[\begin{array}{ll} E(UX_{1}^{\prime }),&E(UX_{2}^{\prime }) \end{array}] \\&=\beta _{o}E(XX^{\prime })+\iota _{p}[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&[\begin{array}{ll} \alpha _{u}E(U^{2})\phi _{u}^{\prime },&0 \end{array}]\end{array}] \end{aligned}$$
exists and are finite.
\((ii.a)\) Given that \(E(X_{1}X_{1}^{\prime })\) and \(E(X_{2}X_{2}^{\prime })\) are nonsingular, we have
$$\begin{aligned} P_{x_{1}}&\equiv E(\epsilon _{x_{1}.x_{2}}\epsilon _{x_{1}.x_{2}}^{\prime })=E(\epsilon _{x_{1}.x_{2}}X_{1}^{\prime })=E(X_{1}X_{1}^{\prime })-\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime })\quad \mathrm{and } \\ P_{x_{2}}&\equiv E(\epsilon _{x_{2}.x_{1}}\epsilon _{x_{2}.x_{1}}^{\prime })=E(\epsilon _{x_{2}.x_{1}}X_{2}^{\prime })=E(X_{2}X_{2}^{\prime })-\pi _{x_{2}.x_{1}}E(X_{1}X_{2}^{\prime }) \end{aligned}$$
exist and are finite. \((ii.b)\) If also \(P_{x_{1}}\) and \(P_{x_{2}}\) are nonsingular, then \(E(XX^{\prime })^{-1}\) exists, is finite, and is given by (e.g., Baltagi 1999, p. 185):
$$\begin{aligned} E(XX^{\prime })^{-1}=\left[ \begin{array}{cc} E(X_{1}X_{1}^{\prime }),&E(X_{1}X_{2}^{\prime }) \\ E(X_{2}X_{1}^{\prime }),&E(X_{2}X_{2}^{\prime }) \end{array} \right] ^{-1}=\left[ \begin{array}{cc} P_{x_{1}}^{-1},&-\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1} \\ -\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1},&P_{x_{2}}^{-1} \end{array} \right] , \end{aligned}$$
with \(P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}=\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}.\) It follows that \(\pi _{y.x}\) exists and is finite. To show that
$$\begin{aligned} E(\epsilon _{y_{1}.x}Y_{2}^{\prime })=E(Y_{1}Y_{2}^{\prime })-E(Y_{1}X^{\prime })E(XX^{\prime })^{-1}E(XY_{2}^{\prime }) \end{aligned}$$
exists and is finite, note that
$$\begin{aligned} E(YY^{\prime })&=E[(\beta _{o}X+\alpha _{u}\iota _{p}U+\alpha _{y}U_{y})(\beta _{o}X+\alpha _{u}\iota _{p}U+\alpha _{y}U_{y})^{\prime }] \\&=\beta _{o}E(XX^{\prime })\beta _{o}^{\prime }+\beta _{o}E(XU)\iota _{p}^{\prime }\alpha _{u}^{\prime }+\alpha _{u}\iota _{p}E(UX^{\prime })\beta _{o}^{\prime }\\&\quad +\iota _{p}\iota _{p}^{\prime }\alpha _{u}^{2}E(U^{2})+\alpha _{y}E(U_{y}U_{y}^{\prime })\alpha _{y}^{\prime }. \end{aligned}$$
Substituting for the diagonal term \(E(Y_{1}Y_{2}^{\prime })\) in the above expression for \(E(\epsilon _{y_{1}.x}Y_{2}^{\prime })\) then gives
$$\begin{aligned} E(\epsilon _{y_{1}.x} Y_{2}^{\prime })&=\beta _{1o} E(XX^{\prime })\beta _{2o}^{\prime } +\beta _{1o} \alpha _{u}E(XU)+\alpha _{u}E(UX^{\prime })\beta _{2o}^{\prime } \\&\quad + \alpha _{u}^{2}E(U^{2})-E(Y_{1}X^{\prime })E(XX^{\prime })^{-1}E(XY_{2}^{\prime }), \end{aligned}$$
and thus \(E(\epsilon _{y_{1}.x} Y_{2}^{\prime })\) exists and is finite given that \(\alpha _{u}E(UX^{\prime })=[\alpha _{u}^{2}E(U^{2}), [\alpha _{u}E(U^{2})\phi _{u}^{\prime }, 0]].\)
\((ii.c)\) Next, we have that
$$\begin{aligned} \Delta _{JR}=[2P_{{x}_{1}}^{-1}E(X_{1} X_{1}^{\prime })-1]^{2}-4P_{{x}_{1}}^{-1}[E(X_{1} X_{2}^{\prime })P_{{x}_{2}}^{-1}E(X_{2} X_{1}^{\prime })+E(\epsilon _{{y}_{1}.x} Y_{2}^{\prime })], \end{aligned}$$
exists and is finite as it is a function of finite moments and coefficients. We now show that \(\Delta _{JR}\) is nonnegative. Given the nonsingularity of \( E(XX^{\prime })\), substituting for
$$\begin{aligned} \beta _{o}=[E(YX^{\prime })-\alpha _{u}\iota _{p}E(UX^{\prime })]E(XX^{\prime })^{-1}, \end{aligned}$$
in the expression for \(E(YY^{\prime })\) gives
$$\begin{aligned} E(YY^{\prime })&=[E(YX^{\prime })-\alpha _{u}\iota _{p}E(UX^{\prime })]E(XX^{\prime })^{-1}E(XX^{\prime })E(XX^{\prime })^{-1}[E(XY^{\prime }) \\&\quad -E(XU^{\prime })\iota _{p}^{\prime }\alpha _{u}^{\prime }]+[E(YX^{\prime })-\alpha _{u}\iota _{p}E(UX^{\prime })]E(XX^{\prime })^{-1}E(XU)\iota _{p}^{\prime }\alpha _{u}^{\prime } \\&\quad +\alpha _{u}\iota _{p}E(UX^{\prime })E(XX^{\prime })^{-1}[E(XY^{\prime })-E(XU)\iota _{p}^{\prime }\alpha _{u}^{\prime }]\\&\quad +\iota _{p}\iota _{p}^{\prime }\alpha _{u}^{2}E(U^{2})+\alpha _{y}E(U_{y}U_{y}^{\prime })\alpha _{y}^{\prime } \\&=E(YX^{\prime })E(XX^{\prime })^{-1}E(XY^{\prime })-\alpha _{u}\iota _{p}E(UX^{\prime })E(XX^{\prime })^{-1}E(XU^{\prime })\iota _{p}^{\prime }\alpha _{u}^{\prime } \\&\quad +\iota _{p}\iota _{p}^{\prime }\alpha _{u}^{2}E(U^{2})+\alpha _{y}E(U_{y}U_{y}^{\prime })\alpha _{y}^{\prime }. \end{aligned}$$
The off-diagonal term then gives
$$\begin{aligned} E(\epsilon _{y_{1}.x}Y_{2}^{\prime })&=E(Y_{1}Y_{2}^{\prime })-E(Y_{1}X^{\prime })E(XX^{\prime })^{-1}E(XY_{2}^{\prime }) \\&=\alpha _{u}^{2}E(U^{2})-\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}E(XU^{\prime })\alpha _{u}^{\prime } \end{aligned}$$
Substituting for \(\alpha _{u}E(UX^{\prime })=[\alpha _{u}^{2}E(U^{2}), [\alpha _{u}E(U^{2})\phi _{u}^{\prime }, 0]] =[\alpha _{u}^{2}E(U^{2}),E(X_{1}X_{2}^{\prime })]\) gives
$$\begin{aligned}&\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}E(XU)\alpha _{u}^{\prime } \\&\qquad =[\alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime })]\left[ \begin{array}{cc}P_{x_{1}}^{-1},&-\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1} \\ -\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1},&P_{x_{2}}^{-1}\end{array}\right] [\alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime })]^{\prime } \\&\qquad =\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}-E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2}) \\&\qquad \qquad -\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime }). \end{aligned}$$
Thus, we expand the term \(E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(\epsilon _{y_{1}.x}Y_{2}^{\prime }) \) in \(\Delta _{JR}\) as:
$$\begin{aligned}&E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(\epsilon _{y_{1}.x}Y_{2}^{\prime }) \\&\qquad =E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+\alpha _{u}^{2}E(U^{2})-\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1} \\&\qquad \qquad +E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})+\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })\\&\qquad \qquad -E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime }) \\&\qquad =-\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}+ \alpha _{u}^{2}E(U^{2})[2P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime })+1]\\&\qquad =-\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}+\alpha _{u}^{2}E(U^{2})[2P_{x_{1}}^{-1}[E(X_{1}X_{1}^{\prime })-P_{x_{1}}]+1] \\&\qquad =-\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}+\alpha _{u}^{2}E(U^{2})[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1] \end{aligned}$$
where we use \(P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}=\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}\) and \(P_{x_{1}}=E(X_{1}X_{1}^{\prime })-\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime })\). Then
$$\begin{aligned} \Delta _{JR}&\equiv [2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1]^{2}-4P_{x_{1}}^{-1}[E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(\epsilon _{y_{1}.x}Y_{2}^{\prime })] \\&=[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1]^{2}+4\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-2}\\&\qquad -4P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1] \\&=\{[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1]-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})\}^{2}\ge 0. \end{aligned}$$
\((iii)\) We begin by showing that
$$\begin{aligned}&\mathrm{Var}(\alpha _{x_{1}}U_{x_{1}})+\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)^{\prime }\nonumber \\&\qquad \qquad {\times }\,[\mathrm{Var}(\phi _{u}U)+\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})]^{-1}\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)-\mathrm{Var}(\alpha _{u}U) \end{aligned}$$
(A.1)
has the same sign as the expression \(2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})\) from \(\Delta _{JR}.\) First, clearly, () can be negative, zero, or positive (e.g., set \(\dim (X_{21})=1\), \(\mathrm{Var}(\alpha _{x_{1}}U_{x_{1}})=1,\) and \(\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})=\mathrm{Var}(\phi _{u}U)=\frac{1}{2}.\) Then () reduces to \(1-\frac{1}{2}\mathrm{Var}(\alpha _{u}U)\) with sign depending on \(\mathrm{Var}(\alpha _{u}U)\)). Next, multiplying this expression by \(P_{x_{1}}\equiv E(\epsilon _{x_{1}.x_{2}}\epsilon _{x_{1}.x_{2}}^{\prime })\) preserves its sign and we obtain
$$\begin{aligned}&2E(X_{1}X_{1}^{\prime })-P_{x_{1}}-2\alpha _{u}^{2}E(U^{2}) \\&=2E(X_{1}X_{1}^{\prime })-[E(X_{1}X_{1}^{\prime })-E(X_{1}X_{2}^{\prime })E(X_{2}X_{2}^{\prime })^{-1}E(X_{2}X_{1}^{\prime })]-2\alpha _{u}^{2}E(U^{2}) \\&=E(X_{1}X_{1}^{\prime })+E(X_{1}X_{2}^{\prime })E(X_{2}X_{2}^{\prime })^{-1}E(X_{2}X_{1}^{\prime })-2\alpha _{u}^{2}E(U^{2}). \end{aligned}$$
But we have
$$\begin{aligned} E(X_{1}X_{1}^{\prime })&=\alpha _{u}^{2}E(U^{2})+\alpha _{x_{1}}E(U_{x_{1}}U_{x_{1}}^{\prime })\alpha _{x_{1}}^{\prime }\;\text{ and}\\ E(X_{2}X_{2}^{\prime })&=\left[\begin{array}{cc} \phi _{u}E(UU^{\prime })\phi _{u}^{\prime }+\alpha _{x_{2}}E(U_{x_{2}}U_{x_{2}}^{\prime })\alpha _{x_{2}}^{\prime },&0 \\ 0,&1\end{array}\right]. \end{aligned}$$
Then using \([\begin{array}{ll}\alpha _{u}E(U^{2})\phi _{u}^{\prime },&0\end{array} ]=E(X_{1}X_{2}^{\prime })\) gives
$$\begin{aligned}&E(X_{1}X_{1}^{\prime })+E(X_{1}X_{2}^{\prime })E(X_{2}X_{2}^{\prime })^{-1}E(X_{2}X_{1}^{\prime })-2\alpha _{u}^{2}E(U^{2}) \\&\qquad =\alpha _{u}^{2}E(U^{2})+\alpha _{x_{1}}E(U_{x_{1}}U_{x_{1}}^{\prime })\alpha _{x_{1}}^{\prime } +\left[\begin{array}{cc}\alpha _{u}E(U^{2})\phi _{u}^{\prime },&0 \end{array}\right]\\&\qquad \qquad {\times }\left[\begin{array}{cc}\phi _{u}E(UU^{\prime })\phi _{u}^{\prime }+\alpha _{x_{2}}E(U_{x_{2}}U_{x_{2}}^{\prime })\alpha _{x_{2}}^{\prime },&0 \\ 0,&1\end{array}\right]^{-1}\left[\begin{array}{c}\phi _{u}E(U^{2})\alpha _{u} \\ 0 \end{array}\right]-2\alpha _{u}^{2}E(U^{2}) \\&\qquad =\mathrm{Var}(\alpha _{x_{1}}U_{x_{1}})+\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)^{\prime }[\mathrm{Var}(\phi _{u}U)+\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})]^{-1}\\&\qquad \qquad {\times }\;\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)-\mathrm{Var}(\alpha _{u}U). \end{aligned}$$
\((iii.a)\) Now, recall from \((ii.c)\) that
$$\begin{aligned} \Delta _{JR}=\{[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1]-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})\}^{2}. \end{aligned}$$
Suppose that (3) is negative, then
$$\begin{aligned} \sqrt{\Delta _{JR}}&= \left|2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})\right|\\&= -2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })+1+2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and we have
$$\begin{aligned} \sigma _{JR}^{\dag }&\equiv E(X_{1}X_{1}^{\prime })+\frac{1}{2}P_{x_{1}}(-1-\sqrt{\Delta _{JR}}) \\&=2E(X_{1}X_{1}^{\prime })-P_{x_{1}}-\alpha _{u}^{2}E(U^{2}) \\&=\mathrm{Var}(\alpha _{x_{1}}U_{x_{1}})+\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)^{\prime }[\mathrm{Var}(\phi _{u}U)+\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})]^{-1}\\&\qquad {\times }\;\mathrm{Cov}(\phi _{u}U,\alpha _{u}U) \\&\qquad <\alpha _{u}^{2}E(U^{2}) \text{(and} \ge 0), \end{aligned}$$
and
$$\begin{aligned} \sigma _{JR}^{*}\equiv E(X_{1}X_{1}^{\prime })+\frac{1}{2}P_{x_{1}}(-1+ \sqrt{\Delta _{JR}})=\alpha _{u}^{2}E(U^{2}). \end{aligned}$$
\((iii.b)\) Suppose instead that () is nonnegative then
$$\begin{aligned} \sqrt{\Delta _{JR}}&=\left|2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2})\right|\\&=2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1-2P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and we have
$$\begin{aligned} \sigma _{JR}^{\dag }=\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and
$$\begin{aligned} \sigma _{JR}^{*}&= \mathrm{Var}(\alpha _{x_{1}}U_{x_{1}})+\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)^{\prime }[\mathrm{Var}(\phi _{u}U)\\&\qquad +\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})]^{-1}\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)\\&\ge \,\alpha _{u}^{2}E(U^{2})\ge 0. \end{aligned}$$
Thus, \(\alpha _{u}^{2}E(U^{2})\) is partially identified in the set \(\{\sigma _{JR}^{\dag },\sigma _{JR}^{*}\}\). It follows from the moment
$$\begin{aligned} E(YX^{\prime })=\beta _{o}E(XX^{\prime })+\iota _{p}[ \alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime }) ], \end{aligned}$$
and the nonsingularity of \(E(XX^{\prime })\) that \(\beta _{o}\) is partially identified in the set \(\{\beta _{JR}^{*},\beta _{JR}^{\dag }\}.\)\(\square \)
Proof of Theorem
6.1 \((i)\) We have that
$$\begin{aligned} E(ZZ^{\prime })&=\left[ \begin{array}{cc} \alpha _{u}^{2}E(U^{2}),&0 \\ 0,&1 \end{array} \right], \\ E(XZ^{\prime })&=E\left( \begin{array}{c} [X_{1}^{\prime },X_{21}^{\prime }]^{\prime }Z^{\prime } \\ Z^{\prime } \end{array} \right)=\left[ \begin{array}{c} \gamma _{o}E(ZZ^{\prime })+\left[ \begin{array}{cc} \eta _{u}E(U^{2})\alpha _{u}^{\prime }&0 \end{array}\right] \\ \left[ 0, \quad 1 \right] \end{array}\right], \end{aligned}$$
$$\begin{aligned} E(XX^{\prime })&=\left[\begin{array}{cc} \gamma _{o}E(ZX^{\prime })+\eta _{u}E(UX^{\prime })+\alpha _{x}E(U_{x}X^{\prime }),&E(X) \\ E(X^{\prime }),&1 \end{array}\right] \\&=\left[\begin{array}{cc} \gamma _{o}E(ZX^{\prime })+[[\eta _{u}E(U^{2})\alpha _{u}^{\prime }, 0 ]\gamma _{o}^{\prime }&\\ \qquad +\eta _{u}E(U^{2})\eta _{u}^{\prime }, 0]+\left[\begin{array}{ll}\alpha _{x}E(U_{x}U_{x})^{\prime }\alpha _{x}^{\prime },&0 \end{array}\right],&\left[\begin{array}{cc} 0^{\prime },&1^{\prime } \end{array} \right]^{\prime } \\ \left[ \begin{array}{cc} 0,&1 \end{array}\right],&1 \end{array}\right], \end{aligned}$$
$$\begin{aligned} E(YX^{\prime })&= \beta _{o}E(XX^{\prime })+\alpha _{u}E(UX^{\prime }) =\beta _{o}E(XX^{\prime }) \\&\quad +[[\alpha _{u}^{2}E(U^{2}), 0]\gamma _{1o}^{\prime }+\alpha _{u}^{2}E(U^{2}), [[\alpha _{u}^{2}E(U^{2}), 0]\gamma _{2o}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime }, 0]], \\ E(YZ^{\prime })&= \beta _{o}E(XZ^{\prime })+[\alpha _{u}^{2}E(U^{2}), 0], \end{aligned}$$
Thus, these moments exist and are finite since they are functions of existing finite coefficients and moments.
\((ii.a)\) Given that \(P_{z_{1}}\equiv E(\epsilon _{z_{1}.z_{2}}Z_{1}^{\prime })=E(Z_{1}Z_{1}^{\prime })\) is nonsingular and \(Z_{2}=1\), we have that
$$\begin{aligned} E(ZZ^{\prime })^{-1}=\left[ \begin{array}{cc} P_{z_{1}}^{-1},&-\pi _{z_{2}.z_{1}}^{\prime }P_{z_{2}}^{-1} \\ -\pi _{z_{1}.z_{2}}^{\prime }P_{z_{1}}^{-1},&P_{z_{2}}^{-1} \end{array} \right] =\left[ \begin{array}{cc} E(Z_{1}Z_{1}^{\prime })^{-1}&0 \\ 0&1 \end{array} \right] \end{aligned}$$
is nonsingular and thus \(\pi _{x.z}\) and \(E(\epsilon _{x_{1}.z}X_{2}^{\prime })=E(X_{1}X_{2}^{\prime })-\pi _{x_{1}.z}E(ZX_{2}^{\prime })\) exist and are finite. With \(E(XX^{\prime })\) also nonsingular, \(\pi _{z.x}\) exists and is finite. Also,
$$\begin{aligned} E(\epsilon _{y.x}Z_{1}^{\prime })&= E(Y\epsilon _{z_{1}.x}^{\prime })\\&= \beta _{o}E(X\epsilon _{z_{1}.x}^{\prime })+\alpha _{u}E(U\epsilon _{z_{1}.x}^{\prime })+\alpha _{y}E(U_{y}\epsilon _{z_{1}.x}^{\prime })\\&= \alpha _{u}E(U\epsilon _{z_{1}.x}^{\prime }). \end{aligned}$$
Using \(E(X_{1}X_{2}^{\prime })=\gamma _{1o}E(ZX_{2}^{\prime })+\alpha _{u}E(UX_{2}^{\prime })\) then gives
$$\begin{aligned} E(\epsilon _{y.x}Z_{1}^{\prime })&=\alpha _{u}E(U\epsilon _{z_{1}.x}^{\prime })=\alpha _{u}E(UZ_{1}^{\prime })-\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}E(XZ_{1}^{\prime }) \\&=\alpha _{u}^{2}E(U^{2})-[[\alpha _{u}^{2}E(U^{2}), 0]\gamma _{1o}^{\prime }\\&\qquad +\alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime })-\gamma _{1o}E(ZX_{2}^{\prime })] \pi _{z_{1}.x}^{\prime } \end{aligned}$$
exists and is finite.
\((ii.b)\) We have that \(\Delta _{PC}\) exists and is finite as it is a function of finite coefficients and moments. Next, we verify that \(\Delta _{PC}\ge 0.\) We begin by expanding the term \(E(\epsilon _{y.x}Z_{1}^{\prime })\) in \(\Delta _{PC}\). For this, we substitute for \(\gamma _{1o}\) with
$$\begin{aligned} \gamma _{1o}=\pi _{x_{1}.z}-[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]E(ZZ^{\prime })^{-1}, \end{aligned}$$
in \(-\alpha _{u}E(UX^{\prime })\pi _{z.x}^{\prime }\) which gives
$$\begin{aligned}&-\alpha _{u} E(UX^{\prime })\pi _{z.x}^{\prime }\\&\quad =-[ [\alpha _{u}^{2}E(U^{2}), \quad 0] \gamma _{1o}^{\prime } + \alpha _{u}^{2} E(U^{2}), \quad E(X_{1}X_{2}^{\prime })-\gamma _{1o}E(ZX_{2}^{\prime })]\pi _{z.x}^{\prime } \\&\quad =-[\alpha _{u}^{2}E(U^{2}), \quad 0 ]\pi _{x_{1}.z}^{\prime }\pi _{z.x_{1}|x_{2}}^{\prime }+[ \alpha _{u}^{2}E(U^{2}), \quad 0]E(ZZ^{\prime })^{-1}[ \alpha _{u}^{2}E(U^{2}), \quad 0]^{\prime }\pi _{z.x_{1}|x_{2}}^{\prime } \\&\qquad -\alpha _{u}^{2}E(U^{2})\pi _{z.x_{1}|x_{2}}^{\prime }-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z.x_{2}|x_{1}}^{\prime }-[ \alpha _{u}^{2}E(U^{2}), \quad 0]\pi _{x_{2}.z}^{\prime }\pi _{z.x_{2}|x_{1}}^{\prime } \\&\quad =-\alpha _{u}^{2}E(U^{2})\pi _{x_{1}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{1}|x_{2}}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z.x_{1}|x_{2}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{z.x_{1}|x_{2}}^{\prime }\\&\qquad -E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z.x_{2}|x_{1}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{2}|x_{1}}^{\prime }, \end{aligned}$$
where we make use of \([\begin{array}{ll}\alpha _{u}^{2}E(U^{2}),&0\end{array}]E(ZZ^{\prime })^{-1}[ \begin{array}{ll}\alpha _{u}^{2}E(U^{2}),&0\end{array}]^{\prime }=\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\). Thus,
$$\begin{aligned} E(\epsilon _{y.x}Z_{1}^{\prime })&=\alpha _{u}^{2}E(U^{2})-\alpha _{u}E(UX^{\prime })\pi _{z_{1}.x}^{\prime } \\&=\alpha _{u}^{2}E(U^{2})-\alpha _{u}^{2}E(U^{2})\pi _{x_{1}.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime } \\&\quad -\alpha _{u}^{2}E(U^{2})\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x_{2}|x_{1}}^{\prime } \\&=\alpha _{u}^{2}E(U^{2})-\alpha _{u}^{2}E(U^{2})\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime } \\&\quad -\alpha _{u}^{2}E(U^{2})\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }. \end{aligned}$$
Then
$$\begin{aligned} \Delta _{PC}&\equiv [-\pi _{x.z_{1}|z_{2}}^{\prime }\ \pi _{z_{1}.x}^{\prime }-\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+1]^{2}+4P_{z_{1}}^{-1}\ \pi _{z_{1}.x_{1}|x_{2}}^{\prime }[E(\epsilon _{y.x}Z_{1}^{\prime })\\&\qquad +E(\epsilon _{x_{1}.z}X_{2}^{\prime })\ \pi _{z_{1}.x_{2}|x_{1}}^{\prime }] \\&=[-\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }-\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+1]^{2} \\&\qquad +4P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }[\alpha _{u}^{2}E(U^{2})-\alpha _{u}^{2}E(U^{2})\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime } \\&\qquad +\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{z_{1}.x_{1}|x_{2}}^{\prime } \\&\qquad -E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }+E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }] \\&=\{[\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1]-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})\}^{2}\ge 0. \end{aligned}$$
\((iii)\) Suppose that
$$\begin{aligned} \pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})<0. \end{aligned}$$
Then
$$\begin{aligned} \sqrt{\Delta _{PC}}&=\left|\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})\right|\\&=-\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }-\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+1+2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and thus
$$\begin{aligned} \sigma _{PC}^{\dag }&\equiv \frac{\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-\sqrt{\Delta _{PC}} }{2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }} \\&=\frac{\pi _{x.z}^{\prime }\pi _{z.x}^{\prime }+\pi _{z.x_{1}|x_{2}}^{\prime }-1-P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})}{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}\\ \quad&<\frac{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2}) }{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}=\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and
$$\begin{aligned} \sigma _{PC}^{*}\equiv \frac{\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1+\sqrt{\Delta _{PC}} }{2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}=\alpha _{u}^{2}E(U^{2}). \end{aligned}$$
Now, with \(E(ZZ^{\prime })\) nonsingular, we have
$$\begin{aligned} E(X_{1}Z^{\prime })&=\gamma _{1o}E(ZZ^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}], \text{ or} \\ \gamma _{1o}&=\pi _{x_{1}.z}-[ \begin{array}{ll} \sigma _{PC}^{*},&0 \end{array}]E(ZZ^{\prime })^{-1}. \end{aligned}$$
Further, with \(E(XX^{\prime })\) nonsingular, we have
$$\begin{aligned} E(YX^{\prime })&=\beta _{o}E(XX^{\prime })+\alpha _{u}E(UX^{\prime }),\text{ or} \\ \beta _{o}&=\{E(YX^{\prime })-[[\alpha _{u}^{2}E(U^{2}), 0]\gamma _{1o}^{\prime }\\&\quad +\alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime })- \gamma _{1o}E(ZX_{2}^{\prime })]\}E(XX^{\prime })^{-1}. \end{aligned}$$
Substituting for \(\gamma _{1o}\) gives
$$\begin{aligned}&\left[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}\right] \gamma _{1o}^{\prime }+\alpha _{u}^{2}E(U^{2}) \\&\quad =\left[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}\right] \pi _{x_{1}.z}^{\prime }-\left[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}\right] E(ZZ^{\prime })^{-1}\left[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}\right]^{\prime }+\alpha _{u}^{2}E(U^{2}) \\&\quad =\left[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}\right]\pi _{x_{1}.z}^{\prime }-\alpha _{u}^{4}E(U^{2})^{2}\ P_{z_{1}}^{-1}+\alpha _{u}^{2}E(U^{2}) \\&\quad =\alpha _{u}^{2}E(U^{2})\ (\pi _{x_{1}.z_{1}|z_{2}}^{\prime }-\alpha _{u}^{2}E(U^{2})\ P_{z_{1}}^{-1}+1), \end{aligned}$$
and
$$\begin{aligned}&E(X_{1}X_{2}^{\prime })-\gamma _{1o}E(ZX_{2}^{\prime })\\&\quad =E(X_{1}X_{2}^{\prime })-[\pi _{x_{1}.z}-[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]E(ZZ^{\prime })^{-1}]E(ZX_{2}^{\prime }) \\&\quad =E(\epsilon _{x_{1}.z}X_{2}^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]\pi _{x_{2}.z}^{\prime }=E(\epsilon _{x_{1}.z}X_{2}^{\prime })+\alpha _{u}^{2}E(U^{2})\ \pi _{x_{2}.z_{1}|z_{2}}^{\prime }, \end{aligned}$$
so that
$$\begin{aligned} \beta _{o}&= \pi _{y.x}-[\sigma _{PC}^{*}\ (\pi _{x_{1}.z_{1}|z_{2}}^{\prime }-\sigma _{PC}^{*}\ P_{z_{1}}^{-1}+1), E(\epsilon _{x_{1}.z}X_{2}^{\prime })\\&+\;\sigma _{PC}^{*}\ \pi _{x_{2}.z_{1}|z_{2}}^{\prime }]E(XX^{\prime })^{-1}. \end{aligned}$$
Also, we have
$$\begin{aligned} E(X_{1}X_{21}^{\prime })&=\gamma _{1o}E(ZX_{21}^{\prime })+\alpha _{u}E(UX_{21}^{\prime }) \\&=\gamma _{1o}E(ZX_{21}^{\prime })+\alpha _{u}E(UZ^{\prime })\gamma _{2o}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime } \\&=\gamma _{1o}E(ZX_{21}^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array} ]\gamma _{2o}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime } \text{ and} \\ E(X_{21}Z^{\prime })&=\gamma _{2o}E(ZZ^{\prime })+[ \begin{array}{ll} \phi _{u}E(U^{2})\alpha _{u}^{\prime },&0 \end{array}]. \end{aligned}$$
Substituting for
$$\begin{aligned} \gamma _{2o}=\pi _{x_{21}.z}-[ \begin{array}{ll} \phi _{u}E(U^{2})\alpha _{u}^{\prime },&0 \end{array} ]E(ZZ^{\prime })^{-1} \end{aligned}$$
in the expression for \(E(X_{1}X_{21}^{\prime })\) gives
$$\begin{aligned} E(X_{1}X_{21}^{\prime })&= \gamma _{1o}E(ZX_{21}^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array} ]\pi _{x_{21}.z}^{\prime } \\&\quad -[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array} ]E(ZZ^{\prime })^{-1}[ \begin{array}{ll} \phi _{u}E(U^{2})\alpha _{u}^{\prime },&0 \end{array} ]^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime } \\&= \gamma _{1o}E(ZX_{21}^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]\pi _{x_{21}.z}^{\prime }\\&\quad -\alpha _{u}^{2}E(U^{2})P_{z_{1}}^{-1}\alpha _{u}E(U^{2})\phi _{u}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime }. \end{aligned}$$
Further substituting for \(\gamma _{1o}\) with \([E(X_{1}Z^{\prime })-[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0\end{array}]]E(ZZ^{\prime })^{-1}\) gives
$$\begin{aligned}&E(X_{1}X_{21}^{\prime })-[E(X_{1}Z^{\prime })-[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array} ]]E(ZZ^{\prime })^{-1}E(ZX_{21}^{\prime })-[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array} ]\pi _{x_{21}.z}^{\prime } \\&\quad =-\alpha _{u}^{2}E(U^{2})P_{z_{1}}^{-1}\alpha _{u}E(U^{2})\phi _{u}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime }, \end{aligned}$$
or
$$\begin{aligned} E(X_{1}\epsilon _{x_{21}.z}^{\prime })=-\alpha _{u}^{2}E(U^{2})P_{z_{1}}^{-1}\alpha _{u}E(U^{2})\phi _{u}^{\prime }+\alpha _{u}E(U^{2})\phi _{u}^{\prime }. \end{aligned}$$
Substituting for
$$\begin{aligned} \phi _{u}E(U^{2})\alpha _{u}^{\prime }=E(X_{21}\epsilon _{x_{1}.z}^{\prime })[1-\alpha _{u}^{2}E(U^{2})P_{z_{1}}^{-1}]^{-1} \end{aligned}$$
in the expression for \(\gamma _{2o}\) gives
$$\begin{aligned} \gamma _{2o}&=\pi _{x_{21}.z}-[ \begin{array}{ll} \phi _{u}E(U^{2})\alpha _{u}^{\prime },&0 \end{array} ]E(ZZ^{\prime })^{-1} \\&=\pi _{x_{21}.z}-[ \begin{array}{ll} E(X_{21}\epsilon _{x_{1}.z}^{\prime })[1-\sigma _{PC}^{*}P_{z_{1}}^{-1}]^{-1},&0 \end{array} ]E(ZZ^{\prime })^{-1}. \end{aligned}$$
\((iii.b)\) Suppose instead that
$$\begin{aligned} \pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})\ge 0. \end{aligned}$$
Then
$$\begin{aligned} \sqrt{\Delta _{PC}}&=\left|\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})\right|\\&=\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and thus
$$\begin{aligned} \sigma _{PC}^{\dag }=\alpha _{u}^{2}E(U^{2}), \end{aligned}$$
and
$$\begin{aligned} \sigma _{PC}^{*}&= \frac{\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1-P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})}{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}\\&\ge \frac{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }\alpha _{u}^{2}E(U^{2})}{P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}=\alpha _{u}^{2}E(U^{2}). \end{aligned}$$
It follows that
$$\begin{aligned} \gamma _{1o}&=\gamma _{1}^{\dag }\equiv \pi _{x_{1}.z}-[ \begin{array}{ll} \sigma _{PC}^{\dag },&0 \end{array} ]E(ZZ^{\prime })^{-1}, \\ \gamma _{2o}&=\gamma _{2}^{\dag }\equiv \pi _{x_{2}.z}-[ \begin{array}{ll} E(X_{21}\epsilon _{x_{1}.z}^{\prime })[1-\sigma _{PC}^{\dag }\ P_{z_{1}}^{-1}]^{-1},&0 \end{array} ]E(ZZ^{\prime })^{-1}, \text{ and} \\ \beta _{o}&=\beta ^{\dag }\equiv \pi _{y.x}-[ \sigma _{PC}^{\dag }\ (\pi _{x_{1}.z_{1}|z_{2}}^{\prime }-\sigma _{PC}^{\dag }\ P_{z_{1}}^{-1}+1), E(\epsilon _{x_{1}.z}X_{2}^{\prime })\\&\qquad \qquad \qquad \quad +\sigma _{PC}^{\dag }\ \pi _{x_{2}.z_{1}|z_{2}}^{\prime }]E(XX^{\prime })^{-1}.\square \end{aligned}$$
Appendix B: Constructive Identification
1.1 B.1 Equiconfounded Cause and Joint Responses: Constructive Identification
We present an argument to constructively demonstrate how the expression for \( \Delta _{JR}\) and the identification of \(\alpha _{u}^{2}E(U^{2})\), and thus \( \beta _{o}\), in the proof of Theorem 5.1 obtain. Recall that in \(\mathcal S _{3}\)
$$\begin{aligned} E(YX^{\prime })=\beta _{o}E(XX^{\prime })+\iota _{p}[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&[ \begin{array}{ll} \alpha _{u}E(U^{2})\phi _{u}^{\prime },&0 \end{array} ] \end{array} ]. \end{aligned}$$
We have that \(\alpha _{u}E(U^{2})\phi _{u}^{\prime }=E(X_{1}X_{2}^{\prime })\). It remains to identify \(\alpha _{u}^{2}E(U^{2})\). For this, recall that the proof of Theorem 5.1 gives
$$\begin{aligned} E(YY^{\prime })&= E(YX^{\prime })E(XX^{\prime })^{-1}E(XY^{\prime })-\alpha _{u}\iota _{p}E(UX^{\prime })E(XX^{\prime })^{-1}\alpha _{u}E(XU)\iota _{p}^{\prime } \\&\quad +\iota _{p}\iota _{p}^{\prime }\alpha _{u}^{2}E(U^{2})+\alpha _{y}E(U_{y}U_{y}^{\prime })\alpha _{y}^{\prime }, \end{aligned}$$
which we rewrite as
$$\begin{aligned}&\iota _{p}\iota _{p}^{\prime }\alpha _{u}^{2}E(U^{2})-\alpha _{u}\iota _{p}E(UX^{\prime })E(XX^{\prime })^{-1}E(XU)\iota _{p}^{\prime }\alpha _{u}^{\prime }\nonumber \\&\qquad -E(\epsilon _{y.x}Y^{\prime })+\alpha _{y}E(U_{y}U_{y}^{\prime })\alpha _{y}^{\prime }=0. \end{aligned}$$
(B.1)
From the proof of Theorem 5.1, we also have
$$\begin{aligned}&\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}E(XU)\alpha _{u}^{\prime }\\&= \alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}-E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2}) \\&\quad -\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime }). \end{aligned}$$
Thus, collecting the off-diagonal terms in Eq. B.1 gives:
$$\begin{aligned}&\alpha _{u}^{2}E(U^{2})-\alpha _{u}^{4}E(U^{2})^{2}P_{x_{1}}^{-1}+E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}\alpha _{u}^{2}E(U^{2}) \\&\quad +\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })-E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })-E(\epsilon _{y_{1}.x}Y_{2}^{\prime })=0. \end{aligned}$$
This is a quadratic equation in \(\alpha _{u}^{2}E(U^{2})\) of the form
$$\begin{aligned} a\alpha _{u}^{4}E(U^{2})^{2}+b\alpha _{u}^{2}E(U^{2})+c=0, \end{aligned}$$
with
$$\begin{aligned} a&=P_{x_{1}}^{-1}, \\ b&=-[1+E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}+\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })] \\&=-[1+E(X_{1}X_{2}^{\prime })\pi _{x_{1}.x_{2}}^{\prime }P_{x_{1}}^{-1}+P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime })] \\&=-[1+2P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime })] \\&=-[1+2P_{x_{1}}^{-1}[E(X_{1}X_{1}^{\prime })-P_{x_{1}}]]=-[2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1]\text{,} \text{ and} \\ c&=E(X_{1}X_{2}^{\prime })P_{x_{2}}^{-1}E(X_{2}X_{1}^{\prime })+E(\epsilon _{y_{1}.x}Y_{2}^{\prime }), \end{aligned}$$
where we make use of \(P_{x_{1}}^{-1}\pi _{x_{1}.x_{2}}=\pi _{x_{2}.x_{1}}^{\prime }P_{x_{2}}^{-1}\) and \(P_{x_{1}}=E(X_{1}X_{1}^{\prime })-\pi _{x_{1}.x_{2}}E(X_{2}X_{1}^{\prime }).\) The discriminant of this quadratic equation gives the expression for \(\Delta _{JR}=b^{2}-4ac\). Theorem 5.1 \((ii.c)\) gives that \(\Delta _{JR}\ge 0\) and \( (iii)\) gives the two roots \(\sigma _{PC}^{\dag }\) and \(\sigma _{PC}^{*}\) of this quadratic equation
$$\begin{aligned} \frac{-b\pm \sqrt{\Delta _{JR}}}{2a}&=\frac{1}{2}P_{x_{1}} \left\{ 2P_{x_{1}}^{-1}E(X_{1}X_{1}^{\prime })-1\pm \sqrt{\Delta _{JR}} \right\} \\&=E(X_{1}X_{1}^{\prime })+\frac{1}{2}P_{x_{1}}\left(-1\pm \sqrt{\Delta _{JR}}\right), \end{aligned}$$
and shows that these are nonnegative. One of these roots identifies \(\alpha _{u}^{2}E(U^{2})\), depending on the sign of
$$\begin{aligned}&\mathrm{Var}(\alpha _{x_{1}}^{\prime }U_{x_{1}})+\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)^{\prime }[\mathrm{Var}(\phi _{u}U)\\&\quad +\mathrm{Var}(\alpha _{x_{2}}U_{x_{2}})]^{-1}\mathrm{Cov}(\phi _{u}U,\alpha _{u}U)-\mathrm{Var}(\alpha _{u}U). \end{aligned}$$
\(\beta _{o}\) is then identified from the moment \(E(YX^{\prime })=\beta _{o}E(XX^{\prime })+\iota _{p}[\alpha _{u}^{2}E(U^{2}), E(X_{1}X_{2}^{\prime })].\)
1.2 B.2 Equiconfounding in Triangular Structures: Constructive Identification
We present an argument to constructively demonstrate how the expression for \( \Delta _{PC}\) and the identification of \(\alpha _{u}^{2}E(U^{2})\) in the proof of Theorem 6.1 obtain. From the proof of Theorem 6.1, we have that
$$\begin{aligned} \beta _{o}=\{E(YX^{\prime })-\alpha _{u}E(UX^{\prime })\}E(XX^{\prime })^{-1}=\pi _{y.x}-\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}. \end{aligned}$$
Substituting for \(\beta _{o}\) in the expression for \(E(YZ^{\prime })\) gives
$$\begin{aligned} E(YZ^{\prime })&= \beta _{o}E(XZ^{\prime })+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}], \\&= \pi _{y.x}E(XZ^{\prime })-\alpha _{u}E(UX^{\prime })E(XX^{\prime })^{-1}E(XZ^{\prime })+[\begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0\end{array}],\quad \text{ or} \\&\quad -E(\epsilon _{y.x}Z^{\prime })-\alpha _{u}E(UX^{\prime })\pi _{z.x}^{\prime }+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]=0. \end{aligned}$$
From the proof of Theorem 6.1, we have
$$\begin{aligned}&-\alpha _{u}E(UX^{\prime })\pi _{z.x}^{\prime }\\&\quad =-\alpha _{u}^{2}E(U^{2})\pi _{x_{1}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{1}|x_{2}}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z.x_{1}|x_{2}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{z.x_{1}|x_{2}}^{\prime } \\&\qquad -E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z.x_{2}|x_{1}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{2}|x_{1}}^{\prime }. \end{aligned}$$
Substituting for \(-\alpha _{u}E(UX^{\prime })\pi _{z.x}^{\prime }\) in the above equality then gives
$$\begin{aligned}&-E(\epsilon _{y.x}Z^{\prime })-\alpha _{u}^{2}E(U^{2})\pi _{x_{1}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{1}|x_{2}}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z.x_{1}|x_{2}}^{\prime } \\&-\alpha _{u}^{2}E(U^{2})\pi _{z.x_{1}|x_{2}}^{\prime }-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z.x_{2}|x_{1}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.z_{1}|z_{2}}^{\prime }\pi _{z.x_{2}|x_{1}}^{\prime } \\&+[ \begin{array}{ll} \alpha _{u}^{2}E(U^{2}),&0 \end{array}]=0. \end{aligned}$$
Collecting the first elements of this vector equality gives
$$\begin{aligned}&-E(\epsilon _{y.x}Z_{1}^{\prime })-\alpha _{u}^{2}E(U^{2})\pi _{x_{1}.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+\alpha _{u}^{4}E(U^{2})^{2}P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime } \\&-\alpha _{u}^{2}E(U^{2})\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }-\alpha _{u}^{2}E(U^{2})\pi _{x_{2}.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x_{2}|x_{1}}^{\prime } \\&+\alpha _{u}^{2}E(U^{2})=0. \end{aligned}$$
This is a quadratic equation in \(\alpha _{u}^{2}E(U^{2})\) of the from
$$\begin{aligned} a\alpha _{u}^{4}E(U^{2})^{2}+b\alpha _{u}^{2}E(U^{2})+c=0, \end{aligned}$$
with
$$\begin{aligned} a&=P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }, \\ b&=-\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }-\pi _{z_{1}.x_{1}|x_{2}}^{\prime }+1, \text{ and} \\ c&=-E(\epsilon _{y.x}Z_{1}^{\prime })-E(\epsilon _{x_{1}.z}X_{2}^{\prime })\pi _{z_{1}.x_{2}|x_{1}}^{\prime }. \end{aligned}$$
The discriminant of this equation gives the expression for \(\Delta _{PC}=b^{2}-4ac\) in Theorem 6.1 where it is shown that \(\Delta _{PC}\ge 0\) and that the solutions to this quadratic equation are \(\sigma _{PC}^{\dag }\) and \(\sigma _{PC}^{*}\):
$$\begin{aligned} \frac{-b\pm \sqrt{\Delta _{PC}}}{2a}=\frac{\pi _{x.z_{1}|z_{2}}^{\prime }\pi _{z_{1}.x}^{\prime }+\pi _{z_{1}.x_{1}|x_{2}}^{\prime }-1\pm \sqrt{\Delta _{PC}}}{2P_{z_{1}}^{-1}\pi _{z_{1}.x_{1}|x_{2}}^{\prime }}. \end{aligned}$$
This then enables the identification of \((\beta _{o},\gamma _{o})\) as shown in the proof of Theorem 6.1.
