Testing for cointegration with threshold adjustment in the presence of structural breaks

Karsten Schweikert

doi:10.1515/snde-2018-0034

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Testing for cointegration with threshold adjustment in the presence of structural breaks

Karsten Schweikert

Veröffentlicht/Copyright: 1. Mai 2019

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Aus der Zeitschrift Studies in Nonlinear Dynamics & Econometrics Band 24 Heft 1

Abstract

In this paper, we develop new threshold cointegration tests with SETAR and MTAR adjustment allowing for the presence of structural breaks in the equilibrium equation. We propose a simple procedure to simultaneously estimate the previously unknown breakpoint and test the null hypothesis of no cointegration. Thereby, we extend the well-known residual-based cointegration test with regime shift introduced by (Gregory, A. W., and B. E. Hansen. 1996a. “Residual-based Tests for Cointegration in Models with Regime Shifts.” Journal of Econometrics 70: 99–126) to include forms of nonlinear adjustment. We derive the asymptotic distribution of the test statistics and demonstrate the finite-sample performance of the tests in a series of Monte Carlo experiments. We find a substantial decrease of power of the conventional threshold cointegration tests caused by a shift in the slope coefficient of the equilibrium equation. The proposed tests perform superior in these situations. An application to the “rockets and feathers” hypothesis of price adjustment in the US gasoline market provides empirical support for this methodology.

Keywords: Cointegration; MTAR; SETAR; structural change; threshold autoregression

Acknowledgement

I thank Karl-Heinz Schild, Karlheinz Fleischer, Robert Jung, Konstantin Kuck, Martin Wagner, Oliver Stypka, Florian Stark and two anonymous referees for valuable comments and suggestions. Further, I thank seminar participants at the University of Marburg, University of Hohenheim, University of Tübingen and Technical University Dortmund, as well as participants of the THE Christmas Workshop in Stuttgart, German Statistical Week in Rostock and the 23rd Spring Meeting of Young Economists in Palma.

Appendix

Proof of Theorem 2.

The asymptotic distribution is derived by adapting the results of Gregory and Hansen (1992) to match the F-statistic process involving a threshold indicator function using results in Maki and Kitasaka (2015). However, Maki and Kitasaka (2015) use a different definition of the threshold parameter space in their SETAR model. The threshold parameter in our model is fixed, i.e. belongs to a trivial compact subset of ℝ whereas the parameter space in Maki and Kitasaka (2015) is data dependent (see the discussion on threshold parameter space in Section 2.2 of their paper). Indicator functions with threshold parameters defined on compact sets are treated in Seo (2008). The proof only refers to model C/S while the results for the remaining models can be deduced from the results obtained for this model. Hence, we consider the cointegrating regression,

(16) y t = α ^ 1 ′ x t + μ ^ 1 + α ^ 2 ′ x t φ t , τ + μ ^ 2 φ t , τ + e ^ t τ ,

where e^tτ is an integrated process under the null hypothesis of no cointegration and zt=(yt,xt′)′ is generated according to (8).

We define the (2m + 3)-vector Xtτ=(yt,xt′,1,xtφt,τ′,φt,τ)′ and partition Xtτ=(X1tτ,X2tτ′)′ where X1tτ=yt and X2tτ contains all regressors of (16). We define δT=diag(T−1/2Im+1,1,T−1/2Im,1), φτ(s)=𝟙{s>τ} and Xτ(s)=(B(s)′,1,Bx(s)φτ(s)′,φτ(s))′. Next, we partition δT=(δ1T,δ2T) in conformity to Xtτ and partition the (m + 1)-vector standard Brownian Motion W as W=(Wy,Wx′)′, where

(17) W y = l 11 − 1 ( B y − ω 21 ′ Ω 22 − 1 B x ) W x = Ω 22 − 1 / 2 B x .

Furthermore, we define

(18) W x τ = ( W x ′ , 1 , W x φ τ ′ , φ τ ) ′

and Wτ=(Wy,Wxτ′)′.

First, we consider the least squares estimator of the parameters of the cointegrating regression. It is shown in Gregory and Hansen (1992) using the FCLT and the continuous mapping theorem (CMT, see Billingsley (1999), Theorem 2.7) that

(19) T − 1 δ T ∑ t = 1 T X t τ X t τ ′ δ T ⇒ ∫ 0 1 X τ X τ ′ ,

where the weak convergence is with respect to the uniform metric over τ∈T. In the remainder of the proof, we refer to weak convergence results involving the break fraction parameter τ as holding uniformly over τ [see also Arai and Kurozumi (2007) for a similar application].

We define the vector θ^τ=(α^1′,μ^1,α^2′,μ^2) as the least squares estimator of (16) for each τ. It follows from (19) and the CMT that

(20) T − 1 / 2 δ 2 T − 1 θ ^ τ = ( T − 1 δ 2 T ∑ t = 1 T X 2 t τ X 2 t τ ′ δ 2 T ) − 1 ( T − 1 δ 2 T ∑ t = 1 T X 2 t τ X 1 t τ δ 1 T ) ⇒ ( ∫ 0 1 X 2 τ X 2 τ ′ ) − 1 ( ∫ 0 1 X 2 τ X 1 τ ) .

When we set η^τ=T−1/2δT−1(1,−θ^τ′)′=(1,−δ2T−1θ^τ′)′, it follows that

(21) η ^ τ ⇒ ( 1 , − ( ∫ 0 1 X 1 τ X 2 τ ′ ) ( ∫ 0 1 X 2 τ X 2 τ ′ ) − 1 ) ′ = η τ .

Next, we state some useful convergence results for the residuals of the cointegrating regression. We define the residual series e^tτ=yt−α^1′xt−μ^1−α^2′xtφt,τ−μ^2φt,τ which is dependent on τ. Note that e^tτ can be expressed as

(22) e ^ t τ = T 1 / 2 η ^ τ ′ δ T X t τ .

Using Lemma 2.2 of Phillips and Ouliaris (1990) yields

(23) T − 1 / 2 e ^ t τ ⇒ η τ ′ X τ = l 11 κ τ ′ W τ = l 11 Q κ τ ,

where

(24) κ τ = ( 1 , − ( ∫ 0 1 W y W x τ ′ ) ( ∫ 0 1 W x τ W x τ ′ ) − 1 ) ′ L η τ = l 11 κ τ Q κ τ = W y − ( ∫ 0 1 W y W x τ ′ ) ( ∫ 0 1 W x τ W x τ ′ ) − 1 W x τ .

The first-differenced residuals are expressed as Δe^tτ=T1/2η^τ′δTΔXtτ, where

(25) Δ X t τ = Δ ( y t , x t ′ , 1 , x t φ t , τ ′ , φ t , τ ) ′ = ( ξ 1 t , ξ 2 t ′ , 0 , x t − 1 Δ φ t , τ ′ + Δ x t φ t , τ ′ , Δ φ t , τ ) ′ = ( ξ 1 t , ξ 2 t ′ , 0 , x t − 1 Δ φ t , τ + ξ 2 t φ t , τ ′ , Δ φ t , τ ) ′

and

(26) Δ φ t , τ = { 1 if t = [ T τ ] 0 if t ≠ [ T τ ] .

The asymptotic counterpart to Δφt,τ is the differential dφτ(s), a Dirac function concentrating the unit mass at the point s = τ so that

∫ a b f d φ τ = lim z ↑ τ f ( z ) , a < τ < b ,

for all functions with left-limits. Then, we can define the differential dXτ by

(27) d X τ ( s ) = ( d B ( s ) ′ , 0 , B x ( s ) ′ d φ τ ( s ) + d B x ( s ) ′ φ τ ( s ) , d φ τ ( s ) ) ′ .

Under Assumption 1, ξ_t is a stationary linear vector process and consequently, the scalar process T1/2η^τ′δTΔXtτ⇒T1/2ητ′δTΔXtτ is also a stationary linear process with an intervention outlier at t = [Tτ]. Moreover, under Assumption 2 the lag truncation parameter K → ∞ for T → ∞. This means that the error of approximating εtτ by a finite AR process becomes small as K grows large. Following Phillips and Ouliaris (1990) we write the infinite order AR representation of the SETAR error term process as εtτ=∑j=0∞Dj(T1/2δTΔXt−jτ)′ητ=D(L)(T1/2δTΔXtτ)′ητ. The lag structure is chosen in a way that εtτ is an orthogonal (0,σ2(η,τ)) sequence with long-run variance σ2(η,τ)=D(1)2ητ′Ωτητ. From Lemma 2.1 of Phillips and Ouliaris (1990), it follows that

(28) T − 1 / 2 ∑ t = 1 [ T s ] ε t τ K = D ( L ) η τ ′ ( T − 1 / 2 ∑ t = 1 [ T s ] T 1 / 2 δ T Δ X t τ ) + o p ( 1 ) ⇒ D ( 1 ) η τ ′ X τ ( s ) ,

where D(1)=∑j=0∞Dj.

Now, we consider the auxiliary regression. We apply the SETAR model to the residuals according to (4) and compute the test statistics F_τ. Note that the estimated adjustment coefficients might be correlated with the estimated coefficients of the additional lagged differences. Therefore, we write the least squares estimator of ρ=(ρ1,ρ2)′ in the breakpoint specific notation under the null hypothesis ρ1=ρ2=0 as ρ^=(Uτ′QKUτ)−1Uτ′QKετ, where

(29) U τ = [ e ^ 0 τ 𝟙 { e ^ 0 τ ≥ λ } e ^ 0 τ 𝟙 { e ^ 0 τ < λ } e ^ 1 τ 𝟙 { e ^ 1 τ ≥ λ } e ^ 1 τ 𝟙 { e ^ 1 τ < λ } ⋮ ⋮ e ^ T − 1 τ 𝟙 { e ^ T − 1 τ ≥ λ } e ^ T − 1 τ 𝟙 { e ^ T − 1 τ < λ } ] ,

ετ=(ε1τ,ε2τ,…,εTτ)′ and QK=I−MK(MK′MK)−1MK′ is the projection matrix onto the space orthogonal to the regressors MK=(Δe^t−1τ,…,Δe^t−Kτ).

We partition the matrix U_τ as Uτ=(U1τ,U2τ), then the t ratio of ρ^1 can be expressed as

(30) t 1 = ρ ^ 1 s e ( ρ ^ 1 ) = ρ ^ 1 ( σ ^ 2 ( U 1 τ ′ Q K U 1 τ ) − 1 ) 1 / 2 = U 1 τ ′ Q K ε τ σ ^ ( U 1 τ ′ Q K U 1 τ ) 1 / 2

and similarly the t ratio of ρ^2 can be expressed as

(31) t 2 = U 2 τ ′ Q K ε τ σ ^ ( U 2 τ ′ Q K U 2 τ ) 1 / 2 .

In the remainder of the proof, we focus on t₁. Scaling the t ratio appropriately yields the numerator

(32) T − 1 U 1 τ ′ Q K ε τ = T − 1 U 1 τ ′ ε τ − T − 1 / 2 ⋅ T − 1 U 1 τ ′ M K ( T − 1 M K ′ M K ) − 1 T − 1 / 2 M K ′ ε τ = T − 1 U 1 τ ′ ε τ + o p ( 1 ) = N T ( λ , τ ) + o p ( 1 )

and the term

(33) T − 2 U 1 τ ′ Q K U 1 τ = T − 2 U 1 τ ′ U 1 τ − T − 1 ⋅ T − 1 U 1 τ ′ M K ( T − 1 M K ′ M K ) − 1 T − 1 M K ′ U 1 τ = T − 2 U 1 τ ′ U 1 τ + o p ( 1 ) = D T ( λ , τ ) + o p ( 1 ) .

Finally, we need convergence results for NT(λ,τ), DT(λ,τ) and σ^2. Since x↦x𝟙{x≥λ} is a regular function, it follows from (23) and Theorem 3.1 of Park and Phillips (2001) that

(34) T − 1 / 2 e ^ t − 1 τ 𝟙 { e ^ t − 1 τ ≥ λ } = η ^ τ ′ δ T X t − 1 τ 𝟙 { T 1 / 2 η ^ τ ′ δ T X t − 1 τ ≥ λ } = η ^ τ ′ δ T X t − 1 τ 𝟙 { η ^ τ ′ δ T X t − 1 τ ≥ T − 1 / 2 λ } ⇒ η τ ′ X τ 𝟙 { η τ ′ X τ ≥ 0 } = l 11 Q κ τ 𝟙 { Q κ τ ≥ 0 } .

Thus, Theorem 2.2 of Kurtz and Protter (1991) combined with results (28) and (34) yields

(35) N T ( λ , τ ) = T − 1 ∑ t = 1 T 𝟙 { e ^ t − 1 τ ≥ λ } e ^ t − 1 τ ϵ t τ = η ^ τ ′ δ T ∑ t = 1 T 𝟙 { δ T η ^ τ ′ X t − 1 τ ≥ T − 1 / 2 λ } X t − 1 τ D ( L ) ( Δ X t τ ) ′ δ T η τ ⇒ D ( 1 ) η τ ′ ∫ 0 1 𝟙 { η τ ′ X τ ≥ 0 } X τ d X τ ′ η τ = D ( 1 ) l 11 2 ∫ 0 1 𝟙 { Q κ τ ≥ 0 } Q κ τ d Q κ τ ,

while (28), (34) and the CMT yield

(36) D T ( λ , τ ) = T − 2 ∑ t = 1 T 𝟙 { e ^ t − 1 τ ≥ λ } e ^ t − 1 τ 2 = η ^ τ ′ δ T T − 1 ∑ t = 1 T 𝟙 { δ T η ^ τ ′ X t − 1 τ ≥ T − 1 / 2 λ } X t − 1 τ X t − 1 τ ′ δ T η ^ τ ⇒ η τ ′ ∫ 0 1 𝟙 { η τ ′ X τ ≥ 0 } X τ X τ ′ η τ = l 11 2 ∫ 0 1 𝟙 { Q κ τ ≥ 0 } Q κ τ 2 .

For the variance estimate, σ^2, we note that ρ^1=Op(T−1) and ρ^2=Op(T−1), but (γ^j−γj)=Op(T−1/2). Using Lemma 2.2 of Phillips and Ouliaris (1990) yields

(37) σ ^ 2 = T − 1 ∑ t = 1 T ( Δ e ^ t τ − ρ ^ 1 e ^ t − 1 τ 𝟙 { e ^ t − 1 τ ≥ λ } − ρ ^ 2 e ^ t − 1 τ 𝟙 { e ^ t − 1 τ < λ } − ∑ j = 1 K γ ^ j Δ e ^ t − j τ ) 2 = T − 1 ∑ t = 1 T ϵ t τ 2 + o p ( 1 ) ⇒ D ( 1 ) 2 η τ ′ Ω τ η τ = D ( 1 ) 2 l 11 2 κ τ ′ D τ κ τ ,

where the long-run covariance matrix is given by

(38) Ω τ = [ ω 11 ω 21 ′ 0 ( 1 − τ ) ω 21 ′ 0 ω 21 Ω 22 0 ( 1 − τ ) Ω 22 0 0 0 0 0 0 ( 1 − τ ) ω 21 ( 1 − τ ) Ω 22 0 ( 1 − τ ) Ω 22 0 0 0 0 0 0 ]

and

(39) D τ = [ 1 0 0 0 0 0 I m 0 ( 1 − τ ) I m 0 0 0 0 0 0 0 ( 1 − τ ) I m 0 ( 1 − τ ) I m 0 0 0 0 0 0 ] .

Similar results can be obtained for t₂ so that the results (35), (36), (37) combine with the CMT to proof the theorem under the null hypothesis.

Under the alternative, the system is cointegrated so that we have η^τ→pητ and

(40) η ^ τ = η τ + O p ( T − 1 )

from Phillips and Durlauf (1986), Theorem 4.1. Thus, for the residual series it holds that

(41) e ^ t τ = η ^ τ ′ z t = η τ ′ z t + O p ( T − 1 / 2 ) = q t η τ + O p ( T − 1 / 2 ) .

By assumption a stationary SETAR representation of qtητ exists and is given by

(42) q t η τ = a 11 q t − 1 η τ 𝟙 { q t − 1 η τ ≥ λ } + a 12 q t − 1 η τ 𝟙 { q t − 1 η τ < λ } + ∑ j = 2 ∞ a j q t − j η τ + ϵ t η τ ∗ ,

where εtητ∗ is an orthogonal (0,σεητ∗) sequence. This can alternatively be written as

(43) Δ q t η τ = ψ 11 q t − 1 η τ 𝟙 { q t − 1 η τ ≥ λ } + ψ 12 q t − 1 η τ 𝟙 { q t − 1 η τ < λ } + ∑ j = 2 ∞ ψ j Δ q t − j η τ + ϵ t η τ ∗ .

If we consider the t ratio of ρ^1 and use the expression

(44) t 1 = 1 σ ^ ( ρ ^ 1 ( U 1 τ ′ Q K U 1 τ ) 1 / 2 ) ,

we find that ρ^1→pψ11≠0 and σ^2→pσεητ∗2. Further, we observe that

(45) U 1 τ ′ Q K U 1 τ = U 1 τ ′ U 1 τ − U 1 τ ′ M K ( M K ′ M K ) − 1 M K ′ U 1 τ = O p ( T )

which yields t1=Op(T1/2) and similarly t2=Op(T1/2). Hence, we immediately see that FSETAR∗→∞ as T → ∞. □

Proof of Theorem 2.

The proof is structured similarly to the proof of Theorem 1. Using the results for the cointegrating regression, we write the AR representation of the MTAR error term process as

(46) ε t τ = ∑ j = 0 ∞ a j ( T − 1 / 2 δ T Δ X t − j τ ) ′ η τ = a ( L ) ( T − 1 / 2 δ T Δ X t − j τ ) ′ η τ

and have εtτ as an orthogonal (0,σ2(η,τ)) sequence with σ2(η,τ)=a(1)2ητ′Ωτητ. From Lemma 2.1 of Phillips and Ouliaris (1990), it follows that

(47) T − 1 / 2 ∑ t = 1 [ T s ] ε t τ K = a ( L ) η τ ′ ( T − 1 / 2 ∑ t = 1 [ T s ] T 1 / 2 δ T Δ X t τ ) + o p ( 1 ) ⇒ a ( 1 ) η τ ′ X τ ,

where a(1)=∑j=0∞aj. Now, we apply the MTAR model to the residuals according to (5) and compute the test statistics F_τ. The t ratio of ρ^1 is written as

(48) t 1 = U 1 τ ′ Q K ε τ σ ^ ( U 1 τ ′ Q K U 1 τ ) 1 / 2

and the t ratio of ρ^2 is written as

(49) t 2 = U 2 τ ′ Q K ε τ σ ^ ( U 2 τ ′ Q K U 2 τ ) 1 / 2 ,

where

(50) U τ = ( U 1 τ , U 2 τ ) = [ e ^ 0 τ 𝟙 { Δ e ^ 0 τ ≥ λ ∗ } e ^ 0 τ 𝟙 { Δ e ^ 0 τ < λ ∗ } e ^ 1 τ 𝟙 { Δ e ^ 1 τ ≥ λ ∗ } e ^ 1 τ 𝟙 { Δ e ^ 1 τ < λ ∗ } ⋮ ⋮ e ^ T − 1 τ 𝟙 { Δ e ^ T − 1 τ ≥ λ ∗ } e ^ T − 1 τ 𝟙 { Δ e ^ T − 1 τ < λ ∗ } ] .

Finally, we need convergence results for NT(λ∗,τ), DT(λ∗,τ) and σ^2. The main difference between the asymptotic distribution for the SETAR and the MTAR models lies in the fact that the indicator variable Δe^tτ has a stationary distribution under the null hypothesis and the alternative. Further, the MTAR decomposition of e^t−1τ is not regular and Theorem 3.1 of Park and Phillips (2001) does not apply. However, from Theorem 1 in Caner and Hansen (2001) it follows that

(51) T − 1 / 2 ∑ t = 1 [ T s ] 𝟙 { Δ e ^ t − 1 τ ≥ λ ∗ } ϵ t τ = T − 1 / 2 ∑ t = 1 [ T s ] 𝟙 { G ( Δ e ^ t − 1 τ ) ≥ G ( λ ∗ ) } ϵ t τ = T − 1 / 2 ∑ t = 1 [ T s ] 𝟙 { U t ≥ G ( λ ∗ ) } ϵ t τ ⇒ Q κ τ ( s , u ) = σ ( η , τ ) W ( s , u ) = a ( 1 ) l 11 ( κ τ ′ D τ κ τ ) 1 / 2 W ( s , u ) ,

where G(⋅) is the marginal distribution of Δe^t−1τ so that G(Δe^t−1τ)=Ut∼U[0,1] and G(λ∗)=u. The standard two-parameter Brownian motion W(s, u) is defined on (s,u)∈[0,1]2. Using Theorem 2.2 of Kurtz and Protter (1991) and (51) yields

(52) N T ( λ ∗ , τ ) = T − 1 ∑ t = 1 T 𝟙 { Δ e ^ t − 1 τ ≥ λ ∗ } e ^ t − 1 τ ϵ t τ = η ^ τ ′ δ T ∑ t = 1 T 𝟙 { G ( Δ e ^ t − 1 τ ) ≥ G ( λ ∗ ) } X t − 1 τ ϵ t τ ⇒ a ( 1 ) l 11 ( κ τ ′ D τ κ τ ) 1 / 2 η τ ′ ∫ 0 1 X τ ( s ) d W ( s , u ) = a ( 1 ) l 11 2 ( κ τ ′ D τ κ τ ) 1 / 2 ∫ 0 1 Q κ τ ( s ) d W ( s , u )

and Theorem 3 of Caner and Hansen (2001) yields

(53) D T ( λ ∗ , τ ) = T − 2 ∑ t = 1 T 𝟙 { Δ e ^ t − 1 τ ≥ λ ∗ } e ^ t − 1 τ 2 = η ^ τ ′ δ T T − 1 ∑ t = 1 T 𝟙 { G ( Δ e ^ t − 1 τ ) ≥ G ( λ ∗ ) } X t − 1 τ X t − 1 τ ′ δ T η ^ τ ⇒ u η τ ′ ∫ 0 1 X τ ( s ) X τ ′ ( s ) d s η τ = u l 11 2 ∫ 0 1 Q κ τ 2 ( s ) d s .

For the variance estimate, σ^2, Lemma 2.2 of Phillips and Ouliaris (1990) yields

(54) σ^2=T−1∑t=1Tεtτ2+op(1)⇒a(1)2l112κτ′Dτκτ.

The results (52), (53), (54) combine with the CMT to proof

(55) t 1 ⇒ ∫ 0 1 Q κ τ ( s ) d W ( s , u ) ( u ∫ 0 1 Q κ τ 2 ( s ) d s ) 1 / 2 .

Analogously, we can show that

(56) t 2 ⇒ ∫ 0 1 Q κ τ ( s ) ( d W ( s , 1 ) − d W ( s , u ) ) ( ( 1 − u ) ∫ 0 1 Q κ τ 2 ( s ) d s ) 1 / 2

holds. Finally, we observe that taking the supremum over all τ∈T is a continuous transformation so that we can use the CMT to proof the theorem under the null hypothesis. The proof of the theorem under the alternative is a straightforward adaptation of the results given in the proof of Theorem 1. □

References

Arai, Y., and E. Kurozumi. 2007. “Testing for the Null Hypothesis of Cointegration with a Structural Break.” Econometric Reviews 26: 705–739.10.1080/07474930701653776Suche in Google Scholar

Aue, A., and L. Horváth. 2013. “Structural Breaks in Time Series.” Journal of Time Series Analysis 34: 1–16.10.1111/j.1467-9892.2012.00819.xSuche in Google Scholar

Bachmeier, L. J., and J. M. Griffin. 2002. “New Evidence on Asymmetric Gasoline Price Responses.” Review of Economics and Statistics 85: 772–776.10.1162/003465303322369902Suche in Google Scholar

Bacon, R. W. 1991. “Rockets and Feathers: The Asymmetric Speed of Adjustment of UK Retail Gasoline Prices to Cost Changes.” Energy Economics 13: 211–218.10.1016/0140-9883(91)90022-RSuche in Google Scholar

Banerjee, A., J. J. Dolado, D. F. Hendry, and G. W. Smith. 1986. “Exploring Equilibrium Relationships in Econometrics through Static Models: Some Monte Carlo Evidence.” Oxford Bulletin of Economics and Statistics 48: 253–278.10.1111/j.1468-0084.1986.mp48003005.xSuche in Google Scholar

Billingsley, P. 1999. Convergence of Probability Measures. 2nd ed. New York: Wiley.10.1002/9780470316962Suche in Google Scholar

Borenstein, S., A. C. Cameron, and R. Gilbert. 1997. “Do Gasoline Prices Respond Asymmetrically to Crude Oil Price Changes.” Quarterly Journal of Economics 112: 305–339.10.3386/w4138Suche in Google Scholar

Caner, M., and B. E. Hansen. 2001. “Threshold Autoregression with a Unit Root.” Econometrica 69: 1555–1596.10.1111/1468-0262.00257Suche in Google Scholar

Carrion-i Silvestre, J. L., and A. Sanso. 2006. “Testing for the Null Hypothesis of Cointegration with Structural Breaks.” Oxford Bulletin of Economics and Statistics 68: 623–646.10.1111/j.1468-0084.2006.00180.xSuche in Google Scholar

Chang, Y., and J. Y. Park. 2002. “On the Asymptotics of ADF Tests for Unit Roots.” Econometric Reviews 21: 431–447.10.1081/ETC-120015385Suche in Google Scholar

Davidson, J., and A. Monticini. 2010. “Tests for Cointegration with Structural Breaks Based on Subsamples.” Computational Statistics and Data Analysis 54: 2498–2511.10.1016/j.csda.2010.01.028Suche in Google Scholar

Enders, W., and C. W. J. Granger. 1998. “Unit-Root Tests and Asymmetric Adjustment with an Example using the Term Structure of Interest Rates.” Journal of Business & Economic Statistics 16: 304–311.10.1080/07350015.1998.10524769Suche in Google Scholar

Enders, W., and P. L. Siklos. 2001. “Cointegration and Threshold Adjustment.” Journal of Business & Economic Statistics 19: 166–176.10.1198/073500101316970395Suche in Google Scholar

Engle, R. F., and C. W. J. Granger. 1987. “Co-Integration and Error Correction: Representation, Estimation and Testing.” Econometrica 55: 251–276.10.2307/1913236Suche in Google Scholar

Frey, G., and M. Manera. 2007. “Econometric Models of Asymmetric Price Transmission.” Journal of Economic Surveys 21: 349–415.10.1111/j.1467-6419.2007.00507.xSuche in Google Scholar

Gregory, A. W., and B. E. Hansen. 1992. “Residual-based Tests for Cointegration in Models with Regime Shifts.” Queen’s Economics Department Working Paper 862: 1–32.10.1016/0304-4076(69)41685-7Suche in Google Scholar

Gregory, A. W., and B. E. Hansen. 1996a. “Residual-based Tests for Cointegration in Models with Regime Shifts.” Journal of Econometrics 70: 99–126.10.1016/0304-4076(69)41685-7Suche in Google Scholar

Gregory, A. W., and B. E. Hansen. 1996b. “Tests for Cointegration in Models with Regime and Trend Shifts.” Oxford Bulletin of Economics and Statistics 58: 555–560.10.1111/j.1468-0084.1996.mp58003008.xSuche in Google Scholar

Gregory, A. W., J. M. Nason, and D. G. Watt. 1996. “Testing for Structural Breaks in Cointegrated Relationships.” Journal of Econometrics 71: 321–341.10.1016/0304-4076(96)84508-8Suche in Google Scholar

Harris, D., S. J. Leybourne, and A. M. R. Taylor. 2016. “Tests of the Co-integration Rank in VAR Models in the Presence of a Possible Break in Trend at an Unknown Point.” Journal of Econometrics 192: 451–467.10.1016/j.jeconom.2016.02.010Suche in Google Scholar

Hatemi-J, A. 2008. “Tests for Cointegration with Two Unknown Regime Shifts with an Application to Financial Market Integration.” Empirical Economics 35: 497–505.10.1007/s00181-007-0175-9Suche in Google Scholar

Kejriwal, M., and P. Perron. 2010. “Testing for Multiple Structural Changes in Cointegrated Regression Models.” Journal of Business & Economic Statistics 28: 503–522.10.1198/jbes.2009.07220Suche in Google Scholar

Kilian, L. 2016. “The Impact of the Shale Oil Revolution on U.S. Oil and Gasoline Prices.” Review of Environmental Economics and Policy 10: 185–205.10.1093/reep/rew001Suche in Google Scholar

Kurtz, T., and P. Protter. 1991. “Weak Limit Theorem for Stochastic Integrals and Stochastic Differential Equations.” The Annals of Probability 19: 1035–1070.10.1214/aop/1176990334Suche in Google Scholar

Lee, J., and M. C. Strazicich. 2003. “Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks.” The Review of Economics and Statistics 85: 1082–1089.10.1162/003465303772815961Suche in Google Scholar

Lee, O., and D. W. Shin. 2000. “On Geometric Ergodicity of the MTAR Process.” Statistics & Probability Letters 48: 229–237.10.1016/S0167-7152(99)00208-4Suche in Google Scholar

Lumsdaine, R. L., and D. H. Papell. 1997. “Multiple Trend Breaks and the Unit-Root Hypothesis.” The Review of Economics and Statistics 79: 212–218.10.1162/003465397556791Suche in Google Scholar

Maki, D. 2012a. “Detecting Cointegration Relationships Under Nonlinear Models: Monte Carlo Analysis and Some Applications.” Empirical Economics 45: 605–625.10.1007/s00181-012-0605-1Suche in Google Scholar

Maki, D. 2012b. “Tests for Cointegration Allowing for an Unknown Number of Breaks.” Economic Modelling 29: 2011–2015.10.1016/j.econmod.2012.04.022Suche in Google Scholar

Maki, D., and S.-i. Kitasaka. 2015. “Residual-based Tests for Cointegration with Three-regime TAR Adjustment.” Empirical Economics 48: 1013–1054.10.1007/s00181-014-0822-xSuche in Google Scholar

Manning, D. N. 1991. “Petrol Prices, Oil Price Rises and Oil Price Falls: Some Evidence for the UK Since 1972.” Applied Economics 23: 1535–1541.10.1080/00036849100000206Suche in Google Scholar

Meyer, J., and S. Cramon-Taubadel. 2004. “Asymmetric Price Transmission: A Survey.” Journal of Agricultural Economics 55: 581–611.10.1111/j.1477-9552.2004.tb00116.xSuche in Google Scholar

Park, J. Y., and P. C. B. Phillips. 2001. “Nonlinear Regression with Integrated Processes.” Econometrica 69: 117–161.10.1111/1468-0262.00180Suche in Google Scholar

Perdiguero, J. 2013. “Symmetric or Asymmetric Oil Prices? A Meta-analysis Approach.” Energy Policy 57: 389–397.10.1016/j.enpol.2013.02.008Suche in Google Scholar

Perron, P. 1989. “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis.” Econometrica 57: 1361–1401.10.2307/1913712Suche in Google Scholar

Perron, P. 2006. “Dealing with Structural Breaks.” In Palgrave Handbook of Econometrics - Volume 1: Econometric Theory, edited by H. Hassani, T. Mills, and K. Patterson, 278–352. UK: Palgrave Macmillan.Suche in Google Scholar

Petruccelli, J. D., and S. W. Woolford. 1984. “A Threshold AR(1) Model.” Journal of Applied Probability 21: 270–286.10.2307/3213639Suche in Google Scholar

Phillips, P. C. B. 1995. “Fully Modified Least Squares and Vector Autoregression.” Econometrica 63: 1023–1078.10.2307/2171721Suche in Google Scholar

Phillips, P. C. B., and S. N. Durlauf. 1986. “Multiple Time Series Regression with Integrated Processes.” Review of Economic Studies 53: 473–495.10.2307/2297602Suche in Google Scholar

Phillips, P. C. B., and S. Ouliaris. 1990. “Asymptotic Properties of Residual Based Tests for Cointegration.” Econometrica 58: 165–193.10.2307/2938339Suche in Google Scholar

Phillips, P. C. B., and V. Solo. 1992. “Asymptotics for Linear Processes.” The Annals of Statistics 20: 971–1001.10.1214/aos/1176348666Suche in Google Scholar

Pippenger, M. K., and G. E. Goering. 2000. “Additional Results on the Power of Unit Root and Cointegration Tests under Threshold Processes.” Applied Economics Letters 7: 641–644.10.1080/135048500415932Suche in Google Scholar

Said, S. E., and D. A. Dickey. 1984. “Testing for Unit Roots in Autoregressive-moving Average Models of Unknown Order.” Biometrika 71: 599–607.10.1093/biomet/71.3.599Suche in Google Scholar

Seo, M. H. 2008. “Unit Root Test in a Threshold Autoregression: Asymptotic Theory and Residual-Based Block Bootstrap.” Econometric Theory 24: 1699–1716.10.1017/S0266466608080663Suche in Google Scholar

Tong, H. 1983. Threshold Models in Non-linear Time Series Analysis. New York: Springer.10.1007/978-1-4684-7888-4Suche in Google Scholar

Tong, H. 1990. Non-linear Time Series: A Dynamical System Approach. Oxford: Oxford University Press.10.1093/oso/9780198522249.001.0001Suche in Google Scholar

Westerlund, J., and D. L. Edgerton. 2007. “New Improved Tests for Cointegration with Structural Breaks.” Journal of Time Series Analysis 28: 188–224.10.1111/j.1467-9892.2006.00504.xSuche in Google Scholar

Zivot, E., and D. W. K. Andrews. 1992. “Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis.” Journal of Business & Economic Statistics 10: 251–270.10.1080/07350015.1992.10509904Suche in Google Scholar

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Schlagwörter für diesen Artikel

Cointegration; MTAR; SETAR; structural change; threshold autoregression

Testing for cointegration with threshold adjustment in the presence of structural breaks

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