Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
72,562 characters · 11 sections · 55 citation commands
A Distributed Lag Approach to the Generalised Dynamic Factor Model
\onehalfspacing
Index terms--- Approximate Factor Model, Generalized Dynamic Factor Model, Weak Factors
We consider a high-dimensional time series panel as double indexed zero mean stationary process $(y_{it}: i \in \mathbb N, t \in \mathbb Z) \equiv (y_{it})$. The Generalised Dynamic Factor Model (GDFM), introduced by forni2000generalized, forni2001generalized, hallin2013factor, relies on a decomposition of the form
where $(\bm{\varepsilon}_t)$ is an orthonormal white noise process of the common innovations, $\ubar \bm{B}_i(L)$ are pervasive causal square summable filters forni2015dynamic, gersing2026existence, $(\chi_{it})$ is the dynamic common component - driven by the same $q$ shocks for all units $i$ and $(\xi_{it})$ is the dynamic idiosyncratic component, which is weakly correlated over time and cross-section and orthogonal to $(\chi_{it})$ at all leads and lags. Note, that in view of hallin2013factor, stationarity is sufficient for obtaining a dynamic factor model decomposition—with possibly an infinite number $q$ of common shocks.
The GDFM has proven to be a powerful tool in many applications forni2005generalized, forni2004generalized, forni2017dynamic, forni2018dynamic, barigozzi2020consistent, barigozzi2021time, barigozzi2024fnets, barigozzi2025general, anderson2008generalized and allows for the presence of weak non-pervasive factors in the dynamically common space gersing2023weak. The available tools for estimation forni2001generalized, hallin2007determining, forni2005generalized, forni2017dynamic, barigozzi2024inferential, barigozzi2024fnets, barigozzi2024algebraic involve spectral density estimation and several steps to achieve a representation that is one-sided in the observed variables and therefore fit for forecasting.
In this paper, we argue that under restricted but reasonable conditions the GDFM decomposition (ref) can be estimated by regressing the observed variables on factors estimated by static principal components and their lags. We propose the restricted model
where $\bm{x}_t := (\bm{F}_t', ..., \bm{F}_{t-p}')'$ is of dimension $r_\chi = r (p+1)$ and $(\xi_{it})$ is dynamically idiosyncratic, orthogonal to $(\bm{F}_t)$ at all leads and lags. Here $\bm{F}_t$ is an $r\times 1$ vector of statically pervasive factors, while the dimension $r$ is uniquely determined by the number of diverging signal eigenvalues in the variance matrix of $(\bm{y}_t^n) = (y_{1t}, ..., y_{nt})'$ for $n\to \infty$ chamberlain1983arbitrage. In addition to the dynamic factor structure (ref), we assume that $(y_{it})$ also has a static factor structure as standard in the literature commencing with chamberlain1983arbitrage, chamberlain1983funds, stock2002forecasting, bai2002determining:
where $C_{it}$ is the static common component and $(e_{it})$ is the static idiosyncratic component, weakly correlated in the cross-section and contemporaneously orthogonal to $(e_{it})$.
The key point is that the dynamic common component $(\chi_{it})$ in model (ref) captures richer dynamics than the static common component $(C_{it})$ in (ref). In particular, a central insight of this paper is that $\chi_{it}$ may incorporate feedback from lagged values of $\bm{F}_t$. Notably, this differs from specifying dynamics for $\bm{F}_t$ while imposing the static decomposition (ref) as e.g. in forni2009opening, doz2011two, barigozzi2019quasi. More broadly, the dynamic common component represents the part that is dynamically common - that is, the projection of observed variables onto the infinite past of the common structural shocks gersing2026existence. By contrast, the static common component captures only contemporaneous co-movement.
The general theory linking (ref) and (ref) is introduced in gersing2023reconciling, gersing2023weak and subsequently considered in the time-domain-framework in barigozzi2026dynamic. Both models can be nested within the following canonical decomposition:
The static common component $(C_{it})$ is driven by statically pervasive, or briefly, strong factors, i.e. associated with thick loading columns, usually expressed in terms of $\bm{\Lambda}^{n'}\bm{\Lambda}^n / n \to \bm{\Gamma}_{\bm{\Lambda}} > \bm{0}$ for $n\to \infty$. On the other hand the weak common component $(e_{it}^\chi)$ is driven by weak non-pervasive factors, i.e. associated with non-divergent loadings $\sup_{n\in \mathbb N} \left(\bm{\Lambda}^{w, n'} \bm{\Lambda}^{w, n} \right) < \infty$ as pioneered in onatski2012asymptotics. All three terms in (ref) are contemporaneously orthogonal for all $i \in \mathbb N$. However, $(\xi_{it})$ is orthogonal to $(\bm{F}_t)$ and $(\bm{F}_t^w)$ at all leads and lags, but $(\bm{F}_t)$ and $(e_{it})$ may correlate at leads and lags via $(\bm{F}_t^w)$.
The weak factors $\bm{F}_t^w$ are to be distinguished from the case of rate-weak factors, which are excluded here. Such rate-weak factors have loadings diverging at a slower rate, i.e. $\bm{\Lambda}^{w, n'} \bm{\Lambda}^{w, n} / n^\alpha \to \bm{\Gamma}_{\Lambda^w} > \bm{0}$ with $\alpha \in (0, 1)$ demol2008forecasting, freyaldenhoven2022factor, bai2023approximate and would be part of $C_{it}$ in our setup. barigozzi2026dynamic show that cross-sectional exchangeability - a natural assumption for factor models - rules out rate weak factors.
To bring the distributed lags representation (ref) into the form of the canonical decomposition (ref) we project out $\bm{F}_t$ from its lags: For this let $\bm{\delta}$ be the population regression parameter from a regression of $\bm{F}_t^- :=(\bm{F}_{t-1}', ..., \bm{F}_{t-p}')'$ onto $\bm{F}_t$ and set $\bm{\beta}_i^-:= (\bm{\beta}_{i1}, ...., \bm{\beta}_{ip})$, then from (ref)
Consequently, we solve onatski2012asymptotics's problem, who shows that principal component based estimates are not consistent in the case of weak factors associated to non-divergent eigenvalues. We can estimate the influence of such weak factors consistently, if they live in the dynamic common space, i.e. are driven by the common shocks $(\varepsilon_t)$ gersing2023weak. In our case the weak factors are simply the influence of lags of strong factors which are estimated with a “strong signal” by static principal components. In this paper, we do not focus on the estimation and inference of the weak factors $\bm{F}_t^w$ themselves, but rather on the common components $\chi_{it}$ and $e_{it}^\chi = \chi_{it} - C_{it}$, for which explicit identification of the weak factors is not required.
An interesting interpretation of (ref) is that weak factors are associated to factor loadings that “taper off” for lags of strong factors. While the first $r$ factors are pervasive as $\bm{\beta}_0^{n'}\bm{\beta}_0^n \to \infty$, the influence of lags becomes weaker going further in the past as $\bm{\beta}^{-, n'}\bm{\beta}^{-, n} = \bm{\Lambda}^{w, n'}\bm{\Lambda}^{w, n}< \infty$. It is plausible that fewer series, are influenced by information from the past, e.g. if they are more (or less) persistent relative to the majority of series in the panel. Implications to impulse response functions have been studied as well in gersing2023weak.
The estimation theory of the GDFM commences with forni2001generalized, forni2004generalized, while forni2005generalized provide the first one-sided estimator. Infinite dimensional factor spaces, i.e. $\dim \operatorname{\overline{\operatorname{sp}}}(\chi_{it}: i \in \mathbb N) = \infty$, are covered in forni2015dynamic, forni2017dynamic, barigozzi2024inferential. In contrast, here we look only at the restricted finite dimensional case.
The contributions of this paper are as follows: First, we establish consistency and derive asymptotic confidence intervals for the dynamic common component, $\chi_{it}$, in the restricted GDFM framework (ref), as well as for its weak common component, $e_{it}^\chi$, in (ref). While regression-based approaches using factors and their lags have been extensively studied commencing with the seminal work of bai2006confidence, bernanke2005measuring, the asymptotic framework in this paper is different: In particular, bai2006confidence relies on the assumption that $(e_{it})$ and $(\bm{F}_t)$ are mutually independent. The GDFM requires accommodating dependence across leads and lags, because the common shocks $(\varepsilon_t)$ drive both, the weak factors $\bm{F}_t^w$ which are absorbed in $e_{it}$ and the statically pervasive factors $\bm{F}_t$. Hence, we need to allow explicitly for the presence of weak factors in view of equation (ref) in order to provide asymptotic confidence intervals for the dynamic- and weak common component.
Second, we propose a novel mathematical approach to consistency and asymptotic normality of the estimated factors and common components. This approach starts from the assumption of consistent sample covariance estimators for factors and idiosyncratic component, and exploits eigen-value and -vector convergence under an eigen-gap condition, drawing on yu2015useful, davis1970rotation, as also employed in barigozzi2022principal. It interprets normalised principal components as a static aggregation - in the spirit of chamberlain1983arbitrage, forni2001generalized, hallin2013factor - that averages out the weakly cross-correlated idiosyncratic component (see the role of $\bm{\mathcal K}$ in the proofs).
Third, we develop the first asymptotic theory for the GDFM that entirely avoids frequency-domain methods and avoids estimation of the number of dynamic factors $q$ in (ref). Our framework also accommodates time-varying heteroskedasticity in the idiosyncratic component, extending the scope of existing GDFM estimation theory.
Finally, in the empirical application we present statistical evidence for a significant share of weak common components in key variables for three different macroeconomic time series panels of the Euro Area.
For a real valued matrix $\bm{A} \in \mathbb R^{n \times m}$ we denote by $\bm{A}'$ the transpose. By $\@ifstar{\oldnorm}{\oldnorm*}{\bm{A}}$ we denote the spectral norm, for a vector $\bm{v} \in \mathbb R^n$ we write $\@ifstar{\oldnorm}{\oldnorm*}{\bm{v}}$ to denote the Euclidean norm. We denote by $\mu_j(\bm{A})$ the $j$-th largest eigenvalue of a square matrix $\bm{A}$ and set $\bm{M}(\bm{A}) \equiv \operatorname{diag}(\mu_1(\bm{A}), ..., \mu_r(\bm{A}))$, while $r$ is the number of static factors (see equation (ref)). By $\bm{P}(\bm{A})$ the $r\times n$ matrix consisting of the first orthonormal row eigenvectors (corresponding to the $r$ largest eigenvalues of $\bm{A}$) and by $\bm{P}^i(\bm{A})$ the $1\times r$ vector consisting of the entries of the $i$-th row of $\bm{P}(\bm{A})'$.
Let $\mathcal P = (\Omega, \mathcal A, \mathbb P)$ be a probability space and $L_2(\mathcal P, \mathbb R)$ be the Hilbert space of square integrable real-valued, zero-mean, random-variables defined on $\Omega$ equipped with the inner product $\langle u, v\rangle = \operatorname{\mathbb E} [u v]$ for $u, v \in L_2(\mathcal P, \mathbb R)$. If $\mathbb M\subset L_2(\mathcal P, \mathbb R)$ is a linear subspace, we denote by $\operatorname{proj}(\bm{u} \mid \mathbb M)$ the orthogonal projection of $\bm{u}$ onto $\mathbb M$ deistler2022modelle.
If $\bm{u}_t, \bm{v}_t$ are stochastic vector processes of zero-mean random variables with fixed dimensions $n_1, n_2$, we denote the $n_1\times n_2$-dimensional covariance matrix by $\operatorname{\mathbb E}\left[\bm{u}_t \bm{v}_{t-h}'\right]=: \bm{\Gamma}_{\bm{u}\bm{v}}(h)$ and the sample covariance by $\hat \bm{\Gamma}_{\bm{u} \bm{v}}(h) := (T-h)^{-1}\sum_{t = h+1}^T \bm{u}_t \bm{v}_{t-h}$ and $\hat \bm{\Gamma}_{\bm{u}\bm{v}}(0)=: \hat \bm{\Gamma}_{\bm{u}\bm{v}}$, $\hat \bm{\Gamma}_{\bm{u}}(-h)' = \hat \bm{\Gamma}_{\bm{u}}(h)$. If either $\bm{u}_t$ or $\bm{v}_t$ is of dimension $n$ with $n\to\infty$ we add a superscript $n$, and write e.g. $\bm{\Gamma}_{\bm{u}\bm{v}}^n$. Also, we abbreviate $\operatorname{\mathbb E}\left[\bm{v}_t \bm{v}_t'\right]:= \bm{\Gamma}_{\bm{v}\bm{v}} = \bm{\Gamma}_{\bm{v}}$. If $\operatorname{\mathbb E}\left[\bm{v}_t \bm{v}_t'\right]$ depends on time we write $\bm{\Gamma}_{\bm{v}_t}(h):=\operatorname{\mathbb E}\left[\bm{v}_t \bm{v}_{t-h}'\right]$ and $\bm{\Gamma}_{\bm{v}_t}:=\operatorname{\mathbb E}\left[\bm{v}_t \bm{v}_t'\right]$. For a stochastic vector $\bm{u}$ with coordinates in $L_2(\mathcal P, \mathbb R)$, we write $\operatorname{\mathbb V} \bm{u} := \operatorname{\mathbb E}\left[ \bm{u} \bm{u}'\right]$ to denote the variance matrix, which is symmetric and always diagonalisable.
We consider real valued stochastic double sequences, i.e. a family of random variables indexed in time and cross-section:
Such a process can also be thought of as a nested sequence of multivariate stochastic processes: $\left(\bm{y}_t^n: t \in \mathbb Z \right) = (\bm{y}_t^n)$, where $\bm{y}_t^n = (y_{1t},..., y_{nt})'$ and $\bm{y}_t^{n+1} = (\bm{y}_t^{n'}, y_{n+1, t})'$ for $n \in \mathbb N \cup \{\infty\}$. In general we will write $(\bm{y}_t: t \in \mathbb Z)=(\bm{y}_t)$ for $n = \infty$. We write $\mathbb H(\bm{y})\equiv \operatorname{\overline{\operatorname{sp}}}\left(y_{it}: i \in \mathbb N, t \in \mathbb Z \right)$ for the “time domain” of $(y_{it})$ and $\mathbb H_t(\bm{y}) \equiv \operatorname{\overline{\operatorname{sp}}}\left(y_{is}: i \in \mathbb N, s \leq t \right)$ for the infinite past, where $\operatorname{\overline{\operatorname{sp}}}(\cdot)$ denotes the closure of the linear span.
In this section, we justify why it is reasonable to assume that the GDFM in (ref) can be represented as in (ref). To elucidate the connection with the GDFM and its difference to the static decomposition, we introduce the classical assumptions. The assumptions on the spectral density are invoked solely for theoretical motivation and are not required for estimation, which proceeds entirely in the time domain and accommodates for time dependent heteroskedasticity in the dynamic idiosyncratic component.
We impose the standard GDFM assumption from forni2001generalized:
where “$\operatorname{ess\,sup}$” denotes the essential supremum of a measurable function. Assumption T(ref) implies that $(y_{it})$ can be represented in terms of (ref) (in general with a two-sided transfer-function) with $(\chi_{it})$ and $(\xi_{it})$ being orthogonal at all leads and lags, with $\sup_n \mu_q\left( \bm{f}_{\bm{\chi}}^n \right) = \infty$ a.e. on $[-\pi, \pi]$ and $\operatorname{ess\,sup}_\theta \sup_n \mu_{1}(\bm{f}_{\bm{\xi}}^n) < \infty$. For the latter property, we say that $(\xi_{it})$ is dynamically idiosyncratic. The essential boundedness of the first spectral eigenvalue restricts the cross-correlation and serial correlation in $(\xi_{it})$. Note that under very general conditions there exists a GDFM-representation such that $\varepsilon_t, \chi_{it} \in \mathbb H_t(y)$ for all $i \in \mathbb N, t \in \mathbb Z$ gersing2026existence.
In addition, we also assume that $(y_{it})$ has a static factor structure as in chamberlain1983arbitrage, gersing2023reconciling:
which implies the existence of representation (ref) with $(\bm{F}_t)$ and $(e_{it})$ being contemporaneously orthogonal, while correlation at leads and lags is allowed, and $\sup_{n\in \mathbb N} \mu_r \left(\bm{\Gamma}_C^n\right) = \infty$ and $\sup_{n\in \mathbb N} \mu_1 \left(\bm{\Gamma}_e^n\right) = \infty$. For the latter property, we say that $(e_{it})$ is statically idiosyncratic.
As shown in gersing2023weak, imposing the typical eigenvalue behavior on the spectral density (T(ref)) vs the variance-covariance matrix (T(ref)) imply two different types of decompositions which are nested in (ref). In particular it can be shown that the weak common component in (ref) is the static idiosyncratic component of the dynamic common component. Consequently, the dynamic common component $\chi_{it}$ in Assumption T(ref) retains in general more variation of the output compared to the static common component $C_{it}$ which is due, as argued below, to the feedback in $(y_{it})$ to lags of the static factors $\bm{F}_t$.
Recall that as shown in forni2001generalized, the dynamic common component can be represented as the orthogonal Hilbert space projection on the space spanned by the common shocks. In view of the one-sided representation (ref) gersing2026existence, it is enough to project on the infinite past of what we may call the structural common shocks $(\bm{\varepsilon}_t)$ of the economy:
Furthermore, since the static factors $(\bm{F}_t)$ of $(y_{it})$ are also the static factors of $(\chi_{it})$, we have $\operatorname{sp}\left(\bm{F}_t\right)\subseteq \operatorname{\overline{\operatorname{sp}}}\left(\chi_{it}: i \in \mathbb N\right) \subset \mathbb H_t(\bm{\varepsilon})$, so both $(\chi_{it})$ and $(\bm{F}_t)$ are completely driven by $(\bm{\varepsilon}_t)$. Now, it is natural to assume that the same common shocks which drive the whole economy, also drive the static factors $(\bm{F}_t)$ and are fundamental for them. This assumption has been implicitly imposed literally everywhere in the literature where the common shocks $(\bm{\varepsilon}_t)$ in combination with $(\bm{F}_t)$ play a role bernanke2005measuring, bai2007determining, stock2011theoxford, forni2009opening, forni2025common among many others.
Formally, we require that there is exists a representation of the form:
where $\ubar \bm{b}_F(L)$ is a causal transferfunction that has a causal left inverse, say $\ubar \bm{c}_F (L) = \ubar \bm{b}_F^{\dag}(L)$, while “\dag” denotes the generalised inverse, which is of dimension $q \times r$ such that
In this case we can retrieve the infinite past of the common shocks from the infinite past of the static factors. Therefore, in order for our procedure to work, we must impose the following assumption:
To further explore the role of T(ref), the prototypical “`dynamic factor model” as e.g. in bai2007determining, stock2011theoxford:
where $\bm{f}_t$ are $q\times 1$ dimensional “dynamic factors” stacked in $\bar \bm{F}_t = (\bm{f}_t', ..., \bm{f}_{t-p_\lambda}')$ with loadings $\bar \bm{\Lambda}_i = (\bm{\lambda}_{i1}, ..., \bm{\lambda}_{ip_\lambda})$ and modelled as an autoregressive process. Setting $\ubar \bm{a}_f(L) = \bm{I}_q - \bm{A}_1 L - ... - \bm{A}_{p_f} L^{p_f}$, where $L$ is the lag operator, we can write (ref), (ref) in terms of the GDFM representation as
while the characteristic eigenvalue behaviour of the GDFM is satisfied if at least one column block of the loadings has $q$ divergent eigenvalues gersing2023weak.
As opposed to what is commonly imposed in the literature, this model - though finite dimensional - can in general not be estimated by means of static principal components (through stacking the $\bm{f}_t$'s) but has weak non-pervasive factors. Considering the vector representation of the dynamic common component $\bm{\chi}_t^n = \bar \bm{\Lambda}^{n} \bar \bm{F}_t$, in general $r < r_\chi = q(p+1)$ eigenvalues of $\bar\bm{\Lambda}^{n'} \bar\bm{\Lambda}^n$ diverge as $n\to \infty$, which implies that the weak common component in (ref) is non-trivial and there exists $\bm{T} \in \mathbb R^{r\times r_\chi}$ such that $\bm{F}_t = \bm{T} \bar \bm{F}_t$ in (ref) gersing2023reconciling.
Rewrite (ref), (ref) as a state space system
The transferfunction of $(\bm{F}_t)$ is
Now consider the parameter space consisting of parameters $(\bm{A}_1, ..., \bm{A}_p, \bm{T}, \bm{b})$ where the autoregressive parameters are stable, i.e. $\det \ubar \bm{a}_f(z) \neq \bm{0}$, $\bm{T}$ is free and $\bm{b}$ is a lower triangular matrix of full rank describing the $q(q-1) /2$ free parameters of the innovation variance matrix of for all $\@ifstar{\oldabs}{\oldabs*}{z} \leq 1$.
For the special case that $r = q$, Theorem (ref).1 establishes A (ref)(i) under a minimum phase condition, which standard in time series analysis deistler2022modelle. For $r > q$, Theorem (ref).2, guarantees A(ref)(i) for a generic (open and dense) set in parameter space, which implies that the set of parameters where this condition does not hold has measure zero.
We assume to work with pre-centered data. If data is not pre-centered then all the following applies to the centered realizations $(y_{it}-T^{-1}\sum_{t=1}^T y_{it} : i=1,\ldots, n,\, t=1,\ldots, T)$. Let $\operatorname{\mathbb E}\left[\bm{y}_t^n \bm{y}_t^{n'}\right] =:\bm{\Gamma}_{\bm{y}}^n$ with $\bm{y}_t^n = (y_{1t}, ..., y_{nt})'$. The normalised principal components of $\bm{y}_t^n$ are
Clearly the normalised principal components of $\bm{y}_t^n$ are determined only up to sign. To resolve this sign indeterminacy, we assume (without loss of generality by changing the cross-sectional ordering) that the first $r$ rows of $\bm{P}'(\bm{\Gamma}_{\bm{y}}^n) \bm{M}^{1/2}(\bm{\Gamma}_{\bm{y}}^n)$ have full rank (from a certain $n$ onwards) and fix the diagonal elements to be positive. This fixes the sign of the eigenvectors and the normalised principal components. We use $\hat{\bm{\mathcal K}}_j$ is the $j$-th row of $\hat{\bm{\mathcal K}} := \bm{M}^{-1/2}(\hat \bm{\Gamma}_{\bm{y}}^n) \bm{P}(\hat \bm{\Gamma}_{\bm{y}}^n)$ and $\bm{\mathcal K}_j$ is the $j$-th row of $\bm{\mathcal K} := \bm{M}^{-1/2}(\bm{\Gamma}_{\bm{y}}^n) \bm{P}(\bm{\Gamma}_{\bm{y}}^n)$.
We employ the following assumptions for estimation which are in line with Assumptions T(ref)-T(ref), but in addition impose rates on the divergence, allow for heteroscedasticity over time in the dynamic idiosyncratic component and don't require the existence of the spectral density.
Assuming that $\bm{\Gamma}_{\bm{x}} > \bm{0}$ is a simplification and excludes the case where the stack $\bm{x}_t$ has a singular variance-covariance matrix anderson2016structure, deistler2019singular. Without loss of generality for the proofs one may use a sub-vector of $\bm{x}_t$ with full rank variance-covariance matrix instead.
First note that the assumption $\bm{\Gamma}_{\bm{F}} = \bm{I}_r$ is without loss of generality for the results in this paper (see remark (ref) in the appendix for details) and is made for simplicity. A(ref)(ii) implies T(ref)(i) together with a rate of divergence. Assumption A(ref)(vi) allows that dynamic idiosyncratic variance to be time dependent but globally bounded in the first eigenvalue and imposes furthermore the non-pervasiveness of the weak factors.
In the following we use the eigen-decomposition $\bm{\Gamma}_{\bm{\Lambda}}^n := \bm{P}_{\bm{\Lambda}}^{n'} \bm{D}_{\bm{\Lambda}}^n \bm{P}_{\bm{\Lambda}}^n$ with eigenvalues sorted from the largest to the smallest in the diagonal matrix $\bm{D}_{\bm{\Lambda}}^n$ and $\bm{P}_{\bm{\Lambda}}^n$ being a matrix of orthonormal row eigen-vectors. Analogously we use $\bm{\Gamma}_{\bm{\Lambda}} = \bm{P}_{\bm{\Lambda}}' \bm{D}_{\bm{\Lambda}} \bm{P}_{\bm{\Lambda}}$. Assumption A(ref) implies also that $\sup_{t\in \mathbb Z} \sup_{n\in \mathbb N} \mu_1 \left( \bm{\Gamma}_{\bm{e}_t}^n \right) \leq \mathcal B_\xi +\mathcal B_{\Lambda^w}=: \mathcal B_e < \infty$.
Let $\bm{\eta}_t^n := n^{-1/2}\bm{\Lambda}^{n'} \bm{\xi}_t^n$ be the $r\times 1$ dimensional aggregate of the idiosyncratic component obtained from averaging with the static factor loadings. Note that by Assumption A(ref), it holds that $\@ifstar{\oldnorm}{\oldnorm*}{\operatorname{\mathbb E}\left[\bm{\eta}_t^n \bm{\eta}_t^{n'}\right]} < \mathcal B_\xi \@ifstar{\oldnorm}{\oldnorm*}{n^{-1/2}\bm{\Lambda}^n}< \mathcal B_\xi \mathcal B_\Lambda$.
These conditions establish mean square convergence of the sample covariances. Note that in the standard setup of chamberlain1983arbitrage, implied by Assumption T(ref), the variance of any cross-sectional average (scaled by $\sqrt{n}$), like $\bm{\eta}_t^n$ is bounded in $n$. Consider an aggregate of dynamic idiosyncratic terms with $\bm{\eta}_t^n := p^{n'}\bm{\xi}_t^n$ with any $p\in \mathbb R^n$, $\@ifstar{\oldnorm}{\oldnorm*}{p}<\infty$ for all $n \in \mathbb N$. Then $\operatorname{\mathbb E}\left[(\bm{\eta}_t^n)^2\right]\leq \sup_{n\in \mathbb N}\@ifstar{\oldnorm}{\oldnorm*}{p^n}\mathcal B_\xi$ independent of $n \in \mathbb N$.
To establish asymptotic normality we need slightly more restrictive assumptions which refer mostly to further limiting the serial- and cross-correlation in the idiosyncratic component.
Assumption A(ref) is similar to the conditions used in barigozzi2022principal who provides an excellent overview on how the usual Assumptions of the factor model literature are nested in Assumption A(ref). Whereas for consistency, we only need $L^2$ boundedness of dynamic and static idiosyncratic variance, for asymptotic normality we need $L^1$ boundedness reflected in A(ref)(i), (ii). Furthermore, additional bounds are needed that restrict serial dependence of the dynamic idiosyncratic part A(ref)(iii), (iv).
Following bai2003inferential, bai2006confidence, barigozzi2022principal for estimating the middle term, robust to heteroskedasticity, we may use either one of the following:
where $\kappa(t, s)$ is a suitable kernel with bandwidth $M_T$.
The final estimator for the asymptotic variance is then given by
Following bai2003inferential, bai2006confidence, under cross-sectional independence, we may use:
which allows heteroskedasticity over time.
On the other hand if $(\bm{\xi}_t^n)$ is stationary, we may account for cross-sectional correlation by bai2006confidence
or fresoli2024dealing:
with $\hat \sigma_{ij, h}^\xi := (T-h)^{-1}\sum_{t = h+1}^T \hat \xi_{it} \hat \xi_{j,t-h}$, $\hat{\operatorname{\mathbb V}}\left[\hat \xi_{it} \hat \xi_{j,t-h}\right] := (T-h)^{-1}\sum_{t = h+1}^T \left[\hat \xi_{it} \hat \xi_{jt} - \hat \sigma_{ij, h}^\xi\right]^2$ and $c_{ij, h} := \delta \left[\hat{\operatorname{\mathbb V}}\left[\hat \xi_{it} \hat\xi_{j,t-h}\right] \log(n) / (T-h)\right]^{1/2}$, while $\delta = [2(2-\gamma)]^{1/2}$ with $\gamma \in (0, 2)$ is a parameter for controlling sparsity.
Using the previous Theorems, we can prove asymptotic normality for the common component estimator for $i = 1, ..., n$ and $t = p+1, ..., T$:
For estimating the standard deviations $U_{it}$ and $V_{it}$ in Theorem (ref), we use:
The covariance terms can be estimated using:
while for $ \savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\bm{P}_{\bm{\Lambda}} \bm{\Theta}_{\bm{\Lambda} \bm{\xi}, \bm{\Lambda} \bm{e}, h}\bm{P}_{\bm{\Lambda}}'}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight} }{0.5ex}} \stackon[1pt]{\bm{P}_{\bm{\Lambda}} \bm{\Theta}_{\bm{\Lambda} \bm{\xi}, \bm{\Lambda} \bm{e}, h}\bm{P}_{\bm{\Lambda}}'}{\tmpbox} $ we use the same formula as in (ref), so $(T-h)^{-1}\sum_{t = h+1}^{T}\hat \xi_{it}\hat \xi_{i, t-h}$ instead $(T-h)^{-1}\sum_{t = h+1}^{T}\hat \xi_{it}\hat e_{i, t-h}$ which is justified by the orthogonality between $\bm{F}_t^w$ and $\xi_{is}$ for all leads and lags and $i \in \mathbb N$. The standard deviations are estimated analogously to (ref) using the formulas of Theorem (ref).
To assess the finite sample properties of the proposed procedures, we consider data generated as
with $a = 0.8$ and one statically pervasive factor and one weak factor in the dynamic common space: For this we set $\lambda_{i1} = 0$ for $1\leq i \leq 10$, $\lambda_{i1} = 1$ for $11 \leq i \leq 20$, and $\lambda_{i1} \sim iidN(1, 1)$ for $i \geq 21$, and we set $\lambda_{i2} = 1$ for $1\leq i \leq 10$ and $\lambda_{i2} = 0$ for $i \geq 11$. For the dynamic idiosyncratic component $(\xi_{it})$ we consider different versions, based on the specification
with $\varepsilon_{it}^\xi \sim iidN(0, 1)$, independent of $(\varepsilon_t)$, with $\operatorname{\mathbb E}(\varepsilon_{it}^\xi \varepsilon_{jt}^\xi) = \tau^{\@ifstar{\oldabs}{\oldabs*}{i-j}}, i, j = 1, ..., n$, $\tau \in \{0, 0.5\}$ if $\@ifstar{\oldabs}{\oldabs*}{i-j}\leq 10$ and $\operatorname{\mathbb E}(\varepsilon_{it}^\xi \varepsilon_{jt}^\xi) = 0$ otherwise; last $\alpha_i = \{0, \delta_i\}$ with $\delta_i \sim iidU(0, \delta)$ and $\delta \in \{0, 0.5\}$. The parameters $\tau$ and $\delta$ are crucial to control the cross-sectional and serial correlation in the dynamic idiosyncratic component, respectively.
We obtain the canonical decomposition (ref) by projecting out the statically pervasive factor $f_t$ from the lag $f_{t-1}$ gersing2023weak. Let $\widetilde \lambda_{i1} = \lambda_{i1} + \lambda_{i2}a$, $\widetilde \lambda_{i2} = \lambda_{i2} \sigma_\varepsilon$, $\widetilde f_{1t} = f_t$, $\widetilde f_{2t} = \sigma_\varepsilon^{-1}(f_{t-1} - a f_t)$, with $\sigma_\varepsilon^2:= (1-a^2)$. Then, we can rewrite model (ref) as
where the factors $\widetilde \bm{F}_t=(\widetilde f_{1t} \; \widetilde f_{2t} )'$ are orthonormal, i.e., $\operatorname{\mathbb E} [\widetilde \bm{F}_t \widetilde \bm{F}_t' ]= \bm{I}_2$. The first factor $\widetilde f_{1t} = F_t$ is pervasive while the second one $\widetilde f_{2t} = F_t^w$ is not. Indeed, letting $\bm{\chi}_t^n = (\chi_{1t}, ..., \chi_{nt})'$, it is easily verified that only one signal eigenvalue in $\bm{\Gamma}_{\bm{\chi}}^n := \operatorname{\mathbb E} [\bm{\chi}_t^n \bm{\chi}_t^{n'}]$ diverges. Therefore $r = 1$ in view of the static decomposition.
The first unit, $i = 1$, decomposes as
We generate data according to (ref). At each replication $j = 1, ..., B$ and for each considered estimator of $\chi_{1t}^{[j]}$, generically denoted as $\widehat \chi_{1t}^{[j]}$, we define $MSE_1 = B^{-1}\sum_{j = 1}^B T^{-1}\sum_{t = 1}^T (\widehat \chi_{1t}^{[j]} - \chi_{1t}^{[j]})^2$ and look at results for different $(n, T)$ combinations with $B = 500$ replications.
The left plot in Figure (ref) shows, that the principal component estimator with $r = 2$ (spca2) is not consistent for $\chi_{1t}$, indeed, $MSE_1$ is not approaching zero as $n$ and $T$ becomes larger onatski2012asymptotics. The other two dynamic estimators (dpca and fdl) are consistent (Theorem (ref)) as $MSE_1$ is monotonically decreasing with increasing $(n,T)$, while the finite distributed lags approach fdl yields better performance. Furthermore, the right plot of figure (ref) shows that we cannot recover $\chi_{1t}$ by static principal components, even if we increase the number of factors $r$, i.e., with spcar for $r = 1, 2, 3, 5, 9$. In other words, the weak factor $\widetilde f_{2t}$ is too weak to be recovered by means of static principal components - though it is important individually, as it explains a large part of the variation for the first ten units.
Analogous results are obtained for other configurations of the sample size and the idiosyncratic component, see tables (ref), (ref) and (ref).
Next we explore the coverage rates of the asymptotic confidence intervals provided by the CLTs of section (ref). For estimating the loadings variance, we use the HAC version from equation (ref), for estimating the factors' variance we use the estimator from (ref) by fresoli2024dealing. Table (ref) shows the results for the case where $T>n$. In all cases the sample coverage approaches the nominal rate of 95% for increasing sample size $(n, T)$, while convergence kicks in about $(n, T) = (120, 240)$ and slightly slower for the case with higher idiosyncratic cross-correlation $(\tau = 0.5)$.
To demonstrate our methods empirically, we use different high-dimensional macroeconomic time series panels for the Euro Area (EA) and major EA countries, based on the open source dataset of barigozzi2024large. We consider three datasets: a) monthly EA and country-level series ($n = 381$, $T = 309$, $2000:01$ to $2025:09$) b) quarterly EA and country-level series ($n = 595$, $T = 103$, $2000:Q1$ to $2025:Q3$) c) quarterly series for Germany ($n = 63$, $T = 103$, $2000:Q1$ to $2025:Q3$). All series are standardised to zero mean and unit sample variance. Following barigozzi2024large, the data are transformed to stationarity. As principal components are sensitive to outliers but the sample includes the COVID-19 and several other crisis periods, we preprocess the data by removing univariate and multivariate outliers using standard methods. Missing values are then interpolated by splines (multivariate outliers) and the EM algorithm for factor models stock2002forecasting (univariate outliers).
The number of factors is estimated using alessi2010improved, which refines bai2002determining in the spirit of hallin2007determining. Following factor selection, model (ref) is estimated by regressing on the estimated factors, with the lag order determined by the modified BICM of groen2013model, specifically designed for regressions involving estimated factors. To retain information from crisis periods, parameter estimation is conducted on the cleaned data, while factor and common component estimation use the standardised raw data combined with eigenvectors and eigenvalues estimated from the cleaned sample covariance matrix. Finally, we obtain the estimates $\hat \chi_{it}$ from model (ref) and $\hat e_{it}^\chi = \hat \chi_{it} - \hat C_{it}$.
Let us comment the results in order: a) For the monthly data, we estimate $\hat r = 4$ factors, with $130$ series exhibiting lag order $p>0$. The weak common component (WCC) explains $3.6\%$ of the total variance, while the static common component (SCC) accounts for $19.6\%$. For $55$ variables, the WCC explains more than $10\%$ of the total variance, reaching up to $25.5\%$ for ESENTIX (Economic Sentiment Indicator) of the Euro Area.
Recall that the SCC captures the contemporaneously common variation, arising from linear combinations across all series at a fixed time $t$. In contrast, the dynamic common component (DCC) reflects responses to common structural shocks, i.e., the projection of observed variables on the infinite past of common shocks $\varepsilon_t$ (see (ref)). The DCC thus accommodates heterogeneous persistence in responses to these shocks, whereas persistence in the SCC derives solely from statically pervasive factors.
For ESENTIX (Figure (ref)), the WCC allows the DCC to track the observed series more closely during the COVID period, while the SCC responds only weakly to the large shocks. Moreover, outside crisis periods, the WCC captures additional dynamics not accounted for by the SCC. Similarly, for the growth rate of French industrial production (Figure (ref)), the DCC exhibits dynamics that differ significantly from those of the SCC.
Next, consider dataset b) of quarterly series. Only 10 out of 595 variables exhibit $p>0$. The WCC explains 0.3% of total variation, whereas the SCC accounts for 38.9%. Estimates for total hours worked (THOURS) in the Euro Area - alongside unemployment, a key labour market indicator - are shown in Figure (ref). Here as well, the WCC appears offset the stronger mean reversion of the SCC relative to the observed series.
Finally, consider dataset c), comprising only German series. Only one variable has $p>0$, namely GNFCIR (Gross Investment Share of Non-Financial Corporations) with $p=2$. Despite this, GNFCIR is a key leading indicator of long-term economic growth, reflecting business confidence about future conditions. In this case, most of the dynamics are captured by the WCC (40%), while the SCC accounts for a smaller share of variation (30%) relative to the observed series (see Figure (ref)).
We consider the generalised dynamic factor model under the assumption that the dynamic common component can be represented in terms of a projection of the output on the statically pervasive factors and a finite number of their lags. First, we justify why this assumption is sensible from a theoretical point of view and show that it holds under general conditions for the prototypical dynamic factor model. We set up an asymptotic framework that allows the factors to correlate at leads and lags with the static idiosyncratic component (in which the weak common component is absorbed) to accommodate for the existence of weak non-pervasive factors in the dynamic common space. We provide asymptotic confidence intervals for the dynamic, the weak and the static common component. In an empirical demonstration on Euro-Area Macroeconomic Data, we find that there are several important variables significantly driven by the weak common component.