Appendix 1
13.1.1 Theorem 3.1.1 of Kollo and Rosen (2005)
Let {x
n
} and \(\{\varepsilon _{n}\}\) be sequences of random p-vectors and positive numbers, respectively, and let \(x_{n} - x_{0} = o_{p}(\varepsilon _{n})\), where \(\varepsilon _{n} \rightarrow 0\) as n → ∞. If the function f(x) from \({\mathbb{R}}^{p}\) to \({\mathbb{R}}^{s}\) has continuous partial derivatives up to the order \((\mathcal{M} + 1)\) in a neighborhood \(\mathcal{D}\) of a point x
0, then the function f(x) can be expanded at the point x
0 into the Taylor series
$$\displaystyle{ f(x) = f(x_{0})+\sum _{k=1}^{\mathcal{M}}\frac{1} {k!}\left (I_{s} \otimes {(x - x_{0})}^{\otimes (k-1)}\right ){^\prime}\left (\frac{{d}^{k}f(x)} {d{x}^{k}} \right ){^\prime}_{x=x_{0}}(x-x_{0})+o{(\rho }^{\mathcal{M}}(x,x_{ 0})), }$$
where the Kroneckerian power A
⊗k for any matrix A is given by \({A}^{\otimes k} =\mathop{\underbrace{ A \otimes \cdots \otimes A}}\limits _{\text{k}\ \text{times}}\) with A
⊗0 = 1, ρ(. , . ) is the Euclidean distance in \({\mathbb{R}}^{p}\), and the matrix derivative for any matrices Y and X is given by \(\frac{{d}^{k}Y } {d{X}^{k}} = \frac{d} {dX}\left (\frac{{d}^{k-1}Y } {d{X}^{k-1}} \right )\) with \(\frac{dY } {dX} \equiv \frac{dvec{^\prime}Y } {dvecX}\); and
$$\displaystyle\begin{array}{rcl} f(x_{n})& =& f(x_{0}) +\sum _{ k=1}^{\mathcal{M}}\frac{1} {k!}\left (I_{s} \otimes {(x_{n} - x_{0})}^{\otimes (k-1)}\right ){^\prime} {}\\ & & \times \left (\frac{{d}^{k}f(x_{n})} {dx_{n}^{k}} \right ){^\prime}_{x_{n}=x_{0}}(x_{n} - x_{0}) + o_{p}(\varepsilon _{n}^{\mathcal{M}}). {}\\ \end{array}$$
Appendix 2
13.2.1 Proofs
Proof of Theorem 1: Write (13.11) as
$$\displaystyle{ DM\mathop{\cong}1_{DM} + 2_{DM} + 3_{DM} + 4_{DM}, }$$
(13.31)
where,
$$\displaystyle\begin{array}{rcl} & & 1_{DM} = vec{^\prime}G_{N}(\hat{\theta }_{N})M_{1}vecG_{N}(\hat{\theta }_{N}), {}\\ & & 2_{DM} = \mathbb{M}_{N}{^\prime}(\hat{\theta }_{N})M_{2}vecD_{N}(\hat{\theta }_{N}), {}\\ & & 3_{DM} = -{N}^{-1/2}vec{^\prime}G_{ N}(\hat{\theta }_{N})M_{3}vecD_{N}(\hat{\theta }_{N}), {}\\ & & 4_{DM} = {N}^{-1}\frac{1} {4}vec{^\prime}D_{N}(\hat{\theta }_{N})M_{4}vecD_{N}(\hat{\theta }_{N}). {}\\ \end{array}$$
Taking Taylor expansions of \(\mathbb{M}_{N}(\hat{\theta }_{N})\), \(vec\,G_{N}(\hat{\theta }_{N})\) and \(vec\,D_{N}(\hat{\theta }_{N})\) about θ
0 and using (13.5) and (13.7), we have
$$\displaystyle{ \begin{array}{ll} \mathop{\mathbb{M}_{N}(\hat{\theta }_{N})}\limits_{k \times 1}& = \mathbb{M}_{N}(\theta _{0}) + G{^\prime}(\hat{\theta }_{N} -\theta _{0}) + \frac{1} {2}[I_{k} \otimes (\hat{\theta }_{N} -\theta _{0}){^\prime}]D{^\prime}(\hat{\theta }_{N} -\theta _{0}) + o_{p}({N}^{-1}) \\ & = -{N}^{-1/2}\bar{q} + {N}^{-1/2}G{^\prime}{B}^{-1}G\bar{q} + {N}^{-1}\frac{1} {2}(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})D{^\prime}{B}^{-1}G\bar{q} + o_{ p}({N}^{-1}),\\ \end{array} }$$
$$\displaystyle{ \begin{array}{ll} \mathop{vec\,G_{N}(\hat{\theta }_{N})}\limits_{pk \times 1}& = vecG + D{^\prime}(\hat{\theta }_{N} -\theta _{0}) + \frac{1} {2}[I_{pk} \otimes (\hat{\theta }_{N} -\theta _{0}){^\prime}]C{^\prime}(\hat{\theta }_{N} -\theta _{0}) + o_{p}({N}^{-1}) \\ & = vecG + {N}^{-1/2}D{^\prime}{B}^{-1}G\bar{q} + {N}^{-1}\frac{1} {2}(I_{pk} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})C{^\prime}{B}^{-1}G\bar{q} + o_{ p}({N}^{-1}),\\ \end{array} }$$
$$\displaystyle{ \begin{array}{ll} \mathop{vec\,D_{N}(\hat{\theta }_{N})}\limits_{{p}^{2}k \times 1}& = vecD + C{^\prime}(\hat{\theta }_{N} -\theta _{0}) + o_{p}({N}^{-1/2}) \\ & = vecD + {N}^{-1/2}C{^\prime}{B}^{-1}G\bar{q} + o_{p}({N}^{-1/2}).\end{array} }$$
Note that we do not need to expand \(vecD_{N}(\hat{\theta }_{N})\) further for our purpose. Substituting these expressions into the terms of (13.31) gives:
$$\displaystyle\begin{array}{rcl} 1_{DM}& =& vec{^\prime}G_{N}(\hat{\theta }_{N})M_{1}vecG_{N}(\hat{\theta }_{N}) \\ & =& vec{^\prime}GM_{1}vecG + {N}^{-1/2}2\bar{q}{^\prime}G{^\prime}{B}^{-1}DM_{ 1}vecG \\ & & +\,{N}^{-1}[\bar{q}{^\prime}G{^\prime}{B}^{-1}DM_{ 1}D{^\prime}{B}^{-1}G\bar{q} +\bar{ q}{^\prime}G{^\prime}{B}^{-1}C(I_{ pk} \otimes {B}^{-1}G\bar{q})M_{ 1}vecG] \\ & & +\,o_{p}({N}^{-1}) \\ & =& \bar{q}{^\prime}P\bar{q} + {N}^{-1/2}u_{ 1}(\bar{q}) + {N}^{-1}v_{ 1}(\bar{q}) + o_{p}({N}^{-1}), {}\end{array}$$
(13.32)
where
$$\displaystyle{ \mathop{P}\limits_{k \times k} \equiv G{^\prime}\mathbb{H}G }$$
is a projection matrix, and
$$\displaystyle\begin{array}{rcl} & & u_{1}(\bar{q}) = 2\bar{q}{^\prime}G{^\prime}{B}^{-1}DM_{ 1}vecG, {}\\ & & v_{1}(\bar{q}) =\bar{ q}{^\prime}G{^\prime}{B}^{-1}DM_{ 1}D{^\prime}{B}^{-1}G\bar{q} +\bar{ q}{^\prime}G{^\prime}{B}^{-1}C(I_{ pk} \otimes {B}^{-1}G\bar{q})M_{ 1}vecG; {}\\ \end{array}$$
$$\displaystyle\begin{array}{rcl} 2_{DM}& =& \mathbb{M}_{N}{^\prime}(\hat{\theta }_{N})M_{2}vecD_{N}(\hat{\theta }_{N}) \\ & =& -{N}^{-1/2}\bar{q}{^\prime}M_{ 2}vecD - {N}^{-1}\bar{q}{^\prime}M_{ 2}C{^\prime}{B}^{-1}G\bar{q} \\ & & +\,{N}^{-1/2}\bar{q}{^\prime}G{^\prime}{B}^{-1}GM_{ 2}vecD + {N}^{-1}\bar{q}{^\prime}G{^\prime}{B}^{-1}GM_{ 2}C{^\prime}{B}^{-1}G\bar{q} \\ & & +\,{N}^{-1}\frac{1} {2}\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes {B}^{-1}G\bar{q})M_{ 2}vecD + o_{p}({N}^{-1}) \\ & =& {N}^{-1/2}(\bar{q}{^\prime}G{^\prime}{B}^{-1}M_{ 2}vecD -\bar{ q}{^\prime}M_{2}vecD) \\ & & +\,{N}^{-1}[\bar{q}{^\prime}G{^\prime}{B}^{-1}GM_{ 2}C{^\prime}{B}^{-1}G\bar{q} -\bar{ q}{^\prime}M_{ 2}C{^\prime}{B}^{-1}G\bar{q} \\ & & \quad \quad \quad \quad + \frac{1} {2}\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes {B}^{-1}G\bar{q})M_{ 2}vecD] + o_{p}({N}^{-1}) \\ & =& {N}^{-1/2}u_{ 2}(\bar{q}) + {N}^{-1}v_{ 2}(\bar{q}) + o_{p}({N}^{-1}), {}\end{array}$$
(13.33)
where
$$\displaystyle\begin{array}{rcl} u_{2}(\bar{q})& =& \bar{q}{^\prime}G{^\prime}{B}^{-1}GM_{ 2}vecD -\bar{ q}{^\prime}M_{2}vecD \\ & =& \bar{q}{^\prime}(G{^\prime}{B}^{-1}G - I_{ k})M_{2}vecD, \\ v_{2}(\bar{q})& =& \bar{q}{^\prime}G{^\prime}{B}^{-1}GM_{ 2}C{^\prime}{B}^{-1}G\bar{q} -\bar{ q}{^\prime}M_{ 2}C{^\prime}{B}^{-1}G\bar{q} \\ & & +\,\frac{1} {2}\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes {B}^{-1}G\bar{q})M_{ 2}vecD \\ & =& \bar{q}{^\prime}(G{^\prime}{B}^{-1}G - I_{ k})M_{2}C{^\prime}{B}^{-1}G\bar{q} + \frac{1} {2}\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes {B}^{-1}G\bar{q})M_{ 2}vecD; \\ 3_{DM}& =& -{N}^{-1/2}vec{^\prime}G_{ N}(\hat{\theta }_{N})M_{3}vecD_{N}(\hat{\theta }_{N}) \\ & =& -{N}^{-1/2}vec{^\prime}GM_{ 3}vecD-{N}^{-1}vec{^\prime}GM_{ 3}C{^\prime}{B}^{-1}G\bar{q} \\ & & -\,{N}^{-1}\bar{q}{^\prime}G{^\prime}{B}^{-1}DM_{ 3}vecD + o_{p}({N}^{-1}) \\ & =& {N}^{-1/2}u_{ 3}(\bar{q}) + {N}^{-1}v_{ 3}(\bar{q}) + o_{p}({N}^{-1}), {}\end{array}$$
(13.34)
and
$$\displaystyle\begin{array}{rcl} u_{3}(\bar{q})& =& -vec{^\prime}GM_{3}vecD, {}\\ v_{3}(\bar{q})& =& -vec{^\prime}GM_{3}C{^\prime}{B}^{-1}G\bar{q} -\bar{ q}{^\prime}G{^\prime}{B}^{-1}DM_{ 3}vecD {}\\ & =& -\bar{q}{^\prime}G{^\prime}{B}^{-1}CM_{ 3}{^\prime}vecG -\bar{ q}{^\prime}G{^\prime}{B}^{-1}DM_{ 3}vecD; {}\\ \end{array}$$
$$\displaystyle\begin{array}{rcl} 4_{DM}& =& {N}^{-1}\frac{1} {4}vec{^\prime}D_{N}(\hat{\theta }_{N})M_{4}vecD_{N}(\hat{\theta }_{N}) \\ & =& {N}^{-1}\frac{1} {4}vec{^\prime}DM_{4}vecD + o_{p}({N}^{-1}) \\ & =& {N}^{-1}v_{ 4}(\bar{q}) + o_{p}({N}^{-1}), {}\end{array}$$
(13.35)
where
$$\displaystyle{ v_{4}(\bar{q}) = \frac{1} {4}vec{^\prime}DM_{4}vecD. }$$
Finally, collecting the terms (13.32)–(13.35) gives Eq. (13.12). □
Proof of Lemma 1: From Theorem 1, if \(u_{i}(\bar{q})\) (i = 1, 2, 3) and \(v_{i}(\bar{q})\) (i = 1, 2, 3, 4) could be rewritten as
$$\displaystyle{ u_{i}(\bar{q}) = vec{^\prime}J_{i}(\bar{q} \otimes \bar{ q} \otimes \bar{ q}), }$$
(13.36)
$$\displaystyle{ v_{i}(\bar{q}) = tr[L_{i}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], }$$
(13.37)
then,
$$\displaystyle\begin{array}{rcl} & & u(\bar{q}) = vec{^\prime}J(\bar{q} \otimes \bar{ q} \otimes \bar{ q}), {}\\ & & v(\bar{q}) = tr[L(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], {}\\ \end{array}$$
where
$$\displaystyle{ vecJ = vecJ_{1} + vecJ_{2} + vecJ_{3}, }$$
and
$$\displaystyle{ L = L_{1} + L_{2} + L_{3} + L_{4}. }$$
Therefore, the proof is reduced to showing (13.36) and (13.37).
Using
$$\displaystyle\begin{array}{rcl} & & (A \otimes C)(B \otimes D) = (AB) \otimes (CD), {}\\ & & K_{p,q}vecA = vec(A{^\prime}), {}\\ & & A \otimes B = K_{p,r}(B \otimes A)K_{s,q}, {}\\ \end{array}$$
for A: p × q and B: r × s where K is the commutation matrix, we can rewrite (13.14):
$$\displaystyle\begin{array}{rcl} u_{1}(\bar{q})& =& 2\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes \mathbb{H}G\bar{q})vec(\bar{q}{^\prime}G{^\prime}\mathbb{H}G) \\ & =& 2\bar{q}{^\prime}G{^\prime}\mathbb{H}G(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})(\bar{q}{^\prime}G{^\prime}{B}^{-1} \otimes I_{ pk})vec(D{^\prime}) \\ & =& 2\bar{q}{^\prime}G{^\prime}\mathbb{H}G(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})(I_{pk} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})vecD \\ & =& 2(\bar{q}{^\prime}G{^\prime}\mathbb{H}G \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})vecD \\ & =& 2(\bar{q}{^\prime} \otimes \bar{ q}{^\prime} \otimes \bar{ q}{^\prime})(G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})vecD \\ & =& vec{^\prime}J_{1}(\bar{q} \otimes \bar{ q} \otimes \bar{ q}), {}\end{array}$$
(13.38)
where
$$\displaystyle{ vecJ_{1} = 2(G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})vecD. }$$
(13.39)
Let
$$\displaystyle{ R_{1} = (\mathbb{H}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1}), }$$
(13.40)
partition vecD as
$$\displaystyle{ \mathop{vecD}\limits_{{p}^{2}k\times 1} = \left [\begin{array}{*{10}c} V _{D1} \\ V _{D2}\\ \vdots \\ V _{Dk} \end{array} \right ] }$$
(13.41)
where each subvector V
Di
is p
2 × 1, and let
$$\displaystyle{ V _{D} = V _{D1}V _{D1}{^\prime} + V _{D2}V _{D2}{^\prime} + \cdots + V _{Dk}V _{Dk}{^\prime}. }$$
(13.42)
Then, since
$$\displaystyle{ \begin{array}{ll} (I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})D{^\prime}{B}^{-1}G\bar{q}& = (I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})(\bar{q}{^\prime}G{^\prime}{B}^{-1} \otimes I_{pk})vec(D{^\prime}) \\ & = (I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})(I_{pk} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})vecD \\ & = (I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})vecD, \end{array} }$$
the first term of \(v_{1}(\bar{q})\) in (13.17) becomes
$$\displaystyle\begin{array}{rcl} & & \bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes \mathbb{H}G\bar{q})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})D{^\prime}{B}^{-1}G\bar{q} \\ & & \quad = vec{^\prime}D(I_{k} \otimes \mathbb{H}G\bar{q} \otimes {B}^{-1}G\bar{q})(I_{ k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}{B}^{-1})vecD \\ & & \quad = vec{^\prime}D(I_{k} \otimes R_{1})vecD \\ & & \quad = \left [\begin{array}{*{10}c} V _{D1}{^\prime} & V _{D2}{^\prime} & \cdots &V _{Dk}{^\prime} \end{array} \right ]\left [\begin{array}{*{10}c} R_{1} & & & 0 \\ &R_{1} & &\\ & & \ddots& \\ 0 & & &R_{1} \end{array} \right ]\left [\begin{array}{*{10}c} V _{D1} \\ V _{D2}\\ \vdots \\ V _{Dk} \end{array} \right ] \\ & & \quad = V _{D1}{^\prime}R_{1}V _{D1} + V _{D2}{^\prime}R_{1}V _{D2} + \cdots + V _{Dk}{^\prime}R_{1}V _{Dk} \\ & & \quad = tr[(V _{D1}V _{D1}{^\prime} + V _{D2}V _{D2}{^\prime} + \cdots + V _{Dk}V _{Dk}{^\prime})R_{1}] \\ & & \quad = tr[V _{D}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})] \\ & & \quad = tr[(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})V _{ D}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})]. {}\end{array}$$
(13.43)
Similarly, let
$$\displaystyle{ R_{2} = (\mathbb{H}G \otimes {B}^{-1}G)(\bar{q} \otimes \bar{ q}), }$$
(13.44)
$$\displaystyle{ R_{3} =\bar{ q}{^\prime}G{^\prime}\mathbb{H}, }$$
(13.45)
partition G′B
−1
C and vecG as
$$\displaystyle{ \mathop{G{^\prime}{B}^{-1}C}\limits_{k\times {p}^{2}k} = \left [\begin{array}{*{10}c} M_{ GC1} & M_{GC2} & \cdots &M_{GCk} \end{array} \right ], }$$
(13.46)
$$\displaystyle{ \mathop{vecG}\limits_{pk\times 1} = \left [\begin{array}{*{10}c} V _{G1} \\ V _{G2}\\ \vdots \\ V _{Gk} \end{array} \right ], }$$
(13.47)
where M
GCi
and V
Gi
are k × p
2 and p × 1 respectively, and let
$$\displaystyle{ M_{V } = M_{GC1}{^\prime} \otimes V _{G1}{^\prime} + M_{GC2}{^\prime} \otimes V _{G2}{^\prime} + \cdots + M_{GCk}{^\prime} \otimes V _{Gk}{^\prime}. }$$
(13.48)
Then, since
$$\displaystyle{ \begin{array}{ll} \bar{q}{^\prime}m\bar{q}{^\prime}M(\bar{q} \otimes \bar{ q})& = m{^\prime}\bar{q}\bar{q}{^\prime}M(\bar{q} \otimes \bar{ q}) \\ & = [(\bar{q} \otimes \bar{ q}){^\prime}M{^\prime} \otimes m{^\prime}]vec(\bar{q}\bar{q}{^\prime}) \\ & = (\bar{q} \otimes \bar{ q}){^\prime}(M{^\prime} \otimes m{^\prime})(\bar{q} \otimes \bar{ q}) \\ & = tr[(M{^\prime} \otimes m{^\prime})(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})] \end{array} }$$
for some vector m and matrix M of appropriate sizes, the second term of \(v_{1}(\bar{q})\) in (13.17) becomes
$$\displaystyle\begin{array}{rcl} & & \bar{q}{^\prime}G{^\prime}{B}^{-1}C(I_{ pk} \otimes {B}^{-1}G\bar{q})(I_{ k} \otimes \mathbb{H}G\bar{q})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecG \\ & & \quad =\bar{ q}{^\prime}G{^\prime}{B}^{-1}C(I_{ k} \otimes R_{2})(I_{k} \otimes R_{3})vecG \\ & & \quad =\bar{ q}{^\prime}\left [\begin{array}{*{10}c} M_{GC1} & M_{GC2} & \cdots &M_{GCk} \end{array} \right ]\left [\begin{array}{*{10}c} R_{2} & & & 0 \\ &R_{2} & &\\ & & \ddots& \\ 0 & & &R_{2} \end{array} \right ]\left [\begin{array}{*{10}c} R_{3} & & & 0 \\ &R_{3} & &\\ & & \ddots& \\ 0 & & &R_{3} \end{array} \right ]\left [\begin{array}{*{10}c} V _{G1} \\ V _{G2}\\ \vdots \\ V _{Gk} \end{array} \right ] \\ & & \quad =\sum _{ i=1}^{k}(\bar{q}{^\prime}M_{ GCi}R_{2}R_{3}V _{Gi}) \\ & & \quad = tr\sum _{i=1}^{k}[\bar{q}{^\prime}M_{ GCi}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q} \otimes \bar{ q})\bar{q}{^\prime}G{^\prime}\mathbb{H}V _{ Gi}] \\ & & \quad = tr\sum _{i=1}^{k}[\bar{q}{^\prime}G{^\prime}\mathbb{H}V _{ Gi}\bar{q}{^\prime}M_{GCi}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q} \otimes \bar{ q})] \\ & & \quad = tr\sum _{i=1}^{k}\left \{\{[(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})M_{ GCi}{^\prime}] \otimes V _{Gi}{^\prime}\mathbb{H}G\}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})\right \} \\ & & \quad = tr\sum _{i=1}^{k}[(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})(M_{ GCi}{^\prime} \otimes V _{Gi}{^\prime})(I_{k} \otimes \mathbb{H}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})] \\ & & \quad = tr[(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})M_{ V }(I_{k} \otimes \mathbb{H}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})]. {}\end{array}$$
(13.49)
From (13.43) and (13.49), (13.17) can be rewritten as
$$\displaystyle{ v_{1}(\bar{q}) = tr[L_{1}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], }$$
(13.50)
where
$$\displaystyle{ L_{1} = (G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})V _{ D}(\mathbb{H}G \otimes {B}^{-1}G) + (G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1})M_{ V }(I_{k} \otimes \mathbb{H}G). }$$
(13.51)
Similar to \(u_{1}(\bar{q})\), \(u_{2}(\bar{q})\) in (13.15) can be rewritten as
$$\displaystyle\begin{array}{rcl} u_{2}(\bar{q})& =& \bar{q}{^\prime}(G{^\prime}{B}^{-1}G - I_{ k})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & =& (\bar{q}{^\prime} \otimes \bar{ q}{^\prime} \otimes \bar{ q}{^\prime})[(G{^\prime}{B}^{-1}G - I_{ k}) \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H}]vecD \\ & =& vec{^\prime}J_{2}(\bar{q} \otimes \bar{ q} \otimes \bar{ q}), {}\end{array}$$
(13.52)
where
$$\displaystyle{ vecJ_{2} = [(G{^\prime}{B}^{-1}G - I_{ k}) \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H}]vecD. }$$
(13.53)
The first term of \(v_{2}(\bar{q})\) in (13.18) can be written as
$$\displaystyle{ \bar{q}{^\prime}G{^\prime}{B}^{-1}C(I_{ k} \otimes \mathbb{H}G\bar{q} \otimes \mathbb{H}G\bar{q})(G{^\prime}{B}^{-1}G - I_{ k})\bar{q}. }$$
Since
$$\displaystyle{ \begin{array}{ll} (G{^\prime}{B}^{-1}G - I_{k})\bar{q}& = vec[\bar{q}{^\prime}(G{^\prime}{B}^{-1}G - I_{k})] \\ & = (I_{k} \otimes \bar{ q}{^\prime})vec(G{^\prime}{B}^{-1}G - I_{k}), \end{array} }$$
and \(vec(G{^\prime}{B}^{-1}G - I_{k})\) can be partitioned as
$$\displaystyle{ vec(G{^\prime}{B}^{-1}G-I_{ k}) = \left [\begin{array}{*{10}c} V _{GI1} \\ V _{GI2}\\ \vdots \\ V _{GIk} \end{array} \right ] }$$
(13.54)
where V
GIi
is k × 1, we may mimic the second term of \(v_{1}(\bar{q})\) and rewrite the first term of \(v_{2}(\bar{q})\) further as
$$\displaystyle{ \begin{array}{ll} tr\sum _{i=1}^{k}&[\bar{q}{^\prime}M_{GCi}(\mathbb{H}G \otimes \mathbb{H}G)(\bar{q} \otimes \bar{ q})\bar{q}{^\prime}V _{GIi}] \\ & = tr[(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})M_{V I}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], \end{array} }$$
(13.55)
where
$$\displaystyle{ M_{V I} = M_{GC1}{^\prime} \otimes V _{GI1}{^\prime} + M_{GC2}{^\prime} \otimes V _{GI2}{^\prime} + \cdots + M_{GCk}{^\prime} \otimes V _{GIk}{^\prime}. }$$
(13.56)
Similar to the first term of \(v_{1}(\bar{q})\), since
$$\displaystyle{ \bar{q}{^\prime}G{^\prime}{B}^{-1}D = vec{^\prime}(\bar{q}{^\prime}G{^\prime}{B}^{-1}D) = vec{^\prime}D(I_{ pk} \otimes {B}^{-1}G\bar{q}), }$$
the second term of \(v_{2}(\bar{q})\) in (13.18) can be rewritten as
$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}vec{^\prime}D(I_{k} \otimes {B}^{-1}G\bar{q} \otimes {B}^{-1}G\bar{q})(I_{ k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & & \quad = \frac{1} {2}tr[V _{D}({B}^{-1}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})] \\ & & \quad = tr[\frac{1} {2}(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}({B}^{-1}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})]. {}\end{array}$$
(13.57)
From (13.55) and (13.57), we have
$$\displaystyle{ v_{2}(\bar{q}) = tr[L_{2}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], }$$
(13.58)
where
$$\displaystyle{ L_{2} = (G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})M_{V I} + \frac{1} {2}(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}({B}^{-1}G \otimes {B}^{-1}G). }$$
(13.59)
Since
$$\displaystyle\begin{array}{rcl} & & vec{^\prime}G(I_{k} \otimes \mathbb{H}G\bar{q}) {}\\ & & \quad = [(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecG]{^\prime} {}\\ & & \quad =\bar{ q}{^\prime}G{^\prime}\mathbb{H}G, {}\\ \end{array}$$
(13.16) becomes
$$\displaystyle\begin{array}{rcl} u_{3}(\bar{q})& =& -\bar{q}{^\prime}G{^\prime}\mathbb{H}G(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & =& -(\bar{q}{^\prime} \otimes \bar{ q}{^\prime} \otimes \bar{ q}{^\prime})(G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})vecD \\ & =& vec{^\prime}J_{3}(\bar{q} \otimes \bar{ q} \otimes \bar{ q}), {}\end{array}$$
(13.60)
where
$$\displaystyle{ vecJ_{3} = -(G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})vecD. }$$
(13.61)
Similar to the second term of \(v_{1}(\bar{q})\), the first term of \(v_{3}(\bar{q})\) in (13.19) can be rewritten as
$$\displaystyle\begin{array}{rcl} & & -\bar{q}{^\prime}G{^\prime}{B}^{-1}C(I_{ k} \otimes \mathbb{H}G\bar{q} \otimes \mathbb{H}G\bar{q})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecG \\ & & \quad = tr\sum _{i=1}^{k}[-\bar{q}{^\prime}M_{ GCi}(\mathbb{H}G \otimes \mathbb{H}G)(\bar{q} \otimes \bar{ q})\bar{q}{^\prime}G{^\prime}\mathbb{H}V _{Gi}] \\ & & \quad = tr[-(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})M_{V }(I_{k} \otimes \mathbb{H}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})].{}\end{array}$$
(13.62)
Similar to the second term of \(v_{2}(\bar{q})\), the second term of \(v_{3}(\bar{q})\) in (13.19) can be rewritten as
$$\displaystyle\begin{array}{rcl} & & -\bar{q}{^\prime}G{^\prime}{B}^{-1}D(I_{ k} \otimes \mathbb{H}G\bar{q})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & & \quad = -vec{^\prime}D(I_{pk} \otimes {B}^{-1}G\bar{q})(I_{ k} \otimes \mathbb{H}G\bar{q})(I_{k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & & \quad = -vec{^\prime}D(I_{k} \otimes \mathbb{H}G\bar{q} \otimes {B}^{-1}G\bar{q})(I_{ k} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H} \otimes \bar{ q}{^\prime}G{^\prime}\mathbb{H})vecD \\ & & \quad = tr[-V _{D}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})] \\ & & \quad = tr[-(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}(\mathbb{H}G \otimes {B}^{-1}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})]. {}\end{array}$$
(13.63)
From (13.62) and (13.63), we have
$$\displaystyle{ v_{3}(\bar{q}) = tr[L_{3}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})], }$$
(13.64)
where
$$\displaystyle{ L_{3} = -(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})M_{V }(I_{k} \otimes \mathbb{H}G) - (G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}(\mathbb{H}G \otimes {B}^{-1}G). }$$
(13.65)
Similar to the first term of \(v_{1}(\bar{q})\), \(v_{4}(\bar{q})\) in (13.20) can be easily rewritten as
$$\displaystyle\begin{array}{rcl} v_{4}(\bar{q})& =& \frac{1} {4}tr[V _{D}(\mathbb{H}G \otimes \mathbb{H}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})] \\ & =& tr[\frac{1} {4}(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}(\mathbb{H}G \otimes \mathbb{H}G)(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q}{^\prime})] \\ & =& tr[L_{4}(\bar{q}\bar{q}{^\prime} \otimes \bar{ q}\bar{q})], {}\end{array}$$
(13.66)
where
$$\displaystyle{ L_{4} = \frac{1} {4}(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H})V _{D}(\mathbb{H}G \otimes \mathbb{H}G). }$$
(13.67)
By using (13.38), (13.50), (13.52), (13.58), (13.60), (13.64) and (13.66), we obtain (13.36) and (13.37), thus finishing the proof. □
Proof of Theorem 2: First, a
i
and b
i
are defined (Phillips and Park 1988) as
$$\displaystyle{ a_{i} = tr(A_{i})\;\;(i = 0,1,2), }$$
(13.68)
where
$$\displaystyle\begin{array}{rcl} & & A_{0} = L[(I + K_{k,k})(\bar{P} \otimes \bar{ P}) + vec\bar{P}vec{^\prime}\bar{P}], {}\\ & & A_{1} = L[(I + K_{k,k})(\bar{P} \otimes P + P \otimes \bar{ P}) + vec\bar{P}vec{^\prime}P + vecPvec{^\prime}\bar{P}], {}\\ & & A_{2} = L[(I + K_{k,k})(P \otimes P) + vecPvec{^\prime}P]; {}\\ \end{array}$$
$$\displaystyle{ b_{i} = vec{^\prime}JB_{i}vecJ\;\;(i = 1,2,3), }$$
(13.69)
where
$$\displaystyle\begin{array}{rcl} B_{0}& =& H(\bar{P} \otimes \bar{ P} \otimes \bar{ P}) + H(\bar{P} \otimes vec\bar{P}vec{^\prime}\bar{P})H {}\\ & & \quad +\bar{ P} \otimes K_{k,k}(\bar{P} \otimes \bar{ P}) + K_{k,k}(\bar{P} \otimes \bar{ P}) \otimes \bar{ P} {}\\ & & \quad + K_{k,{k}^{2}}[\bar{P} \otimes K_{k,k}(\bar{P} \otimes \bar{ P})]K_{{k}^{2},k} = C_{0}(\bar{P}),\;say, {}\\ \end{array}$$
$$\displaystyle\begin{array}{rcl} B_{1}& =& H(P \otimes \bar{ P} \otimes \bar{ P})H {}\\ & & \quad + H(P \otimes vec\bar{P}vec{^\prime}\bar{P} +\bar{ P} \otimes vecPvec{^\prime}\bar{P} +\bar{ P} \otimes vec\bar{P}vec{^\prime}P)H {}\\ & & \quad + P \otimes K_{k,k}(\bar{P} \otimes \bar{ P}) +\bar{ P} \otimes K_{k,k}(P \otimes \bar{ P}) {}\\ & & \quad +\bar{ P} \otimes K_{k,k}(\bar{P} \otimes P) + K_{k,k}(P \otimes \bar{ P}) \otimes \bar{ P} {}\\ & & \quad + K_{k,k}(\bar{P} \otimes \bar{ P}) \otimes \bar{ P} + K_{k,k}(\bar{P} \otimes \bar{ P}) \otimes P {}\\ & & \quad + K_{k,{k}^{2}}\{[P \otimes K_{k,k}(\bar{P} \otimes \bar{ P})] + [\bar{P} \otimes K_{k,k}(P \otimes \bar{ P})] {}\\ & & \quad \quad \quad \quad \quad + [\bar{P} \otimes K_{k,k}(\bar{P} \otimes P)]\}K_{{k}^{2},k} = C_{1}(\bar{P},P),\;say, {}\\ B_{2}& =& C_{1}(P,\bar{P}), {}\\ B_{3}& =& C_{0}(P), {}\\ \end{array}$$
with
$$\displaystyle\begin{array}{rcl} & & H = I + K_{k,{k}^{2}} + K_{{k}^{2},k}, {}\\ & & \bar{P} \equiv I - P. {}\\ \end{array}$$
Secondly, from (13.68),
$$\displaystyle\begin{array}{rcl} a_{0} = tr(A_{0})& =& tr\{L[(I + K_{k,k})(\bar{P} \otimes \bar{ P}) + vec\bar{P}vec{^\prime}\bar{P}]\} \\ & =& tr[(\bar{P} \otimes \bar{ P})L(I + K_{k,k}) + vec{^\prime}\bar{P}Lvec\bar{P}] \\ & =& tr[(\bar{P} \otimes \bar{ P})L(I + K_{k,k})] + tr(vec{^\prime}\bar{P}Lvec\bar{P}).{}\end{array}$$
(13.70)
Using (13.13) and \(\bar{P} \equiv I - P\), we have
$$\displaystyle{ (A{^\prime}{B}^{-1}G)\bar{P} = 0, }$$
(13.71)
$$\displaystyle{ \bar{P}(G{^\prime}{B}^{-1}A) = 0. }$$
(13.72)
Therefore, by (13.21)–(13.25),
$$\displaystyle{ (\bar{P} \otimes \bar{ P})L = 0, }$$
(13.73)
and
$$\displaystyle{ (\mathbb{H}G \otimes {B}^{-1}G)vec\bar{P} = vec({B}^{-1}G\bar{P}G\mathbb{H}) = 0, }$$
(13.74)
$$\displaystyle{ (I_{k} \otimes \mathbb{H}G)vec\bar{P} = vec(\mathbb{H}G\bar{P}) = 0. }$$
(13.75)
Combining (13.74) and (13.75) with (13.22) yields
$$\displaystyle{ L_{1}vec\bar{P} = 0. }$$
(13.76)
Similarly,
$$\displaystyle{ L_{3}vec\bar{P} = 0, }$$
(13.77)
$$\displaystyle{ L_{4}vec\bar{P} = 0, }$$
(13.78)
and
$$\displaystyle{ vec{^\prime}\bar{P}L_{2} = (L_{2}{^\prime}vec\bar{P}){^\prime} = 0. }$$
(13.79)
From (13.76)–(13.79),
$$\displaystyle{ tr(vec{^\prime}\bar{P}Lvec\bar{P}) = 0. }$$
(13.80)
Substituting (13.73) and (13.80) into (13.70) gives
$$\displaystyle{ a_{0} = 0. }$$
(13.81)
Also, from (13.69),
$$\displaystyle\begin{array}{rcl} b_{1}& =& vec{^\prime}JB_{1}vecJ \\ & =& vec{^\prime}JH(P \otimes \bar{ P} \otimes \bar{ P})HvecJ \\ & & +\,vec{^\prime}JH(P \otimes vec\bar{P}vec{^\prime}\bar{P} +\bar{ P} \otimes vecPvec{^\prime}\bar{P}+\bar{P} \otimes vec\bar{P}vec{^\prime}P)HvecJ \\ & & +\,vec{^\prime}J[P \otimes K_{k,k}(\bar{P} \otimes \bar{ P}) +\bar{ P} \otimes K_{k,k}(P \otimes \bar{ P})]vecJ \\ & & +\,vec{^\prime}J[\bar{P} \otimes K_{k,k}(\bar{P} \otimes P) + K_{k,k}(P \otimes \bar{ P}) \otimes \bar{ P}]vecJ \\ & & +\,vec{^\prime}J[K_{k,k}(\bar{P} \otimes \bar{ P}) \otimes \bar{ P} + K_{k,k}(\bar{P} \otimes \bar{ P}) \otimes P]vecJ \\ & & +\,vec{^\prime}JK_{k,{k}^{2}}\{[P \otimes K_{k,k}(\bar{P} \otimes \bar{ P})] + [\bar{P} \otimes K_{k,k}(P \otimes \bar{ P})] \\ & & \quad \quad \quad \quad \quad \quad \quad + [\bar{P} \otimes K_{k,k}(\bar{P} \otimes P)]\}K_{{k}^{2},k}vecJ. {}\end{array}$$
(13.82)
Using
$$\displaystyle\begin{array}{rcl} & & K_{p,q}vecA = vec(A{^\prime}), {}\\ & & A \otimes B = K_{p,r}(B \otimes A)K_{s,q}, {}\\ \end{array}$$
for A: p × q and B: r × s where K is the commutation matrix, the following equations are obtained:
$$\displaystyle{ K_{k,{k}^{2}}vecJ_{1} = 2(G{^\prime}{B}^{-1} \otimes G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H})vec(D{^\prime}), }$$
(13.83)
$$\displaystyle{ K_{k,{k}^{2}}vecJ_{2} = [G{^\prime}\mathbb{H} \otimes (G{^\prime}{B}^{-1}G - I_{ k}) \otimes G{^\prime}\mathbb{H}]vec(D{^\prime}), }$$
(13.84)
$$\displaystyle{ K_{k,{k}^{2}}vecJ_{3} = -(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H}G \otimes G{^\prime}\mathbb{H})vec(D{^\prime}); }$$
(13.85)
$$\displaystyle{ K_{{k}^{2},k}vecJ_{1} = 2(G{^\prime}\mathbb{H} \otimes G{^\prime}{B}^{-1} \otimes G{^\prime}\mathbb{H}G)K_{{ p}^{2},k}vecD, }$$
(13.86)
$$\displaystyle{ K_{{k}^{2},k}vecJ_{2} = [G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H} \otimes (G{^\prime}{B}^{-1}G - I_{ k})]K_{{p}^{2},k}vecD, }$$
(13.87)
$$\displaystyle{ K_{{k}^{2},k}vecJ_{3} = -(G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H} \otimes G{^\prime}\mathbb{H}G)K_{{p}^{2},k}vecD. }$$
(13.88)
Then, substituting (13.83)–(13.88) into (13.82), and using
$$\displaystyle\begin{array}{rcl} & & vec(ABC) = (C{^\prime} \otimes A)vecB, {}\\ & & (A \otimes B){^\prime} = A{^\prime} \otimes B{^\prime}, {}\\ & & (A \otimes C)(B \otimes D) = (AB) \otimes (CD), {}\\ \end{array}$$
together with (13.71) and (13.72) yield
$$\displaystyle{ b_{1} = 0. }$$
(13.89)
Given (13.81) and (13.89), the proof of Theorem 2.4 in Phillips and Park (1988) establishes the conclusion of Theorem 2. □
Appendix 3
13.3.1 Data Description
The earnings data used are drawn from the Panel Study of Income Dynamics (PSID), available at http://psidonline.isr.umich.edu/
The sample consists of men who were heads of household from 1969 to 1974, between the ages of 21 (not inclusive) and 64 (not inclusive), and who reported positive earnings in each year. Individuals with average hourly earnings greater than $100 or reported annual hours greater than 4680 were excluded.
Variables V7492, V7490, V0313, V0794, V7460, V7476, V7491 listed on p.443 of Abowd and Card (1989) are not available now on the PSID website. The variables for sex listed on that page are not consistent with those on the PSID website. The following are the PSID variables used here:
-
ANNUAL EARNINGS: V1196, V1897, V2498, V3051, V3463, V3863;
-
ANNUAL HOURS: V1138, V1839, V2439, V3027, V3423, V3823;
-
SEX: ER32000;
-
AGE: ER30046.
