$\displaystyle\|T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}}_{1}(T^{-\nu_{1}}\mathbf{V}_{1})-T^{-\nu_{1}}\mathbf{F}_{1}^{0\prime}\mathbf{J}_{9}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$
(A.27)

which in turn implies that

$\displaystyle\boldsymbol{\Sigma}_{F_{1}^{0}}\boldsymbol{\Sigma}_{\Gamma_{1}^{0}}(T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}})=(T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}})(T^{-\nu_{1}}\mathbf{V}_{1})+o_{P}(1).$
(A.28)

Note that
$T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}}$ is of
rank $d_{1}$, which then further indicates that $\mathbf{V}_{1}$ has at least
$d_{1}$ non-zero elements on the main diagonal which converge to the
eigenvalues of
$\boldsymbol{\Sigma}_{F_{1}^{0}}\boldsymbol{\Sigma}_{\Gamma_{1}^{0}}$. We now
can conclude that $\mathbf{V}_{1}$ is of rank $d_{1}$ in limit.

We are now ready to investigate $\widehat{\lambda}_{1,d}$ for $d\leq d_{1}$.
Because here $d\leq d_{1}$, we then focus on $\widehat{\mathbf{F}}_{1}^{0}$
and $\mathbf{V}_{1}^{0}$, which are defined in Appendix A.1. Let us write
(A.8) as follows:

$\displaystyle
T^{-(\nu_{1}+\delta)/2}\widehat{\mathbf{F}}_{1}^{0}\mathbf{V}_{1}^{0}-\mathbf{J}_{9}^{0}=(T^{(\nu_{1}-\delta)/2}\widehat{\mathbf{F}}_{1}^{0}-\mathbf{F}_{1}^{0}\mathbf{H}_{1}^{0})(T^{-\nu_{1}}\mathbf{V}_{1}^{0}),$
(A.29)

where $\mathbf{H}_{1}^{0}$ is defined in Appendix A.1. The above expression
implies that (A.8) can be written as

$\displaystyle
T^{(\nu_{1}-\delta)/2}\widehat{\mathbf{F}}_{1}^{0}-\mathbf{F}_{1}^{0}\mathbf{H}_{1}^{0}=\sum_{j=1}^{8}\mathbf{J}_{j}^{0}(T^{-\nu_{1}}\mathbf{V}_{1}^{0})^{-1},$
(A.30)

where $\mathbf{J}_{1}^{0},\ldots,\mathbf{J}_{9}^{0}$ are
$\mathbf{J}_{1},\ldots,\mathbf{J}_{9}$ as defined in (A.6), except that now
$d_{1}$ is taken as known. Note that by the above development

$\displaystyle\sum_{j=1}^{8}T^{-\nu_{1}/2}\|\mathbf{J}_{j}^{0}\|=T^{-(\nu_{1}-\delta)/2}\sum_{j=1}^{8}T^{-\delta/2}\|\mathbf{J}_{j}^{0}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}).$
(A.31)

Moreover, since $T^{-\nu_{1}}\mathbf{V}_{1}^{0}$ converges to a full rank
matrix by the argument under (A.28), we have
$\|(T^{-\nu_{1}}\mathbf{V}_{1}^{0})^{-1}\|=O_{P}(1)$, which in turn implies

$\displaystyle
T^{-\nu_{1}/2}\|T^{(\nu_{1}-\delta)/2}\widehat{\mathbf{F}}_{1}^{0}-\mathbf{F}_{1}^{0}\mathbf{H}_{1}^{0}\|\leq\sum_{j=1}^{8}T^{-\nu_{1}/2}\|\mathbf{J}_{j}^{0}\|\|(T^{-\nu_{1}}\mathbf{V}_{1}^{0})^{-1}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}).$
(A.32)

This is an important result and in what follows we will use it frequently.

Let us now consider $T^{-\nu_{1}}(\widehat{\lambda}_{1,d}-\lambda_{1,d})$. By
the definitions of $\lambda_{1,d}$ and $\widehat{\lambda}_{1,d}$ given in
Section A.1,

$\displaystyle T^{-\nu_{1}}(\widehat{\lambda}_{1,d}-\lambda_{1,d})$
$\displaystyle=(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}+T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})^{\prime}T^{-\nu_{1}}(\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}+\boldsymbol{\Sigma}_{1}^{0})$
$\displaystyle\times(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}+T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})-T^{-2\nu_{1}}\mathbf{h}_{1,d}^{0\prime}\mathbf{F}_{1}^{0\prime}\boldsymbol{\Sigma}_{1}^{0}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}$
$\displaystyle=(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})^{\prime}T^{-\nu_{1}}(\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0})(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})$
$\displaystyle+2(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})^{\prime}T^{-\nu_{1}}(\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0})T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}$
$\displaystyle+(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})^{\prime}T^{-\nu_{1}}\boldsymbol{\Sigma}_{1}^{0}(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})$
$\displaystyle+2(T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})^{\prime}T^{-3\nu_{1}/3}\boldsymbol{\Sigma}_{1}^{0}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}$
$\displaystyle+T^{-2\nu_{1}}\mathbf{h}_{1,d}^{0\prime}\mathbf{F}_{1}^{0\prime}(\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0})\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}$
$\displaystyle=J_{1}+2J_{2}+J_{3}+2J_{4}+J_{5},$ (A.33)

with obvious definitions of $J_{1},\ldots,J_{5}$. From (A.32),

$\displaystyle
T^{-\nu_{1}/2}\|T^{(\nu_{1}-\delta)/2}\widehat{\mathbf{F}}_{1,d}^{0}-\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}\|=\|T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$
(A.34)

which is $o_{P}(1)$ under Assumption 1. This implies $|J_{1}|=o_{P}(|J_{5}|)$,
$|J_{2}|=o_{P}(|J_{5}|)$ and $|J_{3}|=o_{P}(|J_{4}|)$. It remains to consider
$J_{4}$ and $J_{5}$. The order of the first of these terms is given by

$\displaystyle|J_{4}|$
$\displaystyle\leq\|T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}\|T^{-3\nu_{1}/2}\|\boldsymbol{\Sigma}_{1}^{0}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}\|$
$\displaystyle\leq\|T^{-\delta/2}\widehat{\mathbf{F}}_{1,d}^{0}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}\|T^{-\nu_{1}/2}\|\mathbf{F}_{1}^{0}\|\|(N^{-1}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0})\|T^{-\nu_{1}}\|\mathbf{F}_{1}^{0\prime}\mathbf{F}_{1}^{0}\|\|\mathbf{h}_{1,d}^{0}\|$
$\displaystyle=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$ (A.35)

where we have made use of the fact that $\|\mathbf{H}_{g}^{0}\|=O_{P}(1)$ (see
Appendix A.1), which implies that $\|\mathbf{h}_{1,d}^{0}\|$ is of the same
order.

The order of $J_{5}$ is the same as that of $J_{4}$. In order to appreciate
this, we begin by noting

$\displaystyle
T^{-\nu_{1}}(\widehat{\mathbf{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0})$
$\displaystyle=\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}$
$\displaystyle+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\gamma}_{1,i}^{0\prime}\mathbf{F}_{1}^{0\prime}+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}$
$\displaystyle+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})^{\prime}$
$\displaystyle+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}$
$\displaystyle+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})^{\prime}$
$\displaystyle+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})^{\prime}+\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})\boldsymbol{\gamma}_{1,i}^{0\prime}\mathbf{F}_{1}^{0\prime}.$
(A.36)

By the proof for each term of (A.6), it is easy to know that

$\displaystyle
T^{-\nu_{1}}\|\widehat{\mathbf{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$
(A.37)

and so

$\displaystyle|J_{5}|\leq\|\mathbf{h}_{1,d}^{0}\|^{2}T^{-\nu_{1}}\|\mathbf{F}_{1}^{0}\|^{2}T^{-\nu_{1}}\|\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}).$
(A.38)

Hence, by putting everything together,

$\displaystyle
T^{-\nu_{1}}|\widehat{\lambda}_{1,d}-\lambda_{1,d}|\leq|J_{1}|+2|J_{2}|+|J_{3}|+2|J_{4}|+|J_{5}|=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$
(A.39)

which establishes (a).

Consider (b). This proof is based on Lemma A.1. We therefore start by
introducing some notation in order to make the problem here fit the one in
Lemma A.1. Let us therefore denote by $\mathbf{F}_{1}^{\perp}$ a
$T\times(d_{max}-d_{1})$ matrix such that
$T^{-\nu_{1}}(\mathbf{F}_{1}^{\perp},\mathbf{F}_{1}^{0}\mathbf{R})^{\prime}(\mathbf{F}_{1}^{\perp},\mathbf{F}_{1}^{0}\mathbf{R})=\mathrm{diag}(\mathbf{I}_{d_{max}-d_{1}},\mathbf{I}_{d_{1}})$,
where $\mathbf{R}$ is a $d_{1}\times d_{1}$ rotation matrix. The matrices
$T^{-\nu_{1}/2}\mathbf{F}_{1}^{\perp}$,
$T^{-\nu_{1}/2}\mathbf{F}_{1}^{0}\mathbf{R}$, $\boldsymbol{\Sigma}_{1}^{0}$
and $\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}$ correspond
to $\mathbf{Q}_{1}$, $\mathbf{Q}_{2}$, $\mathbf{A}$ and $\mathbf{E}$ of Lemma
A.1. Our counterpart of the matrix $\mathbf{Q}^{0}_{1}$ appearing in this
other lemma is thus given by

$\displaystyle\widehat{\mathbf{F}}^{\perp}=T^{-\nu_{1}/2}(\mathbf{F}_{1}^{\perp}+\mathbf{F}_{1}^{0}\mathbf{R}\mathbf{P})(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2},$
(A.40)

where

$\displaystyle\|\mathbf{P}\|_{2}$
$\displaystyle\leq\frac{4}{\mathrm{sep}(0,T^{-2\nu_{1}}\mathbf{F}_{1}^{0\prime}\boldsymbol{\Sigma}_{1}^{0}\mathbf{F}_{1}^{0})}T^{-\nu_{1}}\|\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|\leq
O_{P}(1)T^{-\nu_{1}}\|\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|$
$\displaystyle=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}).$ (A.41)

Since $\widehat{\mathbf{F}}^{\perp}$ is an orthonormal basis for a subspace
that is invariant for $\widehat{\boldsymbol{\Sigma}}_{1}$, we have
$\widehat{\lambda}_{1,d_{1}+d}=\widehat{\mathbf{F}}_{d}^{\perp\prime}\widehat{\boldsymbol{\Sigma}}_{1}\widehat{\mathbf{F}}_{d}^{\perp}$,
where $d=1,\ldots,d_{max}-d_{1}$ and $\widehat{\mathbf{F}}_{d}^{\perp}$ is the
$d$-th column of $\widehat{\mathbf{F}}^{\perp}$. Consider
$\|\widehat{\mathbf{F}}^{\perp}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{\perp}\|_{2}$.
By the definition of $\widehat{\mathbf{F}}^{\perp}$,

$\displaystyle\|\widehat{\mathbf{F}}^{\perp}-T^{-\nu_{1}/2}\mathbf{F}_{1}^{\perp}\|_{2}$
$\displaystyle=T^{-\nu_{1}/2}\|[\mathbf{F}_{1}^{\perp}+\mathbf{F}_{1}^{0}\mathbf{R}\mathbf{P}-\mathbf{F}_{1}^{\perp}(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{1/2}](\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2}\|_{2}$
$\displaystyle\leq
T^{-\nu_{1}/2}\|\mathbf{F}_{1}^{\perp}(\mathbf{I}_{d_{max}-d_{1}}-(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{1/2})(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2}\|_{2}$
$\displaystyle+T^{-\nu_{1}/2}\|\mathbf{F}_{1}^{0}\mathbf{R}\mathbf{P}(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2}\|_{2}$
$\displaystyle\leq\|(\mathbf{I}_{d_{max}-d_{1}}-(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{1/2})(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2}\|_{2}+\|\mathbf{P}(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{-1/2}\|_{2}$
$\displaystyle\leq\|\mathbf{I}_{d_{max}-d_{1}}-(\mathbf{I}_{d_{max}-d_{1}}+\mathbf{P}^{\prime}\mathbf{P})^{1/2}\|_{2}+\|\mathbf{P}\|_{2}\leq
2\|\mathbf{P}\|_{2}$ $\displaystyle=O_{P}(T^{-(\nu_{1}-\nu_{2})/2}),$ (A.42)

where the second and third inequalities follow from (Magnus and Neudecker,
2007, Exercise 1 on page 231). This last result can be used to show that

$\displaystyle
T^{-\nu_{1}}|\widehat{\lambda}_{1,d_{1}+d}|=|\widehat{\mathbf{F}}_{d}^{\perp\prime}(T^{-\nu_{1}}\widehat{\boldsymbol{\Sigma}}_{1})\widehat{\mathbf{F}}^{\perp}_{d}|$
$\displaystyle=|(\widehat{\mathbf{F}}^{\perp}_{d}-T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp}+T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp})^{\prime}T^{-\nu_{1}}(\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}+\boldsymbol{\Sigma}_{1}^{0})(\widehat{\mathbf{F}}^{\perp}_{d}-T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp}+T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp})|$
$\displaystyle\leq\|\widehat{\mathbf{F}}^{\perp}_{d}-T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp}\|^{2}T^{-\nu_{1}}\|\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|$
$\displaystyle+2\|\widehat{\mathbf{F}}^{\perp}_{d}-T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp}\|T^{-\nu_{1}}\|\widehat{\boldsymbol{\Sigma}}_{1}-\boldsymbol{\Sigma}_{1}^{0}\|T^{-\nu_{1}/2}\|\mathbf{F}_{1,d}^{\perp}\|+\|\widehat{\mathbf{F}}^{\perp}_{d}-T^{-\nu_{1}/2}\mathbf{F}_{1,d}^{\perp}\|^{2}T^{-\nu_{1}}\|\boldsymbol{\Sigma}_{1}^{0}\|$
$\displaystyle=O_{P}(T^{-(\nu_{1}-\nu_{2})}),$ (A.43)

where $\mathbf{F}_{1,d}^{\perp}$ is the $d$-th column of
$\mathbf{F}_{1}^{\perp}$, and the last equality follows from (A.6) and the
proof of part (a). This completes the proof of the lemma.$\blacksquare$

Proof of Lemma A.4.

For (a), we take the same starting point as in the proof of part (a) in Lemma
A.3, which is (A.8) with $\widehat{\mathbf{F}}_{1}$ and $\mathbf{V}_{1}$ based
on treating $d_{1}$ as known. The rationale for doing so is, as already
explained in Appendix A.4, that $\widehat{d}_{1}$ is consistent. Pre-
multiplying this equation through by
$T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}$ gives

$\displaystyle
T^{-(\nu_{1}+\delta)}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}\mathbf{V}_{1}-T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{9}$
$\displaystyle=T^{-(\nu_{1}+\delta)}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}\mathbf{V}_{1}-T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{F}_{1}^{0}(N^{-1}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0})(T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}}_{1})$
$\displaystyle=\sum_{j=1}^{8}T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{j},$
(A.44)

or

$\displaystyle
T^{-\delta}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}-T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{9}(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}$
$\displaystyle=T^{-\delta}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}-T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{F}_{1}^{0}(N^{-1}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0})(T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}}_{1})(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}$
$\displaystyle=T^{-(\nu_{1}-\nu_{2})/2}\sum_{j=1}^{8}T^{-(\delta+\nu_{2})/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{j}(T^{-\nu_{1}}\mathbf{V}_{1})^{-1},$
(A.45)

Under Assumption 5, the orders of $\mathbf{J}_{1},\ldots,\mathbf{J}_{6}$ are
the same as in those in the proof of the first result of Lemma A.3. The stated
orders of $\mathbf{J}_{7}$ and $\mathbf{J}_{8}$ are, however, not sharp and
can be improved upon. The order of
$T^{-(\delta+\nu_{2})/2}\|\mathbf{F}_{2}^{0\prime}\mathbf{J}_{7}\|$ is given
by

$\displaystyle
T^{-(\delta+\nu_{2})/2}\|\mathbf{F}_{2}^{0\prime}\mathbf{J}_{7}\|$
$\displaystyle\leq
O_{P}(1)T^{-\nu_{2}/2}\left\|\frac{1}{NT^{(\nu_{1}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0\prime}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}(\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}+\boldsymbol{\varepsilon}_{i})^{\prime}\right\|$
$\displaystyle\leq
O_{P}(1)\frac{1}{NT^{(\nu_{2}+\nu_{1}+\delta)/2}}\|\mathbf{F}_{2}^{0\prime}\mathbf{F}_{1}^{0}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{+1}^{0}\mathbf{F}_{+1}^{0\prime}\|$
$\displaystyle+O_{P}(1)\frac{1}{NT^{(\nu_{2}+\nu_{1}+\delta)/2}}\|\mathbf{F}_{2}^{0\prime}\mathbf{F}_{1}^{0}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\varepsilon}\|$
$\displaystyle=O_{P}(N^{-(1-p)}T^{q-(\nu_{1}+\delta)/2})+O_{P}(N^{-1/2}T^{q-(\nu_{2}+\nu_{1}+\delta-1)/2}),$
(A.46)

where the last equality follows from Assumption 5 and Lemma A.2. We can
similarly show that

$\displaystyle
T^{-(\delta+\nu_{2})/2}\|\mathbf{F}_{2}^{0\prime}\mathbf{J}_{8}\|$
$\displaystyle\leq
O_{P}(1)T^{-\nu_{2}/2}\left\|\frac{1}{NT^{(\nu_{1}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0\prime}(\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}+\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{+2,i}^{0}+\boldsymbol{\varepsilon}_{i})\boldsymbol{\gamma}_{1,i}^{0\prime}\mathbf{F}_{1}^{0\prime}\right\|$
$\displaystyle\leq
O_{P}(1)\frac{1}{NT^{(\nu_{2}+\nu_{1}+\delta)/2}}\|\mathbf{F}_{2}^{0\prime}\mathbf{F}_{2}^{0}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\Gamma}_{1}^{0}\mathbf{F}_{1}^{0\prime}\|$
$\displaystyle+O_{P}(1)N^{-1}T^{-(\nu_{2}+\nu_{1}+\delta)/2}\|\mathbf{F}_{2}^{0\prime}\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{1}^{0}\mathbf{F}_{1}^{0\prime}\|$
$\displaystyle+O_{P}(1)N^{-1}T^{-(\nu_{2}+\nu_{1}+\delta)/2}\|\mathbf{F}_{2}^{0\prime}\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{1}^{0}\mathbf{F}_{1}^{0\prime}\|$
$\displaystyle=O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{2})/2})+O_{P}(N^{-1/2}T^{-(\delta-1)/2}),$
(A.47)

where $\mathbf{F}_{+2}^{0}$ and $\boldsymbol{\Gamma}_{+2}^{0}$ are defined
analogously to $\mathbf{F}_{+1}^{0}$ and $\boldsymbol{\Gamma}_{+1}^{0}$ in the
proof of Lemma A.3. By using these last two results together with the orders
of $\mathbf{J}_{1},\ldots,\mathbf{J}_{6}$ given in the proof of the first
result of Lemma A.3,

$\displaystyle\left\|\sum_{j=1}^{8}T^{-(\delta+\nu_{2})/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{j}\right\|$
$\displaystyle\leq
O_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|)+O_{P}(T^{\nu_{2}-(\nu_{1}+\delta)/2})+O_{P}(N^{-1/2}T^{1-(\nu_{1}+\delta)/2})$
$\displaystyle+O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{2})/2})+O_{P}(N^{-1/2}T^{-(\delta-1)/2})$
$\displaystyle=O_{P}(T^{-\delta/2}+N^{-1/2}T^{-(\delta-1)/2}+N^{-(1-p)}T^{-(\delta-\nu_{2})/2}),$
(A.48)

where we have used
$\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|=O_{P}(N^{-1/2}\vee
T^{-1/2})$ of Lemma 3.1, and Assumptions 3 and 5. Hence,

$\displaystyle\|T^{-\delta}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}-T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{9}(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}\|$
$\displaystyle\leq
T^{-(\nu_{1}-\nu_{2})/2}\left\|\sum_{j=1}^{8}T^{-(\delta+\nu_{2})/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{j}\right\|\|(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}\|$
$\displaystyle=T^{-(\nu_{1}-\nu_{2})/2}[O_{P}(T^{-\delta/2}+N^{-1/2}T^{-(\delta-1)/2}+N^{-(1-p)}T^{-(\delta-\nu_{2})/2})]$
$\displaystyle=O_{P}(T^{-(\delta+\nu_{1}-\nu_{2})/2}+N^{-1/2}T^{-(\delta+\nu_{1}-\nu_{2}-1)/2}+N^{-(1-p)}T^{-(\delta+\nu_{1}-2\nu_{2})/2}),$
(A.49)

which together with Assumption 5 yields

$\displaystyle\|T^{-\delta}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{1}\|$
$\displaystyle\leq\|T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{9}\|\|(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}\|$
$\displaystyle+T^{-(\nu_{1}-\nu_{2})/2}\left\|\sum_{j=1}^{8}T^{-(\delta+\nu_{2})/2}\mathbf{F}_{2}^{0\prime}\mathbf{J}_{j}\right\|\|(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}\|$
$\displaystyle=O_{P}(T^{q-(\nu_{1}+\delta)/2}+T^{-(\delta+\nu_{1}-\nu_{2})/2}+N^{-1/2}T^{-(\delta+\nu_{1}-\nu_{2}-1)/2}$
$\displaystyle+N^{-(1-p)}T^{-(\delta+\nu_{1}-2\nu_{2})/2}),$ (A.50)

as was to be shown for (a).

Let us now consider (b). Analogously to the proof of (a), by invoking
Assumption 5 we can improve the orders of $\mathbf{J}_{7}$ and
$\mathbf{J}_{8}$. For $\mathbf{J}_{7}$,

$\displaystyle T^{-\delta/2}\|\mathbf{J}_{7}\|$ $\displaystyle\leq
O_{P}(1)N^{-1}T^{-(\nu_{1}+\delta)/2}\|\mathbf{F}_{1}^{0}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{+1}^{0}\mathbf{F}_{+1}^{0\prime}\|+O_{P}(1)N^{-1}T^{-(\nu_{1}+\delta)/2}\|\mathbf{F}_{1}^{0}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\varepsilon}\|$
$\displaystyle=O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{2})/2})+O_{P}(N^{-1/2}T^{-(\delta-1)/2}),$
(A.51)

where the equality follows from Assumption 5 and Lemma A.2. For
$\mathbf{J}_{8}$,

$\displaystyle T^{-\delta/2}\|\mathbf{J}_{8}\|\leq
O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{2})/2})+O_{P}(N^{-1/2}T^{-(\delta-1)/2}).$
(A.52)

This implies that the result in (A.32) changes to (after replacing
$T^{-\nu_{1}/2}$ by $T^{-\delta/2}$)

$\displaystyle
T^{-\delta/2}\|T^{(\nu_{1}-\delta)/2}\widehat{\mathbf{F}}_{1}-\mathbf{F}_{1}^{0}\mathbf{H}_{1}^{0}\|\leq\sum_{j=1}^{8}T^{-\delta/2}\|\mathbf{J}_{j}\|\|(T^{-\nu_{1}}\mathbf{V}_{1})^{-1}\|$
$\displaystyle=O_{P}(T^{-\delta/2}+N^{-1/2}T^{-(\delta-1)/2}+N^{-(1-p)}T^{-(\delta-\nu_{2})/2}).$
(A.53)

Hence, since $\mathbf{H}_{1}^{0}$ is invertible with
$\|\mathbf{H}_{1}^{0}\|=O_{P}(1)$ and
$\mathbf{H}_{1}=T^{-(\nu_{1}-\delta)/2}\mathbf{H}_{1}^{0}$,

$\displaystyle
T^{-\delta/2}\|\widehat{\mathbf{F}}_{1}\mathbf{H}_{1}^{-1}-\mathbf{F}_{1}^{0}\|=O_{P}(T^{-\delta/2}+N^{-1/2}T^{-(\delta-1)/2}+N^{-(1-p)}T^{-(\delta-\nu_{2})/2}).$
(A.54)

We are now ready to consider
$\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0\prime}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}\|^{2}$.

$\displaystyle\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0\prime}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}\|^{2}=\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-T^{-\delta}\widehat{\mathbf{F}}_{1}\widehat{\mathbf{F}}_{1}^{\prime}(\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0})\|^{2}$
$\displaystyle\leq
O(1)\sum_{i=1}^{N}[\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}+\|\mathbf{M}_{\widehat{F}_{1}}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}\|^{2}+\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}\|^{2}+\|\mathbf{P}_{\widehat{F}_{1}}\boldsymbol{\varepsilon}_{i}\|^{2}].$
(A.55)

We now evaluate each of the terms on the right-hand side one by one. Making
use of Assumption 1 and Lemma 3.1, we get

$\displaystyle\sum_{i=1}^{N}\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}$
$\displaystyle\leq\sum_{i=1}^{N}\|\mathbf{X}_{i}\mathbf{D}_{T}\|^{2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}$
$\displaystyle=O_{P}(NT)O_{P}(N^{-1}\vee T^{-1})=O_{P}(N\vee T),$ (A.56)

and by another application of Lemma A.2,

$\displaystyle\sum_{i=1}^{N}\|\mathbf{P}_{\widehat{F}_{1}}\boldsymbol{\varepsilon}_{i}\|^{2}\leq\sum_{i=1}^{N}\|\boldsymbol{\varepsilon}_{i}\|^{2}=O_{P}(N\vee
T).$ (A.57)

For
$\sum_{i=1}^{N}\|\mathbf{M}_{\widehat{F}_{1}}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}\|^{2}$,
we use (A.54) from which it follows that

$\displaystyle\|\mathbf{M}_{\widehat{F}_{1}}\mathbf{F}_{1}^{0}\|^{2}$
$\displaystyle=\|\mathbf{M}_{\widehat{F}_{1}}(\mathbf{F}_{1}^{0}-\widehat{\mathbf{F}}_{1}\mathbf{H}_{1}^{-1})\|^{2}\leq
T^{\delta}(T^{-\delta}\|\mathbf{F}_{1}^{0}-\widehat{\mathbf{F}}_{1}\mathbf{H}_{1}^{-1}\|^{2})$
$\displaystyle=O_{P}(1)+O_{P}(N^{-1}T)+O_{P}(N^{-2(1-p)}T^{\nu_{2}}),$ (A.58)

which in turn implies

$\displaystyle\sum_{i=1}^{N}\|\mathbf{M}_{\widehat{F}_{1}}\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}\|^{2}=O_{P}(N)+O_{P}(T)+O_{P}(N^{-(1-2p)}T^{\nu_{2}}).$
(A.59)

For
$\sum_{i=1}^{N}\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{F}_{+1}^{0}\boldsymbol{\gamma}_{+1,i}^{0}\|^{2}$,
we use the result given in part (a), giving

$\displaystyle\sum_{i=1}^{N}\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\|^{2}$
$\displaystyle=\sum_{i=1}^{N}\|T^{-\delta}\widehat{\mathbf{F}}_{1}\widehat{\mathbf{F}}_{1}^{\prime}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\|^{2}=O_{P}(NT^{\delta})\|T^{-\delta}\widehat{\mathbf{F}}_{1}^{\prime}\mathbf{F}_{2}^{0}\|^{2}$
$\displaystyle=O_{P}(NT^{\delta})[O_{P}(T^{q-(\nu_{1}+\delta)/2})+O_{P}(T^{-(\delta+\nu_{1}-\nu_{2})/2})$
$\displaystyle+O_{P}(N^{-1/2}T^{-(\delta+\nu_{1}-\nu_{2}-1)/2})+O_{P}(N^{-(1-p)}T^{-(\delta+\nu_{1}-2\nu_{2})/2})]^{2}$
$\displaystyle=O_{P}(N)[O_{P}(T^{-(\nu_{1}-\nu_{2})/2})+O_{P}(N^{-1/2}T^{-(\nu_{1}-\nu_{2}-1)/2})$
$\displaystyle+O_{P}(N^{-(1-p)}T^{-(\nu_{1}-2\nu_{2})/2})+O_{P}(T^{q-\nu_{1}/2})]^{2}.$
(A.60)

The order of
$\sum_{i=1}^{N}\|\mathbf{P}_{\widehat{F}_{1}}\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{+2,i}^{0}\|^{2}$
is the same. Hence, by adding the above results, (b) follows after simple
algebra. The proof is now complete.$\blacksquare$

Proof of Lemma A.5.

Let
$\mathbf{U}_{i}=\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{+2,i}^{0}+\mathbf{F}_{1}^{0\prime}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}$.
In this notation,

$\displaystyle T^{-(\nu_{2}+\delta)/2}\widehat{\mathbf{F}}_{2}\mathbf{V}_{2}$
$\displaystyle=\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}[(\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}+\mathbf{U}_{i}+\boldsymbol{\varepsilon}_{i})$
$\displaystyle\times(\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}+\mathbf{U}_{i}+\boldsymbol{\varepsilon}_{i})^{\prime}]\widehat{\mathbf{F}}_{2}$
$\displaystyle=\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{U}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{U}_{i}\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{U}_{i}\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{U}_{i}\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}+\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2}$
$\displaystyle=\sum_{j=1}^{16}\mathbf{K}_{j},$ (A.61)

where $\mathbf{K}_{1},\ldots,\mathbf{K}_{16}$ are implicitly defined.
Analogously to the proof of the first result of Lemma A.3 we move
$\mathbf{K}_{16}=\mathbf{F}_{2}^{0}(N^{-1}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\Gamma}_{2}^{0})(T^{-(\nu_{2}+\delta)/2}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2})$
over to the left, giving

$\displaystyle
T^{-(\nu_{2}+\delta)/2}\widehat{\mathbf{F}}_{2}\mathbf{V}_{2}-\mathbf{F}_{2}^{0}(N^{-1}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\Gamma}_{2}^{0})(T^{-(\nu_{2}+\delta)/2}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2})=\sum_{j=1}^{15}\mathbf{K}_{j}.$
(A.62)

By using the same steps employed in the proof of the first result of Lemma
A.3, we can show that

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{1}+\mathbf{K}_{2}+\mathbf{K}_{3}\|=O_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.63)

For $\mathbf{K}_{4}$,

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{4}\|=T^{-\delta/2}\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}\right\|$
$\displaystyle\leq
O(1)\left(\frac{1}{NT^{\delta}}\sum_{i=1}^{N}\|\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}\right)^{1/2}$
$\displaystyle\times\left(\frac{1}{NT^{\nu_{2}}}\sum_{i=1}^{N}\|\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\|^{2}+\frac{1}{NT^{\nu_{2}}}\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0\prime}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}\|^{2}\right)^{1/2}$
$\displaystyle=O_{P}(1)T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|[O_{P}(T^{-(\nu_{2}-\nu_{3})})+O_{P}(T^{-\nu_{2}})+O_{P}(N^{-1}T^{-(\nu_{2}-1)})$
$\displaystyle+O_{P}(T^{-(\nu_{1}-\nu_{2})})+O_{P}(T^{-(\nu_{2}+\nu_{1}-2q)})+O_{P}(N^{-1}T^{-(\nu_{1}+\nu_{2}-2)})+O_{P}(N^{-2(1-p)})]$
$\displaystyle=o_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|),$
(A.64)

where the second equality follows Lemma A.4, and the third follows from
Assumptions 1, 3, and 5. The same arguments can be used to show that

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{5}\|=o_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.65)

For $\mathbf{K}_{6}$,

$\displaystyle T^{-\delta/2}\|\mathbf{K}_{6}\|$ $\displaystyle\leq
T^{-\delta/2}\|\widehat{\mathbf{F}}_{2}\|\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{X}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{i}^{\prime}\right\|$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\delta+\nu_{2}-2)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|)$
$\displaystyle=o_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|),$
(A.66)

where the development is similar to (A.6). The order of
$T^{-\delta/2}\|\mathbf{K}_{7}\|$ is the same.

For $\mathbf{K}_{8}$,

$\displaystyle T^{-\delta/2}\|\mathbf{K}_{8}\|$
$\displaystyle=T^{-\delta/2}\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}\right\|\leq
T^{-\delta/2}\|\widehat{\mathbf{F}}_{2}\|N^{-1}T^{-(\nu_{2}+\delta)/2}\|\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\varepsilon}\|$
$\displaystyle=O_{P}(T^{1-(\nu_{2}+\delta)/2}(N^{-1/2}\vee T^{-1/2})),$ (A.67)

where the last equality holds by Lemma A.2.

Further use of Lemma A.2 gives

$\displaystyle T^{-\delta/2}\|\mathbf{K}_{9}\|$
$\displaystyle=T^{-\delta/2}\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\boldsymbol{\varepsilon}_{i}^{\prime}\widehat{\mathbf{F}}_{2}\right\|\leq
N^{-1}T^{-(\nu_{2}+\delta)/2}\|\mathbf{F}_{2}^{0}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\varepsilon}\|T^{-\delta/2}\|\widehat{\mathbf{F}}_{2}\|$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\delta-1)/2}),$ (A.68)

and we can show that $T^{-\delta/2}\|\mathbf{K}_{10}\|$ is of the same order.

$\mathbf{K}_{11}$ requires more work. We begin by expanding it in the
following way:

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{11}\|=T^{-\delta/2}\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}(\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}+\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{2,i}^{0})^{\prime}\widehat{\mathbf{F}}_{2}\right\|$
$\displaystyle\leq\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}(\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}T^{-\delta}\widehat{\mathbf{F}}_{1}^{\prime}(\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{2,i}^{0})^{\prime}\right\|T^{-\delta/2}\|\widehat{\mathbf{F}}_{2}\|$
$\displaystyle\leq
O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{P}_{\widehat{F}_{1}}\right\|$
$\displaystyle+O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\gamma}_{1,i}^{0\prime}\mathbf{F}_{1}^{0\prime}\mathbf{M}_{\widehat{F}_{1}}\right\|+O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\gamma}_{+1,i}^{0\prime}\mathbf{F}_{+1}^{0\prime}\mathbf{P}_{\widehat{F}_{1}}\right\|$
$\displaystyle+O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\varepsilon}_{i}^{\prime}\mathbf{P}_{\widehat{F}_{1}}\right\|+O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}\boldsymbol{\gamma}_{+2,i}^{0\prime}\mathbf{F}_{+2}^{0\prime}\right\|$
$\displaystyle=O(1)(K_{111}+K_{112}+K_{113}+K_{114}+K_{115}),$ (A.69)

where, similarly to the analysis of $\mathbf{K}_{6}$ and using
$\|\mathbf{P}_{\widehat{F}_{1}}\|_{2}=1$,

$\displaystyle K_{111}$
$\displaystyle\leq\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\right\|=O_{P}(N^{-1/2}T^{-(\delta+\nu_{2}-2)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|)$
$\displaystyle=o_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.70)

Also, making use of (A.6), we can show that

$\displaystyle K_{112}$ $\displaystyle\leq
N^{-1}T^{-(\nu_{2}+\delta)/2}\|\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\varepsilon}\|\|\mathbf{M}_{\widehat{F}_{1}}\mathbf{F}_{1}^{0}\|$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\nu_{2}+\delta-1)/2})+O_{P}(N^{-1}T^{1-(\nu_{2}+\delta)/2})+O_{P}(N^{-(3/2-p)}T^{-(\delta-1)/2}).$
(A.71)

Similarly, for $K_{113}$, we can show that

$\displaystyle
K_{113}=o_{P}(K_{112})+O_{P}(N^{-1/2}T^{q-(\nu_{1}+\nu_{2}+\delta-1)/2}),$
(A.72)

where we have used Lemmas A.2 and A.4, and Assumption 1. Further use of Lemma
A.2 shows that $K_{114}$ is of the following order:

$\displaystyle K_{114}\leq
N^{-1}T^{-(\nu_{2}+\delta)/2}\|\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\varepsilon}\|=O_{P}(T^{-(\nu_{2}+\delta-2)/2}(N^{-1/2}\vee
T^{-1/2})),$ (A.73)

while the order of $K_{115}$ is

$\displaystyle
K_{115}=N^{-1}T^{-(\nu_{2}+\delta)/2}\|\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{+2}^{0}\mathbf{F}_{+2}^{0\prime}\|=O_{P}(N^{-1/2}T^{-(\nu_{2}+\delta-1-\nu_{3})/2}).$
(A.74)

By inserting the above results into (A.6), we obtain

$\displaystyle T^{-\delta/2}\|\mathbf{K}_{11}\|$
$\displaystyle=O_{P}(N^{-(3/2-p)}T^{-(\delta-1)/2})+O_{P}(N^{-1/2}T^{q-(\nu_{1}+\nu_{2}+\delta-1)/2})$
$\displaystyle+O_{P}(T^{-(\nu_{2}+\delta-2)/2}(N^{-1/2}\vee
T^{-1/2}))+O_{P}(N^{-1/2}T^{-(\nu_{2}+\delta-1-\nu_{3})/2})$
$\displaystyle+o_{P}(T^{-(\delta-1)/2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.75)

The order of $T^{-\delta/2}\|\mathbf{K}_{12}\|$ is the same, which can be
shown using the above steps.

We move on to $\mathbf{K}_{13}$, whose order is given by

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{13}\|\leq\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}(\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{2,i}^{0}+\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i})\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\right\|T^{-\delta/2}\|\widehat{\mathbf{F}}_{2}\|$
$\displaystyle\leq
O(1)N^{-1}T^{-(\nu_{2}+\delta)/2}\|\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{2}^{0}\mathbf{F}_{2}^{0\prime}\|$
$\displaystyle+O(1)\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}(\mathbf{F}_{1}^{0\prime}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i})\boldsymbol{\gamma}_{2,i}^{0\prime}\mathbf{F}_{2}^{0\prime}\right\|$
$\displaystyle=O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{3})/2})+\left(\frac{1}{NT^{\delta}}\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}\|^{2}\right)^{1/2}\left(\frac{1}{NT^{\nu_{2}}}\sum_{i=1}^{N}\|\mathbf{F}_{2}^{0}\boldsymbol{\gamma}_{2,i}^{0}\|^{2}\right)^{1/2}$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\delta-1)/2})+O_{P}(T^{-\delta/2})+O_{P}(T^{q-(\delta+\nu_{1})/2})+O_{P}(N^{-(1-p)}T^{-(\delta-\nu_{2})/2}),$
(A.76)

where the second equality follows from Lemma A.4 and Assumption 1. The order
of $T^{-\delta/2}\|\mathbf{K}_{14}\|$ is the same.

For $\mathbf{K}_{15}$,

$\displaystyle
T^{-\delta/2}\|\mathbf{K}_{15}\|=T^{-\delta/2}\left\|\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\mathbf{U}_{i}\mathbf{U}_{i}^{\prime}\widehat{\mathbf{F}}_{2}\right\|$
$\displaystyle\leq
O_{P}(1)\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}-\widehat{\mathbf{F}}_{1}\widehat{\boldsymbol{\gamma}}_{1,i}\|^{2}+O_{P}(1)\frac{1}{NT^{(\nu_{2}+\delta)/2}}\sum_{i=1}^{N}\|\mathbf{F}_{+2}^{0}\boldsymbol{\gamma}_{+2,i}^{0}\|^{2}$
$\displaystyle=O_{P}(T^{-(\nu_{2}+\delta)/2})+O_{P}(N^{-1}T^{1-(\nu_{2}+\delta)/2})+O_{P}(T^{2q-\nu_{1}-(\nu_{2}+\delta)/2})$
$\displaystyle+O_{P}(N^{-2(1-p)}T^{-(\delta-\nu_{2})/2})+O_{P}(T^{\nu_{3}-(\nu_{2}+\delta)/2}),$
(A.77)

where the second equality follows from Lemma A.4.

We now insert the above results for $\mathbf{K}_{1},\ldots,\mathbf{K}_{15}$
into (A.62). But first we left multiply by
$T^{-\nu_{2}}\mathbf{F}_{2}^{0\prime}$ and
$T^{-(\nu_{2}+\delta)/2}\widehat{\mathbf{F}}_{2}$ respectively. It then gives

$\displaystyle\|T^{-(\nu_{2}+\delta)/2}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2}(T^{-\nu_{2}}\mathbf{V}_{2})-(T^{-\nu_{2}}\mathbf{F}_{2}^{0\prime}\mathbf{F}_{2}^{0})(N^{-1}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\Gamma}_{2}^{0})(T^{-(\nu_{2}+\delta)/2}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2})\|$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\nu_{2}-1)/2})+O_{P}(T^{-\nu_{2}/2})+O_{P}(T^{-(\nu_{2}-\nu_{3})})$
$\displaystyle+O_{P}(T^{q-(\nu_{1}+\nu_{2})/2})+O_{P}(N^{-(1-p)})$ (A.78)

and

$\displaystyle\|T^{-\nu_{2}}\mathbf{V}_{2}-(T^{-(\nu_{2}+\delta)/2}\widehat{\mathbf{F}}_{2}^{\prime}\mathbf{F}_{2}^{0})(N^{-1}\boldsymbol{\Gamma}_{2}^{0\prime}\boldsymbol{\Gamma}_{2}^{0})(T^{-(\nu_{2}+\delta)/2}\mathbf{F}_{2}^{0\prime}\widehat{\mathbf{F}}_{2})\|$
$\displaystyle=O_{P}(N^{-1/2}T^{-(\nu_{2}-1)/2})+O_{P}(T^{-\nu_{2}/2})+O_{P}(T^{-(\nu_{2}-\nu_{3})})$
$\displaystyle+O_{P}(T^{q-(\nu_{1}+\nu_{2})/2})+O_{P}(N^{-(1-p)}).$ (A.79)

The rest of the proofs of (a) and (b) follows from the same arguments used in
Lemma A.3. It is therefore omitted.$\blacksquare$

Before continuing onto the proof of Lemma A.6, we note that the rates given in
Lemma A.5 can actually be improved upon. Suppose that $d_{2}$ in known.
Consider the next term which is a part of $\mathbf{K}_{15}$:

$\displaystyle
N^{-1}T^{-(\nu_{2}/2+\delta})\|\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{+2}^{0}\mathbf{F}_{+2}^{0\prime}\widehat{\mathbf{F}}_{2}\|$
$\displaystyle=N^{-1}T^{-(\nu_{2}/2+\delta})\|\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{+2}^{0}\mathbf{F}_{+2}^{0\prime}(\widehat{\mathbf{F}}_{2}\mathbf{V}_{2}^{0}-\mathbf{F}_{2}^{0}\mathbf{H}_{2}\mathbf{V}_{2}^{0}+\mathbf{F}_{2}^{0}\mathbf{H}_{2}\mathbf{V}_{2}^{0})(\mathbf{V}_{2}^{0})^{-1}\|$
$\displaystyle\leq
N^{-1}T^{-(\nu_{2}/2+\delta})\|\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{+2}^{0}\mathbf{F}_{+2}^{0\prime}(\widehat{\mathbf{F}}_{2}\mathbf{V}_{2}^{0}-\mathbf{F}_{2}^{0}\mathbf{H}_{2}\mathbf{V}_{2}^{0})(\mathbf{V}_{2}^{0})^{-1}\|$
$\displaystyle+N^{-1}T^{-(\nu_{2}/2+\delta})\|\mathbf{F}_{+2}^{0}\boldsymbol{\Gamma}_{+2}^{0\prime}\boldsymbol{\Gamma}_{+2}^{0}\mathbf{F}_{+2}^{0\prime}\mathbf{F}_{2}^{0}\mathbf{H}_{2}\|$
$\displaystyle\leq O_{P}(1)T^{\nu_{3}-\nu_{2}}\cdot
T^{-(\nu_{2}/2+\delta)}\|\widehat{\mathbf{F}}_{2}\mathbf{V}_{2}^{0}-\mathbf{F}_{2}^{0}\mathbf{H}_{2}\mathbf{V}_{2}^{0}\|+O_{P}(1)T^{\nu_{3}/2-(\nu_{2}+\delta/2)}\|\mathbf{F}_{+2}^{0\prime}\mathbf{F}_{2}^{0}\|,$
(A.80)

where $\mathbf{V}_{2}^{0}$ and $\mathbf{H}_{2}$ are defined in Appendix A.1.
Simple algebra shows that the term $T^{-(\nu_{2}-\nu_{3})}$ in $\tau_{NT}$ of
Lemma A.5 can be dropped.

Proof of Lemma A.6.

Consider (a). Let us assume without loss of generality that
$\boldsymbol{\beta}^{0}=\mathbf{0}_{d_{x}\times 1}$, as in Bai (2009). It then
follows that

$\displaystyle(NT)^{-1}[\mathrm{SSR}(\widehat{\boldsymbol{\beta}}_{1},\widehat{\mathbf{F}})-\mathrm{SSR}(\widehat{\boldsymbol{\beta}}_{0},\widehat{\mathbf{F}})]$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}(-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{1}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})^{\prime}\mathbf{M}_{\widehat{F}}(-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{1}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})$
$\displaystyle-\widehat{\boldsymbol{\beta}}_{1}^{\prime}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}$
$\displaystyle-\frac{1}{NT}\sum_{i=1}^{N}(-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})^{\prime}\mathbf{M}_{\widehat{F}}(-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})$
$\displaystyle+\widehat{\boldsymbol{\beta}}_{0}^{\prime}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}-\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}-\frac{1}{NT}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}$
$\displaystyle\geq\frac{1}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle-2\left(\frac{1}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\right)^{1/2}\left(\frac{1}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\right)^{1/2}$
$\displaystyle-\frac{2}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}-(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}.$
(A.81)

Note that by expanding the term $\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}$ as in
the proof for the second result of this lemma, we can obtain that

$\displaystyle\left\|\frac{1}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}\right\|$
$\displaystyle\leq\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|\left\|\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}\right\|$
$\displaystyle=O_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)[O_{P}((NT)^{-1/2})+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)].$
(A.82)

It follows that

$\displaystyle 0$
$\displaystyle\geq(NT)^{-1}[\mathrm{SSR}(\widehat{\boldsymbol{\beta}}_{1},\widehat{\mathbf{F}})-\mathrm{SSR}(\widehat{\boldsymbol{\beta}}_{0},\widehat{\mathbf{F}})]$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle+O_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)[O_{P}((NT)^{-1/2})+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)]$
$\displaystyle\geq\lambda_{min}\left(\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}\mathbf{D}_{T}\right)\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}$
$\displaystyle+O_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)[O_{P}((NT)^{-1/2})+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)],$
(A.83)

where the first term on the right is quadratic in
$\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|$.
Hence, to ensure the right-hand side is non-positive,
$\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|$
cannot converge to zero at a rate faster than
$(NT)^{-1/2}\vee\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|$.
It follows that

$\displaystyle\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|=O_{P}((NT)^{-1/2}\vee\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|),$
(A.84)

as was to be shown.

We now turn to (b). Note that

$\displaystyle\widehat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0}$
$\displaystyle=\mathbf{D}_{T}\left(\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}\mathbf{D}_{T}\right)^{-1}$
$\displaystyle\times\left(\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}+\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}\right)$
$\displaystyle=\mathbf{D}_{T}\mathbf{B}^{-1}\left(\mathbf{L}+\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}\right),$
(A.85)

with obvious definitions of $\mathbf{L}$ and $\mathbf{B}$.

Consider $\mathbf{L}$. Let
$\mathbf{Q}_{g}=(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})(T^{-(\nu_{g}+\delta)/2}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g})$,
such that
$\mathbf{H}_{g}^{-1}=T^{-(\nu_{g}+\delta)/2}\mathbf{V}_{g}^{0}\mathbf{Q}_{g}^{-1}$.
We also introduce $\mathbf{e}_{g,i}$, which is defined to be zero for $g=1$
and
$\mathbf{e}_{g,i}=\sum_{j=1}^{g-1}(\mathbf{F}_{j}^{0}\gamma_{j,i}^{0}-\widehat{\mathbf{F}}_{j}\widehat{\boldsymbol{\gamma}}_{j,i})$
for $g=2,\ldots,G$. In this notation,

$\displaystyle\mathbf{L}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}=-\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}(\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}-\mathbf{F}_{g}^{0})\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=-(\mathbf{L}_{1}+\cdots+\mathbf{L}_{15}),$ (A.86)

where

$\displaystyle\mathbf{L}_{1}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{2}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{3}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{4}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\mathbf{e}_{g,j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{5}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{e}_{g,j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{6}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{7}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{8}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{9}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{10}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{11}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\mathbf{e}_{g,j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{12}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{e}_{g,j}\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{13}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{e}_{g,j}\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{14}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}\mathbf{e}_{g,j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
$\displaystyle\mathbf{L}_{15}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{e}_{g,j}\mathbf{e}_{g,j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}.$

We now evaluate each of these terms. From the analysis of $\mathbf{L}_{2}$
below, it is easy to show that
$\|\mathbf{L}_{1}\|=o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)$.
We therefore start from $\mathbf{L}_{2}$, which we write as

$\displaystyle\mathbf{L}_{2}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}\mathbf{D}_{T}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})(\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0})^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=\frac{1}{N^{2}T}\sum_{g=1}^{G}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\boldsymbol{\gamma}_{g,j}^{0\prime}(T^{-(\nu_{g}+\delta)/2}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g})(T^{-(\nu_{g}+\delta)/2}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g})^{-1}$
$\displaystyle\times(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\boldsymbol{\gamma}_{g,j}^{0\prime}(\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}a_{ji}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0}).$
(A.87)

We will use this expression for $\mathbf{L}_{2}$ later.

Let us now move on to $\mathbf{L}_{3}$.

$\displaystyle\mathbf{L}_{3}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\sum_{g=1}^{G}(\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1})$
$\displaystyle\times\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
(A.88)

where, by using arguments that are similar to those used in the proofs of
Lemmas A.3 and A.5,

$\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\left\|\sum_{g=1}^{G}(\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1})\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|^{2}$
$\displaystyle\leq\frac{1}{NT}\sum_{i=1}^{N}\left(\sum_{g=1}^{G}\|\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}\|\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\|\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{D}_{T}^{-1}\mathbf{D}_{T}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\|\|\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\|\right)^{2}$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}T^{-\nu_{g}}\|\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}\|^{2}\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2}$
$\displaystyle=o_{P}(\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|^{2})$
(A.89)

implying

$\displaystyle\|\mathbf{L}_{3}\|$
$\displaystyle\leq\left(\frac{1}{NT}\sum_{i=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\|^{2}\right)^{1/2}$
$\displaystyle\times\left(\frac{1}{NT}\sum_{i=1}^{N}\left\|\sum_{g=1}^{G}(\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1})\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\gamma}_{g,j}^{0}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})^{\prime}\mathbf{X}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|^{2}\right)^{1/2}$
$\displaystyle=o_{P}(\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.90)

The same steps can be used to show that $\mathbf{L}_{4}$ and $\mathbf{L}_{5}$
are of the same order.

For $\mathbf{L}_{6}$, write

$\displaystyle\mathbf{L}_{6}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle\leq\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{j}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle+\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\varepsilon}_{j}^{\prime}(\widehat{\mathbf{F}}_{g}-\mathbf{F}_{g}^{0}\mathbf{H}_{g})\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
(A.91)

where the first term on the right is bounded by

$\displaystyle\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\boldsymbol{\varepsilon}_{j}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\|\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|$
$\displaystyle\leq
O_{P}(1)\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\left(\sum_{i=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\|^{2}\right)^{1/2}\left(\sum_{i=1}^{N}\|\boldsymbol{\gamma}_{g,i}^{0}\|^{2}\right)^{1/2}$
$\displaystyle\times\left(\sum_{j=1}^{N}\|\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\|^{2}\right)^{1/2}\left(\sum_{j=1}^{N}\|\boldsymbol{\varepsilon}_{j}^{\prime}\mathbf{F}_{g}^{0}\|^{2}\right)^{1/2}\|\mathbf{H}_{g}\|$
$\displaystyle\leq
O_{P}(1)\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|\sum_{g=1}^{G}T^{-(\nu_{g}+\delta)/2-1}O_{P}(\sqrt{T})O_{P}(\sqrt{T})O_{P}(T^{\nu_{g}/2})O_{P}(T^{-(\nu_{g}-\delta)/2})$
$\displaystyle=o_{P}(\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|).$
(A.92)

The second term is of the same order. Thus,
$\|\mathbf{L}_{6}\|=o_{P}(\|\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\|)$.
The same is true for $\|\mathbf{L}_{7}\|$.

We now examine $\mathbf{L}_{8}$. Let us define
$\boldsymbol{\Sigma}_{\varepsilon}=N^{-1}\sum_{i=1}^{N}\boldsymbol{\Sigma}_{\varepsilon,i}$,
in which $\boldsymbol{\Sigma}_{\varepsilon,i}$ has been defined in Assumption
3. $\mathbf{L}_{8}$ can then be written as

$\displaystyle\mathbf{L}_{8}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\Sigma}_{\varepsilon}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle+\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\boldsymbol{\varepsilon}_{j}\boldsymbol{\varepsilon}_{j}^{\prime}-\boldsymbol{\Sigma}_{\varepsilon,j})\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle-\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{P}_{\widehat{F}}(\boldsymbol{\varepsilon}_{j}\boldsymbol{\varepsilon}_{j}^{\prime}-\boldsymbol{\Sigma}_{\varepsilon,j})\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}.$
(A.93)

For the first term on the right,

$\displaystyle\sqrt{NT}\left\|\sum_{g=1}^{G}\frac{1}{NT^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\Sigma}_{\varepsilon}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle\leq\sum_{g=1}^{G}\frac{1}{\sqrt{N}T^{(\nu_{g}+\delta+1)/2}}\sum_{i=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\Sigma}_{\varepsilon,j}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\|$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}N^{-1/2}T^{-(\nu_{g}+\delta+1)/2}O_{P}(NT)O_{P}(T^{(\delta-1)/2})$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}N^{1/2}T^{-\nu_{g}/2}=O_{P}(\sqrt{N}T^{-\nu_{G}/2})=O_{P}(1),$
(A.94)

where the last equality holds by $NT^{-\nu_{G}}<\infty$. Let

$\displaystyle\overline{\mathbf{A}}_{1}=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F_{0}}\boldsymbol{\Sigma}_{\varepsilon}T^{-(\nu_{G}-1)/2}\mathbf{F}_{G}^{0}(T^{-\nu_{G}}\mathbf{F}_{G}^{0\prime}\mathbf{F}_{G}^{0})^{-1}(N^{-1}\boldsymbol{\Gamma}_{G}^{0\prime}\boldsymbol{\Gamma}_{G}^{0})^{-1}\boldsymbol{\gamma}_{G,i}^{0}.$
(A.95)

Note how
$\mathbf{A}_{1}=\operatorname*{p\\!\lim}_{N,T\to\infty}E(\overline{\mathbf{A}}_{1}|\mathcal{C})$.
In this notation,

$\displaystyle\sqrt{NT}\sum_{g=1}^{G}\frac{1}{NT^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\Sigma}_{\varepsilon}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=\sqrt{NT}\sum_{g=1}^{G}\frac{1}{NT^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F_{0}}\boldsymbol{\Sigma}_{\varepsilon}\widehat{\mathbf{F}}_{g}(T^{-(\nu_{g}+\delta)/2}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g})^{-1}(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}(1+o_{P}(1))$
$\displaystyle=\sum_{g=1}^{G}\sqrt{\frac{N}{T^{\nu_{g}}}}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F_{0}}\boldsymbol{\Sigma}_{\varepsilon}T^{-(\nu_{g}-1)/2}\mathbf{F}_{g}^{0}(T^{-\nu_{g}}\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0})^{-1}(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}(1+o_{P}(1))$
$\displaystyle=\sqrt{\frac{N}{T^{\nu_{G}}}}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F_{0}}\boldsymbol{\Sigma}_{\varepsilon}T^{-(\nu_{G}-1)/2}\mathbf{F}_{G}^{0}(T^{-\nu_{G}}\mathbf{F}_{G}^{0\prime}\mathbf{F}_{G}^{0})^{-1}(N^{-1}\boldsymbol{\Gamma}_{G}^{0\prime}\boldsymbol{\Gamma}_{G}^{0})^{-1}\boldsymbol{\gamma}_{G,i}^{0}(1+o_{P}(1))$
$\displaystyle=\sqrt{N}T^{-\nu_{G}/2}\overline{\mathbf{A}}_{1}(1+o_{P}(1)),$
(A.96)

where the second equality follows from arguments similar to those used in
(A.30). Note also that $\lim\sqrt{N}T^{-\nu_{G}/2}<\infty$. Further use of the
same steps used by Jiang et al. (2020) establishes that the second and third
terms of $\mathbf{L}_{8}$ are
$o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)+o_{P}((NT)^{-1/2})$.
Hence,

$\displaystyle\mathbf{L}_{8}=T^{-(\nu_{G}+1)/2}\overline{\mathbf{A}}_{1}+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)+o_{P}((NT)^{-1/2}).$
(A.97)

$\mathbf{L}_{9}$ can be written as

$\displaystyle\mathbf{L}_{9}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}(\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1})\boldsymbol{\gamma}_{g,j}^{0}\boldsymbol{\varepsilon}_{j}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle+\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}(\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1})\boldsymbol{\gamma}_{g,j}^{0}\boldsymbol{\varepsilon}_{j}^{\prime}(\widehat{\mathbf{F}}_{g}-\mathbf{F}_{g}^{0}\mathbf{H}_{g})\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0},$
(A.98)

where the first term on the right-hand side is bounded by

$\displaystyle\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\|\|\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}\|\|T^{-(\nu_{g}-1)/2}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\varepsilon}\mathbf{F}_{g}^{0}\|$
$\displaystyle\times
T^{(\nu_{g}-1)/2}\|\mathbf{H}_{g}\|\|\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\|$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}N^{-1}T^{-(\nu_{g}+\delta)/2-1}O_{P}(\sqrt{T})\|\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}\|O_{P}(\sqrt{NT})O_{P}(T^{(\nu_{g}-1)/2})O_{P}(T^{-(\nu_{g}-\delta)/2})$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}(NT)^{-1/2}T^{-\nu_{g}/2}\|\mathbf{F}_{g}^{0}-\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}\|$
$\displaystyle=o_{P}((NT)^{-1/2}),$ (A.99)

where the first inequality follows from
$\|T^{-(\nu_{g}-1)/2}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\varepsilon}\mathbf{F}_{g}^{0}\|=O_{P}(\sqrt{NT})$.
The second term on the right-hand side of $\mathbf{L}_{9}$ is also
$o_{P}((NT)^{-1/2})$. We therefore conclude that
$\|\mathbf{L}_{9}\|=o_{P}((NT)^{-1/2})$.

$\mathbf{L}_{10}$ can be written more compactly as

$\displaystyle\mathbf{L}_{10}$
$\displaystyle=\frac{1}{NT}\sum_{g=1}^{G}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{j}a_{ij},$
(A.100)

which we will again make use of later.

Let us move on to $\mathbf{L}_{11}$. We begin by rewriting $\mathbf{e}_{g,j}$
as

$\displaystyle\mathbf{e}_{g,j}$
$\displaystyle=\sum_{d=1}^{g-1}[\mathbf{F}_{d}^{0}\boldsymbol{\gamma}_{d,j}^{0}-\widehat{\mathbf{F}}_{d}T^{-\delta}\widehat{\mathbf{F}}_{d}^{\prime}(\mathbf{y}_{j}-\mathbf{X}_{j}\widehat{\boldsymbol{\beta}}_{0}-\widehat{\mathbf{F}}_{d}\widehat{\boldsymbol{\gamma}}_{d,j})]$
$\displaystyle=\sum_{d=1}^{g-1}[-\mathbf{P}_{\widehat{F}_{d}}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{M}_{\widehat{F}_{d}}\mathbf{F}_{d}^{0}\boldsymbol{\gamma}_{d,j}^{0}-\mathbf{P}_{\widehat{F}_{d}}\mathbf{F}_{+d}^{0}\boldsymbol{\gamma}_{+d,j}^{0}-\mathbf{P}_{\widehat{F}_{d}}\mathbf{e}_{d,j}-\mathbf{P}_{\widehat{F}_{d}}\boldsymbol{\varepsilon}_{j}],$
(A.101)

where $\mathbf{F}_{+d}^{0}=(\mathbf{F}_{d+1}^{0},\ldots,\mathbf{F}_{G}^{0})$
and
$\boldsymbol{\gamma}_{+d,j}^{0}=(\boldsymbol{\gamma}_{d+1,j}^{0\prime},\ldots,\boldsymbol{\gamma}_{G,j}^{0\prime})^{\prime}$,
as in Appendix A.1. By inserting this into $\mathbf{L}_{11}$, we get

$\displaystyle\mathbf{L}_{11}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\sum_{d=1}^{g-1}[-\mathbf{P}_{\widehat{F}_{d}}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle+\mathbf{M}_{\widehat{F}_{d}}\mathbf{F}_{d}^{0}\boldsymbol{\gamma}_{d,j}^{0}-\mathbf{P}_{\widehat{F}_{d}}\mathbf{F}_{+d}^{0}\boldsymbol{\gamma}_{+d,j}^{0}-\mathbf{P}_{\widehat{F}_{d}}\mathbf{e}_{d,j}-\mathbf{P}_{\widehat{F}_{d}}\boldsymbol{\varepsilon}_{j}]^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}.$
(A.102)

Hence, $\mathbf{L}_{11}$ can be written as a sum of five terms. There is no
need to study the forth term, the one due to
$\mathbf{P}_{\widehat{F}_{d}}\mathbf{e}_{d,j}$, as we can keep expanding
$\mathbf{e}_{d,j}$ until we cannot. The first and fifth terms are
$o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)$
and $o_{P}((NT)^{-1/2})$, respectively, by the same arguments used for
evaluating $\mathbf{L}_{7}$ and $\mathbf{L}_{8}$. Moreover, the steps used for
evaluating $\mathbf{L}_{9}$ can be used to show that

$\displaystyle\left\|\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\sum_{d=1}^{g-1}\boldsymbol{\gamma}_{d,j}^{0\prime}\boldsymbol{F}_{d}^{0\prime}\mathbf{M}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle=\left\|\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{d=1}^{g-1}\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{d}^{0}\mathbf{F}_{d}^{0\prime}\mathbf{M}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle=\left\|\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{d=1}^{g-1}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{d}^{0}(\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1})^{\prime}\mathbf{M}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle+\left\|\sum_{g=1}^{G}\frac{1}{N^{2}T^{(\nu_{g}+\delta)/2+1}}\sum_{i=1}^{N}\sum_{d=1}^{g-1}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{P}_{\widehat{F}}\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{d}^{0}(\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1})^{\prime}\mathbf{M}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle=o_{P}((NT)^{-1/2}).$ (A.103)

The second term in $\mathbf{L}_{11}$ is therefore negligible. It remains to
consider the third term, which is

$\displaystyle\left\|\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\sum_{d=1}^{g-1}\boldsymbol{\gamma}_{+d,j}^{0\prime}\mathbf{F}_{+d}^{0\prime}\mathbf{P}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle=\left\|\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{d=1}^{g-1}\boldsymbol{\varepsilon}^{\prime}\boldsymbol{\Gamma}_{+d}^{0}\mathbf{F}_{+d}^{0\prime}\mathbf{P}_{\widehat{F}_{d}}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle=o_{P}(\|\mathbf{L}_{10}\|)=o_{P}((NT)^{-1/2})$ (A.104)

where we have used Assumption 5. Therefore,
$\|\mathbf{L}_{11}\|=o_{P}((NT)^{-1/2})$.

For $\mathbf{L}_{12}$,

$\displaystyle\mathbf{L}_{12}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{e}_{g,j}\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{i}^{0}\boldsymbol{\varepsilon}_{j}^{\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\boldsymbol{\gamma}_{g,i}^{0}=\mathbf{L}_{9},$
(A.105)

where the second equality follows from the construction of $\mathbf{e}_{g,j}$.
Hence, since $\mathbf{L}_{9}$ is negligible, $\mathbf{L}_{12}$ is also
negligible.

Next up is $\mathbf{L}_{13}$, whose order can be worked out in the following
way:

$\displaystyle\|\mathbf{L}_{13}\|$
$\displaystyle\leq\sum_{g=1}^{G}\frac{1}{N^{2}T}\sum_{i=1}^{N}\left\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\sum_{d=1}^{g-1}\mathbf{F}_{d}^{0}\boldsymbol{\Gamma}_{d}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle\leq\sum_{g=1}^{G}\frac{1}{N^{2}T}\sum_{i=1}^{N}\left\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\sum_{d=1}^{g-1}(\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1})\boldsymbol{\Gamma}_{d}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}(N^{-1}\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0})^{-1}\boldsymbol{\gamma}_{g,i}^{0}\right\|$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}\frac{1}{N^{2}T}\sum_{d=1}^{g-1}\sum_{i=1}^{N}\|\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\|\|\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1}\|\|\boldsymbol{\Gamma}_{d}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}\|\|\boldsymbol{\gamma}_{g,i}^{0}\|$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}\frac{1}{NT}\sum_{d=1}^{g-1}O_{P}(\sqrt{T})\|\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1}\|O_{P}(N^{p})$
$\displaystyle\leq
O_{P}(1)\sum_{g=1}^{G}\frac{1}{N^{1-p}\sqrt{T}}\sum_{d=1}^{g-1}\|\mathbf{F}_{d}^{0}-\widehat{\mathbf{F}}_{d}\mathbf{H}_{d}^{-1}\|=o_{P}((NT)^{-1/2}),$
(A.106)

where the first inequality follows from the construction of
$\mathbf{e}_{g,j}$, and the last equality follows by going through a
development similar to the first result of Lemma A.5 and further using a
development similar to (A.6). The same arguments show that
$\|\mathbf{L}_{14}\|$ and $\|\mathbf{L}_{15}\|$ are negligible, too.

We now put everything together. This yields

$\displaystyle\mathbf{L}$
$\displaystyle=-(\mathbf{L}_{1}+\cdots+\mathbf{L}_{15})$
$\displaystyle=-\mathbf{L}_{2}-\mathbf{L}_{8}-\mathbf{L}_{10}+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)+o_{P}((NT)^{-1/2})$
$\displaystyle=\left(-\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}a_{ji}+o_{P}(1)\right)\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})-T^{(\nu_{G}+1)/2}\overline{\mathbf{A}}_{1}$
$\displaystyle-\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}a_{ji}\boldsymbol{\varepsilon}_{j}+o_{P}((NT)^{-1/2}),$
(A.107)

which in turn implies

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{B}^{-1}\left(\mathbf{L}+\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}\right)$
$\displaystyle=(\mathbf{B}^{-1}\mathbf{N}+o_{P}(1))\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})+\mathbf{B}^{-1}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle-T^{-(\nu_{G}+1)/2}\mathbf{B}^{-1}\overline{\mathbf{A}}_{1}+o_{P}((NT)^{-1/2}),$
(A.108)

where $\mathbf{Z}_{i}(\mathbf{F})$ is defined in Assumption 2, and

$\displaystyle\mathbf{N}=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}a_{ij}.$

This expression for
$\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0})$
can be inserted into
$\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})$,
giving

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0})-\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle=(\mathbf{B}^{-1}\mathbf{N}+o_{P}(1))\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})+\mathbf{B}^{-1}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle-\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-T^{-(\nu_{G}+1)/2}\mathbf{B}^{-1}\overline{\mathbf{A}}_{1}+o_{P}((NT)^{-1/2}),$
(A.109)

which can be solved for
$\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})$

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\left(\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}-\mathbf{B}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-T^{-(\nu_{G}+1)/2}\overline{\mathbf{A}}_{1}\right)$
$\displaystyle+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)+o_{P}((NT)^{-1/2}).$
(A.110)

Also, making use of Assumption 3, it is not difficult to show that
$(NT)^{-1}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}=O_{P}((NT)^{-1/2})$.
Hence,

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})+\mathbf{B}(\widehat{\mathbf{F}})^{-1}\mathbf{B}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}-T^{-(\nu_{G}+1)/2}\mathbf{B}(\widehat{\mathbf{F}})^{-1}\overline{\mathbf{A}}_{1}$
$\displaystyle+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)+o_{P}((NT)^{-1/2}).$
(A.111)

By using the fact that

$\displaystyle\mathbf{B}(\widehat{\mathbf{F}})^{-1}\mathbf{B}\mathbf{D}_{T}^{-1}=\mathbf{D}_{T}^{-1}\left(\sum_{i=1}^{N}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i},$
(A.112)

the left-hand side of this last equation can be written as

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})+\mathbf{B}(\widehat{\mathbf{F}})^{-1}\mathbf{B}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})$
$\displaystyle=\mathbf{D}_{T}^{-1}\left[\widehat{\boldsymbol{\beta}}_{0}+\left(\sum_{i=1}^{N}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-\boldsymbol{\beta}^{0}\right].$
(A.113)

It follows that

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}\left[\widehat{\boldsymbol{\beta}}_{0}+\left(\sum_{i=1}^{N}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-\boldsymbol{\beta}^{0}\right]$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}-\sqrt{N}T^{-\nu_{G}/2}\mathbf{B}(\widehat{\mathbf{F}})^{-1}\overline{\mathbf{A}}_{1}$
$\displaystyle+o_{P}(\sqrt{NT}\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)+o_{P}(1).$
(A.114)

Consider the
$o_{P}(\sqrt{NT}\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|)$
reminder term. By the first result of this lemma,
$\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|=O_{P}((NT)^{-1/2}\vee\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)$.
This can be inserted into (A.6), giving

$\displaystyle\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}+o_{P}((NT)^{-1/2}\vee\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)$
$\displaystyle=O_{P}((NT)^{-1/2})+o_{P}(\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|),$
(A.115)

which in turn implies

$\displaystyle\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|=O_{P}((NT)^{-1/2}).$
(A.116)

The second result then follows.$\blacksquare$

##  A.7 Proofs of main results

Proof of Lemma 3.1.

Without loss of generality, we assume that
$\boldsymbol{\beta}^{0}=\mathbf{0}_{d_{x}\times 1}$. This implies

$\displaystyle(NT)^{-1}[\mathrm{SSR}(\boldsymbol{\beta},\mathbf{F})-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})]$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}(\mathbf{X}_{i}\boldsymbol{\beta}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})^{\prime}\mathbf{M}_{F}(\mathbf{X}_{i}\boldsymbol{\beta}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})$
$\displaystyle+\boldsymbol{\beta}^{\prime}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}(\mathbf{P}_{F^{0}}-\mathbf{P}_{F})\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}(\mathbf{X}_{i}\mathbf{D}_{T}\mathbf{D}_{T}^{-1}\boldsymbol{\beta}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})^{\prime}\mathbf{M}_{F}(\mathbf{X}_{i}\mathbf{D}_{T}\mathbf{D}_{T}^{-1}\boldsymbol{\beta}+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0})$
$\displaystyle+\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+O_{P}(N^{-1}\vee
T^{-1})$
$\displaystyle=\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\mathbf{B}(\mathbf{F})\mathbf{D}_{T}^{-1}\boldsymbol{\beta}+\boldsymbol{\theta}^{\prime}\mathbf{B}\boldsymbol{\theta}+\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+O_{P}(N^{-1}\vee
T^{-1}),$ (A.1)

where the second equality follows from Lemma A.2, and
$\boldsymbol{\theta}=\boldsymbol{\eta}+\mathbf{B}^{-1}\mathbf{C}\boldsymbol{\beta}$
and
$\mathbf{B}=N^{-1}\boldsymbol{\Gamma}^{0\prime}\boldsymbol{\Gamma}^{0}\otimes\mathbf{I}_{T}$
are defined as on page 1265 of Bai (2009). Note that for
$\boldsymbol{\beta}\in\mathbb{R}^{d_{x}}$, we may have

$\displaystyle\sup_{\boldsymbol{\beta}\in\mathbb{R}^{d_{x}},\mathbf{F}\in\mathbb{D}_{F}}\left|\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}\right|\neq
o_{P}(1).$ (A.2)

In our proof of consistency, we consider two cases; (i)
$\|\mathbf{D}_{T}^{-1}\boldsymbol{\beta}\|\leq C$ and (ii)
$\|\mathbf{D}_{T}^{-1}\boldsymbol{\beta}\|>C$, where $C$ is a large positive
constant. Under (i),

$\displaystyle\sup_{\|\mathbf{D}_{T}^{-1}\boldsymbol{\beta}\|\leq
C,\mathbf{F}\in\mathbb{D}_{F}}\left|\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}\right|=O_{P}(N^{-1/2}\vee
T^{-1/2})$ (A.3)

by Lemma A.2. The expression given in (A.7) for
$(NT)^{-1}[\mathrm{SSR}(\boldsymbol{\beta},F)-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})]$
therefore reduces to

$\displaystyle(NT)^{-1}[\mathrm{SSR}(\boldsymbol{\beta},\mathbf{F})-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})]$
$\displaystyle=\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\mathbf{B}(\mathbf{F})\mathbf{D}_{T}^{-1}\boldsymbol{\beta}+\boldsymbol{\theta}^{\prime}\mathbf{B}\boldsymbol{\theta}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+O_{P}(N^{-1/2}\vee T^{-1/2}),$ (A.4)

where
$\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\mathbf{B}(\mathbf{F})\mathbf{D}_{T}^{-1}\boldsymbol{\beta}$
does not involve $\mathbf{F}^{0}$ and
$\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}$
is independent of $\boldsymbol{\beta}$. Hence, provided $d_{max}\geq d_{f}$,
the consistency of $\mathbf{D}_{T}^{-1}\widehat{\boldsymbol{\beta}}_{0}$ in
case (i) follows from the same arguments as in Bai (2009).

Under (ii), (A.7) can be written as

$\displaystyle(NT)^{-1}[\mathrm{SSR}(\boldsymbol{\beta},\mathbf{F})-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})]$
$\displaystyle=\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\mathbf{B}(\mathbf{F})\mathbf{D}_{T}^{-1}\boldsymbol{\beta}+\boldsymbol{\theta}^{\prime}\mathbf{B}\boldsymbol{\theta}+\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+O_{P}(N^{-1/2}\vee T^{-1/2})$ $\displaystyle\geq
c_{0}\|\mathbf{D}_{T}^{-1}\boldsymbol{\beta}\|^{2}+\boldsymbol{\beta}^{\prime}\mathbf{D}_{T}^{-1}\frac{2}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+\boldsymbol{\theta}^{\prime}\mathbf{B}\boldsymbol{\theta}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+O_{P}(N^{-1/2}\vee T^{-1/2})$
$\displaystyle\geq\frac{c_{0}}{2}C^{2}+\boldsymbol{\theta}^{\prime}\mathbf{B}\boldsymbol{\theta}+\frac{2}{NT}\sum_{i=1}^{N}\boldsymbol{\gamma}_{i}^{0\prime}\mathbf{F}^{0\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}+O_{P}(N^{-1/2}\vee
T^{-1/2}),$ (A.5)

where $c_{0}$ is defined in Assumption 2, and the second inequality follows
from the fact that the quadratic term dominates the linear one for large
values of $C$. Hence,
$(NT)^{-1}[\mathrm{SSR}(\boldsymbol{\beta},\mathbf{F})-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})]>0$,
but from the definition of $\widehat{\boldsymbol{\beta}}_{0}$ we also know
that
$\mathrm{SSR}(\widehat{\boldsymbol{\beta}}_{0},\widehat{\mathbf{F}})-\mathrm{SSR}(\boldsymbol{\beta}^{0},\mathbf{F}^{0})\leq
0$, which means that $\mathbf{D}_{T}^{-1}\widehat{\boldsymbol{\beta}}_{0}$
cannot belong to (ii).

Note that since
$\|(NT)^{-1}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F}\boldsymbol{\varepsilon}_{i}\|=O_{P}(N^{-1/2}\vee
T^{-1/2})$ by Lemma A.2, all we need is $C=C^{0}(N^{-1/2}\vee T^{-1/2})$ for
some large constant $C^{0}$ in order to ensure that the last inequality of
(A.7) holds. This implies
$\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})=O_{P}(N^{-1/2}\vee
T^{-1/2})$, so the proof is complete. $\blacksquare$

Proof of Lemma 3.2.

Suppose first that $d_{1}=0$, such that $d_{f}=0$. In this case,

$\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\|\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}\|^{2}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\|\mathbf{X}_{i}\mathbf{D}_{T}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\boldsymbol{\varepsilon}_{i}\|^{2}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\|\boldsymbol{\varepsilon}_{i}\|^{2}+o_{P}(1).$
(A.6)

This implies $\tau_{N}\asymp 1/\ln(T\vee N)$, which is much larger than
$\widehat{\lambda}_{1,1}/\widehat{\lambda}_{1,0}$. The result then follows
immediately.

Suppose now instead that $d_{1}>0$. Straightforward algebra reveals that

$\displaystyle\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\|\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}\|^{2}$
$\displaystyle=\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\|\mathbf{X}_{i}\mathbf{D}_{T}\mathbf{D}_{T}^{-1}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}^{0}\boldsymbol{\gamma}_{i}^{0}+\boldsymbol{\varepsilon}_{i}\|^{2}$
$\displaystyle=\frac{1}{NT^{\nu_{1}}}\sum_{i=1}^{N}\|\mathbf{F}_{1}^{0}\boldsymbol{\gamma}_{1,i}^{0}\|^{2}+o_{P}(1)$
$\displaystyle=(\mathrm{vec}\,\boldsymbol{\Sigma}_{\Gamma_{1}^{0}})^{\prime}\mathrm{vec}\,\boldsymbol{\Sigma}_{F_{1}^{0}}+o_{P}(1),$
(A.7)

which together with Assumption 1 implies $\tau_{N}\asymp 1/\ln(T\vee N)$. Note
that for $d=1,\ldots,d_{1}$,

$\displaystyle\frac{T^{\nu_{1}}}{\lambda_{1,d}}$
$\displaystyle=\frac{T^{2\nu_{1}}}{\mathbf{h}_{1,d}^{0\prime}\mathbf{F}_{1}^{0\prime}\boldsymbol{\Sigma}_{1}^{0}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}}=\frac{NT^{2\nu_{1}}}{\mathbf{h}_{1,d}^{0\prime}\mathbf{F}_{1}^{0\prime}\mathbf{F}_{1}^{0}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0}\mathbf{F}_{1}^{0\prime}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0}}$
$\displaystyle=\frac{1}{(T^{-\nu_{1}}\mathbf{h}_{1,d}^{0\prime}\mathbf{F}_{1}^{0\prime}\mathbf{F}_{1}^{0})(N^{-1}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0})(T^{-\nu_{1}}\mathbf{F}_{1}^{0\prime}\mathbf{F}_{1}^{0}\mathbf{h}_{1,d}^{0})}$
$\displaystyle=\frac{1}{(T^{-(\nu_{1}+\delta)/2}\widehat{\mathbf{F}}_{1,d}^{0\prime}\mathbf{F}_{1}^{0})(N^{-1}\boldsymbol{\Gamma}_{1}^{0\prime}\boldsymbol{\Gamma}_{1}^{0})(T^{-(\nu_{1}+\delta)/2}\mathbf{F}_{1}^{0\prime}\widehat{\mathbf{F}}_{1,d}^{0})}(1+o_{P}(1))\asymp
1,$ (A.8)

where the fourth equality follows from (A.32) and the last step is dues to
(A.28) and Assumption 1. Using Lemma A.3, (A.7) and (A.7), we obtain that

$\displaystyle\frac{\widehat{\lambda}_{1,1}}{\widehat{\lambda}_{1,0}}$
$\displaystyle=\frac{T^{-\nu_{1}}\widehat{\lambda}_{1,1}}{(NT^{\nu_{1}})^{-1}\sum_{i=1}^{N}\|\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}\|^{2}}\asymp
1,$ (A.9)
$\displaystyle\frac{\widehat{\lambda}_{1,d+1}}{\widehat{\lambda}_{1,d}}$
$\displaystyle=\frac{T^{-\nu_{1}}\widehat{\lambda}_{1,d+1}}{T^{-\nu_{1}}\widehat{\lambda}_{1,d}}\asymp
1$ (A.10)

for $d=1,\ldots,d_{1}-1$, and
$\frac{\widehat{\lambda}_{1,d_{1}+1}}{\widehat{\lambda}_{1,d_{1}}}=O_{P}(T^{-(\nu_{1}-\nu_{2})})$.
Moreover, for $d=d_{1}+1,\ldots,d_{max}$,
$\frac{T^{-\nu_{1}}\widehat{\lambda}_{1,d}}{T^{-\nu_{1}}\widehat{\lambda}_{1,0}}=O_{P}(T^{-(\nu_{1}-\nu_{2})})$,
which is less than $\tau_{N}$ by (A.7) and Lemma A.3. The required result
follows from this and the definition of $\widehat{d}_{1}$. $\blacksquare$

Proof of Lemma 3.3.

Each step in the sequential procedure of Step 2 introduces additional
remainder terms that all converge to zero under Assumptions 3 and 5 by Lemmas
A.3 and A.5. This proves (a). Part (b) follows from the (rotational)
consistency of $\widehat{\mathbf{F}}_{g}$ established as a part of the proofs
of Lemmas A.3 and A.5.$\blacksquare$

Proof of Theorem 3.1.

By Lemma A.6, the rate of convergence given in Lemma 3.1 is therefore not the
best one possible. The improved rate implies that

$\displaystyle\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})\|=O_{P}((NT)^{-1/2}\vee\|\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}_{0}-\boldsymbol{\beta}^{0})\|)=O_{P}((NT)^{-1/2}),$
(A.11)

which can be inserted back into (A.6), leading to

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}\left[\widehat{\boldsymbol{\beta}}_{0}+\left(\sum_{i=1}^{N}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-\boldsymbol{\beta}^{0}\right]$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}-\sqrt{N}T^{-\nu_{G}/2}\mathbf{B}(\widehat{\mathbf{F}})^{-1}\overline{\mathbf{A}}_{1}+o_{P}(1).$
(A.12)

Note that $\mathbf{Z}_{i}(\widehat{\mathbf{F}})$ is $\widehat{\mathbf{Z}}_{i}$
with $\widehat{a}_{ij}$ replaced by $a_{ij}$. We now show that the effect of
the estimation of $a_{ij}$ is negligible. We begin by noting how

$\displaystyle\frac{1}{NT}\sum_{i=1}^{N}[\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\mathbf{D}_{T}-\mathbf{D}_{T}\widehat{\mathbf{Z}}_{i}^{\prime}\widehat{\mathbf{Z}}_{i}\mathbf{D}_{T}]$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\widehat{a}_{ij}-\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}a_{ij}.$
(A.13)

Here,

$\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\widehat{a}_{ij}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\sum_{g=1}^{G}(\mathbf{y}_{j}-\mathbf{X}_{j}\widehat{\boldsymbol{\beta}}_{0}-\widehat{\mathbf{F}}_{-g}\widehat{\boldsymbol{\gamma}}_{-g,j})^{\prime}\widehat{\mathbf{F}}_{g}$
$\displaystyle\times(\widehat{\mathbf{F}}_{g}^{\prime}\widehat{\boldsymbol{\Sigma}}_{g}\widehat{\mathbf{F}}_{g})^{-1}\widehat{\mathbf{F}}_{g}^{\prime}(\mathbf{y}_{i}-\mathbf{X}_{i}\widehat{\boldsymbol{\beta}}_{0}-\widehat{\mathbf{F}}_{-g}\widehat{\boldsymbol{\gamma}}_{-g,i})$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\sum_{g=1}^{G}[\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}+\mathbf{e}_{g,j}+\boldsymbol{\varepsilon}_{j}]^{\prime}\widehat{\mathbf{F}}_{g}$
$\displaystyle\times
T^{-\delta}\mathbf{V}_{g}^{-1}\widehat{\mathbf{F}}_{g}^{\prime}[\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,i}^{0}+\mathbf{e}_{g,i}+\boldsymbol{\varepsilon}_{i}]$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}\sum_{g=1}^{G}[\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,j}^{0}+\mathbf{e}_{g,j}+\boldsymbol{\varepsilon}_{j}]^{\prime}\widehat{\mathbf{F}}_{g}$
$\displaystyle\times
T^{-(\nu_{g}+\delta)}(T^{-\nu_{g}}\mathbf{V}_{g})^{-1}\widehat{\mathbf{F}}_{g}^{\prime}[\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})+\mathbf{F}_{g}^{0}\boldsymbol{\gamma}_{g,i}^{0}+\mathbf{e}_{g,i}+\boldsymbol{\varepsilon}_{i}](1+o_{P}(1))$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{j}\mathbf{D}_{T}a_{ij}(1+o_{P}(1)),$
(A.14)

where the second equality here follows from the definition of
$\mathbf{V}_{g}^{0}$, which is again based on taking $d_{g}$ as known, while
the last equality follows from direct calculation. The effect of the
estimation of $a_{ij}$ in $\widehat{\mathbf{Z}}_{i}$ is therefore negligible,
which in turn implies that the right-hand side of (A.7) becomes

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}\left[\widehat{\boldsymbol{\beta}}_{0}+\left(\sum_{i=1}^{N}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\mathbf{Z}_{i}(\widehat{\mathbf{F}})\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-\boldsymbol{\beta}^{0}\right]$
$\displaystyle=\sqrt{NT}\mathbf{D}_{T}^{-1}\left[\widehat{\boldsymbol{\beta}}_{0}+\left(\sum_{i=1}^{N}\widehat{\mathbf{Z}}_{i}^{\prime}\widehat{\mathbf{Z}}_{i}\right)^{-1}\sum_{i=1}^{N}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\mathbf{X}_{i}(\widehat{\boldsymbol{\beta}}_{1}-\widehat{\boldsymbol{\beta}}_{0})-\boldsymbol{\beta}^{0}\right]+o_{P}(1)$
$\displaystyle=\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})+o_{P}(1).$
(A.15)

It follows that

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{B}(\widehat{\mathbf{F}})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}-\sqrt{N}T^{-\nu_{G}/2}\mathbf{B}(\widehat{\mathbf{F}})^{-1}\overline{\mathbf{A}}_{1}$
$\displaystyle+o_{P}(1).$ (A.16)

Consider
$(NT)^{-1/2}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}$.
From the definition of $\mathbf{Z}_{i}(\mathbf{F})$,

$\displaystyle\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\mathbf{F}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}+\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}[\mathbf{Z}_{i}(\widehat{\mathbf{F}})-\mathbf{Z}_{i}(\mathbf{F}^{0})]^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\mathbf{F}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}-\sqrt{NT}(\mathbf{R}_{1}-\mathbf{R}_{2}),$
(A.17)

where

$\displaystyle\mathbf{R}_{1}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\mathbf{P}_{\widehat{F}}-\mathbf{P}_{F^{0}})\boldsymbol{\varepsilon}_{i},$
(A.18) $\displaystyle\mathbf{R}_{2}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}a_{ij}\mathbf{D}_{T}\mathbf{X}_{j}^{\prime}(\mathbf{P}_{\widehat{F}}-\mathbf{P}_{F^{0}})\boldsymbol{\varepsilon}_{i}.$
(A.19)

Here,

$\displaystyle\mathbf{R}_{1}$
$\displaystyle=\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}^{0}\mathbf{H}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}^{0}[\mathbf{H}\mathbf{H}^{\prime}-T^{\delta}(\mathbf{F}^{0\prime}\mathbf{F}^{0})^{-1}]\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\mathbf{R}_{11}+\mathbf{R}_{12}+\mathbf{R}_{13}+\mathbf{R}_{14},$
(A.20)

with obvious implicit definitions of $\mathbf{R}_{11},\ldots,\mathbf{R}_{14}$
and $\mathbf{H}=\mathrm{diag}(\mathbf{H}_{1},\ldots,\mathbf{H}_{G})$. Let
$\mathbf{R}_{1m,j}$ be the $j$-th row of $\mathbf{R}_{1m}$ for
$m\in\\{1,\ldots,4\\}$. In this notation,

$\displaystyle\|\mathbf{R}_{11,j}\|$
$\displaystyle\leq\left\|\frac{1}{NT^{\delta/2+1}}\sum_{i=1}^{N}(\boldsymbol{\varepsilon}_{i}^{\prime}\mathbf{F}^{0}\mathbf{H})\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i})^{\prime}\right\|\|T^{-\delta/2}\mathrm{vec}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})\|$
$\displaystyle=O_{P}((NT)^{-1/2})T^{-\delta/2}\|\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H}\|=o_{P}((NT)^{-1/2}),$
(A.21)

where the equality follows from Lemma A.2 and the fact that
$T^{(\nu_{g}-\delta)/2}\mathbf{H}_{g}=O_{P}(1)$. Similarly, for
$\mathbf{R}_{12}$,

$\displaystyle\|\mathbf{R}_{12,j}\|$
$\displaystyle\leq\left\|\frac{1}{NT}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i})^{\prime}\right\|\|T^{-\delta}\mathrm{vec}[(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}]\|$
$\displaystyle=O_{P}(N^{-1/2})T^{-\delta}\|\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H}\|^{2}=o_{P}((NT)^{-1/2}),$
(A.22)

where the second equality is due to (A.54).

Consider $\mathbf{R}_{14}$. From
$T^{-\delta/2}\|\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H}\|=o_{P}(1)$ and
$\|T^{(\nu_{g}-\delta)/2}\mathbf{H}_{g}\|=O_{P}(1)$, we obtain

$\displaystyle\|\mathbf{I}_{d_{f}}-T^{\delta}(\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\mathbf{F}^{0}\mathbf{H})^{-1}\|$
$\displaystyle=o_{P}(1),$ (A.23)
$\displaystyle\|T^{\nu_{g}-\delta}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}-(T^{-\nu_{g}}\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0})^{-1}\|$
$\displaystyle=o_{P}(1).$ (A.24)

Together with Lemma A.2 this implies

$\displaystyle\|\mathbf{R}_{14,j}\|$
$\displaystyle=\left\|\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}^{0}[\mathbf{H}\mathbf{H}^{\prime}-T^{\delta}\mathbf{H}(\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\mathbf{F}^{0}\mathbf{H})^{-1}\mathbf{H}^{\prime}]\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}\right\|$
$\displaystyle=\left\|\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}^{0}\mathbf{H}[\mathbf{I}_{d_{f}}-T^{\delta}(\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\mathbf{F}^{0}\mathbf{H})^{-1}]\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}\right\|$
$\displaystyle\leq\left\|\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}(\boldsymbol{\varepsilon}_{i}^{\prime}\mathbf{F}^{0}\mathbf{H})\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}^{0}\mathbf{H})\right\|\|\mathbf{I}_{d_{f}}-T^{\delta}(\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\mathbf{F}^{0}\mathbf{H})^{-1}\|$
$\displaystyle=O_{P}((NT)^{-1/2})\|\mathbf{I}_{d_{f}}-T^{\delta}(\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\mathbf{F}^{0}\mathbf{H})^{-1}\|=o_{P}((NT)^{-1/2}).$
(A.25)

Finally, let us consider $\mathbf{R}_{13}$.

$\displaystyle\mathbf{R}_{13}$
$\displaystyle=\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}^{0}\mathbf{H}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}(\widehat{\mathbf{F}}_{g}-\mathbf{F}_{g}^{0}\mathbf{H}_{g})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}(\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}-\mathbf{F}_{g}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}$
(A.26)

Expanding $\widehat{\mathbf{F}}_{g}\mathbf{H}_{g}^{-1}-\mathbf{F}_{g}^{0}$ as
we did for $\mathbf{L}$ above, we then just need to focus on the leading terms
equivalent to $\mathbf{L}_{2}$, $\mathbf{L}_{8}$, and $\mathbf{L}_{10}$.

$\displaystyle\mathbf{R}_{13}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}\left(\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\mathbf{X}_{j}(\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0})\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}\left(\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}\left(\frac{1}{\sqrt{T^{\nu_{g}+1}}}\boldsymbol{\Sigma}_{\varepsilon}\frac{\mathbf{F}_{g}^{0}}{T^{(\nu_{g}-1)/2}}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+o_{P}\left(\frac{1}{\sqrt{NT}}\right)$
$\displaystyle=\mathbf{R}_{131}+\mathbf{R}_{132}+\mathbf{R}_{133}+o_{P}\left(\frac{1}{\sqrt{NT}}\right).$

It is easy to see that
$\|\mathbf{R}_{131}\|=o_{P}(\|\boldsymbol{\beta}^{0}-\widehat{\boldsymbol{\beta}}_{0}\|)$,
so negligible. We now examine $\mathbf{R}_{133}$. Then we can write

$\displaystyle\mathbf{R}_{133}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\nu_{g}}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Sigma}_{\varepsilon}}{\sqrt{T^{\nu_{g}+1}}}\frac{\mathbf{F}_{g}^{0}}{T^{(\nu_{g}-1)/2}}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{T^{\nu_{g}+1}}\cdot\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\frac{\mathbf{F}_{g}^{0}}{T^{(\nu_{g}-1)/2}}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\frac{\mathbf{F}_{g}^{0\prime}}{T^{(\nu_{g}-1)/2}}\boldsymbol{\Sigma}_{\varepsilon}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=o_{P}\left(\frac{1}{\sqrt{NT}}\right),$ (A.27)

where the first equality follows from (A.24), and the last step follows from
some routine analysis using Assumption 3. Below, we investigate
$\mathbf{R}_{132}$, which is one source of the bias term.

$\displaystyle\mathbf{R}_{132}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime}\left(\frac{1}{NT^{(\nu_{g}+\delta)/2}}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\mathbf{F}_{g}^{0\prime}\widehat{\mathbf{F}}_{g}\mathbf{Q}_{g}^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\nu_{g}}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}(T^{\nu_{g}-\delta}\mathbf{H}_{g}\mathbf{H}_{g}^{\prime})\left(\frac{1}{N}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{1+\nu_{g}}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{1}{N}\sum_{j=1}^{N}\boldsymbol{\varepsilon}_{j}\boldsymbol{\gamma}_{g,j}^{0\prime}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\right)^{\prime}\boldsymbol{\varepsilon}_{i}\cdot(1+o_{P}(1))$
$\displaystyle=\sum_{g=1}^{G}\frac{1}{NT^{(\nu_{g}-1)/2}}\cdot\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}}{T^{(\nu_{g}+1)/2}}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\boldsymbol{\gamma}_{g,j}^{0}\frac{\boldsymbol{\varepsilon}_{j}^{\prime}\boldsymbol{\varepsilon}_{i}}{T}\cdot(1+o_{P}(1)),$
(A.28)

where the second equality follows from the definition of $\mathbf{Q}_{g}$, and
the third equality follows from (A.24). Then we are able to further write

$\displaystyle\sqrt{NT}\mathbf{R}_{32}$
$\displaystyle=\sum_{g=1}^{G}\frac{T}{\sqrt{N}T^{\nu_{g}/2}}\cdot\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{g}^{0}}{T^{(\nu_{g}+1)/2}}\left(\frac{\mathbf{F}_{g}^{0\prime}\mathbf{F}_{g}^{0}}{T^{\nu_{g}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{g}^{0\prime}\boldsymbol{\Gamma}_{g}^{0}}{N}\right)^{-1}\boldsymbol{\gamma}_{g,j}^{0}\frac{\boldsymbol{\varepsilon}_{j}^{\prime}\boldsymbol{\varepsilon}_{i}}{T}\cdot(1+o_{P}(1))$
$\displaystyle=\frac{T}{\sqrt{N}T^{\nu_{G}/2}}\cdot\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{G}^{0}}{T^{(\nu_{G}+1)/2}}\left(\frac{\mathbf{F}_{G}^{0\prime}\mathbf{F}_{G}^{0}}{T^{\nu_{G}}}\right)^{-1}\left(\frac{\boldsymbol{\Gamma}_{G}^{0\prime}\boldsymbol{\Gamma}_{G}^{0}}{N}\right)^{-1}\boldsymbol{\gamma}_{G,j}^{0}\frac{\boldsymbol{\varepsilon}_{j}^{\prime}\boldsymbol{\varepsilon}_{i}}{T}\cdot(1+o_{P}(1)).$
(A.29)

Hence, by adding the results,

$\displaystyle\sqrt{NT}\mathbf{R}_{1}$
$\displaystyle=\sqrt{NT}(\mathbf{R}_{11}+\mathbf{R}_{12}+\mathbf{R}_{13}+\mathbf{R}_{14})$
$\displaystyle=T^{(2-\nu_{G})/2}N^{-1/2}\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}T^{-(\nu_{G}+1)/2}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}_{G}^{0}(T^{-\nu_{G}}\mathbf{F}_{G}^{0\prime}\mathbf{F}_{G}^{0})^{-1}$
$\displaystyle\times(N^{-1}\boldsymbol{\Gamma}_{G}^{0\prime}\boldsymbol{\Gamma}_{G}^{0})^{-1}\boldsymbol{\gamma}_{G,j}^{0}T^{-1}\boldsymbol{\varepsilon}_{j}^{\prime}\boldsymbol{\varepsilon}_{i}+o_{P}(1).$
(A.30)

$\sqrt{NT}\mathbf{R}_{2}$ can be evaluated in exactly the same way, and the
limiting representation is analogous to the one given above for
$\sqrt{NT}\mathbf{R}_{1}$ with $\mathbf{X}_{i}$ replaced by
$-\sum_{j=1}^{N}\mathbf{X}_{j}a_{ij}$. Moreover,
$\|\mathbf{B}(\widehat{\mathbf{F}})-\mathbf{B}(\mathbf{F}^{0})\|=o_{P}(1).$ It
follows that if we define

$\displaystyle\overline{\mathbf{A}}_{2}=\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{N}T^{-(\nu_{G}+1)/2}\mathbf{D}_{T}\mathbf{Z}_{i}(0)^{\prime}\mathbf{F}_{G}^{0}(T^{-\nu_{G}}\mathbf{F}_{G}^{0\prime}\mathbf{F}_{G}^{0})^{-1}(N^{-1}\boldsymbol{\Gamma}_{G}^{0\prime}\boldsymbol{\Gamma}_{G}^{0})^{-1}\boldsymbol{\gamma}_{G,j}^{0}T^{-1}\boldsymbol{\varepsilon}_{j}^{\prime}\boldsymbol{\varepsilon}_{i},$
(A.31)

where $\mathbf{Z}_{i}(0)=\mathbf{X}_{i}-\sum_{j=1}^{N}\mathbf{X}_{j}a_{ij}$,
then (A.17) reduces to

$\displaystyle\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\mathbf{F}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}-T^{(2-\nu_{G})/2}N^{-1/2}\overline{\mathbf{A}}_{2}+o_{P}(1),$
(A.32)

which in turn implies that (A.7) becomes

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})$
$\displaystyle=\mathbf{B}(\mathbf{F}^{0})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\mathbf{F}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}-\sqrt{N}T^{-\nu_{G}/2}\mathbf{B}(\mathbf{F}^{0})^{-1}\overline{\mathbf{A}}_{1}$
$\displaystyle-T^{(2-\nu_{G})/2}N^{-1/2}\mathbf{B}(\mathbf{F}^{0})^{-1}\overline{\mathbf{A}}_{2}+o_{P}(1).$
(A.33)

The required result is now implied by Assumption 7. $\blacksquare$

Proof of Corollary A.1.

Under $NT^{-\nu_{G}}\to 0$, (A.7) in the proof of Theorem 3.1 may be written
as

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})=\mathbf{B}(\mathbf{F}^{0})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\widehat{\mathbf{F}})^{\prime}\boldsymbol{\varepsilon}_{i}+o_{P}(1).$

Consider
$(NT)^{-1}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}$,
which we can write as

$\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{\widehat{F}}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F^{0}}\boldsymbol{\varepsilon}_{i}-\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\mathbf{P}_{\widehat{F}}-\mathbf{P}_{F^{0}})\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{M}_{F^{0}}\boldsymbol{\varepsilon}_{i}-\mathbf{R}_{1}.$
(A.34)

As in the proof of Theorem 3.1,

$\displaystyle\mathbf{R}_{1}$
$\displaystyle=\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(T^{-\delta}\widehat{\mathbf{F}}\widehat{\mathbf{F}}^{\prime}-\mathbf{P}_{F^{0}})\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})\mathbf{H}^{\prime}\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}^{0}\mathbf{H}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\boldsymbol{\varepsilon}_{i}+\frac{1}{NT^{1+\delta}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}\mathbf{F}^{0}[\mathbf{H}\mathbf{H}^{\prime}-(\mathbf{F}^{0\prime}\mathbf{F}^{0})^{-1}]\mathbf{F}^{0\prime}\boldsymbol{\varepsilon}_{i}$
$\displaystyle=\mathbf{R}_{11}+\mathbf{R}_{12}+\mathbf{R}_{13}+\mathbf{R}_{14},$
(A.35)

where $\|\mathbf{R}_{11}\|$, $\|\mathbf{R}_{12}\|$ and $\|\mathbf{R}_{14}\|$
are all $o_{P}((NT)^{-1/2})$, just as before. Let us therefore consider
$\mathbf{R}_{13}$, which is the source of the bias in the asymptotic
distribution of
$\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})$.
The purpose of Assumption A.1 is to control this term. Suppose first that
condition (b) holds. Let $x_{j,k,t}$ denote the $j$-th row of
$\mathbf{x}_{k,t}$. In this notation,

$\displaystyle
E\left\|\frac{1}{NT^{\delta/2+1}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}_{g}^{0}T^{-(\nu_{g}-\delta)/2})\right\|^{2}$
$\displaystyle=\frac{1}{N^{2}T^{\delta+2}}\sum_{s=1}^{T}E\left\|\sum_{i=1}^{N}\varepsilon_{i,s}\sum_{t=1}^{T}T^{-\kappa_{j}/2}x_{j,i,t}T^{-(\nu_{g}-1)/2}\mathbf{f}_{g,t}^{0\prime}T^{(\nu_{g}-1)/2}T^{-(\nu_{g}-\delta)/2}\right\|^{2}$
$\displaystyle=\frac{1}{(NT)^{2}}\sum_{i=1}^{N}\sum_{k=1}^{N}E\left[\left(\sum_{t=1}^{T}T^{-\kappa_{j}/2}x_{j,i,t}T^{-(\nu_{g}-1)/2}\mathbf{f}_{g,t}^{0\prime}\right)\left(\sum_{t=1}^{T}T^{-(\nu_{g}-1)/2}\mathbf{f}_{g,t}^{0}T^{-\kappa_{j}/2}x_{j,k,t}\right)\right]\sigma_{\varepsilon,ik}$
$\displaystyle=O(N^{-1}).$ (A.36)

Moreover, by applying Assumption 7 (b) to the expression given for
$\|T^{-\delta/2}\widehat{\mathbf{F}}_{2,d}-T^{-\nu_{2}/2}\mathbf{F}_{2}^{0}\mathbf{h}_{2,d}^{0}\|$
in the proof of Lemma A.5, we can show that

$\displaystyle\|T^{-\delta/2}\mathrm{vec}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\|=o_{P}(T^{-1/2}).$
(A.37)

Making use of these results, we obtain

$\displaystyle\|\mathbf{R}_{3,j}\|$
$\displaystyle\leq\left\|\frac{1}{NT^{\delta/2+1}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}^{0}\mathbf{H})\right\|\|T^{-\delta/2}\mathrm{vec}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\|$
$\displaystyle=o_{P}((NT)^{-1/2}).$ (A.38)

Alternatively, we may invoke Assumption 7 (a) to arrive at the same result. In
this case,
$\|T^{-\delta/2}\mathrm{vec}(\widehat{\mathbf{F}}-\mathbf{F}^{0}\mathbf{H})^{\prime}\|=o_{P}(1)$,
but we also have

$\displaystyle
E\left\|\frac{1}{NT^{\delta/2+1}}\sum_{i=1}^{N}\boldsymbol{\varepsilon}_{i}^{\prime}\otimes(T^{-\kappa_{j}/2}\mathbf{X}_{j,i}^{\prime}\mathbf{F}_{g}^{0}T^{-(\nu_{g}-\delta)/2})\right\|^{2}$
$\displaystyle=\frac{1}{(NT)^{2}}\sum_{i=1}^{N}\sum_{k=1}^{N}E\left[\left(\sum_{t=1}^{T}T^{-\kappa_{j}/2}x_{j,i,t}T^{-(\nu_{g}-1)/2}\mathbf{f}_{g,t}^{0\prime}\right)\left(\sum_{t=1}^{T}T^{-(\nu_{g}-1)/2}\mathbf{f}_{g,t}^{0}T^{-\kappa_{j}/2}x_{j,k,t}\right)\right]\sigma_{\varepsilon,ik}$
$\displaystyle=\frac{1}{(NT)^{2}}\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{k=1}^{N}E[T^{-\kappa_{j}}x_{j,i,t}x_{j,k,t}T^{-(\nu_{g}-1)}E(\|\mathbf{f}_{g,t}^{0}\|^{2}|\boldsymbol{\varepsilon}_{t})]\sigma_{\varepsilon,ik}$
$\displaystyle+\frac{2}{(NT)^{2}}\sum_{t=2}^{T}\sum_{s<t}\sum_{i=1}^{N}\sum_{k=1}^{N}E[T^{-\kappa_{j}}x_{j,i,t}x_{j,k,s}T^{-(\nu_{g}-1)}E(\mathbf{f}_{g,t}^{0\prime}\mathbf{f}_{g,s}^{0}|\boldsymbol{\varepsilon}_{t},\boldsymbol{\varepsilon}_{s})]\sigma_{\varepsilon,ik}$
$\displaystyle=O(1)\frac{1}{(NT)^{2}}\sum_{t=2}^{T}\sum_{s<t}\sum_{i=1}^{N}\sum_{k=1}^{N}|\phi_{ts}||\sigma_{\varepsilon,ik}|=O((NT)^{-1}),$
(A.39)

and so $\|\mathbf{R}_{3,j}\|$ is of the same order as before. The proof under
Assumption 7 (c) is simpler and is therefore omitted.

Hence, by adding the results,

$\displaystyle\sqrt{NT}\|\mathbf{R}_{1}\|\leq\sqrt{NT}(\|\mathbf{R}_{11}\|+\|\mathbf{R}_{12}\|+\|\mathbf{R}_{13}\|+\|\mathbf{R}_{14}\|)=o_{P}(1).$
(A.40)

We have therefore shown that

$\displaystyle\sqrt{NT}\left\|\frac{1}{NT}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\mathbf{P}_{\widehat{F}}-\mathbf{P}_{F^{0}})\boldsymbol{\varepsilon}_{i}\right\|=o_{P}(1),$
(A.41)

and we can similarly show that

$\displaystyle\sqrt{NT}\left\|\frac{1}{NT}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbf{D}_{T}\mathbf{X}_{i}^{\prime}(\mathbf{P}_{\widehat{F}}-\mathbf{P}_{F^{0}})\mathbf{X}_{j}\mathbf{D}_{T}a_{ij}\right\|=o_{P}(1).$
(A.42)

These results can be inserted into (A.7), giving

$\displaystyle\sqrt{NT}\mathbf{D}_{T}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}^{0})=\mathbf{B}(\mathbf{F}^{0})^{-1}\frac{1}{\sqrt{NT}}\sum_{i=1}^{N}\mathbf{D}_{T}\mathbf{Z}_{i}(\mathbf{F}^{0})^{\prime}\boldsymbol{\varepsilon}_{i}+o_{P}(1).$
(A.43)

The sought result now follows from Assumption 7.$\blacksquare$