Time-varying threshold cointegration with an application to the Fisher hypothesis

Lixiong Yang

doi:10.1515/snde-2018-0101

Article

Time-varying threshold cointegration with an application to the Fisher hypothesis

Lixiong Yang

Published/Copyright: February 22, 2021

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 26 Issue 2

Abstract

This paper extends the threshold cointegration model developed by Gonzalo, J., and J. Y. Pitarakis. 2006. “Threshold Effects in Cointegrating Relationships.” Oxford Bulletin of Economics & Statistics 68: 813–33 and Chen, H. 2015. “Robust Estimation and Inference for Threshold Models with Integrated Regressors.” Econometric Theory 31 (4): 778–810 to allow for a time-varying threshold, which is a function of candidate variables that affect the separation of regimes. We derive the asymptotic distribution of the proposed least-square estimator of the threshold, and study the convergence rate of the threshold estimator. We also suggest test statistics for threshold effect and threshold constancy. Monte Carlo simulations point out that the convergence rate of the threshold estimator is consistent with the asymptotic theory, and the proposed tests have good size and power properties. The empirical usefulness of the proposed model is illustrated by an application to the US data to investigate the Fisher hypothesis.

Keywords: estimation; Fisher hypothesis; testing; threshold cointegration; time-varying threshold

JEL classification: C12; C13; C22; C51

Corresponding author: Lixiong Yang, School of Management, Lanzhou University, 222 South Tianshui Road, Lanzhou 730000, China. Phone: +86 13669327501, E-mail: ylx@lzu.edu.cn

Funding source: National Natural Science Foundation of ChinaNational Natural Science Foundation of China

Award Identifier / Grant number: 71803072

Award Identifier / Grant number: Unassigned

Acknowledgements

The author would like to thank the editor and anonymous referees for very valuable comments and suggestions that result in a substantial improvement in this paper. Remaining errors and omissions are my own. The author acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 71803072).

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This study was supported by the National Natural Science Foundation of China (Grant No. 71803072).
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

This appendix provides the Proof of Theorem 1 in the paper. We first collect some lemmas that are useful in proving Theorem 1. Define 1 t ( u ) = 1 U t ≤ u , in which U _t has a marginal U[0, 1] distribution, the partial-sum process

W t ( u ) = ∑ s = 1 t 1 s − 1 ( u ) e t ,

W T ( s , u ) = 1 σ e T W [ T s ] ( u ) = 1 σ e T ∑ t = 1 [ T s ] 1 t − 1 ( u ) e t ,

where σ e 2 = E ( e t 2 ) < ∞ .

Definition 1

W(s, u) is a two-parameter Brownian motion on (s, u) ∈ [0,1]² if W(s, u) ∼ N(0, su) and

E W ( s 1 , u 1 ) W ( s 2 , u 2 ) ′ = ( s 1 ∧ s 2 ) ( u 1 ∧ u 2 ) .

Lemma 1

Under Assumptions 1–3, W _T(s, u) ⇒ W(s, u) on (s, u) ∈ [0,1]² as T → ∞.

Proof

This result follows Theorem 1 of Caner and Hansen (2001). □

Lemma 2

If the joint probability density function of (q _t, z _t) is f(q, z), then q t * ≡ q t − γ 1 ′ z t has a cumulative distribution function F γ 1 ( q * ) = ∫ − ∞ q * f γ 1 ( u ) d u , where the probability density function f γ 1 ( q * ) = ∫ · · · ∫ − ∞ + ∞ f ( q * + γ 1 ′ z , z ) d z 1 … d z k , in which z = (z ₁, …, z _k)′.

Proof

(A.1) F γ 1 ( q * ) = Pr ( Q * ≤ q * ) = ∫ ∫ · · · ∫ q − γ 1 ′ z ≤ q * f ( q , z ) d q d z 1 … d z k = ∫ · · · ∫ − ∞ + ∞ ∫ − ∞ q * + γ 1 ′ z f ( q , z ) d q d z 1 … d z k = ∫ · · · ∫ − ∞ + ∞ ∫ − ∞ q * f ( u + γ 1 ′ z , z ) d u d z 1 … d z k = ∫ − ∞ q * ∫ · · · ∫ − ∞ + ∞ f ( u + γ 1 ′ z , z ) d z 1 … d z k d u ≡ ∫ − ∞ q * f γ 1 ( u ) d u .

Lemma 3

Under Assumptions 1–4, for any γ ∈ Γ, we have 1 T ∑ t = 1 [ T s ] 1 ( q t ≤ γ t ) e t ⇒ σ e 2 W ( s , F γ 1 ( γ 0 ) ) .

Proof

This result follows Lemma 1 by replacing u with F γ 1 ( γ 0 ) . □

Lemma 4

Under Assumptions 1–4, for γ ∈ Γ, we have

1 T ∑ t = 1 [ T s ] x t 1 ( q t ≤ γ t ) e t ⇒ σ e ∫ B v ( s ) d W ( s , F γ 1 ( γ 0 ) ) ,

in which 1 T ∑ t = 1 [ T s ] v t ⇒ B v ( s ) is a Brownian motion with a positive definite long-run covariance matrix.

Proof

The results follows Theorem 2 of Caner and Hansen (2001), Lemma A.2 Chen (2015). □

Lemma 5

Under Assumptions 1–4, for any γ ∈ Γ, as T → ∞, define x t ( γ t ) = [ x ′ t , x ′ t 1 ( q t ≤ γ t ) ] ′ , we have

1 T 2 ∑ t = 1 T x t ( γ t ) x ′ t ( γ t ) = 1 T 2 x ′ t x t x ′ t x t 1 ( q t ≤ γ t ) x ′ t 1 ( q t ≤ γ t ) x t x ′ t 1 ( q t ≤ γ t ) x t 1 ( q t ≤ γ t ) ⇒ ∫ B v ( s ) B ′ v ( s ) d s F γ 1 ( γ 0 ) ∫ B v ( s ) B ′ v ( s ) d s F γ 1 ( γ 0 ) ∫ B v ( s ) B ′ v ( s ) d s F γ 1 ( γ 0 ) ∫ B v ( s ) B ′ v ( s ) d s ≜ M ( γ 0 , γ 1 ) ,

1 T ∑ t = 1 T x t ( γ t ) e t = 1 T x ′ t e t x ′ t 1 ( q t ≤ γ t ) e t ⇒ σ e ∫ B v ( s ) d W ( s ) ∫ B v ( s ) d W ( s , F γ 1 ( γ 0 ) ) .

Proof

The result follows Lemma 3 of Chen (2015). □

Lemma 6

Under Assumptions 1–5, β = [ β ′₂, β ′₁ − β ′₂]′ = [ β ′₂, δ ′]′, in which δ = δ _T = δ ₀ T ^−1/2−α. When γ t = γ t 0 , we have T β ̂ − β = O ( 1 ) . When γ t ≠ γ t 0 , we have T α + 1 / 2 β ̂ − β = O ( 1 ) .

Proof

When γ t = γ t 0 ,

(A.2) T β ̂ − β = 1 T 2 ∑ t = 1 T x t ( γ t 0 ) x ′ t ( γ t 0 ) 1 T ∑ t = 1 T x t ( γ t 0 ) e t ⇒ M ( γ 0 0 , γ 1 0 ) σ e ∫ B v ( s ) d W ( s ) ∫ B v ( s ) d W ( s , F γ 1 ( γ 0 ) ) .

When γ t ≠ γ t 0 and − 1 2 < α < 1 2 , we note that

(A.3) y t = β 2 ′ x t + δ ′ x t 1 ( γ t 0 ) + e t = β ′ x t ( γ t 0 ) + e t = β ′ x t ( γ t ) + e t − δ ′ ( x t 1 ( γ t ) − x t 1 ( γ t 0 ) ) ,

hence we have

(A.4) T α + 1 / 2 β ̂ − β = T α + 1 / 2 ∑ t = 1 T x t ( γ t ) x ′ t ( γ t ) − 1 ∑ t = 1 T x t ( γ t ) y t = T α + 1 / 2 ∑ t = 1 T x t ( γ t ) x ′ t ( γ t ) − 1 ∑ t = 1 T x t ( γ t ) e t − T α + 1 / 2 ∑ t = 1 T x t ( γ t ) x ′ t ( γ t ) − 1 ∑ t = 1 T x t ( γ t ) ( x ′ t 1 ( γ t ) − x ′ t 1 ( γ t 0 ) ) δ = T α − 1 / 2 O p ( 1 ) − M ( γ t ) − 1 1 T 2 ∑ t = 1 T x t ( γ t ) ( x ′ t 1 ( γ t ) − x ′ t 1 ( γ t 0 ) ) δ 0 + o ( 1 ) ⇒ − M ( γ 0 , γ 1 ) − 1 F γ 1 ( γ 0 ) − F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s F γ 1 ( γ 0 ) − F γ 1 ( γ 0 ) ∧ F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s δ 0 .

□

Lemma 7

Under Assumptions 1–5, ( γ ̂ 0 , γ ̂ 1 ) → ( γ 0 0 , γ 1 0 ) .

Proof

For notational simplicity, we rewrite the model y _t = β ′₂ x _t + δ ′x _t 1(γ _t) + e _t in matrix form Y = β ′₂ X + δ ′X(γ _t) + e, in which X ( γ t ) = X 1 ( γ t ) .

Define X*(γ _t) = [X(γ _t), X − X(γ _t)], and P γ t * = X * ( γ t ) [ X * ( γ t ) ′ X * ( γ t ) ] − 1 X * ( γ t ) ′ . Then Y and X lies in the space spanned by P γ t * , and we have

(A.5) SS R T ( γ t ) = Y ′ ( I − P γ t * ) Y = δ ′ X ( γ t 0 ) ′ ( I − P γ t * ) X ( γ t 0 ) δ + 2 δ ′ X ( γ t 0 ) ′ ( I − P γ t * ) e + e ′ ( I − P γ t * ) e .

When γ t = γ t 0 , we have SS R T ( γ t 0 ) = e ′ ( I − P γ t 0 * ) e . Using Lemmas 3 and 4, we have

(A.6) T 2 α − 1 SS R T ( γ t ) − SS R T ( γ t 0 ) = 1 T δ 0 ′ X ( γ t 0 ) ′ ( I − P γ t * ) X ( γ t 0 ) δ 0 + o p ( 1 ) .

Using a similar argument of Theorem 1 in Chen (2015), we can show, for any γ t ≥ γ t 0 ,

(A.7) 1 T δ 0 ′ X ( γ t 0 ) ′ ( I − P γ t * ) X ( γ t 0 ) δ 0 → p F γ 1 0 ( γ 0 0 ) − F γ 1 0 ( γ 0 0 ) F γ 1 ( γ 0 ) − 1 F γ 1 0 ( γ 0 0 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 ≜ b 1 ( γ 0 , γ 1 ) .

Since F γ 1 0 ( γ 0 0 ) − F γ 1 0 ( γ 0 0 ) F γ 1 ( γ 0 ) − 1 F γ 1 0 ( γ 0 0 ) > 0 , and ∫ B v ( s ) B ′ v ( s ) d s is a positive definite matrix, hence b ₁(γ ₀, γ ₁) ≥ 0 and the equality holds if and only if ( γ 0 , γ 1 ) = ( γ 0 0 , γ 1 0 ) (i.e., γ t = γ t 0 ).

Similarly, for γ t ≤ γ t 0 we have

(A.8) 1 T δ 0 ′ X ( γ t 0 ) ′ ( I − P γ t * ) X ( γ t 0 ) δ 0 → p F γ 1 0 ( γ 0 0 ) − F γ 1 ( γ 0 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 ≜ b 2 ( γ 0 , γ 1 ) ,

in which b ₂(γ ₀, γ ₁) ≥ 0 and the equality holds if and only if γ t = γ t 0 since F γ 1 0 ( γ 0 0 ) − F γ 1 ( γ 0 ) ≥ 0 .

Define b ( γ 0 , γ 1 ) = b 1 ( γ 0 , γ 1 ) 1 ( γ t ≥ γ t 0 ) + b 2 ( γ 0 , γ 1 ) 1 ( γ t ≤ γ t 0 ) . Combining the above results we have T 2 α − 1 SS R T ( γ t ) − SS R T ( γ t 0 ) → p b ( γ 0 , γ 1 ) . Also, noting that γ ̂ t = arg min γ 0 , γ 1 ∈ Γ SS R T ( γ t ) , we have γ ̂ t = γ ̂ 0 + γ ̂ 1 ′ z t minimizes SSR_T(γ _t), and hence γ ̂ 0 + γ ̂ 1 ′ z t → p γ 0 0 + γ 1 0 ′ z t . For any z _t, define λ = ( 1 ‖ z t ‖ 2 + 1 , z t ‖ z t ‖ 2 + 1 ) ′ . Clearly we have λ ′ λ = 1, λ ′ ( γ ̂ 0 , γ ̂ 1 ) ′ → p λ ′ ( γ ̂ 0 0 , γ ̂ 1 0 ) ′ . By the Cramér–Wold device theorem, we have ( γ ̂ 0 , γ ̂ 1 ) → p ( γ 0 0 , γ 1 0 ) . □

Proof of Theorem 1

We first prove the following result a T ( γ ̂ 0 , γ ̂ ′ 1 ) ′ − γ 0 0 , γ 1 0 ′ ′ = O p ( 1 ) , in which a _T = T ^1−2α.

To prove this, we need to prove that, for any v ̄ > 0 , we have

(A.9) lim T → ∞ Pr ‖ ( γ ̂ 0 , γ ̂ ′ 1 ) ′ − γ 0 0 , γ 1 0 ′ ′ ‖ ≤ v ̄ / a T = 1 .

For any B > 0, define V B = { γ 0 , γ 1 ′ : ‖ ( γ 0 , γ 1 ′ ) − ( γ 0 0 , γ 1 0 ′ ) ‖ < B } . When the sample size T is large enough, we have v ̄ / a T < B . Since ( γ ̂ 0 , γ ̂ 1 ′ ) → p ( γ 0 0 , γ 1 0 ′ ) as proved before, we have lim T → ∞ Pr ( γ ̂ 0 , γ ̂ ′ 1 ) ∈ V B → p 1 . Therefore, we only need to consider the limiting behavior of ( γ ̂ 0 , γ ̂ 1 ′ ) in V _B. Define a subset of V _B: V B ( v ̄ ) = { γ 0 , γ 1 ′ : v ̄ / a T < ‖ ( γ 0 , γ 1 ′ ) − ( γ 0 0 , γ 1 0 ′ ) ‖ < B } . Thus, to prove Pr ‖ ( γ ̂ 0 , γ ̂ ′ 1 ) − γ 0 , γ 1 ′ ‖ ≤ v ̄ / a T = 1 , we just need to prove Pr ( γ ̂ 0 , γ ̂ ′ 1 ) ∈ V B ( ν ̄ ) = 0 .

Let S T * ( γ t ) = SSR T ( β ̂ , δ ̂ , γ t ) and S T * ( γ t 0 ) = SSR T ( β ̂ , δ ̂ , γ t 0 ) , where SSR_T(.) is the sum of squared errors function (6) defined in the paper. From the estimation of ( γ ̂ 0 , γ ̂ 1 ′ ) , we have S T * ( γ ̂ t ) ≤ S T * ( γ t 0 ) . Thus, to prove Pr ( γ ̂ 0 , γ ̂ ′ 1 ) ∈ V B ( ν ̄ ) = 0 , it suffices to prove that for any ( γ 0 , γ 1 ′ ) ∈ V B ( ν ̄ ) , we have

(A.10) lim T → ∞ Pr S T ( β ̂ , δ ̂ , γ t ) > S T ( β ̂ , δ ̂ , γ t 0 ) = 1 .

To this end, we first consider the case of γ t ≥ γ t 0 . In this case, it is equivalent to prove

(A.11) S T * ( γ t ) − S T * ( γ t 0 ) a T ( γ t − γ t 0 ) > 0 .

Since Y = X β + X ( γ t 0 ) δ + e , we have

(A.12) S T * ( γ t ) − S T * ( γ t 0 ) = Y − X β − X ( γ t ) δ ′ Y − X β − X ( γ t ) δ − Y − X β − X ( γ t 0 ) δ ′ Y − X β − X ( γ t 0 ) δ = δ ̂ Δ X γ ′ Δ X γ δ ̂ − 2 δ ̂ Δ X γ ′ e + 2 δ ̂ Δ X γ ′ Δ X γ ( β ̂ − β ) = δ n Δ X γ ′ Δ X γ δ n − 2 δ ̂ Δ X γ ′ e + 2 δ ̂ Δ X γ ′ Δ X γ ( β ̂ − β ) + ( δ n + δ ̂ ) Δ X γ ′ Δ X γ ( δ ̂ − δ n ) = S 1 − S 2 + S 3 + S 4 ,

in which Δ X γ = X ( γ t ) − X ( γ t 0 ) . We next prove S 1 − S 2 + S 3 + S 4 a T ( γ t − γ t 0 ) converges to a positive random variable with probability one.

For the first term, we have

(A.13) S 1 a T = δ 0 ′ T − α Δ X γ ′ Δ X γ δ 0 T − α a T = δ 0 ′ X ( γ t 0 ) − X ( γ t ) ′ X ( γ t 0 ) − X ( γ t ) δ 0 T ⇒ δ 0 ′ F γ 1 ( γ 0 ) − F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s δ 0 > 0 .

And the second term

(A.14) S 2 a T ( γ t − γ t 0 ) = 2 δ ̂ 0 T − α Δ X γ ′ e a T ( γ t − γ t 0 ) = 2 δ ̂ 0 T − 1 / 2 − α X ( γ t 0 ) − X ( γ t ) ′ e T 1 − 2 α ( γ t − γ t 0 ) = 2 δ ̂ 0 X ( γ t 0 ) − X ( γ t ) ′ e T T 1 / 2 − α ( γ t − γ t 0 ) = O p 1 a n | γ t − γ t 0 | = o p ( 1 ) .

By Lemma 5, T α + 1 / 2 δ ̂ = O ( 1 ) and T α + 1 / 2 ( β ̂ − β ) = O p ( 1 ) . Hence, for the third and fourth terms we have

(A.15) S 3 a T ( γ t − γ t 0 ) = 2 δ ̂ Δ X γ ′ Δ X γ ( β ̂ − β ) a T ( γ t − γ t 0 ) = 2 T α + 1 / 2 δ ̂ Δ X γ ′ Δ X γ T α + 1 / 2 ( β ̂ − β ) T 2 ( γ t − γ t 0 ) = o p ( 1 ) ,

(A.16) S 4 a T ( γ t − γ t 0 ) = ( δ + δ ̂ ) Δ X γ ′ Δ X γ ( δ ̂ − δ ) T 1 − 2 α ( γ t − γ t 0 ) = T α + 1 / 2 ( δ + δ ̂ ) Δ X γ ′ Δ X γ T α + 1 / 2 ( δ ̂ − δ ) T 2 ( γ t − γ t 0 ) = o p ( 1 ) .

Thus, when γ t ≥ γ t 0 we have

(A.17) lim T → ∞ Pr S T ( β ̂ , δ ̂ , γ t ) > S T ( β ̂ , δ ̂ , γ t 0 ) = 1 .

Using a similar procedure, it is easily to show that, when γ t ≤ γ t 0 and ( γ 0 , γ 1 ′ ) ∈ V B ( ν ̄ ) , we also have lim T → ∞ Pr S T ( β ̂ , δ ̂ , γ t ) > S T ( β ̂ , δ ̂ , γ t 0 ) = 1 . As discussed above, this is sufficient to establish the consistency in Theorem 1.

We next derive the asymptotic distribution:

(A.18) a T γ ̂ 0 − γ 0 0 → d ψ arg max r ∈ ( − ∞ , ∞ ) ( W ( r ) − r 2 ) ,

and

(A.19) a T γ ̂ 1 − γ 1 0 → d ψ argmax r ∈ ( − ∞ , ∞ ) ( Λ ( r ) − r 2 ) .

Since the threshold parameters are consistent with convergence rate T ^1−2α, thus we can study their asymptotic behavior in the neighborhood of the true thresholds.

Let γ t = γ t 0 + w a T . By definition of threshold estimates, we have

(A.20) a T ( γ ̂ t − γ t 0 ) ≡ w * = arg min w ∈ ( − ∞ , ∞ ) S T * γ t 0 + w a n − S T * ( γ t 0 ) .

Since γ _t = γ ₀ + γ ′₁ z _t, we can rewrite the above results as

(A.21) a T γ ̂ 0 − γ 0 0 ≡ w 1 * = arg min w 1 ∈ ( − ∞ , ∞ ) S T * γ t 0 + w 1 a n − S T * ( γ t 0 ) ,

(A.22) a T γ ̂ 1 − γ 1 0 ≡ w 2 * = arg min w 2 ∈ ( − ∞ , ∞ ) k S T * γ t 0 + w 2 ′ a n z t − S T * ( γ t 0 ) .

As in the proof of the consistency in Theorem 1, we have S T * ( γ 0 0 + w 1 a T , γ 1 0 ) − S T * ( γ 0 0 , γ 1 0 ) = S 1 − S 2 + S 3 + S 4 . Then we can consider the limiting behavior of each S _i. We only provide the proof for the case w ₁ > 0, as the proof for the case w ₁ ≤ 0 is similar.

For the first term we have

(A.23) S 1 = δ T ′ X γ t 0 + w 1 a T − X ( γ t 0 ) ′ X γ t 0 + w 1 a T − X ( γ t 0 ) δ T = δ 0 ′ T 1 − 2 α 1 T 2 X γ t 0 + w 1 a T − X ( γ t 0 ) ′ X γ t 0 + w 1 a T − X ( γ t 0 ) δ 0 = T 1 − 2 α δ 0 ′ F γ 1 0 γ 0 0 + w 1 a T − F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s δ 0 + o p ( 1 ) = T 1 − 2 α w 1 a T f γ 1 0 ( γ 0 0 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 + o p ( 1 ) → p f γ 1 0 ( γ 0 0 ) w 1 δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 .

For the second term we have

(A.24) S 2 = 2 δ ̂ 0 ′ T 1 / 2 − α 1 T X γ t 0 + w 1 a T − X ( γ t 0 ) ′ e ⇒ 2 a T σ e δ 0 ′ ∫ B v ( s ) d W s , F γ t 0 + w 1 a T − W ( s , F ( γ 0 0 ) ) = 2 a T σ e δ 0 ′ J 1 γ t 0 + w 1 a T − J 1 ( γ t 0 ) ,

in which J ₁(u) =∫B _v(s)dW(s, u) is a zero-mean Gaussian process with an almost surely continuous sample path and with the covariance kernel

(A.25) E ( J ′ 1 ( u 1 ) J 1 ( u 2 ) ) = ( u 1 ∧ u 2 ) ∫ B ′ v ( s ) B v ( s ) d s .

It is easily to show that S ₂ + S ₃ = o _p(1). Define D T ( w 1 ) = a T J 1 ( γ t 0 + w 1 a n ) − J 1 ( γ t 0 ) . Then, by Lemma A.11 of Hansen (2000), D _T(w ₁) is a vector Brownian motion with covariance matrix f γ 1 0 ( γ 0 0 ) ∫ B ′ v ( s ) B v ( s ) d s .

Combing the above convergence results we have

(A.26) S T * ( γ t ) − S T * ( γ t 0 ) = S T * γ t 0 + w 1 a T − S T * ( γ 0 0 ) = S 1 − S 2 + S 3 + S 4 ⇒ f γ 1 0 ( γ 0 0 ) w 1 δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 − 2 σ e f γ 1 0 ( γ 0 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 W 1 ( w 1 ) ,

in which W ₁(w ₁) is a standard Brownian motion on [0, ∞]. Making the change-of-variables w 1 = σ e 2 f γ 1 0 ( γ 0 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 r , and noting that W(a ² r) = aW(r), we have

(A.27) S T * γ t 0 + w 1 a T − S T * ( γ t 0 ) ⇒ 2 σ e 2 ( r 2 − W 1 ( r ) ) .

Recall that a n γ ̂ 0 − γ 0 0 ≡ w 1 * = arg min w 1 ∈ ( − ∞ , ∞ ) S T * ( γ 0 0 + w 1 a n , γ 1 0 ) − S T * ( γ 0 0 , γ 1 0 ) . Using continuous mapping theorem, the asymptotic distribution of γ ̂ 0 is

(A.28) a T γ ̂ 0 − γ 0 0 ≡ r * → p ψ 0 argmax r ∈ ( 0 , ∞ ) ( W 1 ( r ) − r 2 ) ,

in which ψ 0 = σ e 2 f γ 1 0 ( γ 0 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 .

For w ₁ < 0, we can prove it similarly. Hence, for γ ̂ 0 we have

(A.29) a T γ ̂ 0 − γ 0 0 ≡ r * → p ψ 0 argmax r ∈ ( − ∞ , ∞ ) ( W ( r ) − r 2 ) ,

in which W ( r ) = W 1 ( r ) , r > 0 0 , r = 0 W 2 ( − r ) , r < 0

Using a similar procedure we can show that the asymptotic distribution of γ ̂ 1 = ( γ ̂ 11 , … , γ ̂ 1 k ) ′ . For any γ ̂ 1 j , j = 1 , ‥ , k , we have

(A.30) S T * γ t 0 + w 2 j a T z j t − S T * ( γ t 0 ) = S 1 j − S 2 j + S 3 j + S 4 j .

We only provide the proof for the case w _2j > 0, as the proof for the case with w _2j < 0 is analogous. For the first term, we have

(A.31) S 1 j = δ T ′ X γ t 0 + w 2 j a T z j t − X ( γ t 0 ) ′ X γ t 0 + w 2 j a T z j t − X ( γ t 0 ) δ T = δ 0 ′ T 1 − 2 α 1 T 2 X γ t 0 + w 2 j a T z j t − X ( γ t 0 ) ′ X γ t 0 + w 2 j a T z j t − X ( γ t 0 ) δ 0 = T 1 − 2 α δ 0 ′ F γ 1 0 + w 2 j a T ( γ 0 0 ) − F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s δ 0 + o p ( 1 ) ,

in which (as in Lemma 2)

(A.32) F γ 1 0 + w 2 j a T γ 0 0 − F γ 1 0 ( γ 0 0 ) = ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f u + γ 1 0 ′ z + w 2 j a T z j , z d z 1 … d z k d u − ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f ( u + γ 1 0 ′ z , z ) − d z 1 … d z k d u = ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f u + γ 1 0 ′ z + w 2 j a T z j , z − f ( u + γ 1 0 ′ z , z ) d z 1 … d z k d u → p ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f 1 ′ ( u + γ 1 0 ′ z , z ) w 2 j a T z j d z 1 … d z k d u = w 2 j a T ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f 1 ′ ( u + γ 1 0 ′ z , z ) z j d z 1 … d z k d u ≡ w 2 j a T g j ( γ 0 0 , γ 1 0 ) .

Thus, we have

(A.33) S 1 j = T 1 − 2 α δ 0 ′ F γ 1 0 + w 2 j a n ( γ 0 0 ) − F γ 1 0 ( γ 0 0 ) ∫ B v ( s ) B ′ v ( s ) d s δ 0 + o p ( 1 ) = T 1 − 2 α w 2 j a T g ( γ 0 0 , γ 0 1 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 + o p ( 1 ) → p w 2 j g j ( γ 0 0 , γ 1 0 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 .

For the second term, we have

(A.34) S 2 j = 2 δ ̂ T 1 / 2 − α 1 T X γ t 0 + w 2 j a T z j t − X ( γ t 0 ) ′ e = 2 a T σ e δ ′ J 1 γ t 0 + w 2 j a T z j t − J 1 ( γ t 0 ) + o p ( 1 ) .

Using a similar argument of Lemma 11 of Hansen (2000), we can show D ( w 2 j ) = a T J 1 ( γ t 0 + w 2 j a T z j t ) − J 1 ( γ t 0 ) is vector Brownian motion with covariance matrix g j ( γ 0 0 , γ 1 0 ) ∫ B ′ v ( s ) B v ( s ) d s .

And similarly, we have S _3j + S _4j = o _p(1). Therefore, we have

(A.35) S T * γ t 0 + w 2 j a T z j t − S T * ( γ t 0 ) = S 1 j − S 2 j + S 3 j + S 4 j ⇒ w 2 j g j ( γ 0 0 , γ 1 0 ) δ 0 ′ ∫ B v ( s ) B ′ v ( s ) d s δ 0 − 2 σ e g j ( γ 0 0 , γ 1 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 W 3 j ( w 2 ) .

Making the change-of-variables w 2 j = σ e 2 g j ( γ 0 0 , γ 1 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 r ( j ) , and using continuous mapping theorem, we have

(A.36) a T γ ̂ 1 j − γ 1 j 0 ⇒ ψ j argmax r ∈ ( − ∞ , ∞ ) ( Λ j ( r ( j ) ) − r ( j ) 2 ) ,

where Λ j ( r ( j ) ) = W 3 j ( r ) , r ( j ) > 0 0 , r ( j ) = 0 W 4 j ( − r ( j ) ) , r ( j ) < 0 in which W ₁(r), W ₂(r), W 3 j ( r ) and W 4 j ( r ) are two independent standard Brownian motions on [0, ∞). ψ j = σ e 2 g j ( γ 0 0 , γ 1 0 ) δ 0 ′ ∫ B ′ v ( s ) B v ( s ) d s δ 0 , g j ( γ 0 0 , γ 1 0 ) = ∫ − ∞ γ 0 0 ∫ · · · ∫ − ∞ + ∞ f 1 ′ ( u + γ 1 0 ′ z , z ) z j d z 1 … d z k d u . □

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2018-0101).

Received: 2018-10-12

Revised: 2021-01-22

Accepted: 2021-01-28

Published Online: 2021-02-22

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Keywords for this article

estimation; Fisher hypothesis; testing; threshold cointegration; time-varying threshold