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.
Rendered from LaTeX for readability, not typeset faithfully. Citation keys are highlighted; maths is left as source; figures, tables and equation environments are summarised rather than reproduced; unrecognised commands are greyed out so nothing is silently dropped. Email addresses are removed.
Actually, There is No Rotational Indeterminacy in the Approximate Factor Model
abstractWe show that in the approximate factor model the population normalised principal components converge in mean square (up to sign) under the standard assumptions for $n\to \infty$. Consequently, we have a generic interpretation of what the principal components estimator is actually identifying and existing results on factor identification are reinforced and refined. Based on this result, we provide a new asymptotic theory for the approximate factor model entirely without rotation matrices. We show that the factors space is consistently estimated with finite $T$ for $n\to \infty$ while consistency of the factors a.k.a the $L^2$ limit of the normalised principal components requires that both $(n, T)\to \infty$.
Index terms--- Approximate Factor Model
Introduction
Let $(y_{it}: i \in \mathbb N, t \in \mathbb Z) \equiv (y_{it})$ be a zero mean stationary stochastic process indexed in time $t$ and cross-section $i$. The approximate factor model introduced by chamberlain1983arbitrage, chamberlain1983funds, stock2002forecasting, stock2002macroeconomic, bai2002determining relies on a decomposition of the form
align[align omitted — 203 chars of source]
where $F_t$ are factors of dimension $r$ (usually small), the $\Lambda_i$'s are pervasive loadings with $\Lambda^n = (\Lambda_1', ..., \Lambda_n')'$, $C_{it}$ is the “common component” and $e_{it}$ is the contemporaneously weakly correlated “idiosyncratic component”. We consider a setup where the loadings $\Lambda_i$ are “deterministic parameters”.
Clearly, the “true factors” are latent and can only be identified up to a non-singular transformation. A classical result stock2002forecasting, bai2002determining, bai2003inferential, barigozzi2022estimation, what we may call “convergence to the factor space” is
align[align omitted — 155 chars of source]
while $\hat F_t^n$ denotes factors estimated by the normalised (by square root for the sample eigenvalues) principal components (obtained from the sample variance matrix) $\hat H_n$ is a non-singular transformation depending on $n$. Alternatively, there is also a result where $H_n$ is deterministic barigozzi2022estimation.
Next, bai2013principal assume the identifying restrictions
itemize• $(\Lambda^n)' \Lambda^n \in \mathcal D(r)$ is diagonal for all $n \in \mathbb N$
• $T^{-1}\sum_{t = 1}^T F_t F_t' = I_r$
To maintain the property $(\Lambda^n)' \Lambda^n$ is diagonal for all $n\in \mathbb N$ the loadings $\Lambda_i$ associated with a fixed $i$ need to change with every $n$ in general: For instance, let $\lambda_{ij}$ be the $i,j$ entry of the loadings matrix and $\lambda_j^n$ be the $j$-th column, then in the following case $(\Lambda^2)'\Lambda^2$ is diagonal, but for
align*[align* omitted — 304 chars of source]
So under the restrictions $(i), (ii)$ above, without allowing $\Lambda_i$ to depend on $n$, it would be too restrictive to demand orthogonality in the loadings columns for every $n\in \mathbb N$. The same holds for condition (ii) - every trajectory is associated with a another factor representation. As is shown in bai2013principal, barigozzi2022estimation the identification restrictions above imply that under standard assumptions
align[align omitted — 162 chars of source]
where $\hat F_t^{n, C}$ are the normalised sample principal components of $C_t^n$ for every $n$. We can be shown to converge to $F_t^{n, C}$ the population principal components in probability for $T\to \infty$. However, nothing has been said about whether $\hat F_t^{n,C}$ or $F_t^{n,C}$ converges or not. It is well known that in general, the principal components of a multivariate vector $y_t^n$ or $C_t^n$ change with $n$, since adding new variables changes the principal directions of the data. Therefore we conclude that ((ref)) establishes convergence to sequence of factor representations - not to a specific factor representation. Often we wish to interpret the loadings structurally, e.g. achieve a representation in which $\Lambda^n$ is sparse. In an asymptotic setup which requires $n \to \infty$ for identification, convergence to a sequence of representations is not appropriate for interpretations of the loadings, like e.g. sparsity: After all, as shown above if $\Lambda^n$ is sparse under restrictions (i), (ii), then $\Lambda^{n+1}$ in general is not sparse.
Finally bai2020simpler prove for $\hat H_n$ in ((ref)) that $\hat H_n \overset{P}{\to} H_0$. This is wonderful as combining this with ((ref)) it readily implies that
align[align omitted — 378 chars of source]
where here the factor representation $F_t$ does depend on $n$ as in ((ref)). Therefore the sample normalised principal components $\hat F_t^n$ have a probability limit for $(n, T)\to \infty$, which is $H_0 F_t$. We may say that ((ref)) establishes convergence to a specific factor representation in probability.
The contribution of this paper is to extend/refine this result by proving that under suitable (standard conditions) the normalised population principal components of $C_t^n$ and $y_t^n$, say $F_t^{n, C}$ and $F_t^{n, y}$ converge in mean square for $n\to \infty$ to what we may call the normalised principal components of the statically common space, say $F_t^\infty$, i.e.
align[align omitted — 224 chars of source]
- a representation independent of $n$. On the filp side also the associated loadings $\Lambda_i^\infty$ are independent of $n$.
This result paves the way for a simpler asymptotic theory, presented in the second part of the paper, that operates without “rotation matrices” like $\hat H_n$ or $H_n$ and establishes convergence to factors rather than a sequence of factors. The proofs are approached like in forni2004generalized using perturbation theory from wilkinson1965algebraic. Consequently, we have a generic interpretation for what the principal components estimator is actually targeting and structural interpretations on the estimated loadings, like e.g. sparsity are justified. Furthermore it also provides meaningful interpretations for regression outputs in factor-augmented regressions bernanke2005measuring, bai2006confidence, as we have a meaningful interpretation of each of the factors. In particular we show that
align*[align* omitted — 303 chars of source]
which clarifies the different roles of $T$ and $n$ in the identification. While $T\to\infty$ is needed to estimate the normalised principal components, $n\to\infty$ is needed to average out the idiosyncratic terms.
Finally, we also clarify why consistency of the factor space only requires $n\to \infty$, i.e. we show
align*[align* omitted — 132 chars of source]
Convergence of the factor space under finite $T <\infty$ has also been studied and proved in zaffaroni2019factor, fortin2023eigenvalue, onatski2023comment under slightly different assumptions and the earlier in connor1986performance, bai2003inferential. The proof presented in this paper reveals how and why this result seamlessly fits with the approximate factor model framework: To identify the factor space, we only need to average out the idiosyncratic term by cross-sectional aggregation gersing2023reconciling, gersing2023weak. If we have $n = \infty$, the factor space is obtained whenever, we can find cross-sectional averages that result in $r$ linearly independent factors. This is satisfied even for very imprecisely estimated eigenvectors in the case of finite $T$.
General Setup
We suppose that the observed process $(y_{it})$ has the representation as in ((ref)), ((ref)). Let $A$ be a symmetric $n\times n$ matrix. Denote by $\mu_j(A)$ the $j$-th largest eigenvalue of a matrix $A$ and set $M(A) \equiv \operatorname{diag}(\mu_1(A), ..., \mu_r(A))$. Denote by $P(A)$ the $r\times n$ matrix consisting of the first orthonormal row eigenvectors (corresponding to the $r$ largest eigenvalues of $A$) and by $P^i(A)$ the $1\times r$ vector consisting of the entries of the $i$-th row of $P(A)'$. Let $\operatorname{\mathbb E} y_t^n (y_t^n)' \equiv \Gamma_y^n$ with $y_t^n = (y_{1t}, ..., y_{nt})'$.
The normalised principal components of $y_t^n$ are
align*[align* omitted — 146 chars of source]
For simplicity we suppose that the loadings are such that $\operatorname{\mathbb E} F_t F_t' = I_r$ which can always be achieved by a constant (independent of $n$) rotation matrix applied to the factors. Furthermore the following assumptions are made:
assumption[r-Static Factor Structure]
There exists a natural number $r < \infty$, such that
\begin{itemize}
• $\sup_n \mu_r(\Gamma_C^n) = \infty$ (pervasive loadings)
• $\sup_n \mu_1(\Gamma^n_e) < \infty$.
\end{itemize}
Note that part $(ii)$ is also implied by assumptions made for the idiosyncratic component as in bai2002determining. This is shown in barigozzi2022estimation. Henceforth let us commence from a representation in terms of “true factors” which is such that $\operatorname{\mathbb E} F_t F_t' = I_r$. The existence of such a representation is implied by A(ref) gersing2023reconciling. It also follows from the classical assumptions of bai2002determining, bai2003inferential.
assumption[Asymptotic Properties of the Loadings]\ \\[-2.5em]
\begin{itemize}
• Normalised Representation: Without loss of generality we assume that $\operatorname{\mathbb E} F_t F_t' = I_r$
• Convergence of the “loadings variance”: Set $\frac{(\Lambda^n)'\Lambda^n}{n} = \frac{1}{n} \sum_{i = 1}^n \Lambda_i' \Lambda_i \equiv \Gamma_\Lambda^n$ and suppose that $\left\lVert\Gamma_\Lambda^n- \Gamma_\Lambda\right\rVert = \mathcal O(n^{-1/2})$ for some $\Gamma_\Lambda > 0$.
• The eigenvalues of $\Gamma_\Lambda$ are distinct and contained in the diagonal matrix $D_\Lambda$ sorted from the largest to the smallest.
• Loadings and idiosyncratic variances are globally bounded, i.e. $\left\lVert\Lambda_i\right\rVert < B_\Lambda < \infty$ and $\operatorname{\mathbb E} e_{it}^2 < B_e < \infty$ for all $i\in \mathbb N$.
\end{itemize}
As we will see this is equivalent to supposing that $n^{-1}M(\Gamma_y^n) \to D_\Gamma = D_\Lambda$ are distinct. We can think of A(ref)(i) in the sense of a law of large numbers for the loadings, e.g. if the loadings/cross-sectional units are sampled in a first step from an IID distribution. The eigenvalues divided by the rate converge to diagonal matrix $D_\Gamma = D_\Lambda$, are asymptotically well separated. Convergence and separation are both crucial for $L^2$ convergence of the normalised principal components. In the literature bai2002determining, barigozzi2022estimation, it is usually assumed that $n^{-1} (\Lambda^n)' \Lambda^n \to \Gamma_\Lambda > 0$ and $\Gamma_F > 0$ which implies the convergence of $n^{-1}M(\Gamma_C^n)$ as we will see below. Asymptotically separated eigenvalues assumed e.g. in forni2004generalized (Assumption R, but without convergence) in bai2003inferential (Assumption G) and in bai2013principal.
The Convergence Result
Clearly the normalised principal components of $F_t^{n, y}$ are determined only up to sign. To resolve this indeterminancy, lets assume (without loss of generality) that the first $r$ rows of $P'(\Gamma_y^n) M^{1/2}(\Gamma_y^n)$, call it $\Lambda_r(\Gamma_y^n)$ have full rank (from a certain $n$ onwards) and fix the diagonal elements of $\Lambda_r(\Gamma_y^n)$ to be always positive. This fixes the sign of the eigenvectors and the normlised principal components.
lemmaUnder Assumption A(ref), we have
\begin{itemize}
• $\left\lVertp_j(\Gamma_y^n) - p_j(\Gamma_C^n)\right\rVert = \mathcal O(n^{-1/2})$, for $j = 1, ..., r$.
• $\mu_j(\Gamma_y^n) - \mu_j(\Gamma_C^n) = \mathcal O(1)$ for $1\leq j \leq r$.
• $\frac{M(\Gamma_y^n)}{n}\to D_\Gamma, \frac{M(\Gamma_C^n)}{n}\to D_\Gamma$ for $n\to \infty$ and $|\frac{M(\Gamma_y^n)}{n} - \frac{M(\Gamma_C^n)}{n}| = \mathcal O(n^{-1})$
\end{itemize}
proof(i) Consider for any fixed $n$ a symmetric positive semi-definite matrix $\Gamma^n$. Suppose we perturb $\Gamma^n$ such that $\hat \Gamma^n = \Gamma^n + \varepsilon_n B^n$, where the entries of $B^n$ are in modulus smaller than one. Let $p_l \equiv p_l(\Gamma^n), \mu_j \equiv \mu_j(\Gamma^n)$ for $l = 1,..., n$, suppressing dependence on $n$ in the notation. From wilkinson1965algebraic, section 2,(10.2) we obtain the expansion that if $\varepsilon_n$ is sufficiently small, that
\begin{align*}
p_j(\hat \Gamma^n) - p_j(\Gamma^n) &= \sum_{l \neq j, l = 1}^n \frac{p_l\left(\hat \Gamma^n - \Gamma^n\right)p_j'}{\mu_j - \mu_l} p_l \ \varepsilon_n + \cdots \nonumber \\
therefore \quad \left\lVertp_j(\hat \Gamma^n) - p_j(\Gamma^n)\right\rVert^2 &= \sum_{l \neq j, l = 1}^n \left(\frac{p_l\left(\hat \Gamma^n - \Gamma^n\right)p_j'}{\mu_j - \mu_l} \right)^2 \varepsilon_n^2 + \cdots,
\end{align*}
where the higher order terms are smaller than the first order terms and can be neglected and the second equation follows by orthogonality of the eigenvectors and while the higher order terms are negligent.
Apply this to $\hat \Gamma^n = \Gamma_y^n$ and $\Gamma^n = \Gamma_C^n$, so $p_l\equiv p_l(\Gamma_y^n), \mu_j \equiv \mu_j(\Gamma_y^n)$, we obtain
\begin{align*}
\left\lVertp_j(\Gamma_y^n) - p_j(\Gamma_C^n)\right\rVert^2 &= \sum_{l \neq j, l = 1}^n \left(\frac{p_l \Gamma_e^n p_j'}{\mu_j - \mu_l} \right)^2 \varepsilon_n^2 + \cdots \\[1em]
\sum_{l \neq j, l = 1}^n \left(\frac{p_l \Gamma_e^n p_j'}{\mu_j - \mu_l} \right)^2 & \leq \left(\sup_n \mu_1(\Gamma_e^n)\right)^2 \sum_{l \neq j, l = 1}^n \sup_{l\leq n} \left(\frac{1}{\mu_j - \mu_l}\right)^2 \\
& \leq n^{-2} \ n B_e^2 \ \sup_{l\leq n} \left(\frac{1}{n^{-1} (\mu_j - \mu_l)}\right)^2 \quad by A(ref) \\
& \leq n^{-2} \ n B_e^2 \ \max_{l\neq j, 1\leq l, j \leq r}\left(\frac{1}{c_l^- - c_j^+}\right)^2 = \mathcal O(n^{-1}) \quad by A(ref),(i),
\end{align*}
where the last inequality holds for $n$ large enough.
(ii) For the eigenvalues, we use the expansion from wilkinson1965algebraic, section 2, (5.5) with coefficients $k_1, k_2, ...$ and $\varepsilon = \mathcal O(1)$ such that
\begin{align*}
&\mu_1(\Gamma_y^n) - \mu_1(\Gamma_C^n) = k_1 \varepsilon + k_2 \varepsilon^2 + \cdots\\[0.5em]
&k_1 = p_1(\Gamma_C^n) \left(\Gamma_y^n - \Gamma_C^n\right) p_1' (\Gamma_C^n) = p_1(\Gamma_C^n) \left(\Gamma_e^n\right) p_1' (\Gamma_C^n) \leq \mu_1(\Gamma_e^n) = \mathcal O(1)
\end{align*}
where the higher order terms are negligent.
(iii) It readily follows with $\mu_j(\Gamma_C^n) = \mu_j\left(\Lambda^n (\Lambda^n)'\right) = \mathcal O(n)$ and under A(ref), that $M(\Gamma_y^n) / n \to D_\Gamma$.
The normalised static principal components (NSPC) of any random vector of dimension $n$, supposing that the eigenvalues are different, are unique up to a change in sign. Suppose now, we fix the sign. Clearly, the NSPCs do in general not converge if $n$ is successively increased, since a new cross-sectional unit may always alter the direction of the principal components. However, in the case of the approximate factor model and under the condition of asymptotic stability of the first $r$-eigenvalues (which implies that they are well separated) convergence is ensured.
Consequently, under these assumptions the limit of the NSPC have a meaningful interpretation as “normalised principal components of the static aggregation space”. The limit of the first NSPC is the “strongest direction” in the static aggregation space, the limit of the second NSPC is the “strongest direction orthogonal to the first” and so on...
theoremUnder Assumptions A(ref) and A(ref), there exists a mean square limit of the normalised principal components, say $F_t^\infty$.
\begin{itemize}
• $\left\lVertM^{-1/2}(\Gamma_y^n) P(\Gamma_y^n)y_t^n - F_t^\infty\right\rVert = \mathcal O_{ms}(n^{-1/2})$
• $\left\lVertM^{-1/2}(\Gamma_C^n) P(\Gamma_C^n) C_t^n - F_t^\infty\right\rVert = \mathcal O_{ms}(n^{-1/2})$.
\end{itemize}
We may interpret $F_t^\infty$ as the normalised principal components of the static aggregation space.
proofWe want to show that
\begin{align*}
F_t^{n, y} \equiv M^{-1/2}(\Gamma_y^n) P(\Gamma_y^n) y_t^n = M^{-1/2}(\Gamma_y^n) P(\Gamma_y^n) \Lambda^n F_t + M^{-1/2}(\Gamma_y^n) P(\Gamma_y^n) e_t^n
\end{align*}
converges. Again, set $\mu_j \equiv \mu_j(\Gamma_y^n), p_j^n \equiv p_j(\Gamma_y^n)$. The second term on the RHS converges to zero as
\begin{align*}
\operatorname{\mathbb E} (\mu_j^{-1/2} p_j^n e_t^n)^2 \leq \mu_j^{-1}\lambda_1(\Gamma_e^n), \quad so \ \mu_j^{-1/2} p_j^n e_t^n = \mathcal O_{ms}(n^{-1/2}).
\end{align*}
We are left with showing convergence of the first term. Consider the eigen-decompositions:
\begin{align*}
P_\Lambda^n \Gamma_\Lambda^n (P_\Lambda^n)' = D_\Lambda^n
\quad &and \quad
P_\Lambda \Gamma_\Lambda P_\Lambda' = D_\Lambda \\[0.5em]
\quad and set \quad \tilde P_n \equiv D_\Lambda^{-1/2} P_\Lambda \frac{(\Lambda^n)'}{\sqrt{n}}
\quad &and \quad
P_n \equiv (D_\Lambda^n)^{-1/2} P_\Lambda^n \frac{(\Lambda^n)'}{\sqrt{n}}.
\end{align*}
It follows that $P_n$ are orthonormal row-eigenvectors of $\Gamma_C^n$ since
\begin{align*}
P_n P_n' &= I_r\\
and \quad P_n \Gamma_C^n &= (D_\Lambda^n)^{-1/2} P_\Lambda^n \frac{(\Lambda^n)'}{\sqrt{n}}\frac{\Lambda^n (\Lambda^n)'}{n} n \\
&= (D_\Lambda^n)^{-1/2} P_\Lambda^n \Gamma_\Lambda^n \frac{(\Lambda^n)'}{\sqrt{n}} n = (D_\Lambda^n)^{-1/2} D_\Lambda^n P_\Lambda^n \frac{(\Lambda^n)'}{\sqrt{n}}n = D_\Lambda^n n P_n = M(\Gamma_C^n) P_n.
\end{align*}
In order two show that $F_t^{n, y}$ converges, it is enough to show that
\begin{align*}
P(\Gamma_y^n) \frac{\Lambda^n}{\sqrt{n}} F_t
\end{align*}
converges, which is the case whenever $P(\Gamma_y^n) \frac{\Lambda^n}{\sqrt{n}}$ converges to a constant finite $r\times r$ matrix.
So
\begin{align*}
P(\Gamma_y^n) \frac{\Lambda^n}{\sqrt{n}} - D_\Lambda^{-1/2} P_\Lambda \Gamma_\Lambda = \underbrace{\tilde P_n \frac{\Lambda^n}{\sqrt{n}} - D_\Lambda^{-1/2} P_\Lambda \Gamma_\Lambda}_{(I)} + \underbrace{\left[ P(\Gamma_y^n) - P_n\right] \frac{\Lambda^n}{\sqrt{n}}}_{(II)} + \underbrace{\left[P_n - \tilde P_n \right]\frac{\Lambda^n}{\sqrt{n}}}_{(III)}.
\end{align*}
where $\left\lVert\frac{\Lambda^n}{\sqrt{n}}\right\rVert = \mathcal O(1)$ so $\left\lVert(II)\right\rVert = \mathcal O(n^{-1/2})$ by Lemma (ref). For $(III)$ term note that
\begin{align*}
P_n - \tilde P_n = \left[\left(D_\Lambda^n\right)^{-1/2} P_\Lambda^n -
D_\Lambda^{-1/2} P_\Lambda \right]\frac{(\Lambda^n)'}{\sqrt{n}},
\end{align*}
while again by wilkinson1965algebraic we have with $B_n = n^{1/2}c_n^{-1} (\Gamma_\Lambda^n - \Gamma_\Lambda) = \mathcal O(1)$ by A(ref)(i), where $c_n$ is the modulus of the maximum entry of $n^{1/2}(\Gamma_\Lambda^n - \Gamma_\Lambda)$ plus some small constant (rendering the entries of $B_n$ smalle than one in modulus) and $\varepsilon_n \equiv n^{-1/2}c_n$,
\begin{align}
\left\lVertp_j(\Gamma_\Lambda^n) - p_j(\Gamma_\Lambda)\right\rVert^2 &= \sum_{l \neq j, l = 1}^r \left(\frac{p_l\left(\Gamma_\Lambda\right)B_n p_j'(\Gamma_\Lambda)}{\mu_j(\Gamma_\Lambda) - \mu_l(\Gamma_\Lambda)} \right)^2 \varepsilon_n^2 + \cdots = \mathcal O(n^{-1}),
\end{align}
noting that by A(ref)(ii) the eigenvalues of $\Gamma_\Lambda$ are distinct, so the denominator on the RHS in equation ((ref)) does not vanish, consequently $\left\lVertP_\Lambda^n - P_\Lambda\right\rVert = \mathcal O(n^{-1/2})$.
Next, for the eigenvalues, we have
\begin{align*}
&\mu_j(\Gamma_\Lambda^n) - \mu_j(\Gamma_\Lambda) = k_1 \varepsilon_n + k_2 \varepsilon_n^2 + \cdots\\[0.5em]
with \quad &k_1 = p_j(\Gamma_\Lambda) B_n p_j' (\Gamma_\Lambda) = \mathcal O(1) \quad j = 1, ..., r
\end{align*}
So $|\mu_j(\Gamma_\Lambda^n) -\mu_j(\Gamma_\Lambda)| = \mathcal O(n^{-1/2})$ and therefore
\begin{align*}
|\mu_j(\Gamma_\Lambda^n)^{-1} - \mu_j(\Gamma_\Lambda)^{-1}| \leq |\mu_j\left(\Gamma_\Lambda^n\right)^{-1}| |\mu_j\left(\Gamma_\Lambda^n\right) - \mu_j\left(\Gamma_\Lambda\right)||\mu_j(\Gamma_\Lambda)^{-1}| = \mathcal O(n^{-1/2})
\end{align*}
which implies by $(a^2-b^2) = (a-b)/ (a+b)$ for $a, b \in \mathbb R, a+b\neq 0$ that
\begin{align*}
\left\lVert(D_\Lambda^n)^{-1/2} - D_\Lambda^{-1/2}\right\rVert = \mathcal O(n^{-1/2}).
\end{align*}
As a result $(III) = \mathcal O(n^{-1/2})$ since $\left\lVert\tilde P_n - P_n\right\rVert = \mathcal O(n^{-1/2})$.
Finally we get
\begin{align*}
\left\lVert(I)\right\rVert = \left\lVertD_\Lambda^{-1/2} P_\Lambda(\Gamma_\Lambda^n - \Gamma_\Lambda)\right\rVert = \mathcal O(n^{-1/2}).
\end{align*}
Now obviously we have
\begin{align*}
& P(\Gamma_y^n)\frac{\Lambda^n}{\sqrt{n}} F_t \to D_\Lambda^{-1/2} P_\Lambda \Gamma_\Lambda F_t = D_\Lambda^{1/2} P_\Lambda F_t \\[0.8em]
\mbox{and therefore} \quad & \left\lVert\left(\frac{M(\Gamma_y^n)}{n}\right)^{-1} P(\Gamma_y^n) \frac{\Lambda^n}{\sqrt{n}} F_t - P_\Lambda F_t\right\rVert = \mathcal O_{ms}(n^{-1/2}),
\end{align*}
so $P_\Lambda F_t = F_t^\infty$. The second statement follows by analogous arguments.
theoremUnder Assumptions A(ref) and A(ref), there exists a limit of the loadings $\Lambda_i(\Gamma_y^n) \equiv P^i (\Gamma_y^n) M^{1/2}(\Gamma_y^n)$, say $\Lambda_i^\infty$ with
\begin{itemize}
• $\left\lVert\Lambda_i(\Gamma_y^n) - \Lambda_i^\infty\right\rVert = \mathcal O(n^{-1/2})$
• $\left\lVertP^i (\Gamma_C^n) M^{1/2}(\Gamma_C^n) - \Lambda_i^\infty\right\rVert = \mathcal O(n^{-1/2})$
• $\left\lVert\frac{1}{n}\sum_{i = 1}^n (\Lambda_i^\infty)' \Lambda_i^\infty - D_\Lambda\right\rVert = \mathcal O(n^{-1/2})$
\end{itemize}
proofOne way of proving the convergence of the loadings would be by looking directly at the limit of $\sqrt{n}P^i(\Gamma_y^n) \left(\frac{M(\Gamma_y^n)}{n}\right)^{1/2}$, using the approximation of eigenvectors and eigenvalues from above. Alternatively, note that
\begin{align*}
C_{it} = \operatorname{proj}\left(y_{it}\mid \operatorname{sp}(F_t^\infty)\right) &= \Lambda_i^\infty F_t^\infty\\
and \quad \operatorname{proj}\left(y_{it}\mid \operatorname{sp}(F_t^{n, y})\right) &= \Lambda_i(\Gamma_y^n) F_t^{n, y}.
\end{align*}
since by construction $\operatorname{\mathbb V} F_t^{n, y} = I_r$, we know that $\operatorname{\mathbb E} F_t^{n,y}y_{it} = \Lambda_i(\Gamma_y^n)'$. Now check the limit of $\Lambda_i(\Gamma_y^n)'$ by
\begin{align*}
\operatorname{\mathbb E} F_t^{n, y} y_{it} - (\Lambda_i^\infty)' &= \operatorname{\mathbb E} \left[F_t^{n, y} (\Lambda_i^\infty F_t^\infty + e_{it})'\right] - (\Lambda_i^\infty)'\\
&=\operatorname{\mathbb E} \left[(F_t^{n, y} - F_t^\infty + F_t^\infty) e_{it}\right] + \left\{\operatorname{\mathbb E}\left[(F_t^{n, y} - F_t^\infty + F_t^\infty) (F_t^\infty)'\right] - I_r\right\} (\Lambda_i^\infty)' \\
&= \operatorname{\mathbb E}\left[ (F_t^\infty - F_t^{n, y}) e_{it}\right] + \operatorname{\mathbb E} \left[(F_t^{n, y} - F_t^\infty) (F_t^\infty)'\right] (\Lambda_i^\infty)' \\[0.8em]
so \quad \left\lVert\Lambda_i(\Gamma_y^n) - \Lambda_i^\infty\right\rVert &\leq \left\lVertF_t^{n, y} - F_t^\infty\right\rVert_{L^2} \left\lVerte_{it}\right\rVert_{L^2} + \left\lVertF_t^{n, y} - F_t^\infty\right\rVert_{L^2} \left\lVertF_t^\infty\right\rVert_{L^2} \left\lVert\Lambda_i^\infty\right\rVert \\
&=\mathcal O(n^{-1/2}) \mathcal O(1) + \mathcal O(n^{-1/2})\mathcal O(1) \mathcal O(1) = \mathcal O(n^{-1/2}),
\end{align*}
where we used that $\operatorname{\mathbb V} F_t^\infty = I_r$ and $\operatorname{\mathbb E} F_t^\infty e_{it} = 0$ in the third line and Cauchy-Schwarz' inequality in the fourth line. The second statement is proved with analogous arguments.
It immediately follows that $\frac{1}{n}\sum_{i = 1}^n (\Lambda_i^\infty)' \Lambda_i^\infty$ converges to the diagonal matrix $D_\Lambda$ with rate $n^{-1/2}$.
Asymptotic Theory without Rotation Matrices
assumption[Estimation of Variance Matrices]\ \\[-2.5em]
\begin{itemize}
• $\left\lVert\hat \Gamma_F - \Gamma_F\right\rVert = \mathcal O_{ms}(T^{-1/2})$, where $\hat \Gamma_F = T^{-1}\sum_{t=1}^T F_t F_t'$.
• $\left\lVert\hat \Gamma_{Fe}^n p - \Gamma_{Fe}^n p\right\rVert = \mathcal O_{ms}(T^{-1/2})$ for all $n \in \mathbb N$ with for all $p$ with $\left\lVertp\right\rVert = 1$, where $\Gamma_{Fe}^n \equiv \operatorname{\mathbb E} F_t (e_t^n)'$ and $\hat \Gamma_{Fe}^n \equiv T^{-1}\sum_{t = 1}^T F_t (e_t^n)'$.
• $p(\hat \Gamma_e^n - \Gamma_e^n)p' = \mathcal O_{ms}(T^{-1/2})$ for all $n\in \mathbb N$ with for all $p$ with $\left\lVertp\right\rVert = 1$, where $\operatorname{\mathbb E} e_t^n (e_t^n)' \equiv \Gamma_e^n$ and $\hat \Gamma_e^n = T^{-1}\sum_{t = 1}^T e_t^n (e_t^n)'$
\end{itemize}
Primitive conditions for this assumptions can be given along the lines of barigozzi2022estimation.
lemmaUnder Assumptions A(ref)-A(ref), we have
\begin{itemize}
• $\left\lVert\hat p_j^n - p_j^n\right\rVert = \mathcal O_{ms}(T^{-1/2})$
• $\left\lVert\frac{M(\hat\Gamma_y^n)}{n} - \frac{M(\Gamma_y^n)}{n}\right\rVert = \mathcal O_{ms}(T^{-1/2})$
• $|\mu_j(\hat \Gamma_y^n)^{-1/2} - \mu_j(\Gamma_y^n)^{-1/2}| = \mathcal O_{ms}(n^{-1/2}T^{-1/2})$.
• $|\hat p_{ij}^n - p_{ij}^n| = \mathcal O_{ms}(n^{-1/2}T^{-1/2})$
• $\left\lVert\hat{\mathcal K}_j^{F, n} - \mathcal K_j^{F, n}\right\rVert = \mathcal O_{ms}(T^{-1/2}n^{-1/2})$
\end{itemize}
proof(i) Set $\hat \Gamma^n = \hat \Gamma_y^n = T^{-1} \sum_{t = 1}^T y_t^n (y_t^n)'$ and $\Gamma^n = \Gamma_y^n$. Set $\hat \Gamma_F = T^{-1} \sum_{t = 1}^T F_t F_t'$, $\hat \Gamma_{eF}^n = T^{-1} \sum_{t = 1}^T e_t^n F_t'$ and $\hat \Gamma_e^n = T^{-1} \sum_{t = 1}^T e_t^n (e_t^n)'$, so
{
\begin{align}
t_{lj} &= p_l(\Gamma_y^n) \left(\hat \Gamma_y^n - \Gamma_y^n\right) p_j(\Gamma_y^n)' \nonumber \\[0.5em]
&= p_l(\Gamma_y^n) \left( \Lambda^n \hat \Gamma_F \left(\Lambda^n\right)' - \Lambda^n\left(\Lambda^n\right)' + \Lambda^n \hat \Gamma_{Fe}^n + \hat \Gamma_{eF}^n (\Lambda^n)' + \hat \Gamma_e^n - \Gamma_e^n \right) p_j(\Gamma_y^n)' \nonumber \\[0.5em]
&= p_l(\Gamma_y^n)\Lambda^n \left(\hat \Gamma_F - I_r \right)(\Lambda^n)'p_j(\Gamma_y^n)' + p_l(\Gamma_y^n)\Lambda^n \hat \Gamma_{Fe}^n p_j(\Gamma_y^n)' \nonumber \\[0.5em]
& + p_l(\Gamma_y^n) \hat \Gamma_{eF}^n (\Lambda^n)'p_j(\Gamma_y^n)' + p_l(\Gamma_y^n)\left(\hat \Gamma_e^n - \Gamma_e^n\right) p_j(\Gamma_y^n)' \nonumber \\[0.5em]
&= (I)_{lj} + (II)_{lj} + (III)_{lj} + (IV)_{lj}, \ say.
\end{align}
}
Firstly, setting $p_l \equiv p_l(\Gamma_y^n), p_l^C \equiv p_l(\Gamma_C^n)$ for $1\leq l \leq n$, $n \in \mathbb N$, and $\mu_{1, n}^{1/2} \equiv \sqrt{\mu_1(\Gamma_C^n)}$, we have
\begin{align*}
(I)_{lj} &\leq \left\lVertp_l \Lambda^n\right\rVert \left\lVertp_j \Lambda^n\right\rVert \mu_1(\hat \Gamma_F - I_r) \\
where \quad \left\lVertp_l \Lambda^n\right\rVert &= \left\lVertp_l^C\Lambda^n + \underbrace{(p_l - p_l^C)}_{q_l} \Lambda^n\right\rVert \leq \left\lVertp_l^C \Lambda^n\right\rVert + \left\lVertq_l \Lambda^n\right\rVert \\[0.8em]
&\leq \begin{cases}
\mu_{1, n}^{1/2} + B_p n^{-1/2} \mu_{1, n}^{1/2} \leq \mu_{1, n}^{1/2} + B_p B_\mu \ for \ 1\leq l \leq r \\[0.8em]
0 + \left\lVertq_l \Lambda^n\right\rVert \leq r B_p B_\mu \ for \ l > r,
\end{cases}
\end{align*}
where the second case is obtained as follows: For $l > r$, let $G_n$ be the matrix of orthonormal row eigenvectors of $p_{r+1}(\Gamma_C^n), ..., p_{n}(\Gamma_C^n)$ which are in the kernel of $\Lambda^n$ and let $P_n^C$ the matrix of orthonormal row eigenvectors $p_1(\Gamma_C^n), ..., p_r(\Gamma_C^n)$ and $P_n^y$ analogously. From linear projection on the row space of $G_n$, with residual $v_l$ (dependence on $n$ is omitted in the notation),
\begin{align*}
p_l &= \operatorname{proj}\left(p_l \mid row(G_n)\right) + v_l \\
v_l &= p_l - p_l G_n'G_n = p_l \left(I_n - G_n'G_n\right) = p_l\left[P_n^C\right]' P_n^C \\
&= p_l\left[P_n^C - P_n^y\right]' P_n^C = p_l \begin{bmatrix}
q_1' & \cdots q_r'
\end{bmatrix} P_n^C.
\end{align*}
Consequently
\begin{align*}
q_l \Lambda^n &= \left[p_l - p_l^C\right] \Lambda^n = \left[ \operatorname{proj}\left(p_l \mid row(G_n)\right) + v_l - p_l^C\right] \Lambda^n = v_l \Lambda^n \\[0.8em]
& \mbox{where} \left\lVertv_l\right\rVert = \left\lVert\begin{pmatrix}
p_l q_1' & \cdots & p_l q_r'
\end{pmatrix} P_n^C\right\rVert = \left\lVert\sum_{j = 1}^r p_l q_j' p_j^C\right\rVert \\
& \leq \sum_{j = 1}^r \left\lVertp_l q_j'\right\rVert \left\lVertp_j^C\right\rVert \leq \sum_{j = 1}^r \left\lVertq_j\right\rVert \leq r n^{-1/2} B_p\ .
\end{align*}
This yields
\begin{align*}
\left\lVertq_l \Lambda^n\right\rVert \leq \left\lVertv_l\right\rVert \mu_{1, n}^{1/2} \leq r B_p n^{-1/2} \mu_{1, n}^{1/2} = r B_p B_\mu \quad l > r, n \in \mathbb N.
\end{align*}
Therefore for all $n\in \mathbb N$,
\begin{align*}
(I)_{lj} &\leq \begin{cases}
\left(\mu_{1, n}^{1/2} (1 + B_p n^{-1/2})\right)^2 \mu_1\left(\hat \Gamma_F - I_r\right) \ \mbox{for} \ 1 \leq l \leq r \\[0.8em]
\left(\mu_{1, n}^{1/2} (1 + B_p n^{-1/2})\right) r B_p n^{-1/2} \mu_1\left(\hat \Gamma_F - I_r\right) \ \mbox{for} \ l > r.
\end{cases} \\[0.8em]
n^{-1}(I)_{lj} &\leq \begin{cases}
\left(B_\mu^2 +n^{-1/2} B_\mu^2 B_p + n^{-1}(B_p B_\mu)^2 \right) \hat B_F \ \mbox{for} \ 1 \leq l \leq r \\[0.8em]
n^{-1} r \left(B_p B_\mu + B_p^2 B_\mu \right)\hat B_F \ \mbox{for} \ l > r.
\end{cases}
\end{align*}
Next, by A(ref), we have that
\begin{align*}
(II)_{lj} = p_l(\Gamma_y^n) \Lambda^n \hat \Gamma_{Fe} p_j(\Gamma_y^n)' &\leq \begin{cases}
\left(\mu_{1,n}^{1/2} + \mu_{1, n}^{1/2} B_p n^{-1/2}\right)\hat B_{Fe} \ \mbox{for} \ 1\leq l \leq r \\[0.8em]
\mu_{1, n}^{1/2} r B_p n^{-1/2} \hat B_{Fe}\ \mbox{for} \ l > r,
\end{cases}\\[0.8em]
n^{-1}(II)_{lj} &\leq \begin{cases}
\left(n^{-1/2}B_\mu + n^{-1}B_p B_\mu\right) \hat B_{Fe} \ \mbox{for} \ 1\leq l \leq r \\[0.8em]
n^{-1} r B_p B_\mu \hat B_{Fe}\ \mbox{for} \ l > r,
\end{cases}
\end{align*}
and
\begin{align*}
(III)_{lj} = p_l(\Gamma_y^n) \hat \Gamma_{eF}^n (\Lambda^n)' p_j(\Gamma_y^n)' &\leq \begin{cases}
\left(\mu_{1,n}^{1/2} + \mu_{1, n}^{1/2} B_p n^{-1/2}\right)\hat B_{Fe} \ \mbox{for} \ 1\leq l \leq r \\[0.8em]
\left(\mu_{1,n}^{1/2} + \mu_{1, n}^{1/2} B_p n^{-1/2}\right) \hat B_{Fe} \ \mbox{for} \ l > r.
\end{cases}\\[0.8em]
n^{-1}(III)_{lj} &\leq \begin{cases}
\left(n^{-1/2} B_\mu + n^{-1}B_\mu B_p\right)\hat B_{Fe} \ \mbox{for} \ 1\leq l \leq r \\[0.8em]
\left(n^{-1/2} B_\mu + n^{-1} B_p B_\mu\right) \hat B_{Fe} \ \mbox{for} \ l > r.
\end{cases}
\end{align*}
Finally $(IV)_{lj} \leq \hat B_e$ from A(ref).
Therefore putting things together with $n^{-1/2} \mu_{1, n}^{1/2} \leq B_\mu$, we obtain
{
\scriptsize
\begin{align*}
&\sum_{l \neq j, l = 1}^n \left(\frac{t_{lj}}{\mu_j - \mu_l}\right)^2 = \sum_{l \neq j, l = 1}^r \left(\frac{t_{lj}}{\mu_j - \mu_l}\right)^2 + \sum_{l \neq j, l = r+1}^n \left(\frac{t_{lj}}{\mu_j - \mu_l}\right)^2\\[1em]
&= \sum_{l \neq j, l = 1}^r \left(\frac{n^{-1} \left[(I)_{lj} + (II)_{lj} + (III)_{lj} + (IV)_{lj} \right]}{n^{-1}\left(\mu_j - \mu_l\right)}\right)^2 + \sum_{l \neq j, l = r+1}^n \left(\frac{n^{-1} \left[(I)_{lj} + (II)_{lj} + (III)_{lj} + (IV)_{lj} \right]}{n^{-1}\left(\mu_j - \mu_l\right)}\right)^2 \\[0.8em]
&\leq r \frac{1}{\inf_n\min_{l\neq j} n^{-1} (\mu_j - \mu_l)}\left[
\left(B_\mu^2 + n^{-1/2} B_\mu^2 B_p + n^{-1} (B_p B_\mu)^2\right)\hat B_F + 2 (n^{-1/2} B_\mu + n^{-1} B_p B_\mu) \hat B_{Fe} + n^{-1} \hat B_e
\right]^2 \\
&+(r - n) \frac{1}{\inf_n\min_{l\neq j} n^{-1} (\mu_j - \mu_l)} \left(
n^{-1} r \left(B_p B_\mu + B_p^2 B_\mu \right)\hat B_F +
n^{-1} r B_p B_\mu \hat B_{Fe} + \left(n^{-1/2} B_\mu + n^{-1} B_p B_\mu\right) \hat B_{Fe} + n^{-1} \hat B_e
\right)^2\\
&= \mathcal O_p(T^{-1/2})^2 + (r-n)\left[\mathcal O(n^{-1/2}) \mathcal O_p (T^{-1/2})\right]^2 = \mathcal O_p(T^{-1}).
\end{align*}
}
(ii) Again, we employ wilkinson1965algebraic, section 2, equation (5.5) together with ((ref)):
{
\begin{align*}
&\frac{\mu_j(\hat \Gamma_y^n)}{n} - \frac{\mu_j(\Gamma_y^n)}{n} = k_1 \varepsilon_n + k_2 \varepsilon_n^2 + \cdots \\
\mbox{with}\ &k_1 = p_j(\Gamma_y^n) \left(\hat \Gamma_y^n - \Gamma_y^n \right) p_j' (\Gamma_y^n) = n^{-1}t_{jj} = n^{-1}\left[(I)_{jj} + (II)_{jj} + (III)_{jj} + (IV)_{jj}\right] \quad (\mbox{for} j \leq r) \\
&= r\left[ \left(B_\mu^2 +n^{-1/2} B_\mu^2 B_p + n^{-1}(B_p B_\mu)^2 \right) \hat B_F +
\left(n^{-1/2}B_\mu + n^{-1}B_p B_\mu\right) \hat B_{Fe} + \left(n^{-1/2} B_\mu + n^{-1}B_\mu B_p\right)\hat B_{Fe} +\hat B_e \right] \\
&= \mathcal O_p(T^{-1/2}).
\end{align*}
}
Furthermore, setting $\hat \mu_j \equiv \hat \mu_j(\hat \Gamma_y^n), \mu_j \equiv \mu_j(\Gamma_y^n)$
\begin{align*}
\left|\left(\frac{\hat \mu_j}{n}\right)^{-1} - \left(\frac{\mu_j}{n}\right)^{-1} \right| \leq \left|\left(\frac{\hat \mu_j}{n}\right)^{-1} \right| \left|\frac{\hat \mu_j}{n} - \frac{\mu_j}{n}\right| \left|\left(\frac{\mu_j}{n}\right)^{-1}\right| = \mathcal O_p(1) \mathcal O_p(T^{-1/2}) \mathcal O(1) = \mathcal O_p(T^{-1/2}).
\end{align*}
Finally, since for all $a,b \in \mathbb R$ with $a+b \neq 0$, we have $(a^2 - b^2)/(a+b) = (a-b)$, we conclude
\begin{align*}
|\hat \mu_j^{-1/2} - \mu_j^{-1/2}| &= n^{-1/2} \left | \left(\frac{\hat \mu_j}{n}\right)^{-1/2}- \left(\frac{\mu_j}{n}\right)^{-1/2}\right| \\
&=n^{-1/2} \left|\left(\frac{\hat \mu_j}{n}\right)^{-1} - \left(\frac{\mu_j}{n}\right)^{-1} \right| \left | \left(\frac{\hat \mu_j}{n}\right)^{1/2} +\left(\frac{\mu_j}{n}\right)^{1/2}\right|^{-1} = n^{-1/2} \mathcal O_p(T^{-1/2}).
\end{align*}
(iii) For the last statement observe that
\begin{align*}
\left\lVert\hat{\mathcal K}_j^{F, n} - \mathcal K_j^{F, n}\right\rVert &= \left\lVert\hat \mu_j^{-1/2} \hat p_j^n - \mu_j^{-1/2} p_j^n\right\rVert \\
&= \left\lVert\hat p_j^n - p_j\right\rVert |\mu_j^{-1/2}| + |\hat \mu_j^{-1/2} - \mu_j^{-1/2}| \left\lVertp_j^n\right\rVert + \left\lVert\hat p_j^n - p_j^n\right\rVert|\hat \mu_j^{-1/2} - \mu_j^{-1/2}| \\
&= \mathcal O_p(T^{-1/2})\mathcal O(n^{-1/2}) + \mathcal O_p(T^{-1/2} n^{-1/2}) +
\mathcal O_p(T^{-1}n^{-1/2}) = \mathcal O_p(T^{-1/2}n^{-1/2}).
\end{align*}
theorem[Consistency of Factors, Factor space and Common Component]
Under assumptions A(ref) to A(ref), we have
\begin{itemize}
• $\left\lVert\hat F_t^n - F_t^{\infty}\right\rVert = \mathcal O_{ms}(\max(T^{-1/2}, n^{-1/2}))$
• $\left\lVert\hat F_t^n - \hat H_n F_t\right\rVert = \mathcal O_{ms}(n^{-1/2})$ for $\hat H_n = T^{-1}\sum_{t = 1}^T \hat F_t^n F_t' \left(T^{-1} \sum_{t = 1}^T F_t F_t'\right)^{-1}$ with finite $T < \infty$.
\end{itemize}
proof\begin{align*}
\left\lVert\hat F_t^n - F_t^{\infty}\right\rVert \leq \left\lVert\hat F_t^n - F_t^n\right\rVert + \left\lVertF_t^n - F_t^{\infty}\right\rVert = \mathcal O_{ms}(T^{-1/2}) + \mathcal O_{ms}(n^{-1/2})
\end{align*}
where for the first term we use Cauchy-Schwarz inequality
\begin{align*}
\left\lVert\hat F_{tj}^n - F_{j}^n\right\rVert &= \left\lVert(\hat{\mathcal K}_j^{F, n} - \mathcal K_j^{F,n}) y_t^n\right\rVert \leq \left\lVert\hat{\mathcal K}_j^{F, n} - \mathcal K_j^{F,n}\right\rVert\left\lVerty_t^n\right\rVert \\
&= \mathcal O_{ms}(T^{-1/2}n^{-1/2}) \mathcal O_{ms}(n^{1/2}) = \mathcal O_{ms}(T^{-1/2}).
\end{align*}
For statement (ii), note that
\begin{align*}
\frac{1}{T} \sum_{t = 1}^T \hat F_t^n F_t' = \frac{1}{T} \sum_{t = 1}^T \hat{\mathcal K}^{F, n} \left(\Lambda^n F_t + e_t^n \right) F_t' = \hat{\mathcal K}^{F, n} \Lambda^n \hat \Gamma_F + \hat{\mathcal K}^{F, n} \hat \Gamma_F
\end{align*}
Consequently,
\begin{align*}
\hat F_t^n - \hat H_n F_t &= \hat{\mathcal K}^{F, n} \Lambda^n F_t + \hat{\mathcal K}^{F, n} e_t^n - \hat{\mathcal K}^{F, n} \Lambda^n \hat \Gamma_F \hat \Gamma_F^{-1} F_t - \hat{\mathcal K}^{F, n} \hat \Gamma_{eF} \hat \Gamma_F^{-1} F_t \\
\left\lVert\hat F_t^n - \hat H_n F_t\right\rVert &\leq \left\lVert\hat{\mathcal K}^{F, n} e_t^n\right\rVert + \left\lVert\hat{\mathcal K}^{F, n} \hat \Gamma_{eF}\right\rVert\left\lVert\hat \Gamma_F^{-1}\right\rVert\left\lVertF_t\right\rVert = \mathcal O_{ms}(n^{-1/2}).
\end{align*}
Conclusion
We refine the interpretation of what the principal component estimator is targeting by showing that the population normalised principal components actually have a limit in mean square. This is tied to the assumption that the eigenvalues of the common component variance matrix divided by $n$ are distinct which allows convergence of the eigenvectors.