# High-Dimensional Dynamic Factor Models: A Selective Survey and Lines of
Future Research

Marco Lippi
Einaudi Institute for Economics and Finance E-mail<EMAIL_ADDRESS>Manfred Deistler
Technische Universität Wien and Wirtschaftsuniversität Wien E-mail:
<EMAIL_ADDRESS> Brian Anderson
Australian National University E-mail<EMAIL_ADDRESS>School of
Engineering, Australian National University, Acton, ACT 2601, Australia.

(August 27, 2024)

High-Dimensional Dynamic Factor Models are presented in detail: The main
assumptions and their motivation, main results, illustrations by means of
elementary examples. In particular, the role of singular ARMA models in the
theory and applications of High-Dimensional Dynamic Factor Models is
discussed. The emphasis of the paper is on model classes and their structure
theory, rather than on estimation in the narrow sense. Our aim is not a
comprehensive survey. Rather we try to point out promising lines of research
and applications that have not yet been sufficiently developed.

JEL classification: C50, C55, C53.
Keywords: High-dimensional vector processes. Dynamic factor models. State-
space representations. Singular ARMA vector processes.

##  1 Introduction

Analysis and forecasting of high-dimensional time series recently has
attracted substantial interest, see Hallin et al. (2020). However, “classical”
multivariate time-series models such as autoregressive models suffer from the
so-called “curse of dimensionality”: Unless additional restrictions are
imposed, the parameter spaces grow with the square of the dimension of the
time series, $N$ say; thus in many cases, even for moderate $N$, the sample
size available is not sufficient to guarantee reliable parameter estimation.
This is true in particular for macroeconomic applications. In this case, on
the one hand, the interaction of several variables, disaggregated series in
particular, may be important for modeling and forecasting. On the other hand,
sample size of macroeconomic time series is in many cases rather limited, e.g.
because of structural changes in the underlying economies.

High-dimensional dynamic factor models are one way to overcome this curse of
dimensionality. Factor models for the case of i.i.d. observations have a long
history, dating back to Spearman (1904) and Burt (1909). Factor models in
time-series context have been proposed much later, in particular by Geweke
(1977) and Sargent and Sims (1977), Watson and Engle (1983), Quah and Sargent
(1993). The basic idea is to separately model the comovement between the
variables on the one hand, and the individual, or idiosyncratic movement on
the other hand. Equivalently, the variables are decomposed as the sum of a
latent variable, or common component, and an idiosyncratic component.

The above factor models are “exact” in the sense that the idiosyncratic
components are assumed to be cross-sectionally uncorrelated at all leads and
lags. Inspired by the idea of risk diversification, Chamberlain (1983) and
Chamberlain and Rothschild (1983) introduced the notion of “approximate”
factor models, which are high-dimensional (potentially infinite-dimensional)
models where the idiosyncratic terms are allowed to be cross-sectionally
dependent, although in a weak sense. This idea has been extended to the linear
dynamic case by Forni et al. (2000), Forni and Lippi (2001), Stock and Watson
(2002a, b), Bai and Ng (2002), Bai (2003), leading to the class of linear
High-Dimensional Dynamic Factor Models, Dynamic Factor Models for short (DFM).

The present paper does not contain a comprehensive survey. We go over the main
assumptions and results with the purpose of pointing out lines of research
that have not yet been sufficiently developed.

In Section 2 we describe the model class of DFMs, thus the decomposition of
the observable $N$-dimensional vector $y^{N}_{t}$ into common and idioyncratic
components, $\chi^{N}_{t}$ and $\xi^{N}_{t}$ respectively. We assume that
$\chi_{t}^{N}$ has rational spectral density $f^{N}_{\chi}$. The rank $q$ of
$f^{N}_{\chi}$ does not depend on $N$, for $N$ sufficiently large. The state
dimension of minimal, stable and miniphase state space realization of
$f^{N}_{\chi}$ is independent of $N$. (This can be rephrased as saying, that
there is a finite number of “static” factors, see Section 3.)

In Section 3 we discuss the model for the process of the common components.
Precisely, the common components are represented as linear combinations of a
finite-dimensional vector process, whose coordinates are called the (minimal)
static factors. The latter are modeled as an ARMA system driven by a vector
white noise whose coordinates are called the dynamic factors. Here we
introduce the so-called singular ARMA (or state-space) systems. These generate
$N$-dimensional stationary processes with rational spectral density, whose
rank is less than $N$: we call them singular111Not to be confused with
singular processes in the sense of Kolmogorov.. We argue that, under our
assumptions, this is obviously the case for the common-component vector, and
is very likely for the factor vector. On the other hand, it has been shown
that singular ARMA processes can (generically) be modeled as finite AR’s, see
Section 3.3, which implies a most important simplification in the modeling of
common components and static factors.

In Section 4 we describe techniques to obtain the common components and
factors from the observable vector $y^{N}_{t}$, for $N$ and $T$ (the number of
observations for each time series) tending to infinity. In Section 4.1 we show
how principal components (PCA) can be used. The underlying estimation
procedure is to estimate the static factors by PCA in a first step and then to
estimate an ARMA or AR model in order to describe the dynamics of the static
factors. An alternative approach, see Section 4.3, is to assume a dynamic
factor model with autoregressive static factors and cross-sectionally
uncorrelated idiosyncratic components, thus an exact factor model, and to put
this in a state-space framework. In this framework (once the integer
specification parameters have been fixed) an EM algorithm with the E-step
based on Kalman filtering can be applied. That an exact factor model can be
used to estimate a DFM has been shown in Doz et al. (2012), see 4.3.

In the case where no $N$-independent static factors exist, frequency-domain
methods, as described in Section 4.2, may be applied.

In Section 5.1 we present some applications of the results on singular ARMA
models to empirical macroeconomic analysis, to the so-called fundamentalness
problem in particular. Some results on cointegration for singualar ARIMAs are
presented in Section 5.2.

##  2 High-Dimensional Dynamic Factor Models (DFM). The Model Class

The basic idea is to represent the $N$-dimensional observation vector at time
$t\in\mathbb{Z}$, $y^{N}_{t}$ say, as

$y^{N}_{t}=\chi^{N}_{t}+\xi^{N}_{t},$ (2.1)

where $(\chi^{N}_{t}\;|\;t\in\mathbb{Z})$ and
$(\xi^{N}_{t}\;|\;t\in\mathbb{Z})$ are the $N$-dimensional processes of common
and idiosyncratic components respectively. The one-dimensional processes
$(\chi_{it},\ t\in\mathbb{Z})$ are strongly dependent across the index $i$,
whereas the processes $(\xi_{it},\ t\in\mathbb{Z})$ are weakly dependent. The
precise meaning of strong and weak dependence is given below.

Throughout, except for Section 5.2, we assume that $(\chi^{N}_{t})$ and
$(\xi^{N}_{t})$ are wide-sense stationary. In addition, throughout we assume

$\displaystyle{\rm E}\chi^{N}_{t}={\rm E}\xi^{N}_{t}$ $\displaystyle=0\ \ \
\forall t$ (2.2) $\displaystyle{\rm E}\chi^{N}_{t}\,{\xi^{N}_{s}}^{\prime}$
$\displaystyle=0\ \ \ \forall t,s,$ (2.3)

and that the spectral densities of ($\chi^{N}_{t}$) and ($\xi^{N}_{t}$) exist.
As a consequence, ($y^{N}_{t}$) is stationary and has a spectral density,
which, using an obvious notation, is:

$f_{y}^{N}(\lambda)=f_{\chi}^{N}(\lambda)+f^{N}_{\xi}(\lambda),\ \ \
\lambda\in[-\pi,\ \pi].$ (2.4)

Throughout, $z$ is used for a complex variable as well as for the backward
shift on $\mathbb{Z}$.

The following assumptions constitute the class of DFM’s considered in the
present paper (we follow here Deistler et al. (2010)):

######  Assumption 1.

For all $N$, $f^{N}_{\chi}$ is a rational spectral density.

An obvious consequence of Assumption 1 is that $f_{\chi}^{N}$ has constant
normal rank, i.e. has the same rank almost everywhere on $[-\pi,\ \pi]$.

Here, for asymptotic analysis not only the sample size $T$, but also the
cross-sectional dimension $N$ is tending to infinity—this has an empirical
motivation in the study of high-dimensional vector time series, i.e. vector
time series whose dimension is allowed to be close to or even higher than the
sample size. Thus the underlying process considered is a double-indexed
stochastic process $(y_{it}\;|\;i\in\mathbb{N},\ t\in\mathbb{Z})$,
corresponding, as $N$ varies, to a nested sequence of models (2.1), in the
sense that $y_{it},$ $\chi_{it},$ and $\xi_{it}$ do not depend on $N$, for
$i\leq N$.

######  Assumption 2.

We suppose that there exists $N_{0}\geq q$ such that, from $N_{0}$ onwards,
the rank of $f^{N}_{\chi}$ is independent of $N$. Such rank is denoted by $q$.

As is well known, see e.g. Hannan and Deistler (2012), every rational spectral
density can be described by an ARMA or alternatively a state space system.

######  Assumption 3.

The state dimension of a minimal, stable and miniphase state space realization
corresponding to $f_{\chi}^{N}$ is independent of $N$ from a certain $N_{1}$
onwards. Such state dimension is denoted by $n$.

Of course, without loss of generality, we can assume $N_{0}=N_{1}$. Next we
define weak and strong dependence. We use $\omega^{N}_{\xi,s}(\lambda)$,
$\omega^{N}_{\chi,s}(\lambda)$, $\omega_{y,s}^{N}(\lambda)$ to denote the
$s$-th largest eigenvalue of the Hermitian matrix $f^{N}_{\xi}(\lambda)$,
$f^{N}_{\chi}(\lambda)$, $f^{N}_{y}(\lambda)$, respectively.

######  Assumption 4.

Weak cross-sectional dependence of $(\xi^{N}_{t})$. The eigenvalues
$\omega^{N}_{\xi,1}(\lambda)$ are uniformly bounded, i.e. there exists $B>0$
such that $\omega^{N}_{\xi,1}(\lambda)<B$ for all $N$ and $\lambda$.

######  Assumption 5.

Strong cross-sectional dependence of $(\chi^{N}_{t})$. The eigenvalues
$\omega^{N}_{\chi,s}(\lambda)$, $s=1,\ldots,q$, diverge as $N\to\infty$,
$\lambda$ almost everywhere in $[-\pi,\ \pi]$.

Note that Assumptions 4 and 5 place restrictions on the cross-dependence of
the idiosyncratic and common component respectively, not on their
autocorrelations. Note also that the assumptions defining the common and
idiosyncratic terms are asymptotic, for the number $N$ of observable variables
tending to infinity, so that in empirical application the observable vector
$y^{N}_{t}$ is supposed to be high dimensional.

Forni and Lippi (2001) prove that representation (2.1) is identified. More
precisely, if $\tilde{\chi}^{N}_{t}$, $\tilde{\xi}^{N}_{t}$ and the integer
$\tilde{q}$ fulfill

$y^{N}_{t}=\tilde{\chi}^{N}_{t}+\tilde{\xi}^{N}_{t},\ \ \hbox{for all $N$},$

(2.2), (2.3) and Assumptions 1, 2, 4 and 5, then $\tilde{q}=q$,
$\tilde{\chi}^{N}_{t}=\chi^{N}_{t}$, $\tilde{\xi}^{N}_{t}=\xi^{N}_{t}$.

DFM’s generalize exact dynamic factor models, considered in Geweke (1977),
Sargent and Sims (1977), where $N$ is fixed and$f_{\xi}^{N}(\lambda)$ is
diagonal, i.e. the one-dimensional idiosyncratic processes are mutually
uncorrelated at any lead and lag. Exact dynamic factor models are rather
restrictive, in that the class of spectral densities corresponding to them is
restricted, in particular if $q$ is small relative to $N$ (which is the most
important case), see Scherrer and Deistler (1998). On the other hand, for $q$
small enough in relation to $N$, identifiability results are available for
fixed $N$, see again Scherrer and Deistler (1998).

In general, several interpretations for DFM’s are possible. E.g., the
components $\xi_{t}^{N}$ may be interpreted as “measurement” error and
$\chi^{N}_{t}$ as unobserved “true” variables; this is in line with errors-in-
variables models, which, from an abstract point of view, are the same as
factor models. An alternative interpretation of (2.1) is the decomposition of
observations into a part $(\chi^{N}_{t}$) representing the comovements between
the variables (for instance representing the “market” effect on all stock
prices) and individual movements ($\xi^{N}_{t}$) (representing the firm’s
specific effect). The components $\xi_{it}$ are usually called the
idiosyncratic components, while the latent variables $\chi_{it}$ are referred
to as the common components.

It is easy to see that, given a bijective map $g:\mathbb{N}\to\mathbb{N}$, if
Assumption 5 holds for the process $(\chi_{it}\;|\;i\in\mathbb{N},\
t\in\mathbb{Z})$, then it holds for $(\chi_{g(i),t}\;|\;i\in\mathbb{N},\
t\in\mathbb{Z})$, because the vector $\chi^{N}_{t}$ is nested in the vector
$\chi^{g,M}_{t}=(\chi_{g(1),t}\ \cdots\ \chi_{g(M),t})$ for some $M$. As a
consequence, $\omega^{N}_{\chi,s}(\lambda)\leq\omega^{g,M}_{\chi,s}(\lambda)$,
see e.g. Forni and Lippi (2001), Fact M, (b), p. 1121. Thus Assumption 5 holds
irrespective of the order of the variables $y_{it}$ (and thus $\chi_{it})$.

However, the speed of divergence of the eigenvalues
$\omega^{N}_{\chi,s}(\lambda)$, $s=1,\ldots,q$, depends on that order. A
simple example is the following. Let $(a_{s},\ s\in\mathbb{N})$ be a square
summable sequence of real numbers and $(j_{s},\ s\in\mathbb{N})$ a sequence of
positive integers. Then let

$\chi_{it}=b_{i}v_{t},$

where $v_{t}$ is a unit-variance scalar white noise and the coefficients
$b_{i}$ are the following:

$1,\ a_{1},\ \ldots a_{j_{1}},\ 1,\ a_{j_{1}+1},\ \ldots\ a_{j_{1}+j_{2}},\
1,\ a_{j_{1}+j_{2}+1},\ \ldots$ (2.5)

In this case we have

$\omega^{N}_{\chi,1}(\lambda)=\sum_{h=1}^{N}b_{h}^{2}.$

In particular, if $N=K+j_{1}+\ldots+j_{K}$,

$\omega^{N}_{\chi,1}(\lambda)=K+a_{1}^{2}+a_{2}^{2}+\cdots+a_{j_{1}+\cdots+j_{K}}^{2}.$

Assuming that $j_{s}=1$ for all $s$, we see that, for $N$ odd,

$\lim_{N\to\infty}\frac{\omega^{N}_{\chi,1}(\lambda)}{N}=\lim_{N\to\infty}\frac{K+a_{1}^{2}+\cdots+a_{K}^{2}}{2K}=\frac{1}{2}.$

On the other hand, assuming that $j_{s}=s$, we obtain a reordering of the
coefficients $b_{i}$, with ones appearing at linearly increasing intervals. It
is easy to see that in this case the eigenvalue $\omega^{N}_{\chi,1}(\lambda)$
grows asymptotically at speed $N^{1/2}$. DFM’s in which the eigenvalues are
assumed to diverge with rates $N^{\alpha}$, $0<\alpha<1$, known as models with
“weak factors”, necessarily rely on a particular assumption on the order of
the variables, see Onatski (2012) and related literature.

##  3 Modeling the Latent Process

###  3.1 Singular ARMA and state space systems

By Assumptions 1 and 2, the latent process $(\chi^{N}_{t}$) has a rational
and, for $N>N_{0}\geq q$, singular spectral density. By its rationality, it
can be represented by a singular ARMA or state space system, i.e. an ARMA or
state space system with a singular innovation variance. For regular ARMA or
state space systems, i.e. systems whose output has an a.e. nonsingular
spectral density, we refer to Hannan and Deistler (2012), Deistler and
Scherrer (2019); for the singular case see Section 3.3.

Hereafter, unless strictly necessary, we omit the superscript $N$. For the
ARMA system, assuming that $N\geq N_{0}$, we have

$P(z)\chi_{t}=Q(z)v_{t},\ \ \ P(z)=P_{0}-\sum_{j=1}^{S}P_{j}z^{j},\ \ \
Q(z)=\sum^{S^{\prime}}_{j=0}Q_{j}z^{j},$ (3.1)

where $v_{t}$ is an orthonormal $q$-dimensional white noise, with
$P_{j}\in\mathbb{R}^{N\times N}$, $Q_{j}\in\mathbb{R}^{N\times q}$. Moreover,

$\det P(z)\neq 0\ \ {\rm for}\ \ |z|\leq 1,$ (3.2)

which is the stability condition, and

${\rm rank}\,Q(z)=q,\hbox{ \ \ for $|z|\leq 1$},$ (3.3)

which is the strict miniphase condition.

Thus the steady state solution of (3.1) is

$\chi_{t}=P^{-1}(z)Q(z)v_{t}=k(z)v_{t}{=\sum_{j=0}^{\infty}k_{j}v_{t-j}}.$
(3.4)

Due to the stability and the miniphase conditions, (3.4) is a Wold
representation, $v_{t}$ are innovations and the one-step ahead prediction
error for $\chi_{t}$, given its past $\chi_{s}$, $s<t$, is $k_{0}v_{t}$, see
below. The spectral density of $(\chi_{t})$ is of the form

$f_{\chi}(\lambda)=(2\pi)^{-1}k(e^{-i\lambda})k(e^{-i\lambda})^{*}$ (3.5)

where $*$ denotes the Hermite conjugation. $f_{\chi}(\lambda)$ has rank $q$
for all $\lambda\in[-\pi,\ \pi]$. Given $f_{\chi}$, under conditions (3.2) and
(3.3), $k(z)$ is unique up to postmultiplication by (constant) orthogonal
matrices. Moreover:

######  Assumption 6.

We suppose that $P_{0}=I_{N}$ and that $P(z)$ and $Q(z)$ are left coprime (see
Hannan and Deistler, 2012, p.41)), i.e. the matrix $(P(z)\ Q(z))$ has rank $N$
for all $z\in\mathbb{C}$.

Alternatively, by Assumptions 2 and 3, for $N\geq N_{0}$, to
$f_{\chi}(\lambda)$ there corresponds a minimal state-space realization

$\displaystyle x_{t+1}$ $\displaystyle=Fx_{t}+Gw_{t+1}$ (3.6)
$\displaystyle\chi_{t}$ $\displaystyle=Hx_{t},$ (3.7)

where $x_{t}$ is an $n$-dimensional state, $w_{t}$ is a $q$-dimensional
orthonormal white noise, where $n$ and $q$ are independent of $N$,
$F\in\mathbb{R}^{n\times n}$, $G\in\mathbb{R}^{n\times q}$,
$H\in\mathbb{R}^{N\times n}$ are parameter matrices. The stability condition
is

$\rho(F)<1,$ (3.8)

where $\rho(F)$ denotes the spectral radius of $F$, and

${\rm rank}\,\begin{pmatrix}I-Fz&-G\\\ H&\phantom{-}0\\\ \end{pmatrix}=n+q,\ \
{\rm for}\ \ |z|\geq 1$ (3.9)

is the strict miniphase condition.

Under conditions (3.8) and (3.9), $w_{t}$ is an innovation for $\chi_{t}$.
Moreover, setting

$K(z)=H(z^{-1}I-F)^{-1}G,$

we have

$\chi_{t}=H(I-Fz)^{-1}Gw_{t}=K(z)w_{t}.$ (3.10)

Under our assumptions, the innovations and the transfer function are unique up
to premultiplication and postmultiplication by orthogonal matrices,
respectively, so that we can assume with no loss of generality that
$w_{t}=v_{t}$ and $K(z)=k(z)$, where $v_{t}$ and $k(z)$ are defined in (3.1)
and (3.4) respectively.

Representation (3.6)–(3.7) implies that, for $N\geq N_{0}$, the dimension of
the space spanned by $\chi_{1t},\ \chi_{2t},\ \ldots,\ \chi_{Nt}$, call it
${\cal S}^{N}_{t}$, for any given $t$, does not exceed $n$. Therefore there
exists $\tilde{N}\geq N_{0}$ and $r\leq n$ such that for $N\geq\tilde{N}$,

${\rm dim}({\cal S}^{N}_{t})=r.$ (3.11)

With no loss of generality we can assume that $\tilde{N}=N_{0}$. By (3.11),
$\chi_{N_{0}+k,t}$, $k>0$, is a linear combination of $\chi_{1t},\ \chi_{2t},\
\ldots,\ \chi_{N_{0},t}$, so that $\chi_{N_{0}+k,t}=H_{i}x_{t}$,
$H_{i}\in\mathbb{R}^{1\times n}$, where $x_{t}$ is the state vector of the
minimal state-space realization for $N=N_{0}$. Thus, representation
(3.6)–(3.7) holds for all $N\geq N_{0}$ with the same $x_{t}$, $w_{t}$, $F$
and $G$, and nested matrices $H$. Otherwise stated, (3.6)–(3.7) hold with
(3.7) replaced by $\chi_{it}=H_{i}x_{t}$, where $H_{i}$ is the $i$-row of the
matrix $H$.

Note that there is a Hilbert space (in the Hilbert space $L_{2}$ of square
integrable random variables) construction of a state-space system, obtained by
projecting all future values of the output process $(\chi_{t})$ on the Hilbert
space spanned by its past (see e.g. Akaike 1974). Then, by the rationality of
the spectral density, the space spanned by these projections is finite
dimensional and every basis is a minimal state, see e.g. Hannan and Deistler
(2012), Deistler and Scherrer (2018). We will call this the Kalman-Akaike
realization of a state-space system.

The state space system (3.6)–(3.7) corresponds to a definition of past and
future, respectively, as $\chi_{s}$, $s\leq t$ and $\chi_{s}$, $s\geq t$. If
we, however, define the past as $\chi_{s}$, $s<t$, the Kalman-Akaike
realization leads to a system of the form

$\displaystyle\bar{x}_{t+1}$ $\displaystyle=\bar{F}\bar{x}_{t}+\bar{G}v_{t}$
(3.12) $\displaystyle\chi_{t}$ $\displaystyle=\bar{H}\bar{x}_{t}+B_{0}v_{t}$
(3.13)

and the corresponding representation of the transfer function is

$\bar{H}(Iz^{-1}-\bar{F})^{-1}\bar{G}+B_{0}.$ (3.14)

Note that now we impose a stability and a minimum phase condition analogous to
(3.8) and (3.9) respectively, for $(\bar{F},\bar{G},\bar{H},B_{0})$. Moreover,
$\bar{H}\bar{x}_{t}$ is the one-step ahead forecast for $\chi_{t}$ and
$B_{0}v_{t}$ the corresponding one-step ahead prediction error.

When $N>N_{0}\geq q$, so that $(\chi_{t})$ is a singular stochastic process,
i.e. a process with a singular spectral density, the left inverse of $k(z)$ is
not unique. To see this, consider the Smith-McMillan form (see e.g. Hannan and
Deistler (2012))

$k(z)=u(z)d(z)v(z),$ (3.15)

where $u(z)$ and $v(z)$ are $N\times N$ and $q\times q$, respectively,
unimodular polynomial matrices (i.e. their determinants are non-zero
constants) and

$d(z)=\begin{pmatrix}\frac{\displaystyle\epsilon_{1}(z)}{\displaystyle\psi_{1}(z)}&0&\cdots&0\\\
&&\ddots&\\\
0&0&\cdots&\frac{\displaystyle\epsilon_{q}(z)}{\displaystyle\psi_{q}(z)}\\\
0&0&\cdots&0\\\ \vdots\\\ 0&0&\cdots&0\\\ \end{pmatrix},$

where the matrix of zeros at the bottom is $(N-q)\times q$, $\epsilon_{i}$ and
$\psi_{i}$, for $i=1,\ldots,q$, are relatively prime monic polynomials,
$\epsilon_{i}$ divides $\epsilon_{i+1}$ and $\psi_{i+1}$ divides $\psi_{i}$.
Then a particular causal left inverse is given by

$h^{-}(z)=v^{-1}(z)(d^{\prime}(z)d(z))^{-1}d^{\prime}(z)u^{-1}(z).$ (3.16)

As is easily seen, $h^{-}(z)$ has no poles or zeros for $|z|\leq 1$ so that

$v_{t}=h^{-}(z)\chi_{t}$ (3.17)

is a causal relation and thus $(v_{t})$ are indeed innovations.

In order to uniquely determine $k(z)$ corresponding to the Wold decomposition
from $f_{\chi}$ (compare (3.5)), in addition to stability and the miniphase
assumption, we have to remove its non-uniqueness caused by post-multiplying by
a $q\times q$ orthogonal matrix. This leads to the following assumption
guaranteeing uniqueness of $k(z)$:

######  Assumption 7.

For $q=r$, we assume that $k(0)=I_{N}$, i.e. $Q_{0}=I_{N}$ holds. For $q<r$,
we assume that the top $q\times q$ submatrix of $k(0)$ is (1) non-singular
(which is the case generically) and (2) lower triangular (with non-zero
diagonal elements).

###  3.2 Static and dynamic factors

In Section 3.1 we have argued that by Assumption 3, under a suitable choice of
$N_{0}$, for $N\geq N_{0}$ the dimension of the space $\mathcal{S}^{N}_{t}$
spanned by $\chi_{it}$, $i=1,\ldots,N$ is $r$. Let $(f_{t})$ be an
$r$-dimensional process such that

(i) $f_{t}$ forms a basis in $\mathcal{S}^{N}_{t}$,

(ii) $f_{t}=S\chi_{t}$, with $S$ independent of $t$.

Of course (i) and (ii) imply that $(f_{t})$ is weakly stationary with a
rational spectral density and that

$\chi_{t}=Lf_{t},$ (3.18)

where $L$ is an $N\times r$ matrix independent of $t$. The vector $f_{t}$ is
called a vector of (minimal) static factors and $L$ the corresponding loading
matrix.

As is easy to see, a minimal static factor is unique up to premultiplication
by a constant non-singular matrix $\cal T$ and the factor loading matrix $L$
is unique up to postmultiplication by ${\cal T}^{-1}$. By (3.7), the state
$x_{t}$ is a static factor, though in general not a minimal one.

In particular, let $(f_{t})$ be a process of static factors corresponding to
$N_{0}$. Because ${\rm dim}\,\mathcal{S}^{N}_{t}={\rm dim}\,S^{N_{0}}_{t}$ for
all $N>N_{0}$, $f_{t}$ is a basis in $\mathcal{S}^{N}_{t}$ for all $N>N_{0}$.
Thus there exists a representation (3.18) in which the factors are independent
of $N$ and the matrix $L$ corresponding to $N$ is nested in the matrix $L$
corresponding to $M$, for $M>N$. In other words, there exist an
$r$-dimensional vector of factors $f_{t}$ and a matrix
$L\in\mathbb{R}^{\infty\times r}$ such that $\chi_{it}=L_{i}f_{t},$ for all
$i\in\mathbb{N}$, where $L_{i}$ is the $i$-th row of $L$.

Of course, for estimation uniqueness of $f_{t}$ and $L$ is desirable. A common
normalization is to assume that the top $r\times r$ submatrix of $L$, $L_{1}$
say, is nonsingular (which is the case, generically) and then to impose
$L_{1}=I_{r}$. Clearly this corresponds to the selection of the first $r$
elements of $\chi_{t}$ as (minimal) static factors. In this case the on- and
above-diagonal entries of ${\rm E}f_{t}f^{\prime}_{t}$ are additional free
parameters. We will refer to this normalization as the standard normalization.
Another common normalization is to assume that ${\rm
E}f_{t}f^{\prime}_{t}=I_{r}$. Then $L$ is unique up to multiplication by an
orthogonal matrix, which is made unique by assuming appropriate $Q-R$
decompositions for $L_{1}$.

A special explicit form of the static factors can be obtained in the following
way. Let $R$ be an $N\times r$ matrix such that
$RR^{\prime}=\gamma_{\chi}(0)$, the expected value of
$\chi_{t}{\chi}^{\prime}_{t}$. As $\gamma_{\chi}(0)$ has rank $r$ for $N\geq
N_{0}$, we have ${\rm rank}\,R=r$ for $N\geq N_{0}$ as well. Now define

$\displaystyle f_{t}$ $\displaystyle=(R^{\prime}R)^{-1}R^{\prime}\chi_{t}$
(3.19) $\displaystyle=(R^{\prime}R)^{-1}R^{\prime}k(z)v_{t}=w(z)v_{t},$ (3.20)

(see (3.4) for the first equality in (3.20)). Note that $f_{t}$ is orthonormal
and that $L=R$.

When $R=P$, where $P$ has the first $r$ normalized eigenvectors of
$\gamma_{\chi}(0)$ on the columns, the factors are the first $r$ principal
components of $\chi_{t}$. Note that if we take, for each $N$, $f_{t}$ as the
first $r$ principal components of $\chi^{N}_{t}$, the factors depend on $N$
and the matrices $L$ are not nested. The same occurs if the principal
components are normalized, that is if $R=\sqrt{\Lambda^{-1}}P,$ where
$\Lambda$ is the diagonal matrix with the first $r$ eigenvalues of
$\gamma_{\chi}(0)$ on the diagonal.

Obviously, if the rank of the $r\times r$ spectral density of $(f_{t})$ were
less than $q$, then by (3.18) Assumption 2 would be violated, so that $r\geq
q$ must hold. The spectral density of $(f_{t})$ is then nonsingular or
singular, for $r=q$ or $r>q$ respectively. Thus the dynamics of the static
factor process $(f_{t})$ can be represented by a nonsingular or singular ARMA
process of type (3.1), respectively. By (3.18) and (3.20), the Hilbert spaces
spanned by $\chi_{t}$ and $f_{t}$ are the same and therefore, $v_{t}$ is an
innovation for $f_{t}$ as well and $k(z)=Lw(z)$ is the corresponding transfer
function.

The innovations $(v_{t})$ in (3.4), and (3.20), are called the (minimal)
dynamic factors. As has been stated already, under our assumptions, for given
$(f_{t})$ they are unique up to premultiplication by a non-singular matrix or
by an orthogonal matrix if we assume orthonormality of $v_{t}$.

Piecing together what we have seen here and in the previous section, under
Assumption 3 the latent variables can be represented as follows:

$\displaystyle\chi_{it}$ $\displaystyle=L_{i}f_{t}$ (3.21)
$\displaystyle\alpha(z)f_{t}$ $\displaystyle=\beta(z)v_{t},$ (3.22)

for $i\in\mathbb{N}$, where $L_{i}$ is the $i$-th row of the matrix
$L\in\mathbb{R}^{\infty\times r}$ and

$\alpha(z)=I_{r}-\sum_{j=1}^{p}A_{j}z^{j},\ \ A_{j}\in\mathbb{R}^{r\times r},\
\ \ \ \beta(z)=\sum_{j=0}^{m}B_{j}z^{j},\ \ B_{j}\in\mathbb{R}^{r\times q},$

and where we assume that $\alpha(z)$ satisfies the stability condition,
$\beta(z)$ satisfies the the miniphase condition (that is, respectively, (3.2)
with $P(z)$ replaced by $\alpha(z)$ and (3.3) with $Q(z)$ replaced by
$\beta(z)$) and that $\alpha(z)$ and $\beta(z)$ are left coprime as $P(z)$ and
$Q(z)$ in Assumption 6.

Lastly, let us point out that normally the literature on DFM’s assumes, rather
than our Assumption 3, the existence of representation (3.21)–(3.22), that is
the existence of $N$-independent static factors, see the seminal papers Stock
and Watson (2002a, b), Bai and Ng (2002), Bai (2003).

As is seen below, Assumption 3 and representation (3.21)–(3.22) are
equivalent. That Assumption 3 implies representation (3.21)–(3.22) has been
proved above. Conversely, if $(f_{t})$ fulfills equation (3.22), then it has
the minimal state-space representation

$\displaystyle\tilde{x}_{t+1}$
$\displaystyle=\tilde{F}\tilde{x}_{t}+\tilde{G}v_{t}$ $\displaystyle f_{t}$
$\displaystyle=\tilde{H}\tilde{x}_{t},$

where $\tilde{x}_{t}$ is a $\tilde{n}$-dimensional state vector. Thus the
$N$-dimensional vector $\chi_{t}$ has the minimal state-space representation

$\displaystyle\tilde{x}_{t+1}$ $\displaystyle=\tilde{F}\tilde{x}_{t}+Gv_{t}$
$\displaystyle\chi_{t}$ $\displaystyle=L\tilde{H}\tilde{x}_{t},$

where $\tilde{n}$ is independent of $N$, so that representation (3.21)–(3.22)
implies Assumption 3.

Another assumption on the static factors is usually imposed, i.e. that the
first $r$ eigenvalues of the covariance matrix $\gamma_{\chi}(0)$ diverge as
$N\to\infty$. Its introduction and motivation are better discussed in Section
4.1, where we deal with estimation of DFM’s.

The advantage of a static factor process $(f_{t})$ for $(\chi_{t})$ is that
modeling the dynamics of $(\chi_{t})$ can be done by modeling the dynamics of
$(f_{t})$, so that the dimension of the parameter space can be reduced and is
independent of $N$. However, even for $q<N$ we may have $r=N$, see Section
4.1. This is the case when there is no non-trivial constant (i.e. independent
of $\lambda$) element in the left kernel of $f_{\chi}(\lambda)$, as this is
equivalent to

$\gamma_{\chi}(0)=\int_{-\pi}^{\pi}f_{\chi}(\lambda)d\lambda$

being nonsingular (for more details see Deistler (2019)).

###  3.3 Singular ARMA systems: The genericity of the AR case

In this section we explain that for the case $r>q$ “generically”, in a sense
to be described below, the static factor process $(f_{t})$ in (3.22) is an AR
process. This is important because estimation in the AR case is much simpler
compared to the ARMA case.

Consider the ARMA system (3.22)

$\alpha(z)f_{t}=\beta(z)v_{t},$

where $r>q$ and thus $\beta(z)$ is a “tall” matrix. As is intuitively clear,
$\beta(z)$ is generically zeroless (i.e. in the Smith-McMillan form (3.15) the
$\epsilon_{i}$ are generically constant), since e.g. the zeros of a suitably
chosen nonsingular $q\times q$ submatrix of $\beta(z)$ “typically” can be
compensated by another $q\times q$ submatrix of $\beta(z)$. In other words,
and to be more precise, for given $p$ and $m$ (the orders of the AR and MA
matrix polynomials respectively), generically, i.e. for an open and dense
subset in the parameter space, $\beta(z)$ has no zeros. This implies that
generically the ARMA system (3.22) fulfills the minimum phase condition or
that, equivalently, $v_{t}$ is an innovation process for $f_{t}$.

As is well known, every zeroless $r\times q$ polynomial matrix, with $q<r$,
can be extended to a unimodular $r\times r$ matrix
$\tilde{\beta}(z)=(\beta(z)\ \delta(z))$, say. Now write (3.22) as

$\tilde{\beta}^{-1}(z)\alpha(z)f_{t}=\begin{pmatrix}v_{t}\\\ 0\end{pmatrix}.$
(3.23)

Since $\tilde{\beta}(z)$ is unimodular, $\tilde{\beta}^{-1}(z)$ is unimodular
too and thus (3.23) is a (singular) autoregression, additionally satisfying
the stability condition (3.2). Multiplying both sides of (3.23) by $(\beta(0)\
\delta(0))^{-1}$, which is of course nonsingular, we obtain:

$a(z)f_{t}=bv_{t},\ \ a(z)=I_{r}-\sum_{j=0}^{\tilde{p}}A_{j}z^{j},\ \
b\in\mathbb{R}^{r\times q},$ (3.24)

where $b=\beta(0)$, see e.g. Anderson and Deistler (2008), Anderson et al.
(2016). Thus, in a certain sense, for $r>q$, “almost every” factor process
$(f_{t})$ can be assumed to be generated by a singular AR process.

As opposed to regular AR processes (where $r=q$ and $b=I$ hold), here the
assumption $a(0)=I$ does not guarantee identifiability, unless the assumptions
that $(a(z)\ b)$ is left coprime and that ${\rm rank}\,(A_{\tilde{p}},b)=r$
are imposed.

In this AR setting, the factor process is described by the integer-valued
parameters $r$, $q$ and $\tilde{p}$ (the latter being the autoregression
order) and the parameter space guaranteeing identifiability is

$\displaystyle\Theta$ $\displaystyle=\left\\{{\rm
vec}(A_{1},\ldots,A_{\tilde{p}},b),\ {\rm where}\ \eqref{eq:tre.due}\ {\rm
holds},A_{\tilde{p}}\ {\rm and}\ b\ {\rm are\ left\ coprime}\right.$ (3.25)
$\displaystyle\hskip 17.07182pt\left.\ {\rm and}\ {\rm rank}(A_{\tilde{p}},\
b)=r\ \right\\}.$

However, not every singular AR system can be described in such a parameter
space. For more general parameter spaces see Deistler et al. (2011), where it
has been shown that by prescribing the column degrees of $a(z)$ and by
assuming ${\rm rank}\,(A_{c},b)=r$, where $A_{c}$ denotes the column end
matrix, every singular AR system can be parameterized (see also Section
4.3.2).

It is important to point out that $r>q$ has been invariably observed in
empirical applications of DFM’s to large macroeconomic datasets, see Barigozzi
et al. (2021) for a review. A very interesting consequence of singularity is
briefly accounted for in Section 5.1.

##  4  Separation of the Common Components. Estimation

###  4.1 Principal components

We start by illustrating estimation of the latent variables $\chi_{it}$, given
the observables $y_{it}$, by means of this simplest example. Assume that

$y_{it}=\chi_{it}+\xi_{it},\ \ \ \ \chi_{it}=L_{i}v_{t},$ (4.1)

where $(v_{t})$ and $(\xi_{it})$ are scalar unit-variance white noise
processes, fulfilling (2.2) and (2.3). Moreover, assume that the processes
$(\xi_{it})$ are mutually orthogonal at all leads and lags. We have:

$\displaystyle\gamma_{\chi}(0)$ $\displaystyle={\rm
E}\chi_{t}{\chi}^{\prime}_{t}=\begin{pmatrix}L_{1}&L_{2}&\cdots&L_{N}\end{pmatrix}^{\prime}\begin{pmatrix}L_{1}&L_{2}&\cdots&L_{N}\end{pmatrix}$
$\displaystyle\gamma_{y}(0)$ $\displaystyle={\rm
E}\,y_{t}{y}^{\prime}_{t}=\gamma_{\chi}(0)+I_{N}$

The model is static and $f_{\chi}(\lambda)=(2\pi)^{-1}\gamma_{\chi}(0)$ for
all $\lambda$’s. All assumptions 1 through 4 and 6 are obviously fulfilled.
Regarding Assumption 5, the rank of $f_{\chi}(\lambda)$ is 1 so that $q=1$. We
have

$\omega^{N}_{\chi,1}=(2\pi)^{-1}\sum_{i=1}^{N}L_{i}^{2},$

(which is independent of $\lambda$) so that assuming that $\sum
L_{i}^{2}\to\infty$, Assumption 5 is also fulfilled. Lastly, observe that the
first eigenvalue of the matrix $\gamma_{y}(0)$ is

$\mu^{N}_{y,1}=(2\pi)\omega^{N}_{\chi,1}+1=\sum_{i=1}^{N}L_{i}^{2}+1,$

with left eigenvector $(L_{1}\ L_{2}\ \cdots\ L_{N})$. Now consider the
following average of the observable variables $y_{it}$, $i=1,\cdots,N$:

$p^{N}_{y,t}=\frac{1}{\sqrt{\mu^{N}_{y,1}}}\sum_{i=1}^{N}L_{i}y_{it},$

which is known as the first principal component of the $N$-dimensional vector
with coordinates $y_{it}$, $i=1,\ldots,N$, and define

$A^{N}_{y,t}=\frac{1}{\sqrt{\mu^{N}_{y,1}}}p^{N}_{y,t}.$ (4.2)

We see that

$\displaystyle A^{N}_{y,t}$
$\displaystyle=\frac{1}{{\mu^{N}_{y,1}}}\sum_{i=1}^{N}L_{i}\chi_{it}+\frac{1}{{\mu^{N}_{y,1}}}\sum_{i=1}^{N}L_{i}\xi_{it}=\frac{1}{{\mu^{N}_{y,1}}}\sum_{i=1}^{N}L_{i}^{2}v_{t}+\frac{1}{{\mu^{N}_{y,1}}}\sum_{i=1}^{N}L_{i}\xi_{it}$
$\displaystyle=\frac{\sum_{i=1}L_{i}^{2}}{\sum_{i=1}^{N}L_{i}^{2}+1}v_{t}+\frac{1}{\sum_{i=1}^{N}L_{i}^{2}+1}\sum_{i=1}^{N}L_{i}\xi_{it},$

which implies that

${\rm
E}(A^{N}_{y,t}-v_{t})^{2}=\left[\frac{\sum_{i=1}L_{i}^{2}}{\sum_{i=1}^{N}L_{i}^{2}+1}-1\right]^{2}+\frac{\sum_{i=1}^{N}L_{i}^{2}}{\left[\sum_{i=1}^{N}L_{i}^{2}+1\right]^{2}},$

and therefore that the limit in mean square of $A^{N}_{y,t}$, as $N\to\infty$,
is $v_{t}$. Also, in mean square, the projection of $y_{it}$ on $A^{N}_{y,t}$
converges to $\chi_{it}$ and the regression coefficient to $L_{i}$.

Example (4.1) illustrates the basic features of the estimation techniques used
in DFM’s. The weighted average $A^{N}_{y,t}$, which is the rescaled first
principal component of the observable variables $y_{it}$, does both the
“cleaning” of the $y$’s, in that it averages out the idiosyncratic components,
and the consistent estimation of the common components.

However, the estimate $A^{N}_{y,t}$ is defined using the population
covariances of the $y$’s—as though, so to speak, $T$ were infinite—and is
therefore unfeasible. In empirical situations, such covariances are estimated
and depend both on $N$ and $T$. Moreover, the principal-component technique
must be extended to the general case, in which, firstly, $r$, the number of
static factors, can be greater than $q$ and $q$ can be greater than unity and,
secondly, the static factors depend on the dynamic factors through an ARMA
model. Before we go over the estimation procedure in the general case we must
however introduce another assumption.

######  Assumption 8.

The $r$ largest eigenvalues of the covariance matrix $\gamma_{\chi}(0)$
diverge as $N\to\infty$.

We show below, by means of an example, that Assumption 8 is not a consequence
of Assumption 5 and that it is necessary for consistent estimation of the
space spanned by the factors $f_{t}$.

Note that if the static factors are orthonormal,
$\gamma_{\chi}(0)=LL^{\prime}.$ In any case, as the covariance matrix of
$f_{t}$ is nonsingular, Assumption 8 is equivalent to asssuming that the first
$r$ eigenvalues of $LL^{\prime}$ diverge as $N\to\infty.$

Under the assumptions 1 through 8, the first step of the procedure consists in
the estimation of the integers $q$ and $r$. This is a non-standard problem
because, firstly, both $T$ and $N$ tend to infinity, and, secondly, the
factors are estimated, not observed. Bai and Ng (2002) provide a class of
information criteria allowing to consistently estimate $r$. In the same vein
see Hallin and Liška (2007) for the estimation of $q$.

The second step estimates the static factors $f_{t}$. By (2.1) and (3.21),
setting $L_{i}=(L_{i1}\ L_{i2}\ \cdots\ L_{ir})$,

$y_{it}=L_{i}f_{t}+\xi_{it}=L_{i1}f_{1t}+L_{i2}f_{2t}+\cdots+L_{ir}f_{rt}+\xi_{it}.$

The first $r$ principal components of the observable variables $y_{it}$,
rescaled as in (4.2), are computed, based on their estimated covariances, and
used to estimate the space spanned by the factors $f_{t}$. The common
components $\chi_{it}$ and the loadings $L_{i}$ are estimated by regressing
the $y$’s on the estimated factor space. Bai (2003) proves, under some
additional technical assumptions, that these estimates of the common
components converge in probability to their population counterparts, as both
$N$ and $T$ tend to infinity, with rate

$\max\left(\frac{1}{\sqrt{N}},\frac{1}{\sqrt{T}}\right).$

In the third step an ARMA model for the estimated static factors is estimated,
see equation (3.22), this leading to the estimation of the dynamic factors as
the innovations of the ARMA model. When $r>q$ the ARMA can be replaced by a
singular $AR$, see Section 3.3. For this approach see Forni et al. (2009).

Some observations are in order.

(I) Firstly, let us show by an example that Assumption 8, which is the static
counterpart of Assumption 5, is necessary for consistent estimation by means
of principal components. Let us slightly modify model (4.1) in the following
way:

$\displaystyle y_{1t}$ $\displaystyle=v_{t}+v_{t-1}+\xi_{1t}$ (4.3)
$\displaystyle y_{it}$ $\displaystyle=v_{t}+\xi_{it},\ \ {\rm for}\ \ i>1.$

The space ${\cal S}^{N}_{t}$, spanned by $\chi_{it}$, $i\leq N$, for $N>1$,
has dimension $2$, so that $r=2$, a basis being $f_{1t}=v_{t}$,
$f_{2t}=v_{t-1}$. It is fairly easy to see that both the first eigenvalue of
$\gamma_{\chi}(0)$ and of $f_{\chi}(\lambda)$ diverge at rate $N$. However,
the second eigenvalue of $\gamma_{\chi}(0)$ is bounded. Thus Assumption 5
holds with $q=1$ but Assumption 8 does not. It also easy to see that the first
rescaled principal component converges to $v_{t}$, but the second does not
“clean” the variables $y_{it}$ from the idiosyncratic component and therefore
does not converge to the space spanned by the factors. The consequence is that
the common and idiosyncratic components of $y_{it}$, as estimated by principal
components, are, respectively:

$v_{t}$ and $v_{t-1}+\xi_{1t}$ for $i=1$, $v_{t}$ and $\xi_{1t}$ for $i>1$.

For another example consider

$\chi_{it}=v_{t}+M_{i}v_{t-1}$ (4.4)

with $\sum_{i=1}^{\infty}M_{i}^{2}<\infty$. In this case the common and
idiosyncratic components of $y_{it}$ estimated by the principal components are
$v_{t}$ and $M_{i}v_{t-1}+\xi_{it}$, respectively.

(II) What examples (4.3) and (4.4) show is that in order to “finally” (i.e.
for $N\to\infty$) separate the idiosyncratic from the common components it is
necessary that each factor is loaded infinitely often and that the loading
coefficients, if declining, do not decline too fast. This is usually rendered
by saying that the factors must be “pervasive” and has a precise formulation
in Assumption 8.

###  4.2 Generalized Dynamic Factor Models

Let us conclude this section by mentioning a strand of literature in DFMs in
which Assumption 3 does not necessarily hold. As we have seen, Assumption 3 is
equivalent to assuming that the dynamics of the latent variables $\chi_{it}$
are completely accounted for by the dynamics of the finite-dimensional,
$N$-independent vector $f_{t}$, via the static loadings $L_{i}$. The following
factor model is an elementary example in which no static factors exist. Let

$\chi_{it}=\frac{1}{1-\alpha_{i}z}v_{t},$ (4.5)

where $v_{t}$ is scalar unit-variance white noise, $-0.8\leq\alpha_{i}\leq
0.8$, $\alpha_{i}\neq\alpha_{j}$ for all $i$ and $j$, $i\neq j$. We have
$f_{\chi}(\lambda)=(2\pi)^{-1}H_{N}(\lambda)H^{*}_{N}(\lambda)$, where

$H_{N}(\lambda)=\begin{pmatrix}(1-\alpha_{1}e^{-i\lambda})^{-1}&\cdots&(1-\alpha_{N}e^{-i\lambda})^{-1}\end{pmatrix}^{\prime}.$

As $f_{\chi}(\lambda)$ has rank one for all $\lambda$, the first eigenvalue of
$f_{\chi}(\lambda)$ is its trace, that is

$\omega^{N}_{\chi,1}(\lambda)=(2\pi)^{-1}\sum_{j=1}^{N}|1-\alpha_{i}e^{-i\lambda}|^{-2}.$

Because $\omega^{N}_{\chi,1}(\lambda)$ diverges for all $\lambda$, we have
$q=1$. On the other hand,

$\chi_{it}=v_{t}+\alpha_{i}v_{t-1}+\cdots+\alpha_{i}^{N-1}v_{t-N+1}+\cdots.$

If the matrix $\gamma_{\chi}(0)={\rm E}\chi_{t}\chi_{t}^{\prime}$ were
singular then the $N\times N$ matrix with $\alpha_{i}^{m-1}$ in entry $(i,\
m)$, with $i,m=1,\ldots N$ should be singular. But the determinant of the
latter is the Vandermonde determinant of $\alpha_{1},\ \ldots,\ \alpha_{N-1}$,
which vanishes only if at least two of the $\alpha$’s are equal. Thus the
dimension of the space ${\cal S}^{N}_{t}$ is $N$, not some $N$-independent
$r$, and Assumption 3 does not hold. As a consequence the estimation technique
based on a fixed finite number of principal components does not apply.

The DFM without Assumption 3 has been studied by means of frequency-domain
methods in Forni et al. (2000), Forni and Lippi (2001), Hallin and Lippi
(2013), Forni et al. (2015), Forni et al. (2017), Forni et al. (2018), and
called Generalized Dynamic Factor Model. The main tool is the dynamic
principal component analysis introduced in Brillinger (1981), which consist of
linear combinations of current, past and future values of the observable
variables $y_{it}$ (instead of just current values as in the standard
principal components).

We cannot discuss here the merits of this “dynamic” approach relative to the
one adopted in the present paper. We limit ourselves to observing that by
means of the dynamic principal components the latent variables in model (4.5)
can be consistently estimated. Moreover, by means of the dynamic principal
components, the common and idiosyncratic components of the variable $y_{1t}$
in example (4.3) would be correctly estimated as $v_{t}+v_{t-1}$ and
$\xi_{1t}$ respectively. The same holds for example (4.4), where by means of
the dynamic principal components we estimate the latent variables
$v_{t}+M_{i}v_{t-1}$. Thus the approach based on the dynamic principal
components gives the correct results even when Assumption 3 does not hold, or
when Assumption 3 holds but not Assumption 8.

###  4.3  A State-Space Formulation of a DFM. Generic Identifiability and
Maximum Likelihood Estimation

####  4.3.1 The State-Space Formulation

A different approach to estimation of DFM’s has been introduced in Doz et al.
(2012). The paper employs a maximum likelihood estimator for the DFM resulting
from the assumption that the idiosyncratic components are cross-sectionally
uncorrelated, and shows that this misspecification has no effect on the
estimated common components as $N\to\infty$. See also Bai and Li (2016),
Barigozzi and Luciani (2019) and Poncela et al. (2021). This motivates the
following formulation of a DFM in state space.

To repeat, it is assumed that the underlying model is an exact factor model,
i.e. that the univariate idiosyncratic components are mutually uncorrelated;
in addition we assume that they are of AR(1) type (the latter can easily be
generalized). Both assumptions of course restrict generality, but are
nevertheless appropriate for many applications. An advantage of the state-
space formulation is that an EM algorithm of Shumway-Stoffer type, see Shumway
and Stoffer (2000), can be used for parameter estimation by means of the
Kalman smoother. Clearly in this case identifiability is an important
advantage.

We retain the assumption that $N>r\geq q$ and as earlier, we have the latent
variables and minimal static factors related by (3.18). Further, in case
$r>q$, and relying on an assumption of genericity, there is no loss of
generality in working with an AR model for the minimal static factor process
as given by (3.24). Indeed, even if $r=q$, we shall assume that such a model
is valid. This, of course, is not a consequence of genericity, and is
restrictive.

Next, we shall assume that the $i$-th entry of the idiosyncratic component,
$\xi_{it}$, is the first order AR process:

$\xi_{i,t}=\delta_{i}\xi_{it-1}+\eta_{it}$ (4.6)

where $|\delta_{i}|<1,\quad i=1,2,\dots,N$, and $(\eta_{it})$ are mutually
uncorrelated zero-mean white noise processes, and also uncorrelated with the
process $v_{t}$ driving the factor process model.

These assumptions follow the construction of a state-space model, where the
$(r\tilde{p}+N)$-dimensional state vector is taken to be

$x_{t}=\begin{bmatrix}f_{t}\\\ f_{t-1}\\\ \vdots\\\ f_{t-\tilde{p}+1}\\\
\xi_{t}\end{bmatrix}$ (4.7)

The model is given by

$\displaystyle x_{t+1}=$
$\displaystyle\begin{bmatrix}A_{1}&A_{2}&\dots&A_{\tilde{p}-1}&A_{\tilde{p}}&0\\\
I&0&\dots&0&0&0\\\ 0&I&\dots&0&0&0\\\ \vdots&&&&&\vdots\\\ 0&0&\dots&I&0&0\\\
0&0&\dots&0&0&\delta\end{bmatrix}x_{t}+\begin{pmatrix}b&0\\\ 0&0\\\ 0&0\\\
\vdots\\\ 0&0\\\ 0&I_{N}\end{pmatrix}\begin{pmatrix}v_{t+1}\\\
\eta_{t}\end{pmatrix}$ (4.8) $\displaystyle=Ax_{t}+B\begin{pmatrix}v_{t+1}\\\
\eta_{t}\end{pmatrix}$ $\displaystyle y_{t}$
$\displaystyle=[L\;\;0\;\;I_{N}]x_{t}=Cx_{t}$

Here,

$\delta={\rm{diag}}(\delta_{1},\delta_{2},\dots,\delta_{N}),\ \ \
\eta_{t}=(\eta_{1t}\ \eta_{2t}\ \cdots\ \eta_{Nt})^{\prime},\ \ \
\xi_{t}=(\xi_{1t}\ \xi_{2t}\ \cdots\ \xi_{Nt})^{\prime}$

and, to repeat,

$a(z)f_{t+1}=bv_{t+1},\ \ \ a(z)=I_{r}-\sum_{j=1}^{\tilde{p}}A_{j}z^{j}.$
(4.9)

Of course, we retain the stability requirement that
${\rm{det}}(I_{r}-\sum_{j}A_{j}z^{j})\neq 0$ for $z\leq 1$.

Note that the dimension of the state depends on $N$ and that this may cause
problems in proving consistency, see Bańbura and Modugno (2014).

####  4.3.2 Generic Identifiability

In studying identifiability of such a model, one should eliminate unnecessary
parameters. Hence we shall assume, as in Section 3.2, the standard
normalization for $L$ (the top $r\times r$ submatrix of $L$ is equal to
$I_{r}$), in order to uniquely obtain $L$ from ${\rm
E}\chi_{t}\chi^{\prime}_{t}$.

We shall assume, using a further appeal to genericity, that none of the
quantities $\delta_{i}^{-1}$ is a zero of
${\rm{det}}(I_{r}-\sum_{j}A_{j}z^{j})$.

The first step in establishing generic identifiability, is to explain how the
separation of common and idiosyncratic components can be achieved, or,
equivalently, how we can separate the spectrum matrix $f_{y}$ into its two
additive components $f_{\chi}$ and $f_{\xi}$. There are in fact two ways in
which this can be done.

First, since the power spectrum $f_{y}$ is rational, it has a partial fraction
expansion. By genericity, each pole is simple. Each $\delta_{i}$ gives rise to
a pole $\delta_{i}^{-1}$ and appears in the $i$-th diagonal entry of
$f_{\xi}$, but not in $f_{\chi}$, for which all poles are zeros of
${\rm{det}}(I_{r}-\sum_{j}A_{j}z^{j})\neq 0$. Hence the residue matrix
associated with the pole $\delta^{-1}_{i}$ in the partial fraction expansion
of $f_{y}$ is a diagonal matrix of rank 1. On the other hand, the residue
matrix associated with any pole arising as a zero of
${\rm{det}}(I_{r}-\sum_{j}A_{j}z^{j})\neq 0$ will generically be a matrix with
many, and maybe all, nonzero entries, even should it have rank 1. Hence the
power spectrum $f_{\chi}$ can be determined by adding together those summands
of the partial fraction expansion of $f_{y}$ whose residue matrices are other
than diagonal and of rank 1.

For the alternative procedure, let us suppose that $N$ exceeds $2q$. We can
expect by genericity that $q\times q$ submatrices of $f_{\chi}$ obtained by
deleting an arbitrary set of $N-q$ columns and an arbitrary set of $N-q$ rows
are nonsingular, while $(q+1)\times(q+1)$ submatrices obtained via a like
process are necessarily singular. Now for each diagonal entry of $f_{y}$,
choose a $(q+1)\times(q+1)$ matrix of $f_{y}$ by selecting $(q+1)$ not
necessarily continguous rows and $(q+1)$ not necessarily contiguous columns
containing that diagonal entry but containing no other diagonal entry. Note
that such a choice is possible precisely because $N>2q$. The entries of the
submatrix will be identical with the entries of the corresponding submatrix of
$f_{\chi}$, save for the entry corresponding to the single diagonal entry of
$f^{N}_{y}$. Singularity of the submatrix of $f_{\chi}$ for which all but one
entry are known will allow identification of the remaining entry, which is a
diagonal entry of $f_{\chi}$. Since all diagonal entries of $f_{\chi}$ can be
obtained this way, and the off-diagonal entries of the matrix are identical
with those of $f_{y}$, again the separation is achieved.

The next step in establishing generic identifiability is to construct the
“real-valued” parameters $L,\ A_{1},\ \ldots,\ A_{\tilde{p}},\ b,\ \delta$ and
${\rm E}\eta_{it}^{2}$, for given integral-valued specification parameters
$r,\ q,\ \tilde{p}$ from the given spectral density $f_{\chi}$, or,
equivalently, from the second moments of $(\chi_{t})$. This is done as
follows:

1\. From ${\rm E}\chi_{t}\chi_{t}^{\prime}=LL^{\prime}$, $L$ can be uniquely
determined using the standard normalization introduced in Section 3.2.

2\. The transformation (3.19) then uniquely defines the second moments of the
process $(f_{t})$.

3\. Now consider the autoregression (3.24); then, as well known, the
parameters $A_{1},\ \ldots,\ A_{\tilde{p}},\ b$ are uniquely defined from the
(population) second moments of $(f_{t})$ if the following assumption holds:

######  Assumption 9.

${\rm E}\begin{pmatrix}f_{t-1}\\\ \vdots\\\
f_{t-\tilde{p}}\end{pmatrix}\begin{pmatrix}f_{t-1}\\\ \vdots\\\
f_{t-\tilde{p}}\end{pmatrix}^{\prime}>0$ (4.10)

holds.

4\. Finally, from (2.4) we obtain $f_{\xi}$ from $f_{y}$ and $f_{\chi}$ and
thus the parameters $\delta$ and ${\rm E}\eta_{it}^{2}$.

Note that Assumption 9 is equivalent to controllability of (4.8). Due to our
assumptions, (4.10) is fulfilled, as easily shown, for $r=q$. For $r>q$,
however, which in a certain sense is standard, this may not be the case. As
shown in Deistler et al. (2011), see p. 20, in this case a first basis of
elements of $(f_{t}^{\prime},\ \ldots,f_{t-\tilde{p}+1}^{\prime})$ can be
selected and this corresponds to a prescription of column degrees
$p_{i}\leq\tilde{p}$, $i=1,\ldots,r$, for $a(z)$ in (4.9). With the
corresponding prescription of a state vector, this modified state space system
is controllable (for this argument see also the comment on (3.25) in Section
3.3).

##  5 Macroeconomic Applications: Some Consequences of Singularity

A large literature has used DFM’s as a tool for forecasting key macroeconomic
indicators, see the seminal papers Stock and Watson (2002a, b), see also Forni
et al. (2005) and Stock and Watson (2016). In another important application
DFM’s have been used in structural macroeconomic analysis. It has been shown
that by replacing the macroeconomic variables of interest with their common
components, estimated from a large dataset by the DFM technique, provides a
solution to a much-debated difficulty known among macroeconomists as the
“fundamentalness problem”. Such solution, as we see below, depends on the
singularity of the static factors and the results presented in Section 3.3.

Interesting issues, arising with nonstationarity of the variables $y_{it}$,
which is of course the case for the majority of the macroeconomic variables,
are briefly introduced in Section 5.2.

###  5.1 Applications to Structural Macroeconomic Analysis

We give here a short illustration of this literature by means of a very simple
example. Consider a DFM with $q=1$ and suppose that

$y_{it}=v_{t}+M_{i}v_{t-1}+\xi_{it}.$ (5.1)

Then focus on the vector $(\chi_{1t}\ \chi_{2t})^{\prime}:$

$\displaystyle\chi_{1t}$ $\displaystyle=v_{t}+M_{1}v_{t-1}=(1+M_{1}z)v_{t}$
(5.2) $\displaystyle\chi_{2t}$
$\displaystyle=v_{t}+M_{2}v_{t-1}=(1+M_{2}z)v_{t}.$

This is a singular vector and we see that the $2\times 1$ matrix

$\begin{pmatrix}1+M_{1}z\\\ 1+M_{2}z\end{pmatrix}$

is zeroless unless $M_{1}=M_{2}$, and thus generically zeroless as $(M_{1}\
M_{2})$ varies in an open set of $\mathbb{R}^{2}$. It is convenient to exclude
from the parameter space all points $(M_{1}\ M_{2})$ with $|M_{1}|=1$ or
$|M_{2}|=1$.

Thus generically the minimum phase condition is fulfilled for (5.2), or, in an
alternate terminology, $v_{t}$ is fundamental in (5.2). Note that this does
not imply that $|M_{1}|<1$ or $|M_{2}|<1$. In other words, if $M_{1}\neq
M_{2}$, $v_{t}$ is fundamental for the $2$-dimensional vector $(\chi_{1t})$
even though it is non-fundamental for each of the scalar processes $\chi_{1t}$
and $\chi_{2t}$ taken separately.

Now suppose that an econometrician is interested in $y_{1t}$ and, for the sake
of simplicity, that $y_{1t}$ is observed without error, i.e.
$y_{1t}=\chi_{1t}$. We assume also that $y_{1t}=v_{t}+M_{1}v_{t-1}$ is a
structural equation, i.e. that the parameter $M_{1}$ and the white noise
$v_{t}$ have a structural interpretation.

Standard Var analysis would estimate a VAR for $y_{1t}$, which is just an AR
in this case, then the AR would be inverted. As the generating process is an
MA(1), this procedure estimates consistently an MA(1):

$y_{1t}=w_{t}+N_{1}w_{t-1}.$

Now, $w_{t}$, being the residual of an AR, is an innovation for $y_{1t}$,
which implies that $|N_{1}|<1$. Thus $N_{1}$ is equal to $M_{1}$ only if
$|M_{1}|<1$, otherwise $N_{1}=1/M_{1}$. The so-called fundamentalness problem
in Structural VAR analysis arises because usually the econometrician’s
information is not sufficient to identify the structural model among those
consistent with the spectral density of the observable vector. In our case the
econometrician is not able to decide between $N_{1}$, which is by definition
less than unity in modulus, and $1/N_{1}$.

The solution of the fundamentalness problem based on DFM’s can be presented,
in the case of our simple example, as follows:

1\. We have assumed that $y_{1t}$, the variable of interest, belongs to a
large macroeconomic dataset $(y_{it}),$ $i=1,\ldots,N$.

2\. Assuming that the variables in the dataset have the DFM structure 5.1,
with $q=1$ and $r=2$, we apply the separation-estimation technique outlined in
Section 4, thus obtaining the static factors $f_{t}$, the loadings $L_{i}$ and
the common components $\chi_{it}$.

3\. Now consider any $2$-dimensional vector $(\chi_{1t}\ \chi_{it})=(y_{1t}\
\chi_{it})$, with $i\neq 1$, for example $\chi_{t}=(y_{1t}\ \chi_{2t})$. An
estimate of a singular VAR for $\chi_{t}$ and its inversion will consistently
estimate a vector MA(1) for $\chi_{t}$, with a white noise $w_{t}$ which is
fundamental. On the other hand, $v_{t}$ is generically fundamental in (5.2).
Uniqueness of fundamental representations implies that $w_{t}=v_{t}$ and the
first equation in the estimated vector MA(1) is precisely
$y_{1t}=v_{t}+M_{1}v_{t-1}$.

Note that in step 3 we estimate a VAR for the common components of $\chi_{1t}$
and $\chi_{2t}$. Alternatively, we can estimate a singular VAR for the
factors. For details in the general case and macroeconomic applications see
Forni et al. (2009), Stock and Watson (2016), Forni et al. (2020).

###  5.2 Nonstationary DFMs and cointegration of the factors

In general only some of the processes in an empirical dataset are stationary.
Assuming that the nonstationary processes are I(1), the separation-estimation
procedure described in Section 4 applies to the dataset obtained by taking
first differences of the I(1) processes.

Suppose for simplicity that all the processes $y_{it}$, $\chi_{it}$, $f_{t}$
and $\xi_{it}$ are I(1). We consistently estimate $(1-z)f_{t}$,
$(1-z)\chi_{it}$ and $(1-z)\xi_{it}$. Then the levels are obtained by
integration. This, apart from minor issues regarding the initial conditions,
is a fairly trivial extension. However, if we want to estimate a VAR for the
factors $f_{t}$ or a vector of common components, as in Section 5.1,
cointegration must be taken into account. Indeed, under the assumption $r>q$,
i.e. under singularity of $f_{t}$, the spectral density of $f_{t}$ has rank
$q$ at all frequencies and therefore at frequency zero, so that $f_{t}$ is
cointegrated with cointegration rank at least $r-q$.

For cointegration of singular vector processes and the singular version of the
Granger representation theorem, see Deistler and Wagner (2017), Barigozzi et
al. (2020). See also Barigozzi et al. (2021) for estimation and some empirical
applications.

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