Bootstrap Prediction Intervals for Factor-Augmented Regressions with Cross-Section Averages

Rui Chen; Yimeng Xie; Lizhi Tang; Qing Tao

doi:10.1515/snde-2024-0102

Article

Bootstrap Prediction Intervals for Factor-Augmented Regressions with Cross-Section Averages

Rui Chen , Yimeng Xie , Lizhi Tang and Qing Tao

Published/Copyright: October 10, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Studies in Nonlinear Dynamics & Econometrics

Abstract

This paper revisits the problem of forecasting using factor augmented time series regressions. It is shown that the OLS estimator of the regression coefficients using cross-section average is asymptotically biased, when the time series dimension (T) and the cross-section dimension (N) satisfy T / N → c and 0 < c < ∞. We propose bootstrap algorithms using the OLS estimator to construct prediction intervals, and show they are asymptotically valid under Gaussianity of the innovations as well as under more general conditions. The effectiveness of the bootstrap prediction intervals is further justified by Monte Carlo simulations and an empirical application.

Keywords: bootstrap; cross-section average; factor-augmented regression models; forecast

JEL Classification: C10; C13; C15

Corresponding author: Lizhi Tang, Department of Statistics and Data Science at School of Economics, Xiamen University, 422 Siming South Road, Xiamen, 361005, Fujian, China, E-mail: tanglizhi@xmu.edu.cn

Funding source: National Social Science Fund of China

Award Identifier / Grant number: 21BTJ037

Appendix A. Proofs of Main Results

Proof of Lemma 1:

Write F ̂ t = x ̄ t = λ ̄ ⊤ F t + e ̄ t in the matrix form,

(A.1) F ̂ = x ̄ = F λ ̄ + e ̄ ,

where x ̄ = x ̄ 1 , ⋯ , x ̄ T − h ⊤ is (T − h) × m, and F = F 1 , ⋯ , F T − h ⊤ is (T − h) × r. Let e ̄ = e ̄ r , e ̄ − r . Post-multiplying (A.1) by Λ ̄ , we have

F ̂ Λ ̄ = F λ ̄ Λ ̄ + e ̄ Λ ̄ = F , 0 T − h × m − r + e ̄ r λ ̄ r − 1 , e ̄ − r − e ̄ r λ ̄ r − 1 λ ̄ − r .

By (S.14) in Karabiyik and Westerlund (2021),

(A.2) Q N − 1 T δ ̂ − Q N δ 0 = T − 1 z ̂ 0 T z ̂ 0 + T − 1 / 2 z ̂ 0 ⊤ ε − e ̄ r 0 α ,

where ε = ε h + 1 , ⋯ , ε T ⊤ is (T − h) × 1, z ̂ 0 = z ̂ 1 0 , ⋯ , z ̂ T − h 0 ⊤ is (T − h) × (m + n), and e ̄ 0 = e ̄ r 0 , e ̄ − r 0 = e ̄ r λ ̄ r − 1 , N e ̄ − r − e ̄ r λ ̄ r − 1 λ ̄ − r is (T − h) × m. Define z t 0 = F t ⊤ , e ̄ − r , t 0 ⊤ , g t ⊤ ⊤ and z 0 = F , e ̄ − r 0 , g = z 1 0 , ⋯ , z T − h 0 ⊤ , which are (m + n) × 1 and (T − h) × (m + n), respectively. By (S.19) and (S.39) in Karabiyik and Westerlund (2021), T − 1 2 z ̂ 0 ⊤ ε = T − 1 2 z 0 ⊤ ε + O p N − 1 2 , and T − 1 z ̂ 0 ⊤ z ̂ 0 + → p Σ z 0 − 1 , where

Σ z 0 = Σ F 0 r × m − r Σ F g 0 m − r × r Λ − r ⊤ Σ e Λ − r 0 m − r × n Σ F g ⊤ 0 n × m − r Σ g ,

where Λ _−r is from Λ = Λ r , Λ − r = lim N → ∞ Λ ̄ . Then we consider T − 1 / 2 z ̂ 0 ⊤ e ̄ r 0 ,

T − 1 2 z ̂ 0 ⊤ e ̄ r 0 = T − 1 2 z ̂ 0 − z 0 ⊤ e ̄ r 0 + T − 1 2 z 0 ⊤ e ̄ r 0 = T − 1 2 e ̄ r 0 ⊤ e ̄ r 0 0 n × m + T 0 r × 1 T − 1 e ̄ − r 0 ⊤ e ̄ r 0 0 n × 1 + O p N − 1 2 .

Also,

T − 1 2 e ̄ r 0 ⊤ e ̄ r 0 = T − 1 2 e ̄ Λ ̄ r ⊤ e ̄ Λ ̄ r = T − 1 2 Λ ̄ r ⊤ e ̄ ⊤ e ̄ Λ ̄ r = T N − 1 Λ ̄ r ⊤ N T − 1 e ̄ ⊤ e ̄ Λ ̄ r → p c Λ r ⊤ Σ e Λ r ,

Together with the above results, we have

T Q N − 1 ⁢ δ ̂ − Q N ⁢ δ 0 = T − 1 z ̂ 0 ⊤ z ̂ 0 + ⁢ T − 1 2 ⁢ z ̂ 0 ⊤ ⁢ ε − e ̄ r 0 ⁢ α = T − 1 z ̂ 0 ⊤ z ̂ 0 + ⁢ T − 1 2 ⁢ z ̂ 0 ⊤ ⁢ ε − T − 1 z ̂ 0 ⊤ z ̂ 0 + ⁢ T − 1 2 ⁢ z ̂ 0 ⊤ ⁢ e ̄ r 0 ⁢ α = Σ z 0 − 1 ⁢ T − 1 / 2 ⁢ z 0 ⊤ ⁢ ε − T 0 r × 1 T − 1 ⁢ e ̄ − r 0 ⊤ ⁢ e ̄ r 0 ⁢ α 0 n × 1 − T − 1 / 2 ⁢ e ̄ r 0 ⊤ ⁢ e ̄ r 0 ⁢ α 0 n × m + o p ⁢ 1 .

Finally, by Assumption 4(d), T Q N − 1 δ ̂ − Q N δ 0 → d N − c B 1 − B 2 , Σ z 0 − 1 Σ z 0 ε Σ z 0 − 1 , where

B 1 = Σ z 0 − 1 Λ r ⊤ Σ e Λ r α 0 n × 1 ,

B 2 = T N − 1 2 0 r × 1 Λ − r ⊤ Σ e Λ − r − 1 Λ − r ⊤ Σ e Λ r α 0 n × 1 ,

Σ z 0 ε = lim N , T → ∞ T − 1 ∑ t = 1 T − h E ε t + h 2 z t 0 z t 0 ⊤ .

Proof of Theorem 1:

Similar as (A.2), it follows that

Q N * − 1 T δ ̂ * − Q N * δ 0 * = T − 1 z ̂ 0 * ⊤ z ̂ 0 * + T − 1 / 2 z ̂ 0 * ⊤ ε * − e ̄ r 0 * α ̂ .

We now evaluate T − 1 z ̂ 0 * ⊤ z ̂ 0 * , T − 1 / 2 z ̂ 0 * ⊤ ε * and T − 1 / 2 z ̂ 0 * ⊤ e ̄ r 0 * in turn.

Consider T − 1 z ̂ 0 * ⊤ z ̂ 0 * . First,

z ̂ 0 * = F ̂ 0 * , g = F ̂ * Λ ̄ * D N , g = F ̂ λ ̂ ̄ + e ̄ * Λ ̄ * D N , g = F ̂ + e ̄ 0 * , g ,

where e ̄ 0 * = e ̄ * Λ ̄ * D N . Then it follows that

1 T z ̂ 0 * ⊤ z ̂ 0 * = 1 T F ̂ + e ̄ 0 * ⊤ F ̂ + e ̄ 0 * F ̂ + e ̄ 0 * ⊤ g g ⊤ F ̂ + e ̄ 0 * g ⊤ g ,

where

1 T F ̂ + e ̄ 0 * ⊤ F ̂ + e ̄ 0 * = 1 T F ̂ ⊤ F ̂ + e ̄ 0 * ⊤ F ̂ + F ̂ ⊤ e ̄ 0 * + e ̄ 0 * ⊤ e ̄ 0 * .

Plug in F ̂ , we have

1 T F ̂ ⊤ F ̂ = 1 T F λ ̄ + e ̄ ⊤ F λ ̄ + e ̄ = 1 T λ ̄ ⊤ F ⊤ F λ ̄ + λ ̄ ⊤ F ⊤ e ̄ + e ̄ ⊤ F λ ̄ + e ̄ ⊤ e ̄ .

By (A.11) in Lemma 2 of Pesaran (2006), 1 T F ⊤ e ̄ = O p 1 N T . And by (S.27) in Karabiyik and Westerlund (2021), 1 T e ̄ ⊤ e ̄ = O p N − 1 . Thus, 1 T F ̂ ⊤ F ̂ = 1 T λ ̄ ⊤ F ⊤ F λ ̄ + O p 1 N T + O p N − 1 .

Also,

1 T F ̂ ⊤ e ̄ 0 * = 1 T F ̂ ⊤ e ̄ * Λ ̄ r * , N e ̄ * Λ ̄ − r * ,

1 T F ̂ ⊤ e ̄ * = 1 T ∑ t = 1 T − 1 F ̂ t e ̄ t * ⊤ , consider the l th row of 1 T F ̂ ⊤ e ̄ * ,

var * 1 T ∑ t = 1 T − 1 F ̂ l t e ̄ t * = var 1 T ∑ t = 1 T − 1 F ̂ l t 1 N ∑ i = 1 N e ̂ i t · η i t

= var 1 T ∑ t = 1 T − 1 F ̂ l t 1 N ∑ i = 1 N e ̂ i t .

Using (S.86) in Karabiyik and Westerlund (2021), E 1 N 2 ∑ i = 1 N e ̂ i t e ̂ i t ′ ⊤ = O p N − 1 . Therefore,

var * 1 T ∑ t = 1 T − 1 F ̂ l t e ̄ t * = O p * 1 N ∑ t = 1 T − 1 ∑ t ‘ = 1 T − 1 E F ̂ l t F ̂ l t ′ T 2 = O p * 1 N T .

Hence, 1 T F ̂ ⊤ e ̄ * = O p * 1 N T . 1 T F ̂ ⊤ e ̄ * Λ ̄ r * ≤ 1 T F ̂ ⊤ e ̄ * Λ ̄ r * = O p * 1 N T , 1 T F ̂ ⊤ N e ̄ * Λ ̄ − r * ≤ N 1 T F ̂ ⊤ e ̄ * Λ ̄ − r * = O p * 1 T .

In addition,

1 T e ̄ 0 * ⊤ e ̄ 0 * = 1 T e ̄ r 0 * ⊤ e ̄ r 0 * e ̄ r 0 * ⊤ e ̄ − r 0 * e ̄ − r 0 * ⊤ e ̄ r 0 * e ̄ − r 0 * ⊤ e ̄ − r 0 * ,

where 1 T e ̄ r 0 * ⊤ e ̄ r 0 * = 1 T Λ ̄ r * ⊤ e ̄ * ⊤ e ̄ * Λ ̄ r * ≤ 1 T e ̄ * ⊤ e ̄ * Λ ̄ r * 2 = O p * 1 N , 1 T e ̄ r 0 * ⊤ e ̄ − r 0 * = 1 T Λ ̄ r * ⊤ e ̄ * ⊤ N e ̄ * Λ ̄ − r * ≤ N 1 T e ̄ * ⊤ e ̄ * Λ ̄ r * Λ ̄ − r * = O p * 1 N . Consider 1 T e ̄ − r 0 * ⊤ e ̄ − r 0 * ,

1 T e ̄ − r 0 * ⊤ e ̄ − r 0 * = 1 T N Λ ̄ − r * ⊤ e ̄ * ⊤ N e ̄ * Λ ̄ − r * = Λ ̄ − r * ⊤ N T − 1 e ̄ * ⊤ e ̄ * Λ ̄ − r * .

According to Condition 2, 1 T e ̄ − r 0 * ⊤ e ̄ − r 0 * = Λ − r * ⊤ Σ e * Λ − r * + O p * 1 T , where Λ − r * is from Λ * = Λ r * , Λ − r * = lim N → ∞ Λ ̄ * .

It remains to consider

E * ⁢ 1 T ⁢ e ̄ * ⊤ ⁢ e ̄ * = E * ⁢ 1 N 2 T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e i t * ⁢ e j t * ⊤ ⁢ E * ⁢ 1 N 2 T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e ̂ i t · η i t ⁢ e ̂ j t · η j t ⊤ = 1 N 2 T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e ̂ i t ⁢ e ̂ j t ⊤ ⁢ I ⁡ i = j .

Using (S.94) in Karabiyik and Westerlund (2021), we have 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e ̂ i t e ̂ j t ⊤ = O p 1 . Hence, 1 T e ̄ * ⊤ e ̄ * = O p * 1 N . By adding the results, 1 T F ̂ + e ̄ * Λ ̄ * D N ⊤ F ̂ + e ̄ * Λ ̄ * D N = 1 T λ ̄ ⊤ F ⊤ F λ ̄ + 0 r × r 0 r × m − r 0 m − r × r Λ − r * ⊤ Σ e * Λ − r * + O p * 1 N . In addition, 1 T F ̂ 0 * ⊤ g = 1 T F ̂ + e ̄ 0 * ⊤ g = 1 T F ̂ ⊤ g + 1 T e ̄ 0 * ⊤ g , where the first term is

1 T F ̂ ⊤ g = 1 T F λ ̄ + e ̄ ⊤ g = 1 T λ ̄ ⊤ F ⊤ g + 1 T e ̄ ⊤ g .

Assumptions 2 and 4 imply that 1 T e ̄ ⊤ g = 1 T ∑ t = 1 T − 1 e ̄ t g t ⊤ = 1 T 1 N ∑ i = 1 N ∑ t = 1 T − 1 e i t g t ⊤ = O p 1 N T . Also, 1 T e ̄ 0 * ⊤ g = 1 T e ̄ * Λ ̄ r * , N e ̄ * Λ ̄ − r * ⊤ g , 1 T e ̄ * T g = 1 T ∑ t = 1 T − 1 e ̄ t * g t ⊤ , consider the lth row of 1 T e ̄ * ⊤ g ,

var * 1 T ∑ t = 1 T − 1 g l t e ̄ t * ⊤ = var 1 T ∑ t = 1 T − 1 g l t 1 N ∑ i = 1 N e ̂ i t · η i t = var 1 T ∑ t = 1 T − 1 g l t 1 N ∑ i = 1 N e ̂ i t ,

by (S.86) in Karabiyik and Westerlund (2021), E 1 N 2 ∑ i = 1 N e ̂ i t e ̂ i t ′ ⊤ = O p N − 1 . Thus,

var 1 T ∑ t = 1 T − 1 g l t e ̄ t * ⊤ = O p * 1 N ∑ t = 1 T − 1 ∑ t ′ = 1 T − 1 E g l t g l t ′ T 2 = O p * 1 N T ,

Therefore, 1 T e ̄ * ⊤ g = O p * 1 N T . This implies that 1 T Λ ̄ r * ⊤ e ̄ * ⊤ g ≤ 1 T e ̄ * ⊤ g Λ ̄ r * = O p * 1 N T , and 1 T N Λ ̄ − r * ⊤ e ̄ * ⊤ g ≤ N 1 T e ̄ * ⊤ g Λ ̄ − r * = O p * 1 T .

Together with the above results, we have

1 T ⁢ z ̂ 0 * ⊤ ⁢ z ̂ 0 * = 1 T ⁢ λ ̄ ⊤ ⁢ F ⊤ ⁢ F λ ̄ λ ̄ ⊤ ⁢ F ⊤ ⁢ g g ⊤ ⁢ F λ ̄ g ⊤ ⁢ g + 1 T ⁢ 0 r × r 0 r × m − r 0 r × n 0 m − r × r e ̄ − r 0 * ⊤ ⁢ e ̄ − r 0 * 0 m − r × n 0 n × r 0 n × m − r 0 n × n + O p * ⁢ 1 N + O p * ⁢ 1 T = S z * + S e * + O p ⁢ 1 N + O p * ⁢ 1 T ,

where

S z * = 1 T λ ̄ ⊤ F ⊤ F λ ̄ λ ̄ ⊤ F ⊤ g g ⊤ F λ ̄ g ⊤ g ,

S e * = 1 T 0 r × r 0 r × m − r 0 r × n 0 m − r × r e ̄ − r 0 * ⊤ e ̄ − r 0 * 0 m − r × n 0 n × r 0 n × m − r 0 n × n .

Note that

Q N = Λ ̄ D N 0 m × n 0 n × m I n = λ ̄ r − 1 − λ ̄ r − 1 λ ̄ − r 0 r × n 0 m − r × r N I m − r 0 m − r × n 0 n × r 0 n × m − r I n ,

Let F ⁰ = ( F , 0 _{(T−h)×(m−r)}), then

Q N ⊤ ⁢ S z * ⁢ Q N = Λ ̄ D N 0 m × n 0 n × m I n ⊤ ⁢ 1 T ⁢ λ ̄ ⊤ ⁢ F ⊤ ⁢ F λ ̄ λ ̄ ⊤ ⁢ F ⊤ ⁢ g g ⊤ ⁢ F λ ̄ g ⊤ ⁢ g ⁢ Λ ̄ ⁢ D N 0 m × n 0 n × m I n = 1 T ⁢ D N ⁢ Λ ̄ ⊤ ⁢ λ ̄ ⊤ ⁢ F ⊤ ⁢ F λ ̄ ⁢ Λ ̄ ⁢ D N D N ⁢ Λ ̄ ⊤ λ ̄ ⊤ ⁢ F ⊤ ⁢ g g ⊤ ⁢ F λ ̄ ⁢ Λ ̄ ⁢ D N g ⊤ ⁢ g = 1 T ⁢ F 0 ⊤ ⁢ F 0 F 0 ⊤ ⁢ g g ⊤ ⁢ F 0 g ⊤ ⁢ g = 1 T ⁢ F ⊤ ⁢ F 0 r × m − r F ⊤ ⁢ g 0 m − r × r 0 m − r × m − r 0 m − r × n g ⊤ ⁢ F 0 n × m − r g ⊤ ⁢ g .

Equation (S.81) in Karabiyik and Westerlund (2021) yields

Λ ̄ D N + λ ̂ i = Λ ̄ r λ ̄ r λ i + o p 1 ,

then we obtain

Λ ̄ D N + λ ̂ ̄ = Λ ̄ r λ ̄ r λ ̄ + o p 1 ,

which implies that Λ ̄ * Λ ̄ D N = Λ ̄ + o p * 1 . Hence, Λ ̄ * = diag I r , N − 1 / 2 I m − r + o p * 1 , then we can get that 1 T N e ̄ − r 0 * ⊤ e ̄ − r 0 * = 0 m − r × r , I m − r 1 T e ̄ * ⊤ e ̄ * 0 m − r × r , I m − r ⊤ + o p * 1 = Λ ̄ − r ⊤ 1 T e ̄ ⊤ e ̄ Λ ̄ − r + o p * 1 . It follows that

Q N ⊤ S e * Q N = 1 T 0 r × r 0 r × m − r 0 r × n 0 m − r × r N e ̄ − r 0 * ⊤ e ̄ − r 0 * 0 m − r × n 0 n × r 0 n × m − r 0 n × n = 1 T 0 r × r 0 r × m − r 0 r × n 0 m − r × r e ̄ − r 0 ⊤ e ̄ − r 0 0 m − r × n 0 n × r 0 n × m − r 0 n × n + o p * 1 .

We define S z 0 * as S z * + S e * . Next, we consider T − 1 z ̂ 0 * ⊤ z ̂ 0 * + − S z 0 * − 1 . Using Assumptions 1 and 4, we yield rank λ ̄ = r , rank 1 T F ⊤ F = r and rank 1 T g ⊤ g = n . Since 1 T e ̄ − r 0 * ⊤ e ̄ − r 0 * is random, we have rank S z 0 * = m + n . We also have rank T − 1 z ̂ 0 * ⊤ z ̂ 0 * = m + n , so rank T − 1 z ̂ 0 * ⊤ z ̂ 0 * = rank S z 0 * . According to Andrews (1987),

T − 1 z ̂ 0 * ⊤ z ̂ 0 * + − S z 0 * − 1 = O p * N − 1 / 2 + O p * T − 1 / 2 .

Define

Σ z 0 * = 1 T λ ⊤ F ⊤ F λ λ ⊤ F ⊤ g g ⊤ F λ g ⊤ g + 0 r × r 0 r × m − r 0 r × n 0 m − r × r Λ − r * ⊤ Σ e * Λ − r * 0 m − r × n 0 n × r 0 n × m − r 0 n × n .

And we also have rank Σ z 0 * = m + n . Therefore, rank S z 0 * = rank Σ z 0 * . According to Andrews (1987),

S z 0 * − 1 − Σ z 0 * − 1 = o p * 1 .

Finally,

T − 1 z ̂ 0 * ⊤ z ̂ 0 * + − Σ z 0 * − 1 ≤ T − 1 z ̂ 0 * ⊤ z ̂ 0 * + − S z 0 * − 1 + S z 0 * − 1 − Σ z 0 * − 1 = o p * 1 .

Consider T − 1 2 z ̂ 0 * ⊤ ε * ,

T − 1 2 z ̂ 0 * ⊤ ε * = T − 1 2 F ̂ + e ̄ 0 * g ⊤ ε * = T − 1 2 z 0 * ⊤ ε * + T − 1 2 e ̄ r 0 * ⊤ ε * 0 m + n − r × 1 ,

where z 0 * = F ̂ + 0 T − h × m − r , e ̄ − r 0 * , g , Condition 1 implies that T − 1 2 z 0 * ⊤ ε * → d * N 0 , Σ z ε * . In addition, T − 1 2 e ̄ * ⊤ ε * = 1 T ∑ t = 1 T − 1 ε t + 1 * e ̄ t * = 1 N 1 T ∑ t = 1 T − 1 ε t + 1 * ∑ i = 1 N e i t * . Since

E * T − 1 2 e ̄ * ⊤ ε * 2 = 1 N 2 1 T E * ∑ t = 1 T − 1 ∑ s = 1 T − 1 ε t + 1 * ε s + 1 * ∑ i = 1 N ∑ j = 1 N e i t * ⊤ e j s * = 1 N 1 T ∑ t = 1 T − 1 ε ̂ t + 1 2 1 N ∑ i = 1 N e ̂ i t ⊤ e ̂ i t = O p * 1 N ,

T − 1 2 e ̄ * ⊤ ε * = O p * 1 N . Thus T − 1 2 e ̄ r 0 * ⊤ ε * = T − 1 2 Λ ̄ r * ⊤ e ̄ * ⊤ ε * ≤ Λ ̄ r * T − 1 2 e ̄ * ⊤ ε * = O p * 1 N . Then we have that T − 1 2 z ̂ 0 * ⊤ ε * = T − 1 2 z 0 * ⊤ ε * + O p * 1 N → d * N 0 , Σ z ε * . Moreover, T − 1 2 z ̂ 0 * ⊤ e ̄ r 0 * = T − 1 2 z ̂ 0 * − z 0 * ⊤ e ̄ r 0 * + T − 1 2 z 0 * ⊤ e ̄ r 0 * . We have shown that 1 T F ̂ ⊤ e ̄ r 0 * = O p * 1 N T and 1 T e ̄ r 0 * ⊤ g = O p * 1 N T , then T − 1 2 z ̂ ⊤ e ̄ r 0 * = O p * 1 N T , and

T − 1 2 z 0 * ⊤ e ̄ r 0 * = T − 1 2 z ̂ ⊤ e ̄ r 0 * + 0 r × r T − 1 2 e ̄ − r 0 * ⊤ e ̄ r 0 * 0 n × r = 0 r × r T − 1 2 e ̄ − r 0 * ⊤ e ̄ r 0 * 0 n × r + O p * 1 N T .

It remains to consider T − 1 / 2 z ̂ 0 * − z 0 * ⊤ e ̄ r 0 * , T − 1 / 2 z ̂ 0 * − z 0 * ⊤ e ̄ r 0 * = T − 1 / 2 e ̄ r 0 * ⊤ e ̄ r 0 * 0 m + n − r × r .

Note that

T − 1 / 2 ⁢ e ̄ r 0 * ⊤ e ̄ r 0 * = T − 1 2 ⁢ e ̄ * Λ ̄ r * ⊤ ⁢ e ̄ * ⁢ Λ ̄ r * = T − 1 2 ⁢ Λ ̄ r * ⊤ ⁢ e ̄ * ⊤ ⁢ e ̄ * ⁢ Λ ̄ r * = T N − 1 ⁢ Λ ̄ r * ⊤ ⁢ N T − 1 ⁢ e ̄ * ⊤ ⁢ e ̄ * ⁢ Λ ̄ r * = T N − 1 ⁢ Λ ̄ r * ⊤ ⁢ 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e i t * ⁢ e j t * ⊤ ⁢ Λ ̄ r * .

Condition 2 yields

T − 1 / 2 e ̄ r 0 * ⊤ e ̄ r 0 * = T N − 1 Λ ̄ r * ⊤ 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T − 1 e i t * e j t * ⊤ Λ ̄ r * → p * c Λ r * ⊤ Σ e * Λ r * .

By adding the results, we obtain

Q N * − 1 ⁢ T δ ̂ * − Q N * ⁢ δ 0 * = T − 1 z ̂ 0 * ⊤ z ̂ 0 * + ⁢ T − 1 / 2 ⁢ z ̂ 0 * ⊤ ⁢ ε * − e ̄ r 0 * ⁢ α ̂ = T − 1 z ̂ 0 * ⊤ z ̂ 0 * + ⁢ T − 1 / 2 ⁢ z ̂ 0 * ⊤ ⁢ ε * − T − 1 z ̂ 0 * ⊤ z ̂ 0 * + ⁢ T − 1 / 2 ⁢ z ̂ 0 * ⊤ ⁢ e ̄ r 0 * ⁢ α ̂ = Σ z 0 * − 1 ⁢ T − 1 / 2 ⁢ z 0 * ⊤ ⁢ ε * − T 0 r × 1 T − 1 ⁢ e ̄ − r 0 * ⊤ ⁢ e ̄ r 0 * ⁢ α ̂ 0 n × 1 − T − 1 2 ⁢ e ̄ r 0 * ⊤ ⁢ e ̄ r 0 * ⁢ α ̂ 0 n × 1 + o p * ⁢ 1 .

Finally, using Condition 1, Q N * − 1 T δ ̂ * − Q N * δ 0 * → d * N − c B 1 * − B 2 * , Σ z 0 * − 1 Σ z ε * Σ z 0 * − 1 , where

B 1 * = Σ z 0 * − 1 Λ r * ⊤ Σ e * Λ r * α ̂ 0 n × 1 , B 2 * = T N − 1 2 0 r × 1 Λ − r * ⊤ Σ e * Λ − r * − 1 Λ − r * ⊤ Σ e * Λ r * α ̂ 0 n × 1 , Σ z ε * = lim N , T → ∞ T − 1 ∑ t = 1 T − 1 E * ε t + 1 2 z t 0 * z t 0 * ⊤ .

Proof of Corollary 1.

We have shown Σ z 0 * = Q 0 − 1 ⊤ Σ z 0 Q 0 − 1 in Lemma 1, where Q 0 = lim N → ∞ Q N . Also,

Σ z ε * = lim N , T → ∞ ⁢ var * ⁢ 1 T ∑ t = 1 T − 1 z t 0 * ⁢ ε t + 1 * = lim N , T → ∞ ⁢ var * ⁢ 1 T ∑ t = 1 T − 1 Q N − 1 ⁢ z ̂ t 0 ⁢ ε t + 1 * = Q 0 − 1 ⊤ ⁢ lim N , T → ∞ var 1 T ∑ t = 1 T − 1 z t 0 ⁢ ε t + 1 ⁢ Q 0 − 1 = Q 0 − 1 ⊤ ⁢ Σ z 0 ε ⁢ Q 0 − 1 .

Thus,

Σ z 0 * − 1 Σ z ε * Σ z 0 * − 1 = Q 0 − 1 ⊤ Σ z 0 Q 0 − 1 − 1 Q 0 − 1 ⊤ Σ z 0 ε Q 0 − 1 Q 0 − 1 ⊤ Σ z 0 Q 0 − 1 − 1 = Q 0 Σ z 0 − 1 Q 0 ⊤ Q 0 − 1 ⊤ Σ z 0 ε Q 0 − 1 Q 0 Σ z 0 − 1 Q 0 ⊤ = Q 0 Σ z 0 − 1 Σ z 0 ε Σ z 0 − 1 Q 0 ⊤ .

Proof of Theorem 2.

Using the fact that

z ̂ T 0 * = F ̂ T 0 * g T = D N Λ ̄ * ⊤ F ̂ T * g T = D N Λ ̄ * ⊤ λ ̂ ̄ ⊤ F ̂ T + e ̄ T * g T = F ̂ T g T + D N Λ ̄ * ⊤ e ̄ T * 0 n × 1 = z T 0 * + e ̄ r , T 0 * 0 m + n − r × 1 .

It follows that

y ̂ T + 1 T * − y T + 1 T * = T − 1 / 2 z ̂ T 0 * ⊤ Q N * − 1 T δ ̂ * − Q N * δ 0 * + N − 1 / 2 α ̂ ⊤ N e ̄ r , T 0 * = T − 1 / 2 z ̂ T 0 * ⊤ T Q N * − 1 δ ̂ * − δ 0 * + N − 1 / 2 α ̂ ⊤ N e ̄ r , T 0 * = T − 1 / 2 ⁢ z T 0 * + e ̄ r , T 0 * 0 m + n − r × 1 ⊤ ⁢ T Q N * − 1 δ ̂ * − δ 0 * + N − 1 / 2 ⁢ α ̂ ⊤ ⁢ N e ̄ r , T 0 * = T − 1 / 2 ⁢ z T 0 * ⊤ ⁢ T Q N * − 1 δ ̂ * − δ 0 * + N − 1 / 2 ⁢ α ̂ ⊤ ⁢ N e ̄ r , T 0 * + r T * ,

where

r T * = T − 1 2 e ̄ r , T 0 * 0 m + n − r × 1 ⊤ T Q N * − 1 δ ̂ * − δ 0 * = T − 1 2 Λ ̄ r * ⊤ e ̄ t * 0 m + n − r × 1 ⊤ T Q N * − 1 δ ̂ * − δ 0 * = O P * 1 N T .

The last equality holds for that T Q N * − 1 δ ̂ * − δ 0 * = O P * 1 , and T − 1 2 e ̄ t * = O P * 1 N T , which have been verified in the proof of Theorem 1.

Finally, we obtain

y ̂ T + 1 T * − y T + 1 T * T − 1 z ̂ T ⊤ Σ z 0 − 1 Σ z 0 ε Σ z 0 − 1 z ̂ T + N − 1 α ̂ ⊤ Σ e α ̂ → d * N 0 , 1 .

Proof of Corollary 2.

First, by Corollary 1 and z ̂ t * = Q N * ⊤ − 1 z ̂ t 0 * ,

φ ̂ * = z ̂ T * ⊤ Σ ̂ z * + Σ ̂ z ε * Σ ̂ z * + z ̂ T * = z ̂ T * ⊤ 1 T ∑ t = 1 T − 1 z ̂ t * z ̂ t * ⊤ + 1 T ∑ t = 1 T − 1 ε ̂ t + 1 * 2 z ̂ t * z ̂ t * ⊤ 1 T ∑ t = 1 T − 1 z ̂ t * z ̂ t * ⊤ + z ̂ T * = z ̂ T 0 * ⊤ 1 T ∑ t = 1 T − 1 z ̂ t 0 * z ̂ t 0 * ⊤ + 1 T ∑ t = 1 T − 1 ε ̂ t + 1 * 2 z ̂ t 0 * z ̂ t 0 * ⊤ 1 T ∑ t = 1 T − 1 z ̂ t 0 * z ̂ t 0 * ⊤ + z ̂ T 0 * = z T 0 * ⊤ 1 T ∑ t = 1 T − 1 z t 0 * z t 0 * ⊤ + 1 T ∑ t = 1 T − 1 ε ̂ t + 1 * 2 z t 0 * z t 0 * ⊤ 1 T ∑ t = 1 T − 1 z t 0 * z t 0 * ⊤ + z T 0 * = Q N − 1 ⊤ z ̂ T 0 ⊤ Σ z 0 * − 1 Σ z ε * Σ z 0 * − 1 Q N − 1 ⊤ z ̂ T 0 = z ̂ T 0 ⊤ Q N − 1 Σ z 0 * − 1 Σ z ε * Σ z 0 * − 1 Q N − 1 ⊤ z ̂ T 0 → p * z T 0 ⊤ Q 0 − 1 Q 0 Σ z 0 − 1 Σ z 0 ε Σ z 0 − 1 Q 0 − 1 ⊤ Q 0 ⊤ z T 0 = z T 0 ⊤ Σ z 0 − 1 Σ z 0 ε Σ z 0 − 1 z T 0 ⊤ = φ .

And Condition 2 implies N e ̄ r , T 0 * → d * N 0 m × 1 , Λ r * ⊤ Σ e * Λ r * . Then we have α ̂ * = Σ e * − 1 2 ⊤ M Σ e * 1 / 2 ⊤ Λ − r * Σ e * 1 / 2 ⊤ Λ r * α ̂ , Condition 2 and Λ r * = I r , 0 r × m − r ⊤ ensure that α ̂ * = α ̂ + o P * 1 . Hence, we have that α ̂ * ⊤ Σ ̂ e * α ̂ * → d * Φ + Δ ⊤ Σ e Φ + Δ .

Proof of Theorem 3.

Define the following empirical distribution function,

F T , ε ̂ − ε ̂ ̄ x = 1 T − h ∑ t = 1 T − h I ε ̂ t + h − ε ̂ ̄ ≤ x ,

F T , ε x = 1 T − h ∑ t = 1 T − h I ε t + h ≤ x ,

where ε ̂ ̄ = 1 T − h ∑ t = 1 T − h ε ̂ t + h . Note that F T , ε * x = F T , ε ̂ − ε ̂ ̄ x . It follows that

d 2 F T , ε ̂ − ε ̂ ̄ , F ε ≤ d 2 F T , ε ̂ − ε ̂ ̄ , F T , ε + d 2 F T , ε , F ε .

Using Lemma 8.4 in Bickel and Freedman (1981), d ₂(F _T,ε, F _ε) converges to 0 almost surely. It suffices to prove that d 2 F T , ε ̂ − ε ̂ ̄ , F T , ε = o p 1 . Let I represent the uniform distribution on {1, ⋯, T − h}, and define X I = ε ̂ I + h − ε ̂ ̄ and Y _I = ε _I+h. We have that

d 2 F T , ε ̂ − ε ̂ ̄ , F T , ε 2 ≤ E X I − Y I 2 = E I ε ̂ I + h − ε ̂ ̄ − ε I + h 2 = 1 T − h ∑ t = 1 T − h ε ̂ t + h − ε ̂ ̄ − ε t + h 2 = 1 T − h ∑ t = 1 T − h ε ̂ t + h − ε t + h 2 − 2 T − h ∑ t = 1 T − h ε ̂ t + h − ε t + h ε ̂ ̄ + ε ̂ ̄ 2 .

We consider the first term of the above equality,

ε ̂ t + h − ε t + h = − T − 1 / 2 T δ ̂ − Q N δ 0 ⊤ Q N − 1 ⊤ z ̂ t 0 − N − 1 / 2 α ⊤ N e ̄ r , t 0 ,

This implies that

1 T − h ∑ t = 1 T − h ε ̂ t + h − ε ̂ ̄ − ε t + h 2 ≤ 2 T − h ∑ t = 1 T − h Q N − 1 δ ̂ − Q N δ 0 2 z ̂ t 0 2 + 2 T − h ∑ t = 1 T − h α 2 e ̄ r , t 0 2 = O p 1 T + O p 1 N = o p 1 .

Similarly,

ε ̂ ̄ = 1 T − h ∑ t = 1 T − h ε ̂ t + h = 1 T − h ∑ t = 1 T − h ε ̂ t + h − ε t + h + 1 T − h ∑ t = 1 T − h ε t + h = O p 1 T + O p 1 N + o p 1 = o p 1 .

This means that both 2 T − h ∑ t = 1 T − h ε ̂ t + h − ε t + h ε ̂ ̄ and ε ̂ ̄ 2 are o _p(1). Then we obtain

d 2 F T , ε ̂ − ε ̂ ̄ , F T , ε = o p 1 .

Proof of Condition 1.

Under the bootstrap measure by construction, we have E * ε t + 1 * = 0 ; 1 T ∑ t = 1 T − 1 E * ε t + 1 * 2
= 1 T ∑ t = 1 T − 1 E * ε ̂ t + 1 2 v t + 1 2 = 1 T ∑ t = 1 T − 1 ε ̂ t + 1 2 ≤ 1 T ∑ t = 1 T − 1 ε ̂ t + 1 4 1 2 < ∞ .
Let m t * = Σ z ε * − 1 / 2 z t 0 * ε t + 1 * , E * m t * = Σ z ε * − 1 / 2 z t 0 * E * ε t + 1 * = 0 ;
var * 1 T ∑ t = 1 T − 1 m t * = var * 1 T ∑ t = 1 T − 1 Σ z ε * − 1 2 z t 0 * ε t + 1 * = Σ z ε * − 1 / 2 var * 1 T ∑ t = 1 T − 1 z t 0 * ε t + 1 * Σ z ε * − 1 / 2 = I m + n .

Proof of Condition 2.

E * 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T e i t * e j t * ⊤ − E * e i t * e j t * ⊤

= 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T E * e i t * e j t * ⊤ − 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T E * e i t * e j t * ⊤ = 0 .

Also,

var * 1 N T ∑ i = 1 N ∑ j = 1 N ∑ t = 1 T e i t * e j t * ⊤ − E * e i t * e j t * ⊤ = ∑ t = 1 T ∑ s = 1 T 1 N 2 ∑ i , j , k , l Cov * e i t * e j t * ⊤ , e l s * e k s * ⊤ ≤ κ · 1 T 1 N 1 N T ∑ i = 1 N ∑ t = 1 T e ̂ i t 4 = O p 1 T N = o p 1 .

Hence, N T − 1 ∑ t = 1 T e ̄ t * e ̄ t * ⊤ → p * Σ e * , that is, N T − 1 ∑ t = 1 T e ̄ t * e ̄ t * ⊤ − Σ e * = o p * 1 .

Appendix B. Monte Carlo Simulations

See Tables B.1–B.4

Table B.1:

DGP5, m = 2 > r = 1, e _it is homoskedastic and ε _t+1 is normal.

		N = 50				N = 100				N = 200
		T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400
y _T+1\|T		Coverage rate
	F	0.93	0.95	0.94	0.95	0.95	0.95	0.92	0.96	0.93	0.95	0.95	0.94
	PC	0.77	0.65	0.58	0.46	0.84	0.78	0.68	0.57	0.89	0.81	0.76	0.63
	CA	0.96	0.94	0.94	0.95	0.98	0.96	0.97	0.95	0.97	0.97	0.97	0.96
	PCBS	0.90	0.89	0.90	0.91	0.94	0.92	0.95	0.93	0.94	0.93	0.94	0.92
	CABS	0.93	0.93	0.94	0.91	0.93	0.93	0.94	0.94	0.95	0.93	0.93	0.93
		Length
	F	0.75	0.53	0.38	0.26	0.77	0.53	0.37	0.27	0.76	0.52	0.39	0.26
	PC	0.79	0.56	0.41	0.30	0.78	0.54	0.38	0.28	0.77	0.52	0.39	0.26
	CA	1.44	1.21	1.05	0.99	1.28	1.03	0.87	0.78	1.19	0.92	0.74	0.62
	PCBS	1.20	0.99	0.90	0.86	1.05	0.85	0.73	0.68	0.95	0.71	0.61	0.53
	CABS	1.49	1.23	1.06	1.00	1.31	1.03	0.85	0.77	1.23	0.92	0.72	0.61
y _T+1		Coverage rate
	F	0.95	0.96	0.97	0.95	0.95	0.96	0.94	0.94	0.97	0.95	0.93	0.94
	PC	0.96	0.96	0.97	0.94	0.95	0.95	0.94	0.96	0.97	0.96	0.93	0.93
	CA	0.96	0.96	0.97	0.95	0.96	0.96	0.95	0.96	0.98	0.96	0.93	0.94
	PCBS	0.95	0.95	0.96	0.95	0.96	0.93	0.94	0.95	0.95	0.95	0.93	0.94
	CABS	0.95	0.95	0.96	0.95	0.96	0.95	0.94	0.95	0.96	0.95	0.92	0.93
		Length
	F	3.90	3.89	3.93	3.92	3.91	3.91	3.91	3.92	3.93	3.91	3.92	3.91
	PC	4.02	4.01	4.04	4.03	3.97	3.97	3.97	3.98	3.97	3.94	3.95	3.94
	CA	4.16	4.14	4.15	4.14	4.06	4.05	4.04	4.05	4.04	3.99	3.99	3.98
	PCBS	4.14	4.03	4.03	3.98	4.13	4.02	3.95	3.95	4.08	3.98	3.93	3.90
	CABS	4.16	4.09	4.05	4.02	4.14	4.05	3.98	3.94	4.14	4.01	3.95	3.94

Notes : F, PC, CA, PCBS and CABS refer to the results based on the true factors, PC based on knowing the true number of factors, CA, PC + Bootstrap, and CA + Bootstrap, respectively.

Table B.2:

DGP6, m = 2 > r = 1, e _it is homoskedastic and ε _t+1 is mixture.

		N = 50				N = 100				N = 200
		T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400
y _T+1\|T		Coverage rate
	F	0.91	0.94	0.95	0.94	0.94	0.93	0.96	0.93	0.91	0.92	0.95	0.91
	PC	0.73	0.68	0.55	0.43	0.85	0.77	0.70	0.53	0.87	0.80	0.77	0.62
	CA	0.93	0.94	0.94	0.94	0.96	0.97	0.97	0.95	0.97	0.97	0.97	0.96
	PCBS	0.87	0.90	0.88	0.91	0.93	0.93	0.93	0.93	0.91	0.92	0.95	0.93
	CABS	0.90	0.91	0.93	0.92	0.95	0.93	0.95	0.93	0.95	0.92	0.95	0.95
		Length
	F	0.76	0.54	0.37	0.27	0.75	0.54	0.37	0.27	0.74	0.52	0.38	0.27
	PC	0.80	0.57	0.40	0.30	0.77	0.55	0.39	0.28	0.75	0.52	0.38	0.27
	CA	1.45	1.20	1.05	0.98	1.28	1.02	0.87	0.78	1.19	0.88	0.73	0.62
	PCBS	1.25	1.03	0.91	0.86	1.11	0.86	0.76	0.67	0.99	0.73	0.62	0.54
	CABS	1.51	1.22	1.06	1.00	1.36	1.03	0.86	0.77	1.28	0.89	0.72	0.61
y _T+1		Coverage rate
	F	0.90	0.92	0.92	0.87	0.90	0.93	0.91	0.88	0.93	0.92	0.89	0.88
	PC	0.90	0.93	0.92	0.87	0.90	0.93	0.91	0.89	0.93	0.92	0.89	0.88
	CA	0.91	0.93	0.93	0.87	0.91	0.93	0.91	0.89	0.93	0.92	0.90	0.89
	PCBS	0.95	0.96	0.95	0.94	0.96	0.96	0.94	0.96	0.96	0.97	0.97	0.93
	CABS	0.95	0.96	0.96	0.93	0.96	0.98	0.93	0.96	0.97	0.97	0.97	0.95
		Length
	F	3.86	3.88	3.90	3.93	3.88	3.91	3.92	3.90	3.83	3.86	3.91	3.92
	PC	3.99	4.01	4.01	4.04	3.94	3.97	3.97	3.96	3.87	3.88	3.94	3.95
	CA	4.13	4.13	4.12	4.15	4.03	4.04	4.04	4.02	3.94	3.93	3.98	3.99
	PCBS	4.41	4.39	4.27	4.25	4.53	4.28	4.16	4.11	4.32	4.15	4.16	4.10
	CABS	4.52	4.36	4.31	4.28	4.50	4.34	4.19	4.16	4.40	4.21	4.16	4.08

Table B.3:

DGP7, m = 2 > r = 1, e _it is heteroskedastic and ε _t+1 is normal.

		N = 50				N = 100				N = 200
		T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400
y _T+1\|T		Coverage rate
	F	0.92	0.97	0.95	0.95	0.95	0.94	0.95	0.94	0.96	0.94	0.93	0.96
	PC	0.78	0.65	0.56	0.47	0.84	0.77	0.67	0.58	0.90	0.86	0.76	0.64
	CA	0.96	0.92	0.95	0.95	0.97	0.95	0.97	0.95	0.98	0.97	0.96	0.97
	PCBS	0.91	0.90	0.91	0.91	0.93	0.91	0.94	0.91	0.97	0.94	0.93	0.94
	CABS	0.96	0.91	0.94	0.94	0.93	0.93	0.95	0.93	0.96	0.93	0.94	0.93
		Length
	F	0.75	0.53	0.38	0.26	0.77	0.54	0.37	0.27	0.78	0.52	0.39	0.27
	PC	0.79	0.56	0.41	0.29	0.78	0.55	0.39	0.28	0.79	0.53	0.40	0.27
	CA	1.39	1.14	1.02	0.94	1.27	1.00	0.84	0.75	1.23	0.89	0.72	0.60
	PCBS	1.20	1.00	0.93	0.88	1.06	0.86	0.75	0.69	0.97	0.72	0.61	0.53
	CABS	1.49	1.18	1.06	0.99	1.33	1.03	0.85	0.78	1.27	0.89	0.72	0.60
y _T+1		Coverage rate
	F	0.96	0.95	0.95	0.95	0.95	0.95	0.94	0.96	0.96	0.95	0.94	0.96
	PC	0.96	0.95	0.95	0.95	0.94	0.95	0.94	0.95	0.96	0.95	0.95	0.96
	CA	0.97	0.96	0.96	0.96	0.96	0.95	0.94	0.95	0.96	0.96	0.95	0.97
	PCBS	0.95	0.93	0.96	0.95	0.94	0.93	0.92	0.95	0.94	0.95	0.94	0.96
	CABS	0.95	0.93	0.95	0.95	0.94	0.94	0.94	0.95	0.95	0.94	0.94	0.96
		Length
	F	3.93	3.92	3.92	3.91	3.90	3.93	3.91	3.92	3.91	3.90	3.92	3.90
	PC	4.05	4.04	4.02	4.02	3.96	4.00	3.97	3.98	3.94	3.93	3.95	3.93
	CA	4.17	4.15	4.13	4.12	4.05	4.07	4.03	4.04	4.02	3.98	3.99	3.97
	PCBS	4.17	4.08	4.01	3.98	4.09	4.04	3.92	3.96	4.05	3.94	3.93	3.89
	CABS	4.23	4.09	4.03	4.00	4.17	4.03	3.97	3.97	4.17	3.97	3.93	3.92

Table B.4:

DGP8, m = 2 > r = 1, e _it is heteroskedastic and ε _t+1 is mixture.

		N = 50				N = 100				N = 200
		T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400	T = 50	T = 100	T = 200	T = 400
y _T+1\|T		Coverage rate
	F	0.91	0.95	0.93	0.95	0.91	0.94	0.95	0.95	0.92	0.94	0.95	0.96
	PC	0.75	0.65	0.58	0.46	0.79	0.76	0.60	0.52	0.86	0.82	0.75	0.67
	CA	0.94	0.93	0.92	0.95	0.97	0.97	0.95	0.93	0.97	0.97	0.96	0.97
	PCBS	0.91	0.89	0.92	0.92	0.94	0.92	0.92	0.92	0.94	0.92	0.95	0.94
	CABS	0.94	0.93	0.91	0.94	0.94	0.94	0.93	0.91	0.92	0.92	0.93	0.94
		Length
	F	0.74	0.52	0.37	0.27	0.77	0.52	0.37	0.26	0.75	0.53	0.38	0.27
	PC	0.77	0.55	0.40	0.30	0.79	0.54	0.38	0.27	0.76	0.54	0.39	0.27
	CA	1.36	1.14	1.01	0.94	1.31	1.00	0.84	0.75	1.19	0.90	0.72	0.60
	PCBS	1.25	1.03	0.93	0.89	1.14	0.85	0.74	0.69	1.00	0.75	0.62	0.54
	CABS	1.45	1.19	1.04	0.98	1.40	1.03	0.84	0.76	1.25	0.93	0.72	0.61
y _T+1		Coverage rate
	F	0.93	0.92	0.93	0.91	0.91	0.90	0.89	0.91	0.91	0.92	0.91	0.91
	PC	0.93	0.93	0.93	0.91	0.91	0.90	0.89	0.91	0.91	0.92	0.91	0.91
	CA	0.93	0.93	0.93	0.91	0.92	0.90	0.89	0.91	0.92	0.93	0.91	0.91
	PCBS	0.97	0.97	0.95	0.96	0.98	0.96	0.94	0.95	0.97	0.97	0.97	0.96
	CABS	0.97	0.96	0.97	0.95	0.96	0.95	0.94	0.95	0.97	0.98	0.96	0.96
		Length
	F	3.80	3.88	3.90	3.92	3.87	3.86	3.89	3.90	3.83	3.89	3.90	3.91
	PC	3.92	4.00	4.02	4.03	3.94	3.91	3.95	3.96	3.87	3.92	3.93	3.94
	CA	4.04	4.10	4.11	4.13	4.04	3.98	4.02	4.02	3.94	3.97	3.97	3.98
	PCBS	4.50	4.34	4.24	4.26	4.44	4.19	4.16	4.14	4.41	4.20	4.14	4.09
	CABS	4.51	4.41	4.27	4.24	4.53	4.34	4.22	4.14	4.47	4.26	4.16	4.09

References

Andrews, D. W. 1987. “Asymptotic Results for Generalised Wald Tests.” Econometric Theory 3: 348–58. https://doi.org/10.1017/s0266466600010434.Search in Google Scholar

Bai, J., and S. Ng. 2006. “Confidence Intervals for Diffusion Index Forecasts and Inference for Factor-Augmented Regressions.” Econometrica 74: 1133–50. https://doi.org/10.1111/j.1468-0262.2006.00696.x.Search in Google Scholar

Bai, J., and S. Ng. 2008. “Forecasting Economic Time Series using Targeted Predictors.” Journal of Econometrics 146: 304–17. https://doi.org/10.1016/j.jeconom.2008.08.010.Search in Google Scholar

Bickel, P., and D. Freedman. 1981. “Asymptotic Theory for the Bootstrap.” Annals of Statistics 9: 1196–217. https://doi.org/10.1214/aos/1176345637.Search in Google Scholar

Breitung, J., and S. Eickmeier. 2011. “Testing for Structural Breaks in Dynamic Factor Models.” Journal of Econometrics 163: 71–84. https://doi.org/10.1016/j.jeconom.2010.11.008.Search in Google Scholar

Breitung, J., and U. Pigorsch. 2013. “A Canonical Correlation Approach for Selecting the Number of Dynamic Factors.” Oxford Bulletin of Economics & Statistics 75: 23–36. https://doi.org/10.1111/obes.12003.Search in Google Scholar

Chudik, A., M. H. Pesaran, and E. Tosetti. 2011. “Weak and Strong Cross Section Dependence and Estimation of Large Panels.” The Econometrics Journal 14: C45–90. https://doi.org/10.1111/j.1368-423x.2010.00330.x.Search in Google Scholar

Corradi, V., and N. Swanson. 2014. “Testing for Structural Stability of Factor Augmented Forecasting Models.” Journal of Econometrics 182 (1): 100–18. https://doi.org/10.1016/j.jeconom.2014.04.011.Search in Google Scholar

De Vos, I., and O. Stauskas. 2024. “Cross-Section Bootstrap for CCE Regressions.” Journal of Econometrics 240 (1). https://doi.org/10.1016/j.jeconom.2023.105648.Search in Google Scholar

Djogbenou, A., S. Gonçalves, and B. Perron. 2015. “Bootstrap Inference for Regressions with Estimated Factors and Serial Correlation.” Journal of Time Series Analysis 36: 481–502.10.1111/jtsa.12118Search in Google Scholar

Gonçalves, S., B. Perron, and A. Djogbenou. 2017. “Bootstrap Prediction Intervals for Factor Models.” Journal of Business & Economic Statistics 35 (1): 53–69. https://doi.org/10.1080/07350015.2015.1054492.Search in Google Scholar

Gospodinov, N., and S. Ng. 2013. “Commodity Prices, Convenience Yields and Inflation.” The Review of Economics and Statistics 95 (1): 206–19. https://doi.org/10.1162/rest_a_00242.Search in Google Scholar

Hallin, M., and R. Liska. 2007. “Determining the Number of Factors in the General Dynamic Factor Model.” Journal of the American Statistical Association 102: 603–17. https://doi.org/10.1198/016214506000001275.Search in Google Scholar

Karabiyik, H., and J. Westerlund. 2021. “Forecasting using Cross-Section Average-Augmented Time Series Regressions.” The Econometrics Journal 24 (2): 315–33. https://doi.org/10.1093/ectj/utaa031.Search in Google Scholar

Karabiyik, H., S. Reese, and J. Westerlund. 2017. “On the Role of the Rank Condition in CCE Estimation of Factor-Augmented Panel Regressions.” Journal of Econometrics 197: 60–4. https://doi.org/10.1016/j.jeconom.2016.10.006.Search in Google Scholar

Ludvigson, S., and S. Ng. 2007. “The Empirical Risk Return Relation: A Factor Analysis Approach.” Journal of Financial Econometrics 83: 171–222. https://doi.org/10.1016/j.jfineco.2005.12.002.Search in Google Scholar

Ludvigson, S., and S. Ng. 2009. “Macro Factors in Bond Risk Premia.” Review of Financial Studies 22: 5027–67. https://doi.org/10.1093/rfs/hhp081.Search in Google Scholar

Ludvigson, S., and S. Ng. 2011. “A Factor Analysis of Bond Risk Premia.” In Handbook of Empirical Economics and Finance, 313–72. Chapman and Hall.Search in Google Scholar

Pesaran, M. H. 2006. “Estimation and Inference in Large Heterogeneous Panels with a Multi-Factor Error Structure.” Econometrica 74: 967–1012. https://doi.org/10.1111/j.1468-0262.2006.00692.x.Search in Google Scholar

Shintani, M., and Z.-Y. Guo. 2011. Finite Sample Performance of Principal Components Estimators for Dynamic Factor Models: Asymptotic vs. Bootstrap Approximations. Manuscript. Vanderbilt University.Search in Google Scholar

Stock, J., and M. Watson. 2002a. “Forecasting Using Principal Components from a Large Number of Predictors.” Journal of the American Statistical Association 97: 1167–79. https://doi.org/10.1198/016214502388618960.Search in Google Scholar

Stock, J., and M. Watson. 2002b. “Macroeconomic Forecasting Using Diffusion Indexes.” Journal of Business & Economic Statistics 20: 147–62. https://doi.org/10.1198/073500102317351921.Search in Google Scholar

Stock, J., and M. Watson. 2005. “Implications of Dynamic Factor Models for VAR Analysis.” NBER Working Paper No. 11467. Department of Economics, Harvard University.10.3386/w11467Search in Google Scholar

Westerlund, J., and J.-P. Urbain. 2015. “Cross-Sectional Averages Versus Principal Components.” Journal of Econometrics 85: 372–7. https://doi.org/10.1016/j.jeconom.2014.09.014.Search in Google Scholar

Yamamoto, Y. 2019. “Bootstrap Inference for Impulse Response Functions in Factor-Augmented Vector Autoregressions.” Journal of Applied Econometrics 34: 247–67. https://doi.org/10.1002/jae.2659.Search in Google Scholar

Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2024-0102).

Received: 2024-09-18

Accepted: 2025-09-25

Published Online: 2025-10-10

You are currently not able to access this content.

Supplementary Material Details

https://doi.org/10.1515/snde-2024-0102

Keywords for this article

bootstrap; cross-section average; factor-augmented regression models; forecast