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Corrigendum to “Testing predictive regression models with nonstationary regressors” [J. Econometrics 178 (2014) 4–14]
Journal of Econometrics (
)
–
Contents lists available at ScienceDirect
Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom
Corrigendum to ‘‘Testing predictive regression models with nonstationary regressors’’ [J. Econometrics 178 (2014) 4–14] Zongwu Cai a,b,∗ , Yunfei Wang c a
Department of Economics, University of Kansas, Lawrence, KS 66045, USA
b
Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, Fujian 361005, China
c
Direct Banking Decision Sciences, Capital One 360, 1102 Greystone Ln, Newark, DE 19711, USA
article
info
Article history: Available online xxxx
In the proof of Theorem 1 in the paper by Cai and Wang (2014), the statement on page 12 of Cai and Wang (2014), uˆ t = ut + Op (n−1/2 ), is not enough to show the asymptotic distribution of
distribution of βˆ for θ ̸= 0. Therefore, Theorem 2 should be modified accordingly as below.
βˆ . Therefore, the statement on page 12, ‘‘Hence in the following proof, we will use ut to replace uˆ t without further explanation’’, is
Theorem 2. Under Assumptions A1–A3 and model (6) with c ≤ 0, we have
inaccurate. Indeed, by using the notation in Cai and Wang (2014), 1 n n−1/2 t =1 uˆ t ⇒ Wu (1) − ξc 0 Kc (r )dr by using Cai and Wang
√
1 (2014, properties (a)–(d) on page 6), where ξ = [ 0 Kc (r )dWu (r )+ 1 2 1 Ω1 ]/ 0 Kc (r )dr. Therefore, the term −ξc 0 Kc (r )dr in the above
d nDn (βˆ − β) → N 0,
∗
∗
,
β
where
∗
β = ∗∗
S0∗−1 S0∗∗ S0∗−1 , S0∗ is defined on page 13 of Cai and Wang
equation is missing in the proof of Theorem 1 in Cai and Wang (2014). Then, Theorem 1 can be correctly restated as follows.
(2014), S0 = Var(η2 ) and η2 = (Wε (1), Wuv (1), θ 1)dWε (r )/c )T .
Theorem 1. Under Assumptions A1–A3, we have
Finally, β1 ξc should be added into the right hand side of (10) in 2 Cai and Wang ∗(2014). Also, σu in the ∗(2, 2)th element in matrices ∗ Ω , Ω and β on page 7 and Sn and Sn on page 13 in Cai and Wang
√
d
nDn (βˆ − β) → MN 0,
β
+ η1 ,
where η1 = (0, 0, β1 ξ )T . Similarly, in the proof of Theorem 2 in the paper by Cai and Wang (2014), the statement on page 13 of Cai and Wang (2014), uˆ t = ut + Op (n−1/2 ), is not enough to show the asymptotic
DOI of original article: http://dx.doi.org/10.1016/j.jeconom.2013.08.002. Corresponding author at: Department of Economics, University of Kansas, Lawrence, KS 66045, USA. E-mail addresses: [email protected] (Z. Cai), [email protected] (Y. Wang).
∗
http://dx.doi.org/10.1016/j.jeconom.2014.03.008 0304-4076/Published by Elsevier B.V.
1 0
(exp(rc ) −
(2014) should be Var(ut ), σu2 in (9) in Cai and Wang (2014) should √ be Var(ut )4 and 1/ n in (A.5) should be 1/n. References Cai, Z., Wang, Y., 2014. Testing predictive regression models with nonstationary regressors. J. Econometrics 178, 4–14.
)
–
Contents lists available at ScienceDirect
Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom
Corrigendum to ‘‘Testing predictive regression models with nonstationary regressors’’ [J. Econometrics 178 (2014) 4–14] Zongwu Cai a,b,∗ , Yunfei Wang c a
Department of Economics, University of Kansas, Lawrence, KS 66045, USA
b
Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, Fujian 361005, China
c
Direct Banking Decision Sciences, Capital One 360, 1102 Greystone Ln, Newark, DE 19711, USA
article
info
Article history: Available online xxxx
In the proof of Theorem 1 in the paper by Cai and Wang (2014), the statement on page 12 of Cai and Wang (2014), uˆ t = ut + Op (n−1/2 ), is not enough to show the asymptotic distribution of
distribution of βˆ for θ ̸= 0. Therefore, Theorem 2 should be modified accordingly as below.
βˆ . Therefore, the statement on page 12, ‘‘Hence in the following proof, we will use ut to replace uˆ t without further explanation’’, is
Theorem 2. Under Assumptions A1–A3 and model (6) with c ≤ 0, we have
inaccurate. Indeed, by using the notation in Cai and Wang (2014), 1 n n−1/2 t =1 uˆ t ⇒ Wu (1) − ξc 0 Kc (r )dr by using Cai and Wang
√
1 (2014, properties (a)–(d) on page 6), where ξ = [ 0 Kc (r )dWu (r )+ 1 2 1 Ω1 ]/ 0 Kc (r )dr. Therefore, the term −ξc 0 Kc (r )dr in the above
d nDn (βˆ − β) → N 0,
∗
∗
,
β
where
∗
β = ∗∗
S0∗−1 S0∗∗ S0∗−1 , S0∗ is defined on page 13 of Cai and Wang
equation is missing in the proof of Theorem 1 in Cai and Wang (2014). Then, Theorem 1 can be correctly restated as follows.
(2014), S0 = Var(η2 ) and η2 = (Wε (1), Wuv (1), θ 1)dWε (r )/c )T .
Theorem 1. Under Assumptions A1–A3, we have
Finally, β1 ξc should be added into the right hand side of (10) in 2 Cai and Wang ∗(2014). Also, σu in the ∗(2, 2)th element in matrices ∗ Ω , Ω and β on page 7 and Sn and Sn on page 13 in Cai and Wang
√
d
nDn (βˆ − β) → MN 0,
β
+ η1 ,
where η1 = (0, 0, β1 ξ )T . Similarly, in the proof of Theorem 2 in the paper by Cai and Wang (2014), the statement on page 13 of Cai and Wang (2014), uˆ t = ut + Op (n−1/2 ), is not enough to show the asymptotic
DOI of original article: http://dx.doi.org/10.1016/j.jeconom.2013.08.002. Corresponding author at: Department of Economics, University of Kansas, Lawrence, KS 66045, USA. E-mail addresses: [email protected] (Z. Cai), [email protected] (Y. Wang).
∗
http://dx.doi.org/10.1016/j.jeconom.2014.03.008 0304-4076/Published by Elsevier B.V.
1 0
(exp(rc ) −
(2014) should be Var(ut ), σu2 in (9) in Cai and Wang (2014) should √ be Var(ut )4 and 1/ n in (A.5) should be 1/n. References Cai, Z., Wang, Y., 2014. Testing predictive regression models with nonstationary regressors. J. Econometrics 178, 4–14.