Weak convergence of sample covariance matrices and testing for seasonal unit roots

SFB 823

Weak convergence of sample covariance matrices and

testing for seasonal unit roots

Discussion Paper

Nr. 29/2020

Weak Convergence of Sample Covariance Matrices and Testing for Seasonal Unit Roots

Rafael Kawka

Technische Universität Dortmund, Germany

October 18, 2020

The paper has two main contributions. First, weak convergence results are derived from sampling moments of processes that contains a unit root at an arbitrary frequency, where, in contrast to the previous literature, the proofs are mainly based on algebraic manipula- tions and well known weak convergence results for martingale difference sequences. These convergence results are used to derive the limiting distribution of the ordinary least squares estimator for unit root autoregressions. As as second contribution, a Phillips-Perron type test for a unit root at an arbitrary frequency is introduced and its limiting distributions are derived. This test is further extended to a joint test for multiple unit roots and seasonal integration. The limiting distributions of these test statistics are asymptotically equivalent to various statistics presented earlier in the seasonal unit root literature.

Keywords: Invariance Principle, Weak Convergence, Seasonal Unit Root, Unit Root Test

1. Introduction

Consider then-dimensional stochastic process{x_t}_t∈Nin discrete time generated according to the difference equation

x_t=Axt−1+η_t, t∈N, (1)

A=e^−iωI_n,

for some frequencyω ∈(−π, π], where we assume that the the starting value x0 isO_P(1) and where{η_t}t∈Z is a weakly stationary with mean zero. The process{x_t}t∈N0 is calledintegrated at frequencyω or, sincee^iω is the root of the equation 1−e^−iωz= 0, it is also calledunit root process.

The limiting distributions of the sample covariance matrices _T¹₂ PT

t=1x_tx^∗_t and _T¹ PT

t=1xt−1η^∗_t are important building blocks in the derivation of an asymptotic theory for unit root test statistics as well as for inference in cointegrating systems. If the process {η_t}_t∈Z fulfills a functional central limit theorem the limiting distribution of the former can be easily derived by an application of the continuous mapping theorem. The limiting distribution of the latter is more complicated. In case of ω = 0 Phillips (1988b) showed under very general conditions on the process{η_t}_t∈Z that

1 T

T

X

t=1

xt−1η_t⇒ Z ₁

0

B(r)dB(r) + Λ₀,

asT → ∞, whereB(r)is a vector Brownian motion with covariance matrix given by the long- run variance matrix of {η_t}t∈Z. The additive bias term Λ₀ defined as the sum of all E(η₀η⁰_h) overh∈Nand is therefore also called one sided long-run covariance matrix.

Phillips’ proof, however, is quite long and one needs a very deep understanding of certain concepts from probability theory to be able to follow it. Therefore, Phillips (1988a) presented a much simpler proof under marginally more restrictive assumptions. In particular, he requires {η_t}_t∈Z to be a linear process of the form

ηt=

∞

X

j=−∞

ψjεt−j,

with{ε_t}_t∈Zbeing an i.i.d. sequence with zero mean and finite variance and where the coefficient matrices satisfy

∞

X

j=1







∞

X

k=j

ψ_k

+

∞

X

k=j

ψ−k







<∞. (2)

Gregoir (2010) relaxed the i.i.d. assumption on {ε_t}_t∈Z and extended Phillips’ approach by deriving the limiting distribution for arbitrary values ofω.

The drawback of the proofs of Phillips (1988a) and Gregoir (2010) is that they are based on the martingale approximation theory of Hall and Heyde (1980), with which many researchers are not familiar with. Thus, one of the aims of this paper is to derive the same results, but without making use of this theory. Instead, we use a decomposition of{x_t}t∈N0 which is based on the so-called Beveridge-Nelson decompisition, and derive a functional central limit theorem following the approach of Phillips and Solo (1992). Furthermore, this decomposition allows us to decompose the sample covariance matrix in such a way that we can derive its asymptotic distribution with simple algebraic transformations and apply well known convergence results for martingale difference sequences. As the only additional assumption we demand that the process{η_t}_t∈Z is a causal with respect to {ε_t}_t∈Z.

We derive the asymptotic distribution of the OLS estimator forA in the regression model (1) as a direct application. With this result in place we generalize the approach of Phillips (1987) and Phillips and Perron (1988) and modify the OLS estimator so that the limiting distribution is free of nuisance parameters. We then use this modified estimator to construct a test for unit roots at any given frequency ω. As an extension of this test we present a joint test for multiple unit roots and for seasonal integration, similar to the tests of Hylleberg et al. (1990) and Ghysels et al. (1994).

The remainder of this paper is organized as follows: In Section 2 we state the precise assump- tions and present the decomposition mentioned above. In Section 3 we derive the functional central limit theorem and the limiting distributions of the sample covariance matrices. Section 4 contains the tests for unit roots and seasonal integration. Section 5 concludes. Appendix A contains some auxiliary algebraic results. The proofs of the main mathematical results are relegated to Appendix B.

Throughout the paper we use the following notation: Weak convergence is denoted by ⇒ and convergence in probability is signified by →. For convergence in probability to zero we^P use the small O notation o_P(1) whereas we use O_P(1) to indicate stochastic boundedness.

The integer part of a real number x is given by [x] and the modulus of a complex number x = Re(x) +iIm(x) is denoted by |x|. We use the notation kxk to signify the Frobenius norm. For a (possibly complex valued) matrixA we denote its transpose, complex conjugate and Hermitian transpose by A⁰, A and A^∗, respectively. With L and ∆_ω we denote the lag operator and the seasonal first difference operator, respectively and we use the somewhat sloppy notationsLxt=xt−1 and ∆ωxt=xt−e^−iωxt−1.

2. Setup, Assumptions and Decomposition of Unit Root Processes

As mentioned in the introduction, we consider processes generated according to (1) with x₀ beingO_P(1)and {η_t}_t∈Z satisfying the following assumption.

Assumption 1. The process{η_t}_t∈Z is a linear process of the form ηt= Ψ(L)εt=

∞

X

j=0

ψjεt−j, (3)

wheredet(Ψ(e^iω)) 6= 0and where the coefficient matrices ψj ∈ C^n×n satisfy the summability condition

∞

X

j=0

jkψ_jk<∞. (4)

The innovation process{ε_t}_t∈Zis a martingale difference sequence with respect to its canonical filtration F_t = σ{ε_t−j, j ∈ N0} satisfying E(εtε⁰_t|F_t−1) = In and sup_tE(kε_tk^2+δ|F_t−1) < ∞ with probability one for someδ >0.

Remark 1. The summability condition (4) is common in the unit root literature, as it is, for instance, fulfilled by all causal, stationary and invertible ARMA processes. In particular, since

∞

X

j=1







∞

X

k=j

ψk

+

∞

X

k=j

ψ−k







=

∞

X

j=1

∞

X

k=j

ψk

≤

∞

X

j=1

∞

X

k=j

kψ_kk ≤

∞

X

j=0

jkψ_jk,

it implies the previously mentioned summability condition (2).

Remark 2. The assumptions stated on the sequence {ε_t}_t∈Z are quite general and are widely applied in the literature. However, the restriction on the (conditional) covariance matrix is imposed only for notational simplicity and can of course be relaxed by assuming that E(εtε⁰_t|F_t−1) = Σε whereΣε is positive definite.

Under Assumption 1 the process {η_t}_t∈Z has a continuous spectral density, f(ω) say, and we define

Ω_ω= 2πf(ω) =

∞

X

h=−∞

e^−iωhE(η₀η_h^∗) =

∞

X

h=−∞

e^−iωh

∞

X

j=0

ψ_jψ^∗_j+h. (5) Note thatΩ_ω = Ψ(e^iω)Ψ(e^iω)^∗. Furthermore, it holds that Ω_ω = Σ + Λ_ω+ Λ^∗_ω, where

Σ =E(η0η^∗₀) =

∞

X

j=0

ψjψ^∗_j (6)

and

Λω =

∞

X

h=1

e^−iωhE(η0η_h^∗) =

∞

X

h=1

e^−iωh

∞

X

j=0

ψjψ_j+h^∗ . (7)

If ω = 0 it is well known that the process {x_t}_t∈N0 can be decomposed into a pure random walk, a stationary component and an initial value component. The following result generalizes this decomposition to the arbitrary frequency case.

Proposition 1. Let{x_t}_t∈N0 be a stochastic process in discrete time generated according to the difference equation (1)with Assumption 1 in place. Then, it holds that

x_t=e^−iωt(x₀+ ˜η₀) + Ψ(e^iω)e^−iωt

t

X

j=1

e^iωjε_j−η˜_t, t= 1,2. . . ,

where{η˜_t}_t∈Z is a weakly stationary process with moving average representation

˜

η_t= ˜Ψ(L)ε_t=

∞

X

j=0

ψ˜_jεt−j, ψ˜_j =e^−iωj

∞

X

k=j+1

e^iωkψ_k.

Remark 3. The proof of Proposition 1 is essentially an application of the so-called Beveridge- Nelson decomposition at frequency ω. It states that a matrix polynomial A(z) with matrix coefficientsA_j satisfying P∞

j=0jkA_jk<∞ can be decomposed into A(z) =A(e^iω)−(1−e^−iωz)B(z),

whereB(z) is a matrix polynomial with absolutely summable matrix coefficients (cf. Phillips and Solo, 1992). We present a simple algebraic proof of this decomposition in Appendix A.

3. Convergence of Sample Covariance Matrices

In this section we present a functional central limit theorem as well as several results on the limiting distributions of sample covariance matrices of integrated processes at some arbitrary frequency. As our main contribution we extend the result of Phillips (1988a) for processes that are integrated at some arbitrary frequency. The following lemma is the central building block for the subsequent results.

Lemma 1. Let {ε_t}_t∈Z be a martingale difference sequence that satisfies Assumption 1. Then, asT → ∞, it holds that





√1 T

[rT]

X

t=1

e^iωtε_t, 1 T

[rT]

X

t=1

e^−iωt

t−1

X

j=1

e^iωjε_jε⁰_t



⇒

τ_ωW(r), τ_ω² Z _r

0

W(s)dW(s)^∗

.

where

τ_ω=







1 if ω∈ {0, π},

√1

2, if ω∈(−π,0)∪(0, π)

(8) andW(r) is an n-dimensional standard Brownian motion if ω∈ {0, π} and an n-dimensional standard complex Brownian motion if ω ∈ (−π,0)∪(0, π), i.e. W(r) =W₁(r) +iW₂(r) with independent n-dimensional (real valued) standard Brownian motions W₁(r) and W₂(r).

Our first main result is a functional central limit theorem for processes that are integrated at an arbitrary frequency.

Theorem 1. Let {x_t}_t∈N0 be a stochastic process in discrete time generated according to the difference equation (1)with Assumption 1 in place. Then, as T → ∞, it holds that

e^iω[rT]

√

T x_[rT]⇒τ_ωB(r), r ∈(0,1], whereB(r) = Ψ(e^iω)W(r) with τω andW(r) defined in Lemma 1.

Theorem 1 can be extended to the following joint convergence result without any additional effort. Let{x_t,k}_t∈N0,k= 1, . . . , K, be n-dimensional processes generated according to x_t,k = e^−iωkxt−1,k+ηt withωk 6=ωj for all k6=j. Then, as T → ∞,

"

e^iω1[rT]

√

T x_[rT],1, . . . ,e^iωK[rT]

√

T x_[rT],K

#

⇒[τω1B1(r), . . . , τωKBK(r)],

whereB_k(r) = Ψ(e^iωk)W_k(r) for k= 1, . . . , K and W₁(r), . . . , W_K(r) are independent Brow- nian motions, complex valued if the corresponding frequency ω_k is different from zero or π.

Furthermore, Theorem 1 can be generalized for the weak convergence of the cumulative sum of e^iωtx_t. In particular, it holds that

1 T^3/2

[rT]

X

t=1

e^iωtxt⇒τω

Z r 0

B(s)ds,

asT → ∞, which is a direct consequence of the continuous mapping theorem. This result can be extended to multiple cumulative summation.

Corollary 1. Let {x_t}_t∈N0 be a stochastic process in discrete time generated according to the difference equation (1)with Assumption 1 in place. Then, as T → ∞, it holds that

1 T^(2m+1)/2

[rT]

X

t1=1 t1

X

t2=1

· · ·

tm−1

X

tm=1

e^iωtmxtm ⇒τω

Z r 0

Z s1

0

· · · Z sm−1

0

B(sm)dsmdsm−1 . . . ds1,

for any m∈N, where the process limiting process B(r) is defined in Theorem 1.

The subsequent proposition states the limiting distribution of the sample covariance matrix between two processes that are integrated at the same frequency as well as the asymptotic orthogonality of two processes that are integrated at different frequencies. The former statement follows again from Theorem 1 and the continuous mapping theorem whereas the latter is an algebraic consequence of the fact that PT

t=1e^iθt is bounded if and only if θ is different from zero (cf. Lemma A.1 in the appendix).

Proposition 2. Let {x_t,1}_t∈N0 and{x_t,2}_t∈N0 be two n-dimensional stochastic process, gener- ated according to the difference equations

xt,1 =e^−iω1xt−1,1+ηt, xt,2 =e^−iω2xt−1,2+ηt

for t ∈ N, where {η_t}_t∈Z is a stationary process that satisfies Assumption 1 and the starting valuesx0,1 and x0,2 are O_P(1).

If ω1=ω2 then, as T → ∞ it holds that 1

T²

T

X

t=1

xt,1x^∗_t,2 ⇒τ_ω²₁ Z 1

0

B(r)B(r)^∗dr,

with B(r) = Ψ(e^iω1)W(r) being the limiting process from Theorem 1.

If ω₁6=ω₂ then, as T → ∞, it holds that 1 T²

T

X

t=1

x_t,1x^∗_t,2→^P 0. (9)

Remark 4. Proposition 2 can easily be generalized to covariance matrices of more than two integrated processes as follows. For k = 1, . . . , K let {x_t,k}t∈N0 be n-dimensional processes where for everyk the process {x_t,k}_t∈N0 is generated according to x_t,k =e^−iωkxt−1,k+η_t with x0,k being O_P(1)and where ωk6=ωj for allk6=j. Define

X =









x_1,1 x_1,2 . . . x_1,K ... ... ... x_{T ,1} x_{T ,2} . . . x_{T ,K}







 .

Then, asT → ∞, it holds that

1

T²(X^∗X)⇒











 τ_ω²₁R1

0B₁(r)B₁(r)^∗dr 0 . . . 0

0 τ_ω²₂R1

0B2(r)B2(r)^∗dr . . . 0

... ... . .. ...

0 0 . . . τ_ω²_KR1

0B_K(r)B_K(r)^∗dr











 ,

whereB_k(r) = Ψ(e^iωk)W_k(r) for k= 1, . . . , K and W1(r), . . . , W_K(r) are independent Brown- ian motions, complex valued if the corresponding frequencyωk is different from zero orπ.

Remark 5. The statement of Proposition 2 holds also for processes{x_t,1}_t∈N0 and {x_t,2}_t∈N0 that are generated according to the difference equation (1) but with distinct processes{η_t,1}_t∈Z and{η_t,2}_t∈Z, i.e.

x_t,1 =e^−iω1xt−1,1+η_t,1 x_t,2 =e^−iω2xt−1,2+η_t,2,

fort∈Nwith starting valuesx0,1 and x0,2 being O_P(1). If the stacked process{[η⁰_t,1, η_t,2⁰ ]⁰}_t∈Z is stationary and fulfills Assumption 1 then it holds that

√1 T

"

e^iω1[rT]x_[rT],1 e^iω2[rT]x_[rT],2

#

⇒

"

B₁(r) B₂(r)

#

, r∈(0,1],

and, consequently, ifω₁ =ω₂ we obtain as T → ∞, 1

T²

T

X

t=1

x_t,1x^∗_t,2 ⇒ Z 1

0

B₁(r)B₂(r)^∗dr.

whereas ifω1 6=ω2 it holds that

1 T²

T

X

t=1

xt,1x^∗_t,2→^P 0.

The statement in Remark 4 can be extended in a similar way.

By the same arguments as in the proof of Proposition 2 we can derive the limiting distribution of the sample covariance matrix between a process integrated at some frequency ω and a deterministic sequence.

Corollary 2. Let {x_t}_t∈N0 be generated as in Theorem 1 and let {d_t}_t∈N0 be a p-dimensional deterministic sequence with such thatG⁻¹_D e^iθ[rT]D_[rT]⇒D(r), asT → ∞, for someθ∈(−π, π], whereGD ∈R^p×p is a scaling matrix and D(r) is a càdlàg function.

If θ=ω then, as T → ∞, it holds that 1

T^3/2G⁻¹_D

T

X

t=1

dtx^∗_t ⇒τω

Z 1 0

D(r)B(r)^∗dr.

If θ6=ω then, as T → ∞, it holds that 1 T^3/2G⁻¹_D

T

X

t=1

d_tx^∗_t →^P 0.

An important example for a deterministic sequence that satisfy the Assumptions in the Corol- lary is{d_t}_t∈N0, wheredt=e^−iθtft with

ft= [1, t, t², . . . , t^q]⁰. Then, withG_D = diag(1, T, T², . . . , T^q) it holds that

G⁻¹_D e^iθ[rT]d_[rT]=

"

1,[rT] T ,

[rT] T

2

, . . . , [rT]

T q#0

⇒[1, r, r², . . . , r^q]⁰.

Hence, by setting θ = 0, it follows that the sequence of monomials dt = [1, t, t², . . . , t^q]⁰ is asymptotically orthogonal to any process{x_t}_t∈N0 that is integrated at some frequencyω6= 0.

Next, we discuss the limiting distribution of the sample covariance between xt−1 and η_t in model (1), which is the main contribution of this section. If {x_t}_t∈N0 is scalar Phillips (1987) showed that the limiting distribution can be easily calculated using the identity

x²_t = (xt−1+η_t)²=x²_t−1+η²_t + 2xt−1η_t. In particular, it holds that

1 T

T

X

t=1

xt−1ηt= 1 2T

T

X

t=1

(x²_t−x²_t−1)− 1 2T

T

X

t=1

η_t²= 1

2T(x²_T −x²₀)− 1 2T

T

X

t=1

η_t².

The weak law of large numbers implies that the latter term converges to Σ/2 and it holds thatx²₀/T² converges to zero in probability as the starting value x₀ is O_P(1). Theorem 1, the continuous mapping theorem and Itô’s Lemma yield

1

2Tx²_T ⇒ 1

2B(1)² = Ω0

2 W(1)² = Ω0

2 (W(1)²−1) +Ω0

2 = Ω0

Z 1 0

W(r)dW(r) + Ω0

2 . FromΩ₀ = Σ + 2Λ₀ we conclude that

1 T

T

X

t=1

xt−1η_t⇒Ω₀ Z 1

0

W(r)dW(r) +Ω₀

2 −Σ = Z 1

0

B(r)dB(r) + Λ₀. (10) Similarly, we can derive the limiting distribution for {x_t}_t∈N0 being scalar and generated ac- cording to (1) withω=π. In this case it holds that

x²_t =x²_t−1−2xt−1η_t+η²_t

and, using exactly the same arguments as above, we deduce that 1

T

T

X

t=1

xt−1η_t=− 1

2T x²_T −x²₀

− 1 2T

T

X

t=1

η²_t

!

⇒ − Z 1

0

B(r)dB(r)−Λ_π. (11)

We cannot apply this approach whenω∈(−π,0)∪(0, π) since in this case it holds that xtxt= (e^−iωxt−1+ηt)(e^iωxt−1+η_t) =xt−1xt−1+ηtη_t+e^−iωxt−1η_t+e^iωηtxt−1. Hence, asT → ∞,

1 T

T

X

t=1

(e^−iωxt−1η_t+e^iωη_txt−1) = 1 T

T

X

t=1

(x_tx_t−xt−1xt−1)− 1 T

T

X

t=1

η_tη_t

= 1

TxTxT − 1

Tx0x0− 1 T

T

X

t=1

ηtη_t

⇒B(1)B(1)−Σ.

Without any effort, for multivariate {x_t}_t∈N0 we obtain analogously 1

T

T

X

t=1

(e^−iωxt−1η_t^∗+e^iωηtx^∗_t−1)⇒B(1)B(1)^∗−Σ. (12) By an application of the multivariate integration-by-parts formula for Brownian motions¹ and noting thatΨ(e^iω)Ψ(e^iω)^∗−Σ = Λ_ω+ Λ^∗_ω we can rewrite (12) as

B(1)B(1)^∗−Σ = Z 1

0

B(r)dB(r)^∗+ Z 1

0

dB(r)B(r)^∗+ Λ_ω+ Λ^∗_ω. Whilst the above considerations lead one to expect that

1 T

T

X

t=1

e^−iωxt−1η_t^∗⇒ Z 1

0

B(r)dB(r)^∗+ Λω, (13)

1The integration-by-parts formula also applies for complex Brownian motions. LetV(r) =V1(r) +iV2(r)and Z(r) =Z1(r)+iZ2(r)be two complex Brownian motions. Then, by the definition of the complex Itô-Integral it holds that

Z 1

0

V(r)dZ(r)^∗= Z 1

0

V1(r)dZ1(r) + Z1

0

V2(r)dZ2(r) +i Z 1

0

V2(r)dZ1(r)−i Z 1

0

V1(r)dZ2(r).

The complex integration-by-parts formula follows from an application of the multivariate real integration- by-parts formula for each of the integrals and rearranging the resulting terms.

asT → ∞, this claim cannot be deduced from (12). This is similar to the case where {x_t}_t∈N0 is multivariate withω ∈ {0, π}. In particular, ifω = 0 it holds that

xtx⁰_t= (xt−1+ηt)(xt−1+ηt)⁰ =xt−1x⁰_t−1+ηtη⁰_t+xt−1η_t⁰+ηtx⁰_t−1 and, instead of a multivariate version of (10), we now obtain

1 T

T

X

t=1

(xt−1η⁰_t+ηtx⁰_t−1)⇒B(1)B(1)⁰−Σ. (14) Ifω=π it holds thatx_tx⁰_t=xt−1x⁰_t−1−xt−1η_t−η_txt−1+η⁰_t and, hence,

−1 T

T

X

t=1

(xt−1η_t⁰+η_tx⁰_t−1)⇒B(1)B(1)⁰−Σ. (15) Phillips (1988a,b) has proven (13) forω = 0. The general result for arbitrary frequencies is the main result of this section.

Theorem 2. Let {x_t}_t∈N0 be a stochastic process in discrete time generated according to the difference equation (1)with Assumption 1 in place. Then, as T → ∞, it holds that

1 T

T

X

t=1

xt−1η^∗_t ⇒e^iω

τ_ω² Z 1

0

B(r)dB(r)^∗+ Λ_ω

, (16)

whereτ_ω and Λ_ω are introduced in (8) and (7), respectively and B(r) is defined in Theorem 1.

Remark 6. As mentioned in the introduction this result has also been established by Gregoir (2010). However, his proof is a generalization of the proof of Phillips (1988a) and therefore it crucially relies on the martingale approximation of Hall and Heyde (1980). Our proof of Theorem 2 is much simpler as we only require weak convergence results for martingale difference sequences, presented in Lemma 1, and the decomposition stated in Proposition 1.

By the same arguments as in the proof of Theorem 2 we can also derive the limiting distribution of the sample covariance matrix between{η_t}_t∈Z and a deterministic sequence.

Corollary 3. Let {η_t}_t∈Z be a stochastic process that fulfills Assumption 1 and let {d_t}_t∈N0 be a deterministic sequence that satisfies the assumptions stated in Corollary 2. Then, asT → ∞, it holds that

1 T^1/2G⁻¹_F

T

X

t=1

dtη_t^∗⇒τω

Z 1 0

D(r)dB(r)^∗.

Note that there is no additive bias appearing in the limiting distribution which is due to the obvious independence between deterministic sequences and stochastic processes.

At the end of this section we present the limiting distribution of the ordinary least squares estimator (OLS) forA in (1), given by

Aˆ=

T

X

t=1

xtx^∗_t−1

! _T X

t=1

xt−1x^∗_t−1

!−1

, (17)

which is an important building block for the asymptotic theory of seasonal unit root tests discussed in the next section.

Theorem 3. Let {x_t}_t∈N0 be a stochastic process in discrete time generated according to the difference equation (1)with Assumption 1 in place. Then, as T → ∞, it holds that

T( ˆA−A)⇒e^−iω

τ_ω² Z 1

0

dB(r)B(r)^∗+ Λ^∗_ω τ_ω² Z 1

0

B(r)B(r)^∗dr −1

, (18)

whereΛω and τω are introduced in (7) and (8), respectively and B(r) is defined in Theorem 1.

We can extend this result for unit root processes that contain a deterministic component. In particular, consider then-dimensional stochastic process {y_t}_t∈N0 generated according to

yt=Bddt+xt, t∈N, (19) where the process{x_t}_t∈N0 is generated according to (1) and{d_t}_t∈N0is a deterministic sequence satisfying the assumptions stated in Corollary 2 and Corollary 3. Clearly, (19) is equivalent to

y_t=B_dd_t+B_xxt−1+η_t, t∈N, (20) whereBx=e^−iωIn. Setting zt= [d⁰_t, x⁰_t−1]⁰ the OLS estimator forB = [Bd, Bx]is given by

Bˆ =

T

X

t=1

ytz_t^∗

! _T X

t=1

ztz_t^∗

!⁻¹ .

The limiting distribution of the scaled and centered OLS estimator follows now from several results presented previously in this section. Note that from the different convergence rates required in Proposition 2 and Corollary 2 as well as in Theorem 2 and Corollary 3 we deduce that the coefficient estimates must also converge at different rates. We therefore define the scaling matrix

G=

"

G_d 0 0 T^1/2In

# , whereGd is defined in Corollary 2.