Regulated seasonal unit root process

Proof of Lemma 1

The proof of Lemma 1 resembles the proofs in Cavaliere (2005) and Cavaliere and Xu (2014), thus Assumption A.2, which is required to use the results of the authors, is in force throughout the proof. However, we extend their proofs for multivariate processes. First, we start proving the results for i.i.d. random variable case, then investigate the impact of serial correlation later. Now, from Assumption A.1 S-dimensional process V _t = E _t defined in Eqs. (8) and (11) satisfies the conditions so that S N ( ⋅ ) = N − 1 / 2 ∑ t = 1 ⌊ N ⋅ ⌋ V t ⇒ B ( ⋅ ) where V t = [ V 1 , t , V 2 , t , … , V S , t ] ⊤ for all t ∈ {1, …, N}. Notice that for this case we set V t = ∑ j = 0 ∞ Φ j E t − j such that Φ₀ = I _S and Φ _j = 0 _S×S for all j ≥ 1 where is a vector of the i.i.d. random variables with zero mean and identity variance-covariance matrix. Next define the S-dimensional process X ̃ t = [ X ̃ 1 , t , X ̃ 1 , t , … , X ̃ S , t ] ⊤ , where for each j ∈ {1, …, S} and t ∈ {1, …, N} as,

With the above construction, X ̃ t is a vector bounded regulated process within the nominal bounds [ b ̲ , b ̄ ] where b ̲ = b ̲ 1 ⃗ and b ̄ = b ̄ 1 ⃗ . Using these nominal bounds and Assumption A.3, we define the asymptotic lower and upper bounds as c ̲ * = ( b ̲ / N ) 1 ⃗ and c ̄ * = ( b ̄ / N ) 1 ⃗ , respectively. In this specification, we have σ = 1 and Φ(1) = I _S . Note that we concatenate the independent bounded unit root process in the vector X ̃ t . The idea in such construction is to show that the asymptotic properties of the regulated process does not depend on regulation principle. In that sense, we first generate a regulated time series with absorbing bounds, which enable us to easily find the asymptotic distribution, then demonstrate that this process has the same asymptotic behavior as any time series satisfying Eqs. (1) and (2), sharing the same innovation process and being restricted in the same interval. To achieve this, we utilize the following lemma.

Lemma 3

Let X ̃ N ( t ) = N − 1 / 2 ∑ i = 1 ⌊ N t ⌋ X ̃ i be the partial sum process for the vector sequence { X ̃ i } i = 1 N , where X ̃ i = [ X ̃ 1 , i , … , X ̃ S , i ] ′ for all i ∈ {1, …, N} and X ̃ j , i is defined in Eq. (14) for each j ∈ {1, …, S}. Under the conditions of Lemma 1, X ̃ N ( t ) = S N ( t ) + L N ( t ) − U N ( t ) ⇒ B c ̲ * , c ̄ * ( t ) , where S _N (t), L _N (t) and U _N (t) are the partial sum processes defined in the proof and B c ̲ * , c ̄ * ( t ) is the S-dimensional regulated Brownian motion with the asymptotic bounds c ̲ * and c ̄ * .

Proof of Lemma 3

First, we define the following S-dimensional partial sum processes for t ∈ [0, 1]:

where for all i ∈ 1 , … , N and j ∈ {1, …, S}, Ξ ̄ j , i = X ̃ i , i − 1 + V j , i − b ̄ and Ξ ̲ j , i = b ̄ − X ̃ i , i − 1 − V j , i are the upper and lower regulators of X ̃ j , i , respectively. The processes in Eq. (15) are not continuous, thus, following Cavaliere (2005) we construct the continuous approximant of these processes on C [ 0,1 ] with t ∈ [0, 1] as,

for all j ∈ {1, …, S},

Note that we can write S N * ( t ) = [ S 1 , N * ( t ) , … , S S , N * ( t ) ] ⊤ , L N * ( t ) = [ L 1 , N * ( t ) , … , L S , N * ( t ) ] ⊤ and U N * ( t ) = [ U 1 , N * ( t ) , … , U S , N * ( t ) ] ⊤ for all t ∈ [0, 1]. Using these objects, we can write the continuous approximation of the partial sum process for the regulated series { X ̃ i } i = 1 N as

At the beginning of the proof, we mentioned that S _N (t) ⇒ B(t). What needs to be demonstrated for the proof to follow however, is that S N * ( t ) ⇒ B ( t ) . This can be obtained by noting that

The above inequality follows from the proof of Theorem 6 of Cavaliere (2005) for each j ∈ {1, …, S}. Note that the rightmost equality in Eq. (16) is a direct implication of Assumption A.1, which indicates each element of V _t is a martingale difference sequence. Invoking Theorem 4.1 of Billingsley (1968) then implies that S j , N * ( t ) ⇒ B j ( t ) where for each j ∈ {1, …, S} B _j (t) is a univariate standard Brownian motion. Consequently, the Cartesian product S N * ( t ) ⇒ B ( t ) where B ( t ) = [ B 1 ( t ) , … , B S ( t ) ] ⊤ .

The main part of the proof now follows by noting that X ̃ N * ( t ) satisfies the construction for the regulated vector Brownian motion. In particular, this construction is same as the formulation for the regulated vector Brownian motions in Harrison and Reiman (1981). First, we define the function X ̃ N * ( t ) = g c ̲ * c ̄ * ( S N * ( t ) ) regulates S N * ( t ) as an integrated process which lies between [ c ̲ * , c ̄ * ] . Further, this process satisfies the following four conditions in Definition 1.

1. The limits of L j , N * ( s ) and U j , N * ( s ) are continuous and non-decreasing with L j , N * ( 0 ) = U j , N * ( 0 ) = 0 for all j ∈ {1, …, S} by construction.

2. X ̃ N * ( t ) = S N * ( t ) + L N * ( t ) − U N * ( t ) ∈ [ c ̲ * , c ̄ * ] for all t ∈ [0, 1].

3. L j , N * ( t ) only increases when X ̃ j , N * ( t ) = c ̲ * and U j , N * ( t ) increases only when X ̃ j , N * ( t ) = c ̄ * for all j ∈ {1, …, S} t ∈ [0, 1], because, they are continuous approximations of the regulators, which only increases at the associated boundaries.

4. The reflection matrix R in the formulation in Harrison and Reiman (1981) is set to identity matrix I on this case, thus each regulated process are regulated separately when they hit the bound.

From Theorem 1 of Harrison and Reiman (1981), since X ̃ N * ( t ) satisfies the above four condition, X ̃ N * ( t ) is a unique solution to regulation problem for vector Brownian motion process. The continuous mapping theorem (CMT) and the convergence S N * ( t ) ⇒ B ( t ) entail that

What remains to be shown is that X ̃ N ( t ) converges to X ̃ N * ( t ) for all t ∈ [0, 1]. A first step toward attaining this result is to check the condition whether each element of the vector process X ̃ N ( t ) pairwisely converges to the associated element of the process X ̃ N ( t ) . We can verify this condition by using the arguments in the proof of Theorem 6 of Cavaliere (2005). Now, notice that for t ∈ [0, 1]

To show the inequality above holds, consider the following objects

Note that these components belong to a univariate bounded integrated process, thus we can use the finding of Cavaliere (2005), which states that both objects are smaller than N 1 / 2 − 1 V j , [ N t ] + 1 for each j ∈ {1, …, S} and due to the fact that the non-zero sets of L N * ( t ) and U N * ( t ) are clearly disjoint for all t ∈ [0, 1]. The convergence follows from the same arguments used in the convergence of S N * ( t ) . That is,

This completes the proof of Lemma 3 which demonstrates that X ̃ N ( t ) ⇒ X ̃ N * ( t ) ⇒ B c ̲ * , c ̄ * ( t ) for t ∈ [0, 1]. □

Proof of Lemma 2

First note that ∀j ∈ {0, …, ⌊S/2⌋}, X _j,t = C _j X _t and for all j ∈ {1, ‥, S*} we have X j , t * = C j * X t . As in del Barrio Castro, Osborn, and Taylor (2012), we have C ₀Φ(1) = ϕ(1)C ₀, C _⌊S/2⌋Φ(1) = ϕ(−1)C _⌊S/2⌋, C j Φ ( 1 ) = b j C j + a j C j * and C j * Φ ( 1 ) = − a j C j + b j C j * for j = 1, …, S*. Further, we have following identities: C ₀ C ₀ = SC ₀, C _⌊S/2⌋ C _⌊S/2⌋ = SC _⌊S/2⌋, C j C j = S 2 C j , C j C j * = S 2 C j * and C j * C j * = S 2 C j for j ∈ {1, …, S*}. It is also stated in del Barrio Castro, Osborn, and Taylor (2012) that the other products of the circulant matrices are all zero.

Note that for j ∈ {0, ⌊S/2⌋} X _j,t = C ₀ X _t we can write the partial sum process as

where the first convergence immediately follows Lemma 1 and the last equation stems form the property of the circulant matrix introduced above. With similar arguments, the other results follow from manipulating the associated circulant matrices and CMT. □

Proof of Theorem 1

In this proof, we will use the construction of del Barrio Castro, Osborn, and Taylor (2012). To save space, we will skip the parts that are exactly same as in del Barrio Castro, Osborn, and Taylor (2012). Consider the augmented regression in Eq. (5). Under Assumption A.1, del Barrio Castro, Osborn, and Taylor (2012) show that

First consider the denominators. For j ∈ {0, ⌊S/2⌋}, we have

The first equality stems from the properties of the circulant matrices C ₀ and C _S/2. The weak convergence is a result of Lemma 1 and the CMT. But also note that the term 1/S appears since the partial sum convergence is over N, but not T = NS, thus we have an extra term 1/S ². Similarly, for j ∈ {1, …, S*} we can show

Now we can focus on the numerators. del Barrio Castro, Osborn, and Taylor (2012) prove that T − 1 ∑ t = 1 N ∑ s = 1 − S 0 x j , S t + s ( e S t + s k − ϵ S t + s ) = O p ( T ) and T − 1 ∑ t = 1 N ∑ s = 1 − S 0 x j , S t + s * ( e S t + s k − ϵ S t + s ) = O p ( T ) , then we can write the numerators as

From the CMT and Lemma 1, we have

Note that the t statistic can be written as

The rest of the proof follows from the application of the CMT and the consistency of σ ̂ 2 . The innovation variance σ ² can be estimated by σ ̂ 2 = T − 1 ∑ t = 1 N ∑ s = 1 − S 0 e ̂ k , S t + s where e ̂ k , S t + s is the residuals from the augmented HEGY Equation. The consistency of σ ̂ 2 directly follows the results of Lemma A.2 in Cavaliere and Xu (2014) and del Barrio Castro, Osborn, and Taylor (2012), thus we skip the full proof. □