# Reconstruction of functional data via factor models of increasing rank

Maximilian Ofner  Graz University of Technology  Siegfried Hörmann  Graz
University of Technology

###### Abstract

This paper studies linear reconstruction of partially observed functional data
which are recorded on a discrete grid. We propose a novel estimation approach
based on approximate factor models with increasing rank. Whereas alternative
reconstruction procedures commonly involve some preliminary smoothing, our
method separates the signal from noise and reconstructs missing fragments at
once. We establish uniform convergence rates of our estimator and introduce a
new method for constructing simultaneous prediction bands for the missing
trajectories. A simulation study examines the performance of the proposed
methods in finite samples. Finally, a real data application of temperature
curves demonstrates that our theory provides a simple and effective method to
recover missing fragments.

Keywords: approximate factor models, partially observed, preprocessing,
uniform consistency.

##  1 Introduction

Functional data analysis (Ramsay and Silverman, (2005); Bosq, (2000); Ferraty
and Vieu, (2006); Hsing and Eubank, (2015)) gained widespread popularity in
the last decades not least because of modern devices’ capability to measure
data on dense grids. A major branch of works is devoted to the reconstruction
of imperfectly observed random functions $X=(X(u):u\in[0,1])$. Yao et al.,
(2005), among others, illuminate the case of sparsely observed functions. More
recently, Kraus, (2015) and Kneip and Liebl, (2020) aim to recover functions
which are only observed on subsets of their domain. In the underlying work, we
focus on the latter setting and assume that functions are measured with
additive noise on a grid $0\leq u_{1}<\dots<u_{N}\leq 1$, that is to say, one
observes $Y_{ti}=X_{t}(u_{i})+e_{ti}$ for $t\in\\{1,\dots,T\\}$ and
$i\in\\{1,\dots,N\\}$. A generic function $X$, not accessible for $u_{i}$ in
some subset $\textsc{M}=[0,1]\setminus\textsc{O}$, is treated as partially
observed. We illustrate this setting by a sample of temperature data measured
half-hourly in Graz (Austria) between July 1 and August 31, 2022 (Land
Steiermark, (2023)). Figure 1 shows $N=48$ intraday temperature values of the
corresponding $T=62$ days. As it can be seen, 8 out of 62 days (around 13%)
are only partially observed. Our main interest is then to estimate the best
linear reconstruction of $X^{\textsc{m}}=(X(u):u\in\textsc{M})$ from
$X^{\textsc{o}}=(X(u):u\in\textsc{O})$ by utilizing information of a
completely observed sub-sample of the data.

We remark that this task can be viewed as a matrix completion problem, which
has been recently studied on the basis of factor models (see Bai and Ng,
(2021); Cahan et al., (2023); Xiong and Pelger, (2023)). In the context of
functional data, low rank matrix completion is studied by Descary and
Panaretos, (2019). In this work we employ the framework of approximative
factor models (Bai, (2003); Fan et al., (2013); Bai and Ng, (2023)), which is
rather popular in macroeconometrics, and amend the related imputation methods
in Cahan et al., (2023) for functional data. While in the classical factor
models literature the number of factors (denoted by $r$) is fixed and small
(more than two or three factors can hardly be interpreted) a much larger
number of factors is generally required to describe functional data
sufficiently well. In our asymptotic setup, we therefore allow $r$ to diverge
with the sample size which poses a considerable technical challenge (see also
Li et al., (2017) and Hörmann and Jammoul, (2022)). Another complication is
due to the reconstruction problem itself and concerns the local variability of
$X$ on $\textsc{O}\subset[0,1]$. In general, the information contained in
$X^{\textsc{o}}$ is not sufficient for an errorless reconstruction of
$X^{\textsc{m}}$. Roughly speaking, if the dimension of $X^{\textsc{o}}$ is
smaller than the dimension of $X$, reconstruction of $X^{\textsc{m}}$ is only
possible up to a reconstruction error. In contrast to the above contributions
to factor models, we provide an explicit treatment of the corresponding
reconstruction error and use it for establishing uniform error bands.

The remainder of this work is structured as follows. In Section 2, we motivate
the reconstruction problem and discuss its relation to approximate factor
models. Thereafter, we introduce a reconstruction estimator and derive its
asymptotic convergence rate in Section 3. Confidence bands are discussed in
Section 4. We then analyze their performance in finite samples in Section 5,
where a simulation study and our real data problem are considered. Proofs are
shifted to the Appendix.

Figure 1: Intraday temperature values measured half-hourly in the east of Graz
(Austria) between July 1 and August 31, 2022. Highlighted curves reflect 8
days with missing observations.

##  2 Preliminaries

###  2.1 Notation and functional setup

Let $H=L^{2}([0,1])$ denote the separable Hilbert space of square integrable
functions defined on the unit interval. For elements $f,g\in H$, the
corresponding inner product $\langle
f,g\rangle=\int_{0}^{1}f(u)g(u)\,\textup{d}u$ induces the norm $\lVert
f\rVert^{2}=\langle f,f\rangle=\int_{0}^{1}f(u)^{2}\,\textup{d}u$. In the
following, we consider a continuous random function $X=(X(u):u\in[0,1])$
identified with an $H$-valued random element. Define
$\mu(u)=\mathbb{E}\,(X(u))$ to be its mean function. Without loss of
generality, we assume throughout Section 2 that $X$ has zero mean, that is,
$\mu(u)=0$ for all $u\in[0,1]$.

Now suppose that $X$ were only observed on a given sub-interval
$\textsc{O}\subset[0,1]$. The theory of the following pages, however, remains
true in the more general case that O constitutes a finite union of sub-
intervals. No information of $X$ is available on the set
$\textsc{M}=[0,1]\setminus\textsc{O}$. Let
$X^{\textsc{O}}=(X(u):u\in\textsc{O})$ denote the random function $X$
restricted to O. Note that $X^{\textsc{o}}$ maps to the space
$L^{2}(\textsc{O})$ of square integrable functions defined on O. This is a
Hilbert space with inner product $\langle
f,g\rangle_{\textsc{o}}=\int_{\textsc{O}}f(u)g(u)\,\textup{d}u$ and norm
$\lVert f\rVert_{\textsc{o}}^{2}=\int_{\textsc{O}}f(u)^{2}\,\textup{d}u$.
Fluctuations of $X^{\textsc{o}}$ around its mean are captured by the
covariance function
$\sigma^{\textsc{o}}(u,v)=\mathbb{E}\,[X^{\textsc{o}}(u)X^{\textsc{o}}(v)]$. A
relevant object is the integral operator
$\Gamma^{\textsc{o}}:L^{2}(\textsc{O})\to L^{2}(\textsc{O})$ with kernel
$\sigma^{\textsc{o}}$,

$(\Gamma^{\textsc{o}}f)(u)=\int_{\textsc{O}}\sigma^{\textsc{o}}(u,v)f(v)\,\textup{d}v,\qquad
f\in L^{2}(\textsc{O}).$

Let $(\lambda_{k}^{\textsc{o}},\phi_{k}^{\textsc{o}})$ denote the $k$th
eigenvalue and eigenfunction of the operator $\Gamma^{\textsc{o}}$. The
$\phi_{k}^{\textsc{o}}$ are also referred to as “functional principal
components” and have corresponding counterparts in multivariate statistics.
Throughout the underlying work, we suppose that the sequence of eigenvalues is
ordered, that is to say,
$\lambda_{1}^{\textsc{o}}\geq\lambda_{2}^{\textsc{o}}\geq\dots\geq 0$. Its
rate of decay plays a crucial role in functional data analysis and relates to
the smoothness of the covariance function $\sigma^{\textsc{o}}$. In case
$\sigma^{\textsc{o}}$ is continuous on $\textsc{O}\times\textsc{O}$, we may
express $X^{\textsc{o}}$ via the Karhunen-Loève theorem as

$X^{\textsc{o}}(u)=\sum_{k=1}^{\infty}\xi^{\textsc{o}}_{k}\phi_{k}^{\textsc{o}}(u),\qquad
u\in\textsc{O},$

where convergence is in mean square and uniform over $u\in\textsc{O}$. The
“scores” are defined by $\xi_{k}^{\textsc{o}}=\langle
X^{\textsc{o}},\phi_{k}^{\textsc{o}}\rangle_{\textsc{o}}$ and constitute
uncorrelated random variables with zero mean and variance
$\lambda_{k}^{\textsc{o}}$.

It is then natural to ask how to infer $X$ from the information contained in
$X^{\textsc{o}}$. To this end, we introduce the following assumption.

  1. (A1)

There exists some $r_{\textsc{o}}<\infty$ such that
$X^{\textsc{o}}(u)=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}^{\textsc{o}}\phi_{k}^{\textsc{o}}(u)$
for $u\in\textsc{O}$.

Assumption (A1) restricts $X$ to be driven by only finitely many scores on the
set O. While this constitutes a theoretical restriction, it opens the door to
the machinery of factor models, which proves to be highly relevant in
practice. In our asymptotic setup, we let ${r_{\textsc{o}}}$ diverge with the
sample size to account for high dimensionality and the functional nature of
the data (see Hörmann and Jammoul, (2022)).

###  2.2 Linear reconstruction operator

We seek a linear operator $\mathcal{L}:L^{2}(\textsc{O})\to L^{2}([0,1])$
which minimizes

$\mathbb{E}\,\Big{[}\left(X(u)-\mathcal{L}(X^{\textsc{o}})(u)\right)^{2}\Big{]},$
(1)

at any $u\in[0,1]$. Invoking a linear regression (see also Section 2 in Kneip
and Liebl, (2020) for a more detailed discussion), the solution to the above
problem is given by the finite-rank operator

$\mathcal{L}(X^{\textsc{o}})(u)=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}^{\textsc{o}}\tilde{\phi}^{\textsc{o}}_{k}(u),\qquad
u\in[0,1],$ (2)

where the “extrapolated” basis functions $\tilde{\phi}^{\textsc{o}}_{k}$ are
defined as

$\tilde{\phi}^{\textsc{o}}_{k}(u)=\frac{\mathbb{E}\,[X(u)\xi_{k}^{\textsc{o}}]}{\lambda_{k}^{\textsc{o}}},\qquad
u\in[0,1].$ (3)

If $\sigma(u,v)=\mathbb{E}\,[X(u)X(v)]$ is continuous on $[0,1]\times[0,1]$,
then $\tilde{\phi}^{\textsc{o}}_{k}$ and the reconstructed curve
$\mathcal{L}(X^{\textsc{o}})$ are continuous as well. Moreover, for
$u\in\textsc{O}$, we have $\mathcal{L}(X^{\textsc{o}})(u)=X^{\textsc{o}}(u)$.

Kneip and Liebl, (2020) consider the more general setting of unbounded linear
reconstruction operators. In contrast, the reconstruction operator given in
(2) is bounded due to Assumption (A1). Since we allow ${r_{\textsc{o}}}$ to
grow, this induces only a small theoretical limitation which is compensated by
a gain in the empirical performance (see Section 5) as well as explicit and
clean asymptotic error bounds (see Theorem 1).

One cannot hope to reconstruct $X$ from $X^{\textsc{o}}$ without error as $X$
is not identifiable from $X^{\textsc{o}}$ in general. Rather, it will be the
case that

$X(u)=\mathcal{L}(X^{\textsc{o}})(u)+Z^{\textsc{o}}(u),\qquad u\in[0,1],$ (4)

where $Z^{\textsc{o}}=(Z^{\textsc{o}}(u):u\in[0,1])$ denotes the
“reconstruction error” which vanishes on the set O. The following result
provides a simple condition for perfect reconstruction.

######  Proposition 1 (Identifiability).

Suppose that Assumption (A1) holds and let $\Gamma$ denote the integral
operator on $L^{2}([0,1])$ with kernel $\sigma$. If
$\textup{rank}(\Gamma)={r_{\textsc{o}}}$, then
$\mathcal{L}(X^{\textsc{o}})(u)=X(u)$ almost surely for every $u\in[0,1]$.

######  Remark.

The reconstruction error resembles the fact that the problem in (1) relates to
extrapolation. Contrary to extrapolation, interpolation or smoothing is well
understood and has been covered by Yao et al., (2005) among others. A subtle
difference between the setting in the underlying work and the literature on
smoothing lies in the fact that we only require information to be “dense” in O
as opposed to the whole unit interval.

###  2.3 Data generating process

Let $(X_{t}:t\leq T)$ be $T$ independent copies of $X$ and assume, for the
moment, that all observations are available. Adopting a common framework, we
suppose that functions are recorded on a discrete grid and contaminated with
additive noise. Accordingly, we set

$Y_{ti}=X_{t}(u_{i})+e_{ti},\qquad t\in\\{1,\dots,T\\},\quad
i\in\\{1,\dots,N\\},$ (5)

where $0=u_{1}<u_{2}<\dots<u_{N}=1$ defines a grid of equispaced points. In
our asymptotic setup, we assume that the number of grid points diverges, that
is, $N\to\infty$. If $N$ remains bounded, there is no hope of recovering $X$
even in the absence of noise, as there exist infinitely many random functions
which marginal distributions coincide on $u_{1},\dots,u_{N}$; see Hall et al.,
(2006) for a related discussion on this topic.

######  Remark.

The assumption of equispaced grid points is often satisfied in practice, when
measurements are taken by electronic instruments; see our real data
application in Section 5.4. If instead each $X_{t}$ were measured at irregular
but sufficiently dense points $0=u_{t,1}<u_{t,2}<\dots<u_{t,N_{t}}=1$, one may
first define
$\widetilde{Y}_{ti}=\sum_{j=1}^{N_{t}-1}Y_{tj}\mathbbm{1}\\{u_{i}\in[u_{t,j},u_{t,j+1})\\}$
for some equispaced grid $u_{1},\dots,u_{N}$, and then proceed in a similar
way.

Suppose now that some curve $X_{t}$ is only _observable_ on the set
$\textsc{O}_{t}\subset[0,1]$. Then, $Y_{ti}=X_{t}(u_{i})+e_{ti}$ is observed
whenever $u_{i}\in\textsc{O}_{t}$. Throughout this work, we consider the
following condition.

  1. (A2)

The intervals $(\textsc{O}_{t}:t\leq T)$ are independent from all other
quantities.

Assumption (A2) is a missing-completely-at-random condition (Rubin, (1976))
and similarly imposed in the related works of Kraus, (2015) and Kneip and
Liebl, (2020). It prevents the sub-sample corresponding to missing
observations from following a different law (as it would for instance be the
case if values beyond a certain threshold were systematically missing). A
relaxation of the assumption is discussed in Liebl and Rameseder, (2019).

In the following, we assume for simplicity that the first $T_{\textsc{c}}\leq
T$ functions are completely observable, that is, $\textsc{O}_{t}=[0,1]$ and
$Y_{ti}$ is available for every $t\leq T_{\textsc{c}}$ and $i\leq N$. The
corresponding matrix of dimension $(T_{\textsc{c}}\times N)$ is denoted by
$\bm{Y}_{\textsc{c}}$. Furthermore, for some generic $\textsc{O}\subset[0,1]$,
$\bm{Y}_{\textsc{o}}$ denotes the $(T_{\textsc{c}}\times N_{\textsc{o}})$
matrix which consists of elements $Y_{ti}$ with $t\leq T_{\textsc{c}}$ and
$u_{i}\in\textsc{O}$. Matrices
$\bm{X}_{\textsc{c}},\bm{X}_{\textsc{o}},\bm{e}_{\textsc{c}}$ and
$\bm{e}_{\textsc{o}}$ are defined in the same vein.

###  2.4 Relation to factor analysis

Assumption (A1) entails a fundamental relationship between functional data and
factor models. Plugging (2) and (4) into (5) yields

$Y_{ti}=\begin{cases}\begin{aligned}
&\sum_{k=1}^{r_{\textsc{o}}}\xi_{tk}^{\textsc{o}}\phi^{\textsc{o}}_{k}(u_{i})+e_{ti},&&\qquad\text{for
$u_{i}\in\textsc{O}_{t}$,}\\\
&\sum_{k=1}^{r_{\textsc{o}}}\xi_{tk}^{\textsc{o}}\tilde{\phi}^{\textsc{o}}_{k}(u_{i})+Z^{\textsc{o}}_{t}(u_{i})+e_{ti},&&\qquad\text{else.}\end{aligned}\end{cases}$
(6)

We may thus rewrite $\bm{Y}_{\textsc{o}}$ as

$\bm{Y}_{\textsc{o}}=\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}+\bm{e}_{\textsc{o}},$
(7)

where $\bm{F}_{\textsc{o}}$ is a $(T_{\textsc{c}}\times{r_{\textsc{o}}})$
matrix with elements $\xi^{\textsc{o}}_{tk}/\sqrt{\lambda^{\textsc{o}}_{k}}$
and $\bm{\Lambda}_{\textsc{o}}$ an $(N_{\textsc{o}}\times{r_{\textsc{o}}})$
matrix with elements
$\sqrt{\lambda^{\textsc{o}}_{k}}\phi^{\textsc{o}}_{k}(u_{i})$. In addition,
let $\bm{\Lambda}^{*}_{\textsc{o}}$ denote the $(N\times{r_{\textsc{o}}})$
matrix containing the “extrapolated loadings”
$\sqrt{\lambda^{\textsc{o}}_{k}}\tilde{\phi}_{k}^{\textsc{o}}(u_{i})$. If
$\bm{e}_{\textsc{o}}$ is independent of $\bm{X}_{\textsc{o}}$ and if the
largest eigenvalue of
$\mathbb{E}\,[\bm{e}_{\textsc{o}}{\bm{e}_{\textsc{o}}}^{\prime}]$ grows at
most linear in $T_{\textsc{c}}$ (short
$\lambda_{\text{max}}(\mathbb{E}\,[\bm{e}_{\textsc{o}}{\bm{e}_{\textsc{o}}}^{\prime}])=O(T_{\textsc{c}})$),
then (7) constitutes an approximate factor model of rank ${r_{\textsc{o}}}$.
Such models have been extensively covered in the literature; major
contributions to the theory of large factor models, where both
$T_{\textsc{c}}$ and $N_{\textsc{o}}$ tend to infinity, include Bai, (2003)
and Fan et al., (2013).

######  Remark.

Consider the diagonal matrix
$\bm{B}_{\textsc{o}}=\text{diag}(\sqrt{\lambda_{1}^{\textsc{o}}},\dots,\sqrt{\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$.
Under the assumptions of this paper, it then holds that
$\bm{B}_{\textsc{o}}^{-1}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}/N_{\textsc{o}}\to\bm{I}_{r_{\textsc{o}}}$
(identity matrix of rank ${r_{\textsc{o}}}$). Accordingly, our model can be
seen as an approximate factor model with heterogeneous loadings; see Bai and
Ng, (2023). On this account, we borrow results from the existing literature on
factor models and incorporate the case of increasing
$\text{rank}(\bm{\Lambda}_{\textsc{o}})={r_{\textsc{o}}}\to\infty$, when
proving results of the subsequent pages.

##  3 Estimation of the reconstructions

So far we have assumed that the random functions are centered, that is,
$\mu(u)=\mathbb{E}\,(X(u))=0$ for any $u\in[0,1]$. For the sake of simplicity,
we keep this assumption for the remainder of this work. In the case
$\mu(u)\neq 0$, one might proceed as follows. By (5),
$Y_{ti}=\mu(u_{i})+(X_{t}(u_{i})-\mu(u_{i}))+e_{ti}$ for $t\leq
T_{\textsc{c}}$. Accordingly, a simple estimator of $\mu(u_{i})$ is given by
$\widehat{\mu}(u_{i})=\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Y_{ti}$
which satisfies
$\lvert\widehat{\mu}(u_{i})-\mu(u_{i})\rvert=O_{p}(\sqrt{T_{\textsc{c}}})$.
The reconstruction procedure proposed in the subsequent pages may then simply
be performed on the matrix with elements $Y_{ti}-\widehat{\mu}(u_{i})$.

Let $\textsc{O}\subset[0,1]$ be a given sub-interval and $t\leq T$ such that
$X_{t}$ is observable on O. In the next two sections, we introduce an
estimator for $\mathcal{L}(X_{t}^{\textsc{o}})$ and discuss uniform
consistency.

###  3.1 Construction of the estimator

The intrinsic functional nature opens a straightforward path for estimation.
We follow a general approach and estimate the factors $\bm{F}_{\textsc{o}}$
via principal components. To this end, consider the singular value
decomposition

$\frac{1}{\sqrt{N_{\textsc{o}}T_{\textsc{c}}}}\bm{Y}_{\textsc{o}}\approx\bm{U}_{\textsc{o}}\bm{D}_{\textsc{o}}\bm{V}_{\textsc{o}}^{\prime},$

where $\bm{D}_{\textsc{o}}$ is the diagonal matrix containing the first
${r_{\textsc{o}}}$ singular values of
$\frac{1}{\sqrt{N_{\textsc{o}}T_{\textsc{c}}}}\bm{Y}_{\textsc{o}}$ in
descending order. The columns of $\bm{U}_{\textsc{o}}$ and
$\bm{V}_{\textsc{o}}$ contain the first ${r_{\textsc{o}}}$ left and right
eigenvectors, respectively. We then set

$\widehat{\bm{F}}_{\textsc{o}}=\sqrt{T_{\textsc{c}}}\bm{U}_{\textsc{o}},\qquad\widehat{\bm{\Lambda}}_{\textsc{o}}=\sqrt{N_{\textsc{o}}}\bm{V}_{\textsc{o}}\bm{D}_{\textsc{o}},\qquad\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}=\frac{1}{T_{\textsc{c}}}\bm{Y}_{\textsc{c}}^{\prime}\widehat{\bm{F}}_{\textsc{o}},$
(8)

adopting the conventions that
$\frac{1}{T_{\textsc{c}}}{\widehat{\bm{F}}_{\textsc{o}}}{}^{\prime}{\widehat{\bm{F}}}_{\textsc{o}}=\bm{I}_{r_{\textsc{o}}}$
and
$\widehat{\bm{\Lambda}}_{\textsc{o}}{}^{\prime}\widehat{\bm{\Lambda}}_{\textsc{o}}$
diagonal. Note that all quantities in (8) are obtained from the completely
observed subblock $\bm{Y}_{\textsc{c}}$ of the data matrix. The estimators
$\widehat{\bm{F}}_{\textsc{o}}$ and $\widehat{\bm{\Lambda}}_{\textsc{o}}$
consist of scaled eigenvectors of $\bm{Y}_{\textsc{o}}$. In addition,
$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}=\bm{Y}_{\textsc{c}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}(\widehat{\bm{F}}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}})^{-1}$
is obtained by regressing columns of $\bm{Y}_{\textsc{c}}$ onto columns of
$\widehat{\bm{F}}_{\textsc{o}}$. Because
$\frac{1}{T_{\textsc{c}}}\bm{Y}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}=\sqrt{N_{\textsc{o}}}\bm{V}_{\textsc{o}}\bm{D}_{\textsc{o}}=\widehat{\bm{\Lambda}}_{\textsc{o}}$,
$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}$ contains
$\widehat{\bm{\Lambda}}_{\textsc{o}}$ as a sub-matrix.

######  Remark.

Our method for estimating the extrapolated loading matrix
$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}$ relates to the factor-based
imputation procedure for multivariate data introduced in Cahan et al., (2023);
see also Bai and Ng, (2021). In fact,
$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}$ coincides with their loading
estimator when analysis is restricted to the completely observed sub-matrix
$\bm{Y}_{\textsc{c}}$. However, the authors essentially assume in our notation
that $\bm{X}_{\textsc{c}}$ and $\bm{X}_{\textsc{o}}$ have equal rank
$r_{\textsc{o}}$. By careful analysis of the reconstruction error, we
generalize the concept to the practically relevant case of
$\text{rank}(\bm{X}_{\textsc{c}})\gg\text{rank}(\bm{X}_{\textsc{o}})=r_{\textsc{o}}$,
which is a result of its own interest.

To reconstruct an incomplete function $X_{t}$ (for $t>T_{\textsc{c}}$), we set
$Y_{\textsc{o},t}=(Y_{ti}:u_{i}\in\textsc{O})$ and define

$\widehat{F}_{\textsc{o},t}=\frac{1}{N_{\textsc{o}}}Y_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\bm{D}_{\textsc{o}}^{-2},$

which coincides with the $t$-th row of $\widehat{\bm{F}}_{\textsc{o}}$ in (8)
for $t\leq T_{\textsc{c}}$. Define $\widehat{\Lambda}^{*}_{\textsc{o},i}$ to
be the $i$-th row of $\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}$. We then
estimate the reconstructed curve $\mathcal{L}(X_{t}^{\textsc{o}})$ by linear
interpolation,

$\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)=\sum_{i=1}^{N-1}\widehat{F}_{\textsc{o},t}\left(\widehat{\Lambda}^{*}_{\textsc{o},i}+\frac{u-u_{i}}{u_{i+1}-u_{i}}(\widehat{\Lambda}^{*}_{\textsc{o},i+1}-\widehat{\Lambda}^{*}_{\textsc{o},i})\right)^{\prime}\,\mathbbm{1}\\{u\in[u_{i},u_{i+1})\\},\qquad
u\in[0,1].$ (9)

Unless otherwise stated, we tacitly assume that the corresponding estimators
have rank $r_{\textsc{o}}$. The rank is typically unknown and must be
estimated from the data. Further details on this topic can be found in Section
5.

###  3.2 Convergence rates

As a next step, we discuss consistency of the estimated reconstructions. To
this end, let $C>1$ be some large absolute constant which may have a different
value at each appearance. We consider the following assumptions:

  1. (A3)

$(X_{t}:t\leq T)$ are i.i.d. zero mean with $X=(X(u):u\in[0,1])$; there exists
some $\kappa\in(0,2]$ such that $\mathbb{E}\,[(X(u)-X(v))^{2}]\leq C\lvert
u-v\rvert^{\kappa}$ for all $u,v\in[0,1]$;

  2. (A4)

$(e_{t}:t\leq T)$ are i.i.d. $N$-variate random variables with zero mean,
independent of $(X_{t})$ satisfying (a)
$\lambda_{\text{max}}(\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime})=O_{p}(\max\\{N_{\textsc{o}},T_{\textsc{c}}\\})$,
(b)
$\sum_{\begin{subarray}{c}\\{i,j\\}:u_{i},u_{j}\in\textsc{O}\end{subarray}}\lvert\mathbb{E}\,[e_{ti}e_{tj}]\rvert=O(N_{\textsc{o}})$
and (c) $\max_{i\leq N}\mathbb{E}\,[e_{ti}^{2}]<C$;

  3. (A5)

$\mathbb{E}\,[\sup_{u\in[0,1]}X(u)^{2}]$,
$\mathbb{E}\,[\sup_{u\in[0,1]}Z^{\textsc{o}}(u)^{2}]$,
$\mathbb{E}\,[X(u)^{4}]$, $\mathbb{E}\,[Z^{\textsc{o}}(u)^{4}]<C$ for any
$u\in[0,1]$; moreover, $\mathbb{E}\,[(\xi_{k}^{\textsc{o}})^{4}]\leq
C(\lambda_{k}^{\textsc{o}})^{2}$ for any $k\leq r_{\textsc{o}}$;

  4. (A6)

there exists some $M>0$ such that for any $m\geq 2$, $u\in\textsc{O}$,
$v\in[0,1]$ and $i\leq N$, $\mathbb{E}\,[\lvert
X_{t}^{\textsc{o}}(u)Z_{t}^{\textsc{o}}(v)\rvert^{m}],\mathbb{E}\,[\lvert
X_{t}^{\textsc{o}}(u)e_{ti}\rvert^{m}],\mathbb{E}\,[\lvert
e_{ti}^{2}-\mathbb{E}\,{[e_{ti}^{2}]}\rvert^{m}]\leq Cm!M^{m-2}/2$;

  5. (A7)

the $0=u_{1}<u_{2}<\dots<u_{N}=1$ define a grid of equispaced points;

  6. (A8)

$\sup_{k\leq
r_{\textsc{o}}}\sup_{u\in\textsc{O}}\lvert\phi_{k}^{\textsc{o}}(u)\rvert<C$
and $\max_{k,\ell\leq
r_{\textsc{o}}}\lvert\frac{1}{N_{\textsc{o}}}\sum_{i:u_{i}\in\textsc{O}}\phi_{k}^{\textsc{o}}(u_{i})\phi_{\ell}^{\textsc{o}}(u_{i})-\delta_{k\ell}\rvert=O(r_{\textsc{o}}/N_{\textsc{o}})$;

  7. (A9)

$r_{\textsc{o}}/\lambda_{r_{\textsc{o}}}^{\textsc{o}}=o(\min\\{\sqrt{N_{\textsc{o}}},\sqrt{T_{\textsc{c}}/\log(N)}\\})$,
$1/\sqrt{N^{\kappa}}=O(1/\lambda_{r}^{\textsc{o}}\sqrt{1/N_{\textsc{o}}+\log(N)/T_{\textsc{c}}})$,
as well as $\log(N)=o(T_{\textsc{c}})$.

Assumption (A3) entails that $X$ is mean-square continuous and is required to
control the discretization error. See also Kato, (2012) for a discussion of
this assumption under a regression framework. The perspective of processes
endows $X(u)$ at time $u$ with a meaning (whereas $L^{2}([0,1])$ is in
principle only comprised of equivalence classes; see Hsing and Eubank, (2015)
Section 7.4). The conditions on the noise term in Assumption (A4) are similar
to Bai and Ng, (2023). Assumption (A4)(a) is automatically satisfied in the
special case where $e_{ti}$ are i.i.d. (time and cross-section) with bounded
fourth moments (see Theorem 2 in Latał a, (2005) and Moon and Weidner, (2017)
for a further discussion). In general, Assumption (A4) also allows weak
dependence among the errors in the cross-section. Moment conditions in
Assumption (A5) are satisfied in the special case of Gaussian processes with
continuous sample paths. The conditions in Assumption (A6) are needed for
applying a Bernstein inequality in the proof of Theorem 1. Instead of moment
bounds, Fan et al., (2013) assume exponential tails. Assumption (A8) entails
heterogeneous loadings and is a mild condition on convergence of Riemann sums.
Consider for an illustration $\textsc{O}=[0,1]$ and functions $\phi_{k}$ with
$\sup_{u\in[0,1]}\lvert\phi_{k}(u)\rvert<C$ for every $k\leq r_{\textsc{o}}$.
If the $\phi_{k}$ are Lipschitz continuous with constants $L_{k}$, that is
$\lvert\phi_{k}(u)-\phi_{k}(v)\rvert\leq L_{k}\lvert u-v\rvert$ for any
$u,v\in[0,1]$, one can show that
$\lvert\frac{1}{N-1}\sum_{i=1}^{N-1}\phi_{k}(u_{i})\phi_{\ell}(u_{i})-\delta_{k\ell}\rvert\leq\frac{C}{N-1}\\{\max_{k\leq{r_{\textsc{o}}}}L_{k}\\}$.
Consequently, Assumption (A8) holds when
$\max_{k\leq{r_{\textsc{o}}}}L_{k}=O(r_{\textsc{o}})$ which is true for a sine
basis $\phi_{k}(u)=\sqrt{2}\sin(2k\pi u)$ with $L_{k}=\sqrt{8}k\pi$.
Assumption (A9) puts restrictions on the joint asymptotic behavior of the
variables $r_{\textsc{o}}$, $N_{\textsc{o}}\leq N$ and $T_{\textsc{c}}$. A
sufficient condition for the second rate is $\kappa\geq 1$.

######  Theorem 1.

Suppose that Assumptions (A1)-(A9) are satisfied. Then it holds that

$\sup_{u\in[0,1]}\,\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert=O_{p}\left(\frac{r_{\textsc{o}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{\log(N)}{T_{\textsc{c}}}}\right),$
(10)

as ${r_{\textsc{o}}}\to\infty$, $N_{\textsc{o}}\to\infty$ with $N\to\infty$
and $T_{\textsc{c}}\to\infty$ with $T\to\infty$.

Related to a discussion in Bai, (2003, p.140-141), we remark that the limit in
Theorem 1 is simultaneous in the sense that
$(r_{\textsc{o}},N_{\textsc{o}}(N),T_{\textsc{c}}(T))$ is allowed to grow
along all paths which satisfy the restrictions in Assumption (A9). The
$r_{\textsc{o}}$-th eigenvalue $\lambda_{r_{\textsc{o}}}^{\textsc{o}}$
measures pervasiveness of the weakest factor score. For this reason, a faster
decay of eigenvalues entails a more difficult reconstruction problem and the
factor $1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}$ therefore shows up in the
rate (10). The term $\sqrt{1/N_{\textsc{o}}+\log(N)/T_{\textsc{c}}}$ is
similarly derived by Fan et al., (2011) in approximate factor models with
fixed rank.

Let us now compare Theorem 1 with the related Theorem 4.2 in Kneip and Liebl,
(2020). From a theoretical perspective, Kneip and Liebl, (2020) consider a
more general model of _infinite_ dimensional functional data and derive a
nonparametric estimator of the approximation
$\mathcal{L}(X^{\textsc{o}})(u)=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}^{\textsc{o}}\tilde{\phi}_{k}^{\textsc{o}}(u)$
to $\sum_{k=1}^{\infty}\xi_{k}^{\textsc{o}}\tilde{\phi}_{k}^{\textsc{o}}(u)$
(employing our notation in the sequel). We note that in our setting
$r_{\textsc{o}}$ represents the growing dimension of the function space, while
it serves as a truncation parameter in Kneip and Liebl, (2020). Moreover,
Kneip and Liebl, (2020) assume almost surely twice continuously differentiable
functions $X_{t}$ which helps to control the discretization error and is
particularly useful in settings where the number of observation points
$N_{\textsc{o}}$ is comparably small. Since we impose no smoothness
assumptions, we compare the rates when $N_{\textsc{o}}$ is relatively large
compared to $T_{\textsc{c}}$. Assuming $\lambda_{k}=k^{\alpha_{\textsc{o}}}$
for some $\alpha_{\textsc{o}}>1$, the estimator
$\widehat{\mathcal{L}_{\text{KL}}(X^{\textsc{o}}_{t})}$ of Kneip and Liebl,
(2020) satisfies
$\lvert\widehat{\mathcal{L}_{\text{KL}}(X^{\textsc{o}}_{t})}(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert=O_{p}(r_{\textsc{o}}^{\alpha_{\textsc{o}}/2+5/2}\sqrt{1/T_{\textsc{c}}})$
for any fixed $u\in[0,1]$. In contrast, our rate is uniform in the sense of
converging suprema. Inspection of the proof of Theorem 1 actually shows that
the $\log(N)$ term in (10) is introduced by taking the supremum over
$u\in[0,1]$. Without the supremum, (10) reduces to
$\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert=O_{p}(r_{\textsc{o}}^{\alpha_{\textsc{o}}+1}\sqrt{1/T_{\textsc{c}}})$.
This shows that for large $N_{\textsc{o}}$, our rate improves Kneip and Liebl,
(2020) when rough functions with $\alpha_{\textsc{o}}\in(1,3)$ are considered.

For $\textsc{O}=[0,1]$, it holds $\mathcal{L}(X_{t})(u)=X_{t}(u)$ for any
$u\in[0,1]$ and the corresponding estimator
$\widehat{X}_{t}(u)=\widehat{\mathcal{L}(X_{t})}(u)$ can thus be seen as a
smoother of the function $X_{t}=(X_{t}(u):u\in[0,1])$. The following corollary
is immediate.

######  Corollary 1.

Suppose that Assumptions (A1)-(A9) are satisfied for $\textsc{O}=[0,1]$. Then
it holds that

$\sup_{u\in[0,1]}\,\lvert\widehat{X}_{t}(u)-X_{t}(u)\rvert=O_{p}\left(\frac{r_{\textsc{o}}}{\lambda^{\textsc{o}}_{r_{\textsc{o}}}}\sqrt{\frac{1}{N}+\frac{\log(N)}{T_{\textsc{c}}}}\right),$

as $r_{\textsc{o}}\to\infty$, $N\to\infty$ and $T_{\textsc{c}}\to\infty$ with
$T\to\infty$.

The above result is of independent interest for preprocessing noisy functional
data and should be compared to a corresponding rate derived in Hörmann and
Jammoul, (2022). While focusing on uniform convergence rates for the maximum
over $t\leq T$, the authors show in our notation that $\max_{i\leq
N}\lvert\widehat{X}_{t}(u_{i})-X_{t}(u_{i})\rvert=O_{p}(1/T^{1/4}+T^{1/4}/\sqrt{N})$
(for simplicity, we consider the rank to be fixed). As indicated by Corollary
1, careful analysis of the involved quantities allows us to improve this bound
to $O_{p}(\sqrt{1/N+\log(N)/T_{\textsc{c}}})$.

##  4 Simultaneous confidence bands

Once convergence of the reconstruction estimator is understood, questions
concerning inferential procedures arise. In practical applications, confidence
or prediction bands are desired to assess the reconstruction accuracy. A
method for constructing such bands in the context of partially observed
functional data has been suggested by Kraus, (2015). However, as Kraus,’
method assumes measurements without error, the construction of confidence
bands in the noisy observation regime (5) remains an open question. We address
it in the following and propose simultaneous confidence bands for the
unobserved functions $X_{t}^{\textsc{m}}=(X_{t}(u):u\in\textsc{M})$.

Motivated by Theorem 1 and Corollary 1, we estimate $Z^{\textsc{o}}_{t}$ for
$t\leq T_{\textsc{c}}$ by

$\widehat{Z}{}^{\textsc{o}}_{t}(u)=\begin{cases}\begin{aligned}
&\widehat{X}_{t}(u)-\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u),&&\qquad
u\in\textsc{M},\\\ &0,&&\qquad u\in\textsc{O}.\end{aligned}\end{cases}$

Note that consistency of the estimator $\widehat{X}_{t}$ requires Assumptions
(A1)-(A9) be true for $\textsc{O}=[0,1]$, which is tacitly assumed henceforth.
Let us define $r:=\text{rank}(\Gamma)$. For any $\textsc{O}\subset[0,1]$, the
rank $r_{\textsc{o}}$ of $\mathcal{L}(X_{t}^{\textsc{o}})(u)$ then satisfies
$r_{\textsc{o}}\leq r$ as
$\text{rank}(\Gamma^{\textsc{o}})\leq\text{rank}(\Gamma)$. Therefore, we let
$r_{\textsc{o}}$ diverge with $r\to\infty$ from now on. Invoking Theorem 1 and
Corollary 1, it follows that
$\sup_{u\in[0,1]}\,\lvert\widehat{Z}{}^{\textsc{o}}_{t}(u)-Z^{\textsc{o}}_{t}(u)\rvert=o_{p}(1)$
for any $t\leq T_{\textsc{c}}$. In addition, we estimate
$\sigma_{Z}(u,v)=\mathbb{E}\,[Z^{\textsc{o}}(u)Z^{\textsc{o}}(v)]$ by

$\widehat{\sigma}_{Z}(u,v)=\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}\widehat{Z}^{\textsc{o}}_{t}(u)\widehat{Z}^{\textsc{o}}_{t}(v),\qquad
u,v\in[0,1].$

Simultaneous confidence bands for
$X_{t}^{\textsc{m}}=(X_{t}(u):u\in\textsc{M})$ are then based on the following
result.

######  Theorem 2.

Let $\alpha\in(0,1)$ and suppose that Assumptions (A1)-(A9) are satisfied. If
$\sigma_{Z}$ is bounded from above and below,

$\lim\,\mathbb{P}\,\left(\sup_{u\in\textsc{M}}\,\frac{\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)-X_{t}(u)\rvert}{\sqrt{\widehat{\sigma}_{Z}(u,u)}}>q_{\alpha}\right)\leq\alpha,$

where $\mathbb{P}\,(\sup_{u\in\textsc{M}}\\{\lvert
Z^{\textsc{o}}(u)\rvert/\sqrt{\sigma_{Z}(u,u)}\\}>q_{\alpha})=\alpha$ and the
limit is taken as $r_{\textsc{o}}\to\infty$ with $r\to\infty$,
$N_{\textsc{o}}\to\infty$ with $N\to\infty$ and $T_{\textsc{c}}\to\infty$ with
$T\to\infty$.

The above result is based on the unknown value of $q_{\alpha}$ which
estimation is typically a challenging task. Kraus, (2015) proposes to estimate
$q_{\alpha}$ in a similar setting by sampling from Gaussian processes. Here,
we pursue a different approach. To this end, let us define the random
variables $\zeta_{t}=\sup_{u\in\textsc{M}}\\{\lvert
Z_{t}^{\textsc{o}}(u)\rvert/\sqrt{\sigma_{Z}(u,u)}\\}$ and
$\widehat{\zeta}_{t}=\sup_{u\in\textsc{M}}\\{\lvert\widehat{Z}_{t}^{\textsc{o}}(u)\rvert/\sqrt{\widehat{\sigma}_{Z}(u,u)}\\}$.
We then estimate $q_{\alpha}$ by the empirical $\alpha$-quantile
$\widehat{q}_{\alpha}$ of
$(\widehat{\zeta}_{1},\dots,\widehat{\zeta}_{T_{\textsc{c}}})$.

######  Proposition 2.

Let the assumptions of Theorem 2 be satisfied and suppose that the cdf
$F_{\zeta}$ of $\zeta$ is continuous. Then,
$F_{\zeta}(\widehat{q}_{\alpha})\to\alpha$ in probability as
$r_{\textsc{o}}\to\infty$ with $r\to\infty$, $N_{\textsc{o}}\to\infty$ with
$N\to\infty$ and $T_{\textsc{c}}\to\infty$ with $T\to\infty$.

Using Theorem 2 and Proposition 2, we are finally able to state simultaneous
confidence bands for $X_{t}^{\textsc{m}}$,

$\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)\pm\widehat{q}_{\alpha}\sqrt{\widehat{\sigma}_{Z}(u,u)},\qquad
u\in\textsc{M}.$ (11)

Performance of the described procedure in finite samples is discussed in the
next section.

##  5 Finite sample properties

The purpose of this section is to analyze performance of the previously
introduced estimator in finite samples. To distinguish the proposed procedure
from alternative methods, we henceforth refer to it as AFM estimator,
highlighting its relation to approximate factor models. Simulations are
carried out in R (R Core Team, (2022)) using the companion package FDFM
(downloadable from the first author’s GitHub account). We consider the
following process,

$X_{t}(u)=\mu(u)+\sum_{k=1}^{10}\xi_{tk}\phi_{k}(u),\qquad u\in[0,1],$ (12)

where $\mu(u)=\sin(\pi u)$, $\phi_{2\ell-1}(u)=\sqrt{2}\sin(2\ell\pi u)$ and
$\phi_{2\ell}(u)=\sqrt{2}\cos(2\ell\pi u)$ for $\ell=1,\dots,5$. Furthermore,
we consider i.i.d. standard normal distributed variables $\eta_{tk}\sim
N(0,1)$ and define $\xi_{tk}=\sqrt{\lambda_{k}}\eta_{tk}$ with
$\lambda_{k}=k^{-3}$. With this choice for the rate of decay, we obtain
essentially the same theoretical error bound as Kneip and Liebl, (2020); see
the discussion following Theorem 1. The process is recorded on a grid
$0=u_{1}<u_{2}<\dots<u_{N}=1$ of $N=51$ equispaced points and measured with
error,

$Y_{ti}=X_{t}(u_{i})+e_{ti},$ (13)

where $e_{ti}$ are i.i.d. $N(0,0.01)$ distributed random variables. The data
generating process (DGP) produces the $(T_{\textsc{c}}\times N)$ matrix
$\bm{Y}_{\textsc{c}}$ of complete observations, henceforth referred to as
_training set_. To assess the performance of the proposed reconstruction
procedure, we additionally generate _test sets_ as follows.

DGPA. For $s=1,\dots,100$, set $\textsc{O}_{s}=[0,D_{s}]$ where $D_{s}\sim
U(1/2,2/3)$. We then define targets (with slight abuse of notation)
$Y_{si}=X_{s}(u_{i})+e_{si}$ as in (13) except that observations
$u_{i}\notin\textsc{O}_{s}$ are considered to be missing.

DGPB. For $s=1,\dots,100$, set $\textsc{O}_{s}=[0,A_{s}]\,\cup\,[B_{s},1]$
where $A_{s}\sim U(0,1/2)$ and $B_{s}=A_{s}+1/2$. Similar to before, we set
$Y_{si}=X_{s}(u_{i})+e_{si}$ where observations $u_{i}\notin\textsc{O}_{s}$
are treated as missing.

The intervals $\textsc{O}_{s}$ in DGPA have mean length $7/12$ and are
generally larger than the sets $\textsc{O}_{s}$ in DGPB with length $1/2$.
Figure 2 shows examples of curves along with their reconstructions for both
data generating processes. All estimators are based on an unknown rank which
estimation is discussed in the next section. To assess empirical performance,
we generate a training and a test set in $B=100$ simulation runs indexed by
$b=1,\dots,B$. The _maximum absolute error_ ($\text{MAE}_{b}$) in iteration
$b$ is then estimated by

$\text{MAE}_{b}=\frac{1}{100}\sum_{s=1}^{100}\max_{i\leq N}\,\lvert
X_{b,s}(u_{i})-\widehat{\mathcal{L}_{\widehat{r}_{\textsc{o}}}(X_{b,s}^{\textsc{o}})}(u_{i})\rvert,$

where
$\widehat{\mathcal{L}_{\widehat{r}_{\textsc{o}}}(X_{b,s}^{\textsc{o}})}(u)$
denotes the estimated reconstruction of $X_{b,s}^{\textsc{O}}$ computed on the
training set $b$ for some estimator $\widehat{r}_{\textsc{o}}$ and
$\textsc{O}=\textsc{O}_{b,s}$. We further define

$\text{MAE}=\frac{1}{B}\sum_{b=1}^{B}\text{MAE}_{b},\qquad\text{SD}=\sqrt{\frac{1}{B}\sum_{B=1}^{B}(\text{MAE}_{b}-\text{MAE})^{2}},$
(14)

measuring the mean and standard deviation of maximum absolute errors over all
simulation runs.

(a)  (b)

Figure 2: Examples of reconstructed curves with 95% prediction bands for the
data generating processes in Section 5 and a training set of size
$T_{\textsc{c}}=200$. Solid lines refer to reconstructions, dotted lines to
true functions, and points to noisy observations. The number of factors
$\widehat{r}_{\textsc{o}}$ was chosen to be 7.

###  5.1 Choosing the number of factors

Several methods for estimating the number of factors in approximate factor
models have been suggested in the literature. In the following, we restrict
ourselves to two prominent methods which are due to Bai and Ng, (2002) and
Onatski, (2010). From the perspective of partially observed functional data, a
GCV-type criterion has been introduced by Kraus, (2015) and adopted by Kneip
and Liebl, (2020). We refer to the original works for a deeper insight. Note
that the estimates of ${r_{\textsc{o}}}$ are O-specific and computed on the
training set. The methods depend on a maximum value $r_{\text{max}}$ of
$r_{\textsc{o}}$ which needs to be chosen by the practitioner and is set to
$r_{\text{max}}=15$ throughout this section.

Figure 3 summarizes the estimation procedures’ performance under various
settings. As can be seen, all procedures perform arguably well without a clear
favorite. This indicates that the proposed reconstruction estimator is
compatible with standard procedures for choosing the number of factors. For
the remainder of this work, we stay with the GCV-type criterion of Kraus,
(2015) which also leads to comprehensible results in the real data study.

Figure 3: MAE (14) of different estimation procedures for $r_{\textsc{o}}$.

###  5.2 Comparison of reconstruction estimators

Next, we compare our AFM estimator to alternative reconstruction procedures.
First, we examine the PACE method of Yao et al., (2005) which in principle
assumes sparse functional data and recovers functions from irregular
observations inside the (complete) interval $[0,1]$. In addition, we consider
a reconstruction procedure proposed by Kneip and Liebl, (2020) which is
specifically intended for the partial observation regime as in the underlying
work. In fact, the latter is very similar to our AFM method since both are
based on the linear reconstruction problem (1). However, unlike our method,
Kneip and Liebl, (2020) as well as Yao et al., (2005) base on smoothing and
thus require smooth functions.

The linear reconstruction estimator of Kneip and Liebl, (2020) comes with
different options. More specifically, Kneip and Liebl, (2020) suggest to
either estimate the function’s observed and unobserved part at once using a
linear reconstruction estimator, or perform smoothing on the observed part and
use the reconstruction estimator solely for the missing part. In the latter
case, some alignment is required to ensure continuous reconstructions.
Moreover, the authors estimate principal scores either by approximating
integrals or using conditional expectations (CE). As methods involving
integral scores are found to be inferior in our simulation study, we restrict
our attention to the CE method. Following Kneip and Liebl, (2020), we use the
acronyms ANoCE (CE scores, without alignment) and AYesCE (CE scores, with
alignment) to distinguish different options. For computation, we rely on the
R-package ReconstPoFD which companions the work of Kneip and Liebl, (2020).

Table 1 summarizes simulation results for DGPA reporting MAE and SD (14). The
proposed AFM procedure is seen to clearly outperform its competitors in every
scenario. ANoCE involving alignment performs second while PACE and AYesCE
exhibit a similar performance with PACE being preferable for
$T_{\textsc{c}}=200$. Results for DGPB are reported in Table 2. Here, the
observation sets constitute a union of two intervals
$\textsc{O}_{s}=[0,A_{s}]\,\cup\,[B_{s},1]$. While the methodology of Kneip
and Liebl, (2020) in principle also works for such observation regimes, it is
not supported in the R-package ReconstPoFD and thus excluded from the
analysis. As it can be seen, AFM is again found to be superior to PACE with
errors being generally larger than in DGPA.

DGP | $T_{\textsc{c}}$ | AFM | PACE | ANoCE | AYesCE
---|---|---|---|---|---
A | 50 | 0.37 (0.02) | 0.61 (0.12) | 0.55 (0.08) | 0.60 (0.06)
| 100 | 0.33 (0.01) | 0.55 (0.10) | 0.50 (0.05) | 0.56 (0.04)
| 200 | 0.31 (0.01) | 0.49 (0.07) | 0.45 (0.03) | 0.54 (0.03)
Table 1: MAE and SD (in parentheses) for reconstruction estimators AFM (this work), PACE (Yao et al., (2005)), and estimators proposed by (Kneip and Liebl, (2020)). DGP | $T_{\textsc{c}}$ | AFM | PACE
---|---|---|---
B | 50 | 0.47 (0.02) | 0.71 (0.10)
| 100 | 0.42 (0.02) | 0.65 (0.08)
| 200 | 0.41 (0.02) | 0.57 (0.05)
Table 2: MAE and SD (in parentheses) for reconstruction estimators AFM (this
work) and PACE (Yao et al., (2005)).

###  5.3 Adequacy of confidence bands

To analyze the performance of simultaneous confidence bands (11), we simulate
training as well as test data and compute the proportion of bands covering the
corresponding test targets in each simulation run. For estimating the global
rank $r=\text{rank}(\Gamma)$, we resort to the method of Onatski, (2010).
Table 3 reports mean coverage probabilities along with standard deviations,
where the quantities are taken over estimated coverages in $B=100$ simulation
runs.

DGP | $T_{\textsc{c}}$ | nominal coverage
---|---|---
75% | 90% | 95%
A | 50 | 0.73 (0.08) | 0.86 (0.06) | 0.90 (0.06)
| 100 | 0.76 (0.05) | 0.89 (0.04) | 0.93 (0.03)
| 200 | 0.78 (0.04) | 0.90 (0.03) | 0.94 (0.02)
B | 50 | 0.75 (0.07) | 0.87 (0.07) | 0.91 (0.05)
| 100 | 0.78 (0.04) | 0.91 (0.03) | 0.94 (0.02)
| 200 | 0.80 (0.04) | 0.92 (0.03) | 0.95 (0.02)
Table 3: Mean coverage probabilities and standard deviations (in parentheses).

###  5.4 Real data analysis

We return to the temperature data set from the introduction to further
illustrate our methodology. The data consists of $N=48$ half-hourly recorded
temperature values from the east of Graz (Austria) between July 1 and August
31, 2022 (Land Steiermark, (2023)). See Figure 1 for a plot of daily
temperature curves. Out of $T=62$ curves in total, only $T_{\textsc{c}}=54$
are observed at all discretization points. Reconstruction of the remaining
partially observed curves is our immediate goal. To this end, we resort to our
proposed AFM procedure and the GCV-type criterion of Kraus, (2015) for
determining the number of factors $r_{\textsc{o}}$. Figure 4 plots the
reconstructed temperature curves. For a comparative purpose, we include
temperature curves of a different measuring site located in the south of Graz.
With one exception, the proposed AFM method recovers the missing fragments
very well. On August 18 we see a sudden drop in temperature around 4pm caused
by a storm. Since this storm occurred during the outage of the measuring
device, the reconstruction significantly deviates from the observed
temperature curve at the benchmarking site.

##  6 Acknowledgment

This research was partly funded by the Austrian Science Fund (FWF) [P 35520].

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Figure 4: Reconstructions (solid) of imperfectly observed temperature curves
in the east of Graz (Austria) with 95% prediction bands. Points refer to noisy
observations and dotted lines to a different measuring site located in the
south.

##  Appendix A Main proofs

In the following, we heavily rely on notation and arguments from Bai and Ng,
(2023). Regarding norms, let $\lVert\cdot\rVert_{2}$ and
$\lVert\cdot\rVert_{\textup{F}}$ denote the spectral and Frobenius norm of a
matrix, respectively, and $\lVert\cdot\rVert$ the euclidean norm of a vector.
Recall that $\lVert\bm{A}x\rVert_{2}\leq\lVert\bm{A}\rVert_{2}\lVert
x\rVert,\lVert\bm{A}x\rVert_{\textup{F}}\leq\lVert\bm{A}\rVert_{\textup{F}}\lVert
x\rVert,\lVert\bm{A}\bm{B}\rVert_{2}\leq\lVert\bm{A}\rVert_{2}\lVert\bm{B}\rVert_{2},\lVert\bm{A}\bm{B}\rVert_{\textup{F}}\leq\lVert\bm{A}\rVert_{\textup{F}}\lVert\bm{B}\rVert_{\textup{F}}$
and $\lVert\bm{A}\rVert_{2}\leq\lVert\bm{A}\rVert_{\textup{F}}$, for any
matrix $\bm{A}$, $\bm{B}$ and vector $x$. Additionally, we define the
following vectors. For any $t\leq T$ and $i\leq N$, let
$Y_{\textsc{c},i}=(Y_{1i},\dots,Y_{Ti})^{\prime}$,
$e_{\textsc{c},i}=(e_{1i},\dots,e_{Ti})^{\prime}$ and
$Z_{\textsc{o},i}=(Z_{1}^{\textsc{o}}(u_{i}),\dots,Z_{T}^{\textsc{o}}(u_{i}))^{\prime}$
be $(T\times 1)$ vectors. Moreover let
$Y_{\textsc{o},t}=(Y_{ti}:u_{i}\in\textsc{O})$ and
$e_{\textsc{o},t}=(e_{ti}:u_{i}\in\textsc{O})$ be $(1\times N_{\textsc{o}})$
vectors and
$F_{\textsc{o},t}=(\xi_{t1}^{\textsc{o}}/\sqrt{\lambda_{1}^{\textsc{o}}},\dots,\xi_{t{r_{\textsc{o}}}}^{\textsc{o}}/\sqrt{\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$,
$\widehat{F}_{\textsc{o},t}=\frac{1}{N_{\textsc{o}}}Y_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\bm{D}_{\textsc{o}}^{-2}$,
$\Lambda^{*}_{\textsc{o},i}=(\sqrt{\lambda^{\textsc{o}}_{1}}\tilde{\phi}_{1}^{\textsc{o}}(u_{i}),\dots,\sqrt{\lambda^{\textsc{o}}_{r_{\textsc{o}}}}\tilde{\phi}_{r_{\textsc{o}}}^{\textsc{o}}(u_{i}))$,
as well as
$\widehat{\Lambda}^{*}_{\textsc{o},i}=\frac{1}{T_{\textsc{c}}}Y_{\textsc{c},i}^{\prime}\widehat{\bm{F}}_{\textsc{o}}$
be $(1\times r_{\textsc{o}})$ vectors.

Rewrite $\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})$ in (2) as

$\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})=\sum_{k=1}^{r_{\textsc{o}}}\xi_{tk}^{\textsc{o}}\tilde{\phi}_{k}^{\textsc{o}}(u_{i})=\Lambda^{*}_{\textsc{o},i}F_{\textsc{o},t}^{\prime},$
(15)

Combining the above with (6), we get

$Y_{ti}=\Lambda^{*}_{\textsc{o},i}F_{\textsc{o},t}^{\prime}+Z_{\textsc{o},ti}+e_{ti}.$
(16)

Accordingly, $Y_{ti}$ equals
$\Lambda^{*}_{\textsc{o},i}F_{\textsc{o},t}^{\prime}$ plus additional error
terms attributed to reconstruction and noise.

###  A.1 Proof of Theorem 1

To prove Theorem 1, we need to recover $F_{\textsc{o},t}$ and
$\bm{\Lambda}_{\textsc{o}}^{*}$ from $\bm{Y}$ which is only possible up to
rotation. Let us define

  * •

$\bm{G}_{\textsc{o}}=\bm{\Lambda}_{\textsc{o}}^{\prime}\widehat{\bm{\Lambda}}_{\textsc{o}}(\widehat{\bm{\Lambda}}_{\textsc{o}}{}^{\prime}\widehat{\bm{\Lambda}}_{\textsc{o}})^{-1}=\frac{\bm{\Lambda}_{\textsc{o}}^{\prime}\widehat{\bm{\Lambda}}_{\textsc{o}}}{N_{\textsc{o}}}\bm{D}_{\textsc{o}}^{-2}$,

  * •

$\bm{H}_{\textsc{o}}=(\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}})^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}=(\frac{\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}}}{T_{\textsc{c}}})^{-1}$;

the involved quantities have already been introduced in Section 2. As can be
seen, $\bm{G}_{\textsc{o}}$ results from the regression of columns of
$\bm{\Lambda}_{\textsc{o}}$ onto columns of
$\widehat{\bm{\Lambda}}_{\textsc{o}}$. In the same vein,
$\bm{H}_{\textsc{o}}^{-1}$ resembles a regression of $\bm{F}_{\textsc{o}}$
onto $\widehat{\bm{F}}_{\textsc{o}}$. Appendix D shows that the matrices
$\bm{G}_{\textsc{o}}$ and $\bm{H}_{\textsc{o}}$ are asymptotically equivalent
in the sense that
$\lVert\bm{G}_{\textsc{o}}-\bm{H}_{\textsc{o}}\rVert_{2}=o_{p}(1)$. See also
Section 3.2 in Bai and Ng, (2023) for a discussion on equivalence of rotation
matrices. The proofs of the following lemmas are given in Appendix B and C,
respectively.

######  Lemma 1.

Under the assumptions of Theorem 1, for every $t\leq T$,

$\lVert\widehat{F}_{\textsc{o},t}-F_{\textsc{o},t}\bm{G}_{\textsc{o}}\rVert=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right).$

######  Lemma 2.

Under the assumptions of Theorem 1,

$\max_{i\leq
N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}-\Lambda^{*}_{\textsc{o},i}(\bm{H}_{\textsc{o}}^{\prime})^{-1}\rVert=O_{p}\left(\sqrt{\frac{r_{\textsc{o}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{1}{N_{\textsc{o}}}+\frac{\log(N)}{T_{\textsc{c}}}\right)}\right).$

###### Proof of Theorem 1.

We deduce from decomposition (15) of $\mathcal{L}(X_{t}^{\textsc{o}})$,

$\displaystyle\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u_{i})-\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})=\widehat{\Lambda}^{*}_{\textsc{o},i}\widehat{F}_{\textsc{o},t}{}^{\prime}-\Lambda^{*}_{\textsc{o},i}F_{\textsc{o},t}^{\prime}$
$\displaystyle\qquad=(\widehat{\Lambda}^{*}_{\textsc{o},i}-\Lambda^{*}_{\textsc{o},i}(\bm{H}_{\textsc{o}}^{\prime})^{-1})(F_{\textsc{o},t}\bm{H}_{\textsc{o}})^{\prime}+\widehat{\Lambda}^{*}_{\textsc{o},i}(\widehat{F}_{\textsc{o},t}-F_{\textsc{o},t}\bm{G}_{\textsc{o}})^{\prime}+\widehat{\Lambda}^{*}_{\textsc{o},i}(\bm{G}_{\textsc{o}}-\bm{H}_{\textsc{o}})^{\prime}F_{\textsc{o},t}^{\prime}.$

Taking norms, we observe

$\max_{i\leq
N}\,\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u_{i})-\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})\rvert\leq
a+b+c,$

where

$\displaystyle a=\max_{i\leq
N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}-\Lambda^{*}_{\textsc{o},i}(\bm{H}_{\textsc{o}}^{\prime})^{-1}\rVert\lVert
F_{\textsc{o},t}\rVert\lVert\bm{H}_{\textsc{o}}\rVert_{2}$
$\displaystyle=O_{p}\left(\frac{{r_{\textsc{o}}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{\log(N)}{T_{\textsc{c}}}}\right),$
$\displaystyle b=\max_{i\leq
N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}\rVert\lVert\widehat{F}_{\textsc{o},t}-F_{\textsc{o},t}\bm{G}_{\textsc{o}}\rVert$
$\displaystyle=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right),$
$\displaystyle c=\max_{i\leq
N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}\rVert\lVert\bm{G}_{\textsc{o}}-\bm{H}_{\textsc{o}}\rVert_{2}\lVert
F_{\textsc{o},t}\rVert$
$\displaystyle=O_{p}\left(\frac{\sqrt{{r_{\textsc{o}}}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{{r_{\textsc{o}}}}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right).$

The above rates follow from Lemma 1, Lemma 2, Lemma 5, Lemma 6 and Lemma
8(ii), proven in the Appendix, as well as the fact $\lVert
F_{\textsc{o},t}\rVert=O_{p}(\sqrt{{r_{\textsc{o}}}})$. We see that the first
term dominates the others. As a next step, we treat the discretization error.
To this end, observe that for $u,v\in[0,1]$ by Assumption (A3),

$\displaystyle\lvert\mathcal{L}(X_{t}^{\textsc{o}})(u)-\mathcal{L}(X_{t}^{\textsc{o}})(v)\rvert$
$\displaystyle\leq\sum_{k=1}^{r_{\textsc{o}}}\lvert\xi_{tk}^{\textsc{o}}\rvert\frac{\mathbb{E}\,{[\lvert
X(u)-X(v)\rvert\lvert\xi_{k}^{\textsc{o}}\rvert}]}{\lambda_{k}^{\textsc{o}}}\leq\sum_{k=1}^{r_{\textsc{o}}}\frac{\lvert\xi_{tk}^{\textsc{o}}\rvert}{\sqrt{\lambda_{k}^{\textsc{o}}}}\mathbb{E}\,{[(X(u)-X(v))^{2}]}^{1/2}$
(17) $\displaystyle\leq
C\sum_{k=1}^{r_{\textsc{o}}}\frac{\lvert\xi_{tk}^{\textsc{o}}\rvert}{\sqrt{\lambda_{k}^{\textsc{o}}}}\lvert
u-v\rvert^{\kappa/2}=O_{p}\left({r_{\textsc{o}}}\lvert
u-v\rvert^{\kappa/2}\right).$

Now, define $\mathcal{L}^{*}:L^{2}(\textsc{O})\to L^{2}([0,1])$ to be the
operator which results form $\mathcal{L}$ by linear interpolation, that is,
for $f\in L^{2}(\textsc{O})$ and $u\in[0,1]$,

$\mathcal{L}^{*}(f)(u)=\sum_{i=1}^{N-1}\left(\mathcal{L}(f)(u_{i})+\frac{u-u_{i}}{u_{i+1}-u_{i}}(\mathcal{L}(f)(u_{i+1})-\mathcal{L}(f)(u_{i}))\right)\,\mathbbm{1}\\{u\in[u_{i},u_{i+1})\\}.$

We get from (17) and Assumption (A7) on the equidistant spacing of grid
points,

$\displaystyle\sup_{u\in[0,1]}\,\lvert\mathcal{L}^{*}(X_{t}^{\textsc{o}})(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert=\max_{i\leq
N}\,\sup_{u\in[u_{i},u_{i+1})}\,\lvert\mathcal{L}^{*}(X_{t}^{\textsc{o}})(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert$
$\displaystyle\qquad\leq\max_{i\leq
N}\,\left(\sup_{u\in[u_{i},u_{i+1})}\,\lvert\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert+\lvert\mathcal{L}(X_{t}^{\textsc{o}})(u_{i+1})-\mathcal{L}(X_{t}^{\textsc{o}})(u_{i})\rvert\right)$
$\displaystyle\qquad=O_{p}\left({r_{\textsc{o}}}\max_{i\leq N}\,\lvert
u_{i+1}-u_{i}\rvert^{\kappa/2}\right)=O_{p}\left(\frac{{r_{\textsc{o}}}}{\sqrt{N^{\kappa}}}\right).$

Combining the above, we conclude

$\displaystyle\sup_{u\in[0,1]}\,\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert$
$\displaystyle\leq\max_{i\leq
N}\,\lvert\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u_{i})-\mathcal{L}^{*}(X_{t}^{\textsc{o}})(u_{i})\rvert+\sup_{u\in[0,1]}\,\lvert\mathcal{L}^{*}(X_{t}^{\textsc{o}})(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)\rvert$
$\displaystyle=O_{p}\left(\frac{{r_{\textsc{o}}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{\log(N)}{T_{\textsc{c}}}}\right)+O_{p}\left(\frac{{r_{\textsc{o}}}}{\sqrt{N^{\kappa}}}\right).$

The second term is dominated by the first due to Assumption (A9). ∎

###  A.2 Proof of Theorem 2

The definition
$Z_{t}^{\textsc{o}}(u)=X_{t}(u)-\mathcal{L}(X_{t}^{\textsc{o}})(u)$ yields

$\displaystyle\mathbb{P}\,\left(\sup_{u\in\textsc{M}}\,\frac{\lvert
X_{t}(u)-\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)\rvert}{\sqrt{\widehat{\sigma}_{Z}(u,u)}}>q_{\alpha}\right)$
$\displaystyle\quad\leq\mathbb{P}\,\left\\{\left(\sup_{u\in\textsc{M}}\,\frac{\lvert\mathcal{L}(X_{t}^{\textsc{o}})(u)-\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)\rvert}{\sqrt{\sigma_{Z}(u,u)}}+\sup_{u\in\textsc{M}}\,\frac{\lvert
Z_{t}^{\textsc{o}}(u)\rvert}{\sqrt{\sigma_{Z}(u,u)}}\right)\sup_{u\in\textsc{M}}\sqrt{\frac{\sigma_{Z}(u,u)}{\widehat{\sigma}_{Z}(u,u)}}>q_{\alpha}\right\\}.$

Next, let us define
$A=\sup_{u\in\textsc{M}}\,\frac{\lvert\mathcal{L}(X_{t}^{\textsc{o}})(u)-\widehat{\mathcal{L}(X_{t}^{\textsc{o}})}(u)\rvert}{\sqrt{\sigma_{Z}(u,u)}}$,
$B=\sup_{u\in\textsc{M}}\,\frac{\lvert
Z_{t}^{\textsc{o}}(u)\rvert}{\sqrt{\sigma_{Z}(u,u)}}$ and
$C=\sup_{u\in\textsc{M}}\sqrt{\frac{\sigma_{Z}(u,u)}{\widehat{\sigma}_{Z}(u,u)}}$.
For $\epsilon>0$, we get

$\mathbb{P}\,\left\\{(A+B)C>(\epsilon+q_{\alpha})(1+\epsilon)\right\\}\leq\mathbb{P}\,(A>\epsilon)+\mathbb{P}\,(B>q_{\alpha})+\mathbb{P}\,(C>1+\epsilon).$

By Theorem 1 and boundedness of $\sigma_{Z}(u,u)$, it holds
$\mathbb{P}\,(A>\epsilon)\to 0$. Furthermore,
$\mathbb{P}\,(B>q_{\alpha})=\alpha$ by the definition of $q_{\alpha}$. In
addition, using similar arguments as in the proof of Theorem 1, it can be
shown that $\mathbb{P}\,(C>1+\epsilon)\to 0$. Taking $\epsilon\to 0$ concludes
the proof of the theorem.

###  A.3 Proof of propositions

###### Proof of Proposition 1.

The assumption $\textup{rank}(\Gamma)={r_{\textsc{o}}}$ implies
$X(u)=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}\phi_{k}(u)$ a.s. for every
$u\in[0,1]$. Moreover,
$\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}\phi_{k}(u)=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}^{\textsc{o}}\phi_{k}^{\textsc{o}}(u)$
for any $u\in\textsc{O}$, whence
$\xi_{\ell}^{\textsc{o}}=\sum_{k=1}^{r_{\textsc{o}}}\xi_{k}\langle\phi_{k},\phi_{\ell}^{\textsc{o}}\rangle_{\textsc{o}}$
by orthogonality of the $\phi_{k}^{\textsc{o}}$ on O. Now, the rank assumption
implies that the matrix
$\Phi:=\\{\langle\phi_{k},\phi_{\ell}^{\textsc{o}}\rangle_{\textsc{o}}\\}_{k,\ell=1}^{r_{\textsc{o}}}$
is invertible, so
$\xi_{k}=\sum_{\ell=1}^{r_{\textsc{o}}}a_{k\ell}\xi_{\ell}^{\textsc{o}}$ for
some coefficients $a_{k\ell}$. Define the linear operator
$\widetilde{\mathcal{L}}:L^{2}(\textsc{O})\to L^{2}([0,1])$ by
$f\mapsto\sum_{k,\ell=1}^{r_{\textsc{o}}}a_{k\ell}\langle
f,\phi^{\textsc{o}}_{\ell}\rangle_{\textsc{o}}\phi_{k}(u)$. Because
$\mathcal{L}$ is the linear operator minimizing (1) and
$\widetilde{\mathcal{L}}(X^{\textsc{o}})(u)=X(u)$, we conclude
$\mathcal{L}(X^{\textsc{o}})(u)=X(u)$ a.s. for every $u\in[0,1]$. ∎

###### Proof of Proposition 2.

Let
$\widehat{F}_{\zeta}(z)=\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}\mathbbm{1}\\{\zeta_{t}\leq
z\\}$ denote the ecdf of $\zeta$ and additionally define
$\widetilde{F}_{\zeta}(z)=\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}\mathbbm{1}\\{\widehat{\zeta}_{t}\leq
z\\}$. Observe $\lvert F_{\zeta}(\widehat{q}_{\alpha})-\alpha\rvert\leq\lvert
F_{\zeta}(\widehat{q}_{\alpha})-\widehat{F}_{\zeta}(\widehat{q}_{\alpha})\rvert+\lvert\widehat{F}_{\zeta}(\widehat{q}_{\alpha})-\widetilde{F}_{\zeta}(\widehat{q}_{\alpha})\rvert+\lvert\widetilde{F}_{\zeta}(\widehat{q}_{\alpha})-\alpha\rvert$.
Invoking the theorem of Glivenko–Cantelli, $\lvert
F_{\zeta}(\widehat{q}_{\alpha})-\widehat{F}_{\zeta}(\widehat{q}_{\alpha})\rvert=o_{p}(1)$.
Moreover, using $\lvert\zeta_{t}-\widehat{\zeta}_{t}\rvert=o_{p}(1)$ and
continuity of the distribution function, it can be shown that
$\sup_{z}\lvert\widehat{F}(z)-\widetilde{F}(z)\rvert=o_{p}(1)$, implying
$\lvert\widehat{F}_{\zeta}(\widehat{q}_{\alpha})-\widetilde{F}_{\zeta}(\widehat{q}_{\alpha})\rvert=o_{p}(1)$.
Finally,
$\lvert\widetilde{F}_{\zeta}(\widehat{q}_{\alpha})-\alpha\rvert=o_{p}(1)$ by
the definition of empirical quantiles. We thus conclude that $\lvert
F_{\zeta}(\widehat{q}_{\alpha})-\alpha\rvert=o_{p}(1)$. ∎

##  Appendix B Factors: Proof of Lemma 1

###### Proof of Lemma 1.

Using
$\widehat{F}_{\textsc{o},t}=\frac{1}{N_{\textsc{o}}}Y_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\bm{D}_{\textsc{o}}^{-2}$
and
$Y_{\textsc{o},t}=F_{\textsc{o},t}\bm{\Lambda}_{\textsc{o}}^{\prime}+e_{\textsc{o},t}$,
we get

$\widehat{F}_{\textsc{o},t}-F_{\textsc{o},t}\bm{G}_{\textsc{o}}=\frac{1}{N_{\textsc{o}}}(F_{\textsc{o},t}\bm{\Lambda}_{\textsc{o}}^{\prime}+e_{\textsc{o},t})\widehat{\bm{\Lambda}}_{\textsc{o}}\bm{D}_{\textsc{o}}^{-2}-F_{\textsc{o},t}\bm{G}_{\textsc{o}}=\frac{1}{N_{\textsc{o}}}e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\bm{D}_{\textsc{o}}^{-2}.$

Taking norms,

$\lVert\widehat{F}_{\textsc{o},t}-F_{\textsc{o},t}\bm{G}_{\textsc{o}}\rVert\leq\frac{1}{N_{\textsc{o}}}\lVert
e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\rVert\lVert\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right),$

where we used $\lVert
e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\rVert=O_{p}(N_{\textsc{o}}/\sqrt{T_{\textsc{c}}}+\sqrt{N_{\textsc{o}}})$
by Lemma 10(iii) and
$\lVert\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}(1/\lambda^{\textsc{o}}_{r_{\textsc{o}}})$
by Lemma 9(iii). ∎

##  Appendix C Extrapolated Loadings: Proof of Lemma 2

Before proving Lemma 2, let us rewrite the matrix $\bm{G}_{\textsc{o}}$. Since

$\widehat{\bm{\Lambda}}_{\textsc{o}}=\frac{1}{T_{\textsc{c}}}\bm{Y}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}=\frac{1}{T_{\textsc{c}}}(\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}+\bm{e}_{\textsc{o}}^{\prime})\widehat{\bm{F}}_{\textsc{o}},$

we get

$\bm{G}_{\textsc{o}}=\frac{\bm{\Lambda}_{\textsc{o}}{{}^{\prime}}\widehat{\bm{\Lambda}}_{\textsc{o}}}{N_{\textsc{o}}}\bm{D}_{\textsc{o}}^{-2}=\left(\frac{\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}}{N_{\textsc{o}}}\frac{\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}}{T_{\textsc{c}}}+\frac{\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}}{N_{\textsc{o}}T_{\textsc{c}}}\right)\bm{D}_{\textsc{o}}^{-2}.$

The above representation of $\bm{G}_{\textsc{o}}$ turns out to be useful in
the subsequent pages. Note however that establishing a tight bound on
$\bm{G}_{\textsc{o}}$ is not trivial because
$\lVert\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}})$
and thus direct application of the spectral norm’s submultiplicativity only
yields a weak bound.

######  Lemma 3.

Under the assumptions of Theorem 1,

$\lVert\widehat{\bm{F}}_{\textsc{o}}-\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}}\rVert_{2}=O_{p}\left(\sqrt{\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{{r_{\textsc{o}}}T_{\textsc{c}}}{N_{\textsc{o}}}+1\right)}\right).$

###### Proof.

Similar to (24) in Bai and Ng, (2023), it can be shown that

$\widehat{\bm{F}}_{\textsc{o}}-\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}}=\left(\frac{\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}T_{\textsc{c}}}+\frac{\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}T_{\textsc{c}}}\right)(\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2})^{-1}\bm{B}_{\textsc{o}}^{-1}.$

Consequently,

$\displaystyle\lVert\widehat{\bm{F}}_{\textsc{o}}-\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}}\rVert_{2}$
$\displaystyle\leq\left(\frac{\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}\lVert\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}}{N_{\textsc{o}}T_{\textsc{c}}}+\frac{\lVert\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\rVert_{2}\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{B}_{\textsc{o}}^{-1}\rVert}{N_{\textsc{o}}T_{\textsc{c}}}\right)\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}\rVert_{2}\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}$
$\displaystyle=O_{p}\left(\sqrt{\frac{{r_{\textsc{o}}}T_{\textsc{c}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}N_{\textsc{o}}}}\right)+O_{p}\left(\frac{N_{\textsc{o}}+T_{\textsc{c}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}N_{\textsc{o}}\sqrt{T_{\textsc{c}}}}\right)=O_{p}\left(\sqrt{\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{{r_{\textsc{o}}}T_{\textsc{c}}}{N_{\textsc{o}}}+1\right)}\right),$

since
$\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(\sqrt{{r_{\textsc{o}}}N_{\textsc{o}}T_{\textsc{c}}})$
by Lemma 10(i),
$\lVert\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(T_{\textsc{c}})$
by Lemma 11(i),
$\lVert\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\rVert_{2}=O_{p}(N_{\textsc{o}}+T_{\textsc{c}})$
by Assumption (A4)(a),
$\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}^{2}=T_{\textsc{c}}$,
$\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$,
$\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}(1)$ by
Lemma 9(ii) and
$1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}=O(\min\\{N_{\textsc{o}},T_{\textsc{c}}\\})$
by Assumption (A9). ∎

######  Lemma 4.

Under the assumptions of Theorem 1,

  1. (i)

$\max_{i\leq
N}\,\Big{\lVert}\frac{(Z_{\textsc{o},i}+e_{c,i})^{\prime}\bm{F}_{\textsc{o}}}{T_{\textsc{c}}}\Big{\rVert}=O_{p}\Big{(}\sqrt{{r_{\textsc{o}}}\log(N)/T_{\textsc{c}}}\,\Big{)}$,

  2. (ii)

$\max_{i\leq N}\lVert e_{c,i}\rVert=O_{p}(\sqrt{T_{\textsc{c}}})$,

  3. (iii)

$\max_{i\leq N}\lVert Z_{\textsc{o},i}\rVert=O_{p}(\sqrt{T_{\textsc{c}}})$.

###### Proof.

  1. (i)

As a first step, we show

$\mathbb{E}\,\left[\lvert
Z_{\textsc{o},ti}F_{\textsc{o},tk}\rvert^{m}\right]\leq\frac{C_{0}}{2}m!M_{0}^{m-2}\qquad\text{for
any $m\geq 2$},$ (18)

where $C_{0}$ is some constant and $M_{0}$ satisfies
$M_{0}=O(\sqrt{T_{\textsc{c}}/\log(N)})$. For $m=2$, (18) follows from the
Cauchy-Schwarz inequality,

$\mathbb{E}\,[\lvert
Z_{\textsc{o},ti}F_{\textsc{o},tk}\rvert^{2}]\leq\left(\mathbb{E}\,[Z_{\textsc{o},ti}^{4}]\mathbb{E}\,[F_{\textsc{o},tk}^{4}]\right)^{1/2}<C,$

since $\mathbb{E}\,[Z_{\textsc{o},ti}^{4}]<C$ and
$\mathbb{E}\,[F_{\textsc{o},tk}^{4}]<C$ by Assumption (A5). Second, we note
that by the Hölder inequality,

$\mathbb{E}\,\left[\lvert
Z_{\textsc{o},ti}\xi^{\textsc{o}}_{tk}\rvert^{m}\right]=\mathbb{E}\,\left[\lvert\langle
X_{t}^{\textsc{o}}Z_{\textsc{o},ti},\phi_{k}^{\textsc{o}}\rangle_{\textsc{o}}\rvert^{m}\right]\leq\int_{\textsc{O}}\mathbb{E}\,\left[\lvert
X_{t}^{\textsc{o}}(u)Z_{t}^{\textsc{o}}(u_{i})\rvert^{m}\right]\lvert\phi_{k}^{\textsc{o}}(u)\rvert^{m}\,\textup{d}u.$

Moreover, $\mathbb{E}\,\left[\lvert
X_{t}^{\textsc{o}}(u)Z_{t}^{\textsc{o}}(u_{i})\rvert^{m}\right]\leq
Cm!M^{m-2}/2$ by Assumption (A6) and
$\sup_{u\in\textsc{o}}\lvert\phi_{k}^{\textsc{o}}(u)\rvert^{m}<C^{m}$ by
Assumption (A8). Combining the above with the definition
$F_{\textsc{o},tk}=\xi_{tk}^{\textsc{o}}/\sqrt{\lambda_{k}^{\textsc{o}}}$, we
get

$\mathbb{E}\,\left[\lvert
Z_{\textsc{o},ti}F_{\textsc{o},tk}\rvert^{m}\right]\leq
Cm!(CM)^{m-2}\left(\frac{1}{\lambda_{k}^{\textsc{o}}}\right)^{m/2}.$

For $m\geq 4$, we have $m/2\leq m-2$ and the above inequality yields

$\mathbb{E}\,\left[\lvert
Z_{\textsc{o},ti}F_{\textsc{o},tk}\rvert^{m}\right]\leq
Cm!\left(\frac{CM}{\lambda_{k}^{\textsc{o}}}\right)^{m-2}.$

So a suitable choice of $C_{0}$ and $M_{0}=CM/\lambda_{k}^{\textsc{o}}$ show
(18) as $M_{0}=O(\sqrt{T_{\textsc{c}}/\log(N)})$ by Assumption (A9). The case
$m=3$ of (18) can be obtained from the cases $m=2$ and $m\geq 4$. This
completes the proof of (18).

Now let $\kappa>0$. We deduce from the Bonferroni inequality that

$\displaystyle\mathbb{P}\,\Big{(}\max_{k\leq{r_{\textsc{o}}},i\leq
N}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Z_{\textsc{o},ti}F_{\textsc{o},tk}\Big{|}>\kappa\sqrt{\log(N)/T_{\textsc{c}}}\Big{)}$
$\displaystyle\qquad\qquad\leq
N{r_{\textsc{o}}}\max_{k\leq{r_{\textsc{o}}},i\leq
N}\,\mathbb{P}\,\Big{(}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Z_{\textsc{o},ti}F_{\textsc{o},tk}\Big{|}>\kappa\sqrt{\log(N)/T_{\textsc{c}}}\Big{)}.$

Using (18), the Bernstein inequality (Lemma 2.2.11 in van der Vaart and
Wellner, (2000)) implies

$\mathbb{P}\,\Big{(}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Z_{\textsc{o},ti}F_{\textsc{o},tk}\Big{|}>\kappa\sqrt{\log(N)/T_{\textsc{c}}}\Big{)}\leq
2\exp\left(-\frac{1}{2}\frac{\kappa^{2}\log(N)T_{\textsc{c}}}{C_{0}T_{\textsc{c}}+M_{0}\kappa\sqrt{\log(N)T_{\textsc{c}}}}\right).$

Hence,

$\mathbb{P}\,\Big{(}\max_{k\leq{r_{\textsc{o}}},i\leq
N}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Z_{\textsc{o},ti}F_{\textsc{o},tk}\Big{|}>\kappa\sqrt{\log(N)/T_{\textsc{c}}}\Big{)}\\\
\leq
2N{r_{\textsc{o}}}\exp\left\\{-\frac{1}{2}\frac{\kappa^{2}\log(N)T_{\textsc{c}}}{C_{0}T_{\textsc{c}}+M_{0}\kappa\sqrt{\log(N)T_{\textsc{c}}}}\right\\}$

which is $o(1)$ provided that $\kappa$ is large enough. Consequently,

$\max_{k\leq{r_{\textsc{o}}},i\leq
N}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}Z_{\textsc{o},ti}F_{\textsc{o},tk}\Big{|}=O_{p}(\sqrt{\log(N)/T_{\textsc{c}}}).$

A bound for $\max_{k\leq{r_{\textsc{o}}},i\leq
N}\lvert\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}e_{ti}F_{\textsc{o},tk}\rvert$
can be derived in a similar way using Assumption (A6). The assertion of the
lemma thus follows from

$\displaystyle\max_{i\leq
N}\,\Big{\lVert}\frac{(Z_{\textsc{o},i}+e_{i})^{\prime}\bm{F}_{\textsc{o}}}{T_{\textsc{c}}}\Big{\rVert}^{2}=\max_{i\leq
N}\,\sum_{k=1}^{r_{\textsc{o}}}\Big{(}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}(Z_{\textsc{o},ti}+e_{ti})F_{\textsc{o},tk}\Big{)}^{2}$
$\displaystyle\leq{r_{\textsc{o}}}\max_{k\leq{r_{\textsc{o}}},i\leq
N}\Big{\lvert}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}(Z_{\textsc{o},ti}+e_{ti})F_{\textsc{o},tk}\Big{\rvert}^{2}$
$\displaystyle=O_{p}({r_{\textsc{o}}}\log(N)/T_{\textsc{c}}).$

  2. (ii)

Again, the Bonferroni and Bernstein inequality yield

$\displaystyle\mathbb{P}\,\Big{(}\max_{i\leq
N}\Big{|}\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}(e_{ti}^{2}-\mathbb{E}\,{[e_{ti}^{2}]})\Big{|}>\kappa\sqrt{\log(N)/T_{\textsc{c}}}\Big{)}\leq
2N\exp\left(-\frac{1}{2}\frac{\kappa^{2}\log(N)T_{\textsc{c}}}{CT_{\textsc{c}}+M\kappa\sqrt{\log(N)T_{\textsc{c}}}}\right),$

where we have used Assumption (A6). Since the above term is $o(1)$ for large
enough $\kappa$, we get $\max_{i\leq
N}\lvert\frac{1}{T_{\textsc{c}}}\sum_{t=1}^{T_{\textsc{c}}}(e_{ti}^{2}-\mathbb{E}\,{[e_{ti}^{2}]})\rvert=O_{p}(\sqrt{\log(N)/T_{\textsc{c}}})$.
Together with $\max_{i\leq N}\mathbb{E}\,{[e_{ti}^{2}]}=O(1)$ by Assumption
(A4)(c),

$\max_{i\leq N}\,\lVert e_{i}\rVert^{2}=\max_{i\leq
N}\,\sum_{t=1}^{T_{\textsc{c}}}e_{ti}^{2}\leq\max_{i\leq
N}\,\Big{\lvert}\sum_{t=1}^{T_{\textsc{c}}}(e_{ti}^{2}-\mathbb{E}\,{[e_{ti}^{2}]})\Big{\rvert}+T_{\textsc{c}}\max_{i\leq
N}\mathbb{E}\,{[e_{ti}^{2}]}=O(T_{\textsc{c}}),$

where we have used $\log(N)=O(T_{\textsc{c}})$ by Assumption (A9).

  3. (iii)

Concerning the final assertion, we deduce from Assumption (A5),

$\max_{i\leq N}\,\lVert Z_{\textsc{o},i}\rVert^{2}=\max_{i\leq
N}\,\sum_{t=1}^{T_{\textsc{c}}}Z_{t}^{\textsc{o}}(u_{i})^{2}\leq\sum_{t=1}^{T_{\textsc{c}}}\sup_{u\in[0,1]}\,Z_{t}^{\textsc{o}}(u)^{2}=O_{p}(T_{\textsc{c}}).$

∎

###### Proof of Lemma 2.

Plugging (16) into
$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}=\frac{1}{T_{\textsc{c}}}\bm{Y}_{\textsc{c}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}$
implies

$\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}=\bm{\Lambda}^{*}_{\textsc{o}}\,\frac{\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}}{T_{\textsc{c}}}+\frac{(\bm{Z}_{\textsc{o}}+\bm{e}_{\textsc{c}})^{\prime}\widehat{\bm{F}}_{\textsc{o}}}{T_{\textsc{c}}}.$

Rearranging terms, we get

$\displaystyle\widehat{\bm{\Lambda}}^{*}_{\textsc{o}}-\bm{\Lambda}^{*}_{\textsc{o}}(\bm{H}_{\textsc{o}}^{\prime})^{-1}=\frac{(\bm{Z}_{\textsc{o}}+\bm{e}_{\textsc{c}})^{\prime}\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}}}{T_{\textsc{c}}}+\frac{(\bm{Z}_{\textsc{o}}+\bm{e}_{\textsc{c}})^{\prime}(\widehat{\bm{F}}_{\textsc{o}}-\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}})}{T_{\textsc{c}}}.$

Taking norms,

$\displaystyle\max_{i\leq
N}\,\lVert\Lambda^{*}_{\textsc{o},i}-\Lambda^{*}_{\textsc{o},i}(\bm{H}_{\textsc{o}}^{\prime})^{-1}\rVert$
$\displaystyle\qquad\leq\max_{i\leq
N}\,\Big{\lVert}\frac{(Z_{\textsc{o},i}+e_{c,i})^{\prime}\bm{F}_{\textsc{o}}}{T_{\textsc{c}}}\Big{\rVert}\lVert\bm{G}_{\textsc{o}}\rVert_{2}+\left(\max_{i\leq
N}\,\frac{\lVert Z_{\textsc{o},i}\rVert}{\sqrt{T_{\textsc{c}}}}+\max_{i\leq
N}\,\frac{\lVert
e_{c,i}\rVert}{\sqrt{T_{\textsc{c}}}}\right)\frac{\lVert\widehat{\bm{F}}_{\textsc{o}}-\bm{F}_{\textsc{o}}\bm{G}_{\textsc{o}}\rVert_{2}}{\sqrt{T_{\textsc{c}}}}$
$\displaystyle\qquad=O_{p}\left(\sqrt{\frac{{r_{\textsc{o}}}\log(N)}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}T_{\textsc{c}}}}\right)+O_{p}(1)O_{p}\left(\sqrt{\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{{r_{\textsc{o}}}}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}\right)}\right)$
$\displaystyle\qquad=O_{p}\left(\sqrt{\frac{r_{\textsc{o}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{1}{N_{\textsc{o}}}+\frac{\log(N)}{T_{\textsc{c}}}\right)}\right).$

where we have used Lemma 3, Lemma 4 and Lemma 5(i). ∎

##  Appendix D Asymptotic equivalence of $\bm{G}_{\textsc{o}}$ and
$\bm{H}_{\textsc{o}}$

######  Lemma 5.

Under the assumptions of Theorem 1,

  1. (i)

$\lVert\bm{G}_{\textsc{o}}\rVert_{2}=O_{p}(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$,

  2. (ii)

$\lVert\bm{H}_{\textsc{o}}\rVert_{2}=O_{p}(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$,

  3. (iii)

$\lVert\bm{H}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(\sqrt{1/r_{\textsc{o}}})$.

###### Proof.

  1. (i)

Follows from the next lemma and
$\lVert\bm{H}_{\textsc{o}}\rVert_{2}=O_{p}(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$.

  2. (ii)

We deduce from Lemma 11(ii),

$\displaystyle\lVert\bm{H}_{\textsc{o}}\rVert_{2}$
$\displaystyle=\Big{\lVert}\Big{(}\frac{\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}}}{T_{\textsc{c}}}\Big{)}^{-1}\Big{\rVert}_{2}\leq\lVert\bm{B}_{\textsc{o}}\rVert_{2}\Big{\lVert}\Big{(}\frac{\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}}\bm{B}_{\textsc{o}}}{T_{\textsc{c}}}\Big{)}^{-1}\Big{\rVert}_{2}\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}$
$\displaystyle=O_{p}(1)O_{p}(1)O_{p}\left(\frac{1}{\sqrt{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}}\right)=O_{p}\left(\frac{1}{\sqrt{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}}\right).$

  3. (iii)

We observe

$\lVert\bm{H}_{\textsc{o}}^{-1}\rVert_{2}\leq\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{F}_{\textsc{o}}\rVert_{2}/T_{\textsc{c}}=O_{p}(\sqrt{{r_{\textsc{o}}}}).$

∎

######  Lemma 6.

Under the assumptions of Theorem 1,

$\lVert\bm{G}_{\textsc{o}}-\bm{H}_{\textsc{o}}\rVert_{2}=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{{r_{\textsc{o}}}}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right).$

###### Proof.

It can be shown that

$\bm{I}_{r_{\textsc{o}}}-\bm{H}_{\textsc{o}}^{-1}\bm{G}_{\textsc{o}}=\Big{(}\frac{\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-2}}{N_{\textsc{o}}T_{\textsc{c}}^{2}}+\frac{\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-2}}{N_{\textsc{o}}T_{\textsc{c}}^{2}}\Big{)}(\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2})^{-1}.$

The first term in the braces is bounded by

$\frac{\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}\lVert\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}}{N_{\textsc{o}}T_{\textsc{c}}^{2}}=O_{p}\left(\sqrt{\frac{{r_{\textsc{o}}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}N_{\textsc{o}}}}\right),$

as $\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert^{2}_{2}=T_{\textsc{c}}$,
$\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(\sqrt{r_{\textsc{o}}N_{\textsc{o}}T_{\textsc{c}}})$
by Lemma 10(i),
$\lVert\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(T_{\textsc{c}})$
by Lemma 11(i), and
$\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$.
The second term in the braces is bounded by

$\frac{\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\rVert_{2}\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{B}_{\textsc{o}}^{-2}\rVert_{2}}{N_{\textsc{o}}T_{\textsc{c}}^{2}}=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{1}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}\right)\right),$

where we have used the additional fact that
$\lVert\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}\rVert_{2}=O_{p}(N_{\textsc{o}}+T_{\textsc{c}})$.
Combining the above with
$\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}(1)$ by
Lemma 9(ii) and
$1/\lambda^{\textsc{o}}_{r_{\textsc{o}}}=O(\min\\{N_{\textsc{o}},T_{\textsc{c}}\\})$
by Assumption (A9), we obtain

$\bm{I}_{r_{\textsc{o}}}-\bm{H}_{\textsc{o}}^{-1}\bm{G}_{\textsc{o}}=O_{p}\left(\sqrt{\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\left(\frac{{r_{\textsc{o}}}}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}\right)}\right).$

Finally, the above together with
$\lVert\bm{H}_{\textsc{o}}\rVert_{2}=O_{p}(\sqrt{1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$
by Lemma 5(ii) conclude the proof. ∎

##  Appendix E Supplementary lemmas

######  Lemma 7.

Under the assumptions of Theorem 1,

$\Big{\lVert}\bm{I}_{r_{\textsc{o}}}-\frac{\bm{B}_{\textsc{o}}^{-1}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}}\Big{\rVert}_{\text{F}}=O\Big{(}\frac{{r_{\textsc{o}}}^{2}}{N_{\textsc{o}}}\Big{)}.$

###### Proof.

The statement follows directly from Assumption (A8),

$\Big{\lVert}\bm{I}_{r_{\textsc{o}}}-\frac{\bm{B}_{\textsc{o}}^{-1}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}}\Big{\rVert}_{\text{F}}^{2}=\sum_{k,\ell=1}^{r_{\textsc{o}}}\Big{(}\frac{1}{N_{\textsc{o}}}\sum_{i:u_{i}\in\textsc{O}}\phi_{k}^{\textsc{o}}(u_{i})\phi_{\ell}^{\textsc{o}}(u_{i})-\delta_{k\ell}\Big{)}^{2}=O\Big{(}\frac{{r_{\textsc{o}}}^{4}}{N_{\textsc{o}}^{2}}\Big{)}.$

∎

######  Lemma 8.

Under the assumptions of Theorem 1,

  1. (i)

$\max_{i\leq N}\,\lVert\Lambda^{*}_{\textsc{o},i}\rVert=\textsc{o}(1)$,

  2. (ii)

$\max_{i\leq N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}\rVert=O_{p}(1)$.

###### Proof.

  1. (i)

The first assertion follows from

$\max_{i\leq N}\,\lVert\Lambda^{*}_{\textsc{o},i}\rVert^{2}=\max_{i\leq
N}\,\mathbb{E}\,\Big{[}\mathcal{L}(X^{\textsc{o}})(u_{i})^{2}\Big{]}\leq\mathbb{E}\,\Big{[}\sup_{u\in[0,1]}\mathcal{L}(X^{\textsc{o}})(u)^{2}\Big{]}\leq
2\,\mathbb{E}\,\Big{[}\sup_{u\in[0,1]}X(u)^{2}+\sup_{u\in[0,1]}Z^{\textsc{o}}(u)^{2}\Big{]}$

which is $O(1)$ due to Assumption (A5).

  2. (ii)

Observe that by Assumption (A5) and Lemma 4(ii),

$\max_{i\leq N}\,\lVert Y_{c,i}\rVert^{2}=\max_{i\leq
N}\sum_{t=1}^{T_{\textsc{c}}}(X_{t}(u_{i})+e_{ti})^{2}=O_{p}(T_{\textsc{c}}).$

Consequently,

$\max_{i\leq
N}\,\lVert\widehat{\Lambda}^{*}_{\textsc{o},i}\rVert\leq\max_{i\leq
N}\frac{\lVert
Y_{c,i}\rVert\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}}{T_{\textsc{c}}}=O_{p}(1),$

since $\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert^{2}_{2}=T_{\textsc{c}}$.

∎

######  Lemma 9.

Under the assumptions of Theorem 1,

  1. (i)

$\lVert\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}=o_{p}(1)$,

  2. (ii)

$\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}=o_{p}(1)$,

  3. (iii)

$\lVert\bm{D}_{\textsc{o}}^{-2}\rVert_{2}=O_{p}(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}})$.

###### Proof.

  1. (i)

Recall that both $\bm{B}_{\textsc{o}}$ and $\bm{D}_{\textsc{o}}$ are diagonal
matrices. Define the diagonal entries of $\bm{D}_{\textsc{o}}^{2}$ by
$\widehat{\gamma}_{1}^{\textsc{o}}\geq\widehat{\gamma}_{2}^{\textsc{o}}\geq\dots\geq\widehat{\gamma}_{r_{\textsc{o}}}^{\textsc{o}}$.
Then,

$\lVert\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}=\max_{k\leq{r_{\textsc{o}}}}\,\Big{|}\frac{\widehat{\gamma}_{k}^{\textsc{o}}}{\lambda_{k}^{\textsc{o}}}-1\Big{|}\leq\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\max_{k\leq{r_{\textsc{o}}}}\,\lvert\widehat{\gamma}_{k}^{\textsc{o}}-\lambda_{k}^{\textsc{o}}\rvert=O_{p}\left(\frac{1}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\sqrt{\frac{1}{N_{\textsc{o}}}+\frac{1}{T_{\textsc{c}}}}\right),$

where the last bound can be shown similar to Lemma 1 in Hörmann and Jammoul,
(2022). The assertion then follows from
$1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}=o(\min\\{\sqrt{N_{\textsc{o}}},\sqrt{T_{\textsc{c}}}\\})$
by Assumption (A9).

  2. (ii)

Observe

$\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}\leq\frac{\lVert\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}}{1-\lVert\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}}=o_{p}(1),$

by the previous result.

  3. (iii)

We get

$\lVert\bm{D}_{\textsc{o}}^{-2}\rVert_{2}\leq\lVert\bm{B}_{\textsc{o}}^{-2}\rVert_{2}(\lVert\bm{B}_{\textsc{o}}^{2}\bm{D}_{\textsc{o}}^{-2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}+\lVert\bm{I}_{r_{\textsc{o}}}\rVert_{2})=O(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}})(o_{p}(1)+O(1))=O_{p}(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}).$

∎

######  Lemma 10.

Under the assumptions of Theorem 1,

  1. (i)

$\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{\textup{F}}^{2}=O_{p}({r_{\textsc{o}}}N_{\textsc{o}}T_{\textsc{c}})$,

  2. (ii)

$\lVert\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}^{\prime}\rVert_{\textup{F}}^{2}=O_{p}(N_{\textsc{o}}T_{\textsc{c}}^{2})$,

  3. (iii)

$\lVert
e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\rVert^{2}=O_{p}(N_{\textsc{o}}^{2}/T_{\textsc{c}}+N_{\textsc{o}})$.

###### Proof.

  1. (i)

Observe that

$\displaystyle\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{\textup{F}}^{2}=\sum_{t=1}^{T_{\textsc{c}}}\sum_{k=1}^{r_{\textsc{o}}}\Big{(}\sum_{i:u_{i}\in\textsc{O}}e_{ti}\phi_{k}^{\textsc{o}}(u_{i})\Big{)}^{2}=\sum_{t=1}^{T_{\textsc{c}}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{O}\end{subarray}}e_{ti}e_{tj}\sum_{k=1}^{r_{\textsc{o}}}\phi_{k}^{\textsc{o}}(u_{i})\phi_{k}^{\textsc{o}}(u_{j}).$

Taking expectation and using Assumption (A4)(b) together with
$\sup_{u\in\textsc{O}}\lvert\phi_{k}^{\textsc{o}}(u)\rvert<C$ by Assumption
(A8),

$\displaystyle\mathbb{E}\,[\lVert\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{\textup{F}}^{2}]$
$\displaystyle\leq
T_{\textsc{c}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{O}\end{subarray}}\lvert\mathbb{E}\,[e_{ti}e_{tj}]\rvert\sum_{k=1}^{r_{\textsc{o}}}\lvert\phi_{k}^{\textsc{o}}(u_{i})\phi_{k}^{\textsc{o}}(u_{j})\rvert\leq
C{r_{\textsc{o}}}T_{\textsc{c}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{o}\end{subarray}}\lvert\mathbb{E}\,[e_{ti}e_{tj}]\rvert$
$\displaystyle=O({r_{\textsc{o}}}N_{\textsc{o}}T_{\textsc{c}}).$

  2. (ii)

The factor structure entails
$\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}=\bm{X}_{\textsc{o}}$.
Therefore, independence of $X_{s}(u_{i})$ and $e_{ti}$ due to Assumption (A4)
yields

$\mathbb{E}\,\left[\lVert\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}e_{\textsc{o},t}^{\prime}\rVert^{2}\right]=\mathbb{E}\,\left[\lVert\bm{X}_{\textsc{o}}e_{\textsc{o},t}^{\prime}\rVert^{2}\right]=\sum_{s=1}^{T_{\textsc{c}}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{O}\end{subarray}}\mathbb{E}\,[X_{s}(u_{i})X_{s}(u_{j})]\mathbb{E}\,[e_{ti}e_{tj}].$

Furthermore, since
$\mathbb{E}\,[X_{s}(u_{i})X_{s}(u_{j})]=\sigma(u_{i},u_{j})$ is bounded, we
finally deduce from Assumption (A4)(b),

$\mathbb{E}\,\left[\lVert\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}e_{\textsc{o},t}^{\prime}\rVert^{2}\right]\leq
CT_{\textsc{c}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{O}\end{subarray}}\lvert\mathbb{E}\,[e_{ti}e_{tj}]\rvert=O_{p}(N_{\textsc{o}}T_{\textsc{c}}).$
(19)

  3. (iii)

Recall that
$\widehat{\bm{\Lambda}}_{\textsc{o}}=\bm{Y}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}/T_{\textsc{c}}$,
whence

$e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}=\frac{1}{T_{\textsc{c}}}\left(e_{\textsc{o},t}\bm{X}_{\textsc{o}}{}^{\prime}+e_{\textsc{o},t}\bm{e}_{\textsc{o}}^{\prime}\right)\widehat{\bm{F}}_{\textsc{o}}.$
(20)

From (19), we get $\lVert
e_{\textsc{o},t}\bm{X}_{\textsc{o}}{}^{\prime}\rVert^{2}=O_{p}(N_{\textsc{o}}T_{\textsc{c}})$.
In addition,

$\displaystyle\lVert
e_{\textsc{o},t}\bm{e}_{\textsc{o}}^{\prime}\rVert^{2}=\sum_{s=1}^{T_{\textsc{c}}}\Big{(}\sum_{i:u_{i}\in\textsc{O}}e_{ti}e_{si}\Big{)}^{2}\leq\Big{(}\sum_{i:u_{i}\in\textsc{O}}e_{ti}^{2}\Big{)}^{2}+\sum_{\begin{subarray}{c}s=1\\\
s\neq
t\end{subarray}}^{T_{\textsc{c}}}\Big{(}\sum_{i:u_{i}\in\textsc{O}}e_{ti}e_{si}\Big{)}^{2}.$

Due to Assumption (A4)(c), the first term is $O_{p}(N_{\textsc{o}}^{2})$. It
is relevant only if $t\leq T_{\textsc{c}}$ meaning that $Y_{t}$ is completely
observed. For the second term, we deduce from Assumption (A4)(b),

$\mathbb{E}\,\left[\sum_{\begin{subarray}{c}s=1\\\ s\neq
t\end{subarray}}^{T_{\textsc{c}}}\Big{(}\sum_{i:u_{i}\in\textsc{O}}e_{si}e_{ti}\Big{)}^{2}\right]\leq
T_{\textsc{c}}\sum_{\begin{subarray}{c}i:u_{i}\in\textsc{O}\\\
j:u_{j}\in\textsc{O}\end{subarray}}\lvert\mathbb{E}\,[e_{si}e_{sj}]\rvert^{2}=O_{p}(T_{\textsc{c}}N_{\textsc{o}}).$

Combining the above with (20) and
$\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}^{2}=O_{p}(T_{\textsc{c}})$, we
conclude that $\lVert
e_{\textsc{o},t}\widehat{\bm{\Lambda}}_{\textsc{o}}\rVert^{2}=O_{p}(N_{\textsc{o}}+N_{\textsc{o}}^{2}/T_{\textsc{c}})$.

∎

######  Lemma 11.

Under the assumptions of Theorem 1,

  1. (i)

$\lVert\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}\rVert_{2}=O_{p}(T_{\textsc{c}})$,

  2. (ii)

$\lVert(\bm{B}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1})^{-1}\rVert_{2}=O_{p}(1/T_{\textsc{c}})$.

###### Proof.

  1. (i)

Observe that

$\frac{1}{N_{\textsc{o}}T_{\textsc{c}}^{2}}\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{Y}_{\textsc{o}}\bm{Y}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}=\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}.$
(21)

Furthermore,
$\lVert\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}^{\prime}\rVert_{\textup{F}}=O_{p}(\sqrt{N_{\textsc{o}}}T_{\textsc{c}})$
by Lemma 10(ii) and
$\lVert\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}\rVert_{2}=O_{p}(N_{\textsc{o}}+T_{\textsc{c}})$
by Assumption (A4)(a). Therefore,

$\displaystyle\bm{Y}_{\textsc{o}}\bm{Y}_{\textsc{o}}^{\prime}$
$\displaystyle=\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}+\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{e}_{\textsc{o}}^{\prime}+\bm{e}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}+\bm{e}_{\textsc{o}}\bm{e}_{\textsc{o}}^{\prime}$
$\displaystyle=\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}+O_{p}(\sqrt{N_{\textsc{o}}}T_{\textsc{c}}+N_{\textsc{o}}).$

Since
$\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}O_{p}(\sqrt{N_{\textsc{o}}}T_{\textsc{c}}+N_{\textsc{o}})\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}=O_{p}(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}}(\sqrt{N_{\textsc{o}}}T_{\textsc{c}}^{2}+N_{\textsc{o}}T_{\textsc{c}}))=o_{p}(N_{\textsc{o}}T_{\textsc{c}}^{2})$
by Assumption (A9),

$\frac{1}{N_{\textsc{o}}T_{\textsc{c}}^{2}}\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{Y}_{\textsc{o}}\bm{Y}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}=\frac{1}{N_{\textsc{o}}T_{\textsc{c}}^{2}}\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{F}_{\textsc{o}}^{\prime}\widehat{\bm{F}}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}+o_{p}(1).$
(22)

Let us now define the $({r_{\textsc{o}}}\times{r_{\textsc{o}}})$ matrix
$\mathbb{A}=\bm{B}_{\textsc{o}}^{-1}\widehat{\bm{F}}_{\textsc{o}}^{\prime}\bm{F}_{\textsc{o}}\bm{B}_{\textsc{o}}/T_{\textsc{c}}$
to ease notation. A trivial bound of $\mathbb{A}$ can be obtained by observing
$\lVert\mathbb{A}\rVert_{2}\leq\lVert\bm{B}_{\textsc{o}}^{-1}\rVert_{2}\lVert\widehat{\bm{F}}_{\textsc{o}}\rVert_{2}\lVert\bm{F}_{\textsc{o}}\rVert_{2}\lVert\bm{B}_{\textsc{o}}\rVert_{2}/T_{\textsc{c}}=O_{p}(\sqrt{r_{\textsc{o}}/\lambda_{r_{\textsc{o}}}^{\textsc{o}}})$.
Then, combining (21) with (22),

$\mathbb{A}\,\frac{\bm{B}_{\textsc{o}}^{-1}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}}\,\mathbb{A}^{\prime}=\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}+o_{p}(1).$

Consequently,

$\displaystyle\lVert\mathbb{A}\mathbb{A}^{\prime}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}$
$\displaystyle\leq\lVert\mathbb{A}\rVert_{2}^{2}\Big{\lVert}\bm{I}_{r_{\textsc{o}}}-\frac{\bm{B}_{\textsc{o}}^{-1}\bm{\Lambda}_{\textsc{o}}^{\prime}\bm{\Lambda}_{\textsc{o}}\bm{B}_{\textsc{o}}^{-1}}{N_{\textsc{o}}}\Big{\rVert}_{2}+\lVert\bm{B}_{\textsc{o}}^{-2}\bm{D}_{\textsc{o}}^{2}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}+o_{p}(1)$
$\displaystyle=O_{p}\left(\frac{{r_{\textsc{o}}}}{\lambda_{r_{\textsc{o}}}^{\textsc{o}}}\right)O\left(\frac{{r_{\textsc{o}}}^{2}}{N_{\textsc{o}}}\right)+o_{p}(1)=o_{p}(1),$

where we have used Lemma 7 and Lemma 9(i). Note that summability of
$\lambda_{r_{\textsc{o}}}^{\textsc{o}}$ implies
${r_{\textsc{o}}}=o(1/\lambda_{r_{\textsc{o}}}^{\textsc{o}})$ and hence
$O({r_{\textsc{o}}}/\lambda_{r_{\textsc{o}}}^{\textsc{o}})O({r_{\textsc{o}}}^{2}/N_{\textsc{o}})=o(1)$
by Assumption (A9). This finally gives the stronger bound
$\lVert\mathbb{A}\rVert_{2}=O_{p}(1)$.

  2. (ii)

It suffices to observe

$\lVert\mathbb{A}^{-1}\rVert_{2}\leq\frac{\lVert\mathbb{A}\rVert_{2}}{1-\lVert\mathbb{A}\mathbb{A}^{\prime}-\bm{I}_{r_{\textsc{o}}}\rVert_{2}}=O_{p}(1).$

∎