Multiple structural breaks in cointegrating regressions: a model selection approach

Proof of Theorem 1

According to Assumption 1, the scalar partial sum process in Eq. (2) satisfies the functional central limit theorem (FCLT). For s ∈ [0, 1] and as T → ∞, it holds that

where B(s) is a Brownian motion process with variance σ v 2 . This was shown by Herrndorf (1984) and extended to the vector case by Phillips and Durlauf (1986).

Next, we define the objective function V _T ( b ) by

where b = ( b ₁′, b ₂′)′, δ T = diag ( T 1 / 2 I p * + 1 , T I m * + 1 ) and

is the minimizer of V _T with b ̂ 1 i = T ( μ ̂ T , i − μ i * ) and b ̂ 2 j = T ( β ̂ T , j − β j * ) .

First, we consider the asymptotic counterparts to the least squares terms

We use the decomposition X = ( X ₁, X ₂) to express the weak convergence result

where

Using (A.4) in Gregory and Hansen (1996a) and the continuous mapping theorem (CMT, see Billingsley (1999), Theorem 2.7), we observe that

where the weak convergence is uniform over the vector ( τ 1,1 , … , τ 1 , p * , τ 2,1 , … , τ 1 , m * ) ∈ T . Further, using (A.3) in Gregory and Hansen (1996a) and Theorem 3.1 in Hansen (1992), we have the weak convergence to a stochastic integral

where U ∼ N(0, ϒσ ²) and Λ = ∑ t = 0 ∞ E ( v t u 0 ) .

Under Assumptions 2 and 3, the maximum number of breakpoints is limited and initial least squares estimates are available for the weights of the adaptive lasso estimator. We investigate the consistency of the individual coefficients and distinguish between the true coefficients being zero or nonzero:

1. If μ i * ≠ 0 , we have

   (26) λ T w 1 i γ 1 | μ i * + b 1 i / T | − | μ i * | = λ T T 1 μ ̂ I , i γ 1 T | μ i * + b 1 i / T | − | μ i * | → p 0 ,

   since (i) λ T T → 0 , (ii) 1 μ ̂ I , i γ 1 → p 1 μ i * γ 1 if the initial estimator is consistent and (iii) T | μ i * + b 1 i / T | − | μ i * | → p b 1 i sgn ( μ i * ) as in Zou (2006).

2. If μ i * = 0 , we have

   λ T w 1 i γ 1 | μ i * + b 1 i / T | − | μ i * | = λ T T 1 μ ̂ I , i γ 1 | b 1 i |

   (27) = λ T T 1 / 2 − γ 1 / 2 1 T μ ̂ I , i γ 1 | b 1 i |

   ⇒ ∞   if b 1 i ≠ 0 0   if b 1 i = 0   ,

   since (i) λ T T 1 / 2 − γ 1 / 2 → ∞ and (ii) the initial least squares estimator is tight and converges to a normal distribution, T μ ̂ I , i ⇒ W 1 i ∼ N 0 , σ 2 τ 1 , i ( 1 − τ 1 , i ) .

3. If β j * ≠ 0 , we have

   (28) λ T w 2 j γ 2 | β j * + b 2 j / T | − | β j * | = λ T T 1 β ̂ I , j γ 2 T | β j * + b 2 j / T | − | β j * | → p 0 ,

   since (i) λ T T → 0 , (ii) 1 β ̂ I , j γ 2 → p 1 β j * γ 2 if the initial estimator is consistent and (iii) T | β j * + b 2 j / T | − | β j * | → p b 2 j sgn ( β j * ) .

4. If β j * = 0 , we have

   λ T w 2 j γ 2 | β j * + b 2 j / T | − | β j * | = λ T T 1 β ̂ I , j γ 2 | b 2 j |

   (29) = λ T T 1 − γ 2 1 T β ̂ I , j γ 2 | b 2 j |

   ⇒ ∞   if b 2 j ≠ 0 0   if b 2 j = 0   ,

   since (i) λ T T 1 − γ 2 → ∞ , (ii) the least squares estimator is tight and has the following nonstandard distribution

   (30) T β ̂ I , j ⇒ W 2 j = ∫ 0 1 B τ 2 , j ( s ) d U ( s ) + ( 1 − τ 2 , j ) Λ ∫ 0 1 B τ 2 , j 2 ( s ) d s ,

   and (iii) P ( W 2 j = 0 ) → a . s . 0 .

Thus, V _T ( b ) ⇒ V( b ), where

with

Since V _T is a convex function and V has a unique minimum, it follows from Knight and Fu (2000) that

From these results, we can deduce that

where δ ₀ denotes the Dirac measure at 0. Correspondingly, we have

It remains to show that coefficients of inactive variables are set to zero with probability approaching one. We begin with a proof of P ( μ ̂ T , A 1 c = 0 ) → 1 . Consider the event that μ ̂ T , i ≠ 0 although i ∈ A 1 c . We know from the Karush–Kuhn–Tucker (KKT) optimality conditions that the first order condition for a minimum is given by

Note that

since (i) λ T T 1 / 2 − γ 1 / 2 → ∞ and (ii) T μ ̂ I , i is tight. The left hand side of the equation is equivalent to

For the first term, we have the weak convergence

and for the second term, we have the weak convergence of φ τ 1 , i ′ x δ T − 1 T , which depends on the timing of the break fraction τ _1,i relative to all other possible break fractions. Say τ 1 , i = τ 1 , p * > τ 2 , m * holds, then

Further, we have already shown the weak convergence of δ T ( μ ̂ T ′ − μ * ′ , β ̂ T ′ − μ * ′ ) ′ . Hence, the distribution of the first term is tight and

Next, we show that P ( β ̂ T , A 2 c = 0 ) → 1 . Again, we consider the event that β ̂ T , j ≠ 0 although j ∈ A 2 c . The KKT optimality condition in this case is given by

where the factor T substitutes the factor T in Eq. (35). For the right hand side of the equation, we observe that

since T β ̂ I , j is tight. For the left hand side,

we have the weak convergence of the first term using

The expression of the weak convergence result for the second term depends on the timing of the break fraction τ _2,j . Say τ 2 , j = τ 1 , m * > τ 1 , p * holds, we have

and as before δ T ( μ ̂ T ′ − μ * ′ , β ̂ T ′ − μ * ′ ) ′ is tight. Finally, we have shown that

and this completes the proof. □

Proof of Theorem 2

For ease of exposition, we assume that the maximum number of breaks is m* = 2 and the true intercept is known to be μ _t = 0 for all t. In this case, we obtain three possible model selection outcomes regarding the number of breaks for the post-lasso regressions:

1. All coefficients of break indicator regressors are shrunk to zero

   (47) y t = β 1 x t + e t τ 0 .

2. One structural break is (falsely) detected

   (48) y t = β 1 x t + β 2 x t φ t , τ 2,1 + e t τ 1 .

3. Two structural breaks are (falsely) detected

   (49) y t = β 1 x t + β 2 x t φ t , τ 2,1 + β 3 x t φ t , τ 2,2 + e t τ 2 .

Note that the specification of indicator terms in Eqs. (48) and (49), i.e., the timing of (falsely) detected breaks, depends on the tuning parameter λ. We continue the proof for the bias-corrected test statistic, Z ₂, corresponding to the case of two (falsely) detected structural breaks with unknown timing. The asymptotic distribution of the test statistics for the two remaining cases can be easily deduced from our derivations. We decompose the cumulative sum into S _t = (S _1t , S _2t )′. Further, we define the break fraction vector τ ₂ = (τ _2,1, τ _2,2)′ as a compact set on (0, 1) × (0, 1) and define the matrix X t τ 2 = ( S t ′ , S 2 t φ t , τ 2,1 , S 2 t φ t , τ 2,2 ) ′ = ( S 1 t , X 2 t τ 2 ′ ) ′ .

Using the result

and the CMT yields the weak convergence of

Further, (50) and Theorem 4.1 of Hansen (1992) yield the weak convergence of

The result in (51) can straightforwardly be extended to

where X τ 2 = ( B 1 , B 2 , B 2 φ τ 2,1 , B 2 φ τ 2,2 ) ′ = ( B 1 , X 2 τ 2 ′ ) ′ and

Define β ̂ τ 2 = ( β ̂ 1 , β ̂ 2 , β ̂ 3 ) ′ as the post-lasso least squares estimator and set η ̂ τ 2 = ( 1 , − β ̂ τ 2 ′ ) ′ so that

We partition

in conformity with S _t and define Λ_2⋅ = (Λ₂₁, Λ₂₂) and Λ_⋅2 = (Λ₁₂, Λ₂₂)′.

For each element of S 2 t τ 2 = ( S 2 t φ t , τ 2,1 , S 2 t φ t , τ 2,2 ) it holds that

and Δ φ t , τ 2 , i = φ t , τ 2 , i − φ t − 1 , τ 2 , i = 1 { t = [ τ 2 , i T ] } . Since φ t − 1 , τ 2 , i Δ φ t , τ 2 , i = 0 and

for the asymptotic counterpart, we have the identities

and

Consequently, we can state the following important weak convergence results

where d B 2 τ 2 = ( d ( B 2 ( s ) φ τ 2,1 ( s ) ) , d ( B 2 ( s ) φ τ 2,2 ( s ) ) ) and

where

Under the null hypothesis, the cointegration residuals can be written as e ̂ t τ 2 = η ̂ τ 2 ′ X t τ 2 and we can show weak convergence of the sample moments. It holds that

where W τ 2 ( s ) = W 1 ( s ) − ∫ 0 1 W 1 W 2 τ 2 ′ ∫ 0 1 W 2 τ 2 W 2 τ 2 ′ − 1 W 2 τ 2 and

Next, we consider the bias-correction term for the first-order serial correlation coefficient. We denote the kernel weights as w(j/M) = w _j and can show that

Hence, we have the weak convergence result

For the long-run variance, we obtain the result

where