Covariance Matrix Estimation in Time-Varying Factor Models

Proof of Theorem 1.

As mentioned in Fan et al. (2024), the rate of convergence for the estimator Σ _e is related to the sparsity (quantified by s₁) as well as the order of estimators for factor loadings and common factors (i.e., ω(T, N, h)). It is worth noting that this paper uses the Lppca method to estimate the common factors and the time-varying factor loadings. Under Assumptions 1–3, the order of estimators for factor loadings and common factors are obtained by following from Lemmas C.4 and C.12 of Jung (2021). Further, we obtain ω ( T , N , h ) = 1 N + log ⁡ N T T h + h 2 ⁡ log ⁡ T + J − η , which is different from Fan et al. (2024). The solution to ℓ₁-penalized log-likelihood estimation problem (6) could also be treated as a particular type of M-estimator for the covariance matrix Σ _e , based on minimizing a Bregman divergence between positive definite matrices. Similar to Fan et al. (2024), we have Σ ̂ e − Σ e F ≤ N + s 1 ω ( T , N , h ) . Conversely, we have Σ ̂ e − Σ e ≤ Σ ̂ e − Σ e ∞ = O p d ω ( T , N , h ) , so Σ ̂ e − Σ e = O p min { N + s 1 , d } ω ( T , N , h ) . Under Assumption 4 and the assumption that Σ ̂ e − Σ e max = O p ω ( T , N , h ) = o p ( 1 ) , the order of precision matrix estimator Σ ̂ e − 1 − Σ e − 1 max = Σ ̂ e − 1 Σ e − Σ ̂ e Σ e − 1 max < ‖ Σ e ‖ 2 Σ ̂ e − Σ e max = O p ω ( T , N , h ) . Similarly, we can derive the rates of convergence for the precision matrix estimator Σ ̂ e − 1 under the spectral norm and Frobenius norm, respectively. And then the proof of Theorem 1 is completed.

Proof of Theorem 2.

Following the lead of Fan et al. (2024), we evaluate the norm of the term Σ ̂ y , t − Σ y , t in the following way.

For t = 1, 2, …, T, we have the following equalities

Define a new covariance matrix

where S F = 1 T ∑ t = 1 T F ̃ t F ̃ t ⊤ , and F ̃ t = F t K t , h 1 / 2 .

To estimate the convergence rate of Σ ̂ y , t − Σ y , t , we can perform the following scaling

Here, the norm  ‖ ⋅ ‖ Σ y , t 2 is defined as follows: For any N-dimensional matrix A,

where C is a constant.

For the first term on the right side of (A.1), we decompose it again

According to (A.2) and Theorem 1, we can prove that

Similar to Fan et al. (2024), we have

It follows from (A.4) and (A.5) that

For the second term Σ ̃ y , t − Σ y , t , following arguments analogous to the proof of Lemma B.10 of Jung (2021), we have

Therefore, the final evaluation of Σ ̂ y , t − Σ y , t is derived by combining (A.6) and (A.7),

Next, we show the order of Σ ̂ y , t − Σ y , t under the ℓ_∞ norm. We decompose this term in the same way as (A.1)

The first term on the right side of the inequality is

By Theorem 1 we have sup t Σ ̂ e − Σ e max = O p ω ( T , N , h ) and by the same argument of Wang et al. (2021), we have sup t G ̂ t G ̂ t ⊤ − G t Σ F G t ⊤ max = O p ω ( T , N , h ) , we know that

Using a similar proof to (A.7), we have

So it follows from (A.9) and (A.10) that the convergence rate of Σ ̂ y , t − Σ y , t under the ℓ_∞ norm,

Finally, we evaluate the rate of convergence for the precision matrix Σ ̂ y , t − 1 . Similarly, we utilize the same decomposition by the triangle inequality,

By applying the Sherman-Morrison-Woodbury formula to Σ ̃ y , t − 1 and Σ y , t − 1 , we have

Then

where Q = S F − 1 + G t ⊤ Σ e − 1 G t − 1 − Σ F − 1 + G t ⊤ Σ e − 1 G t − 1 ≕ Q 1 − 1 − Q 2 − 1 .

Then we have

By (A.7), we can obtain

Then from (A.12) and (A.16), we obtain

Next, we consider Σ ̃ y , t − 1 − Σ ̂ y , t − 1 . By applying the Sherman-Morrison-Woodbury formula, we obtain

where G t H = G t H t − 1 and Σ F H = H t Σ F H t ⊤ .

Through the same decomposition as in the proof of Theorem 3 in Wang et al. (2021), we have

with Q ̃ = I m + G ̂ t ⊤ Σ ̂ e − 1 G ̂ t − 1 − Σ F H − 1 + G t H ⊤ Σ e − 1 G t H − 1 ≕ Q ̃ 1 − 1 − Q ̃ 2 − 1 .

According to Theorem 1, we have

For D₂,

The last equality uses the fact that sup t I m + G ̂ t Σ ̂ e − 1 G ̂ t ⊤ − 1 = O p 1 N .

We can derive the order of sup _t ‖D₃‖ in a similar way,

For D₄,

The last equality uses Lemma C.12 in Jung (2021), that is max 1 ≤ i ≤ N sup t G ̂ t − H t − 1 ⊤ G t 2 = o p ω 2 ( T , N , h ) .

The order of sup t D 5 is obtained similarly,

For Q ̃ , by using (A.14), we can obtain

From (S15), (S21) of Wang et al. (2021) and Theorem 1, we can conclude that

Thus,

In summary, we can get

Combining (A.8), (A.11) and (A.17), the proof of Theorem 2 is complete.