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Inference of seasonal cointegrationwith linear restrictionsByeongchan Seong a , Sinsup Cho b & Sung K. Ahn aa Department of Management and Operations , Washington StateUniversity Pullman , WA, 99164-4736, USAb Department of Statistics , Seoul National University , Seoul,151-747, KoreaPublished online: 01 Aug 2007.

To cite this article: Byeongchan Seong , Sinsup Cho & Sung K. Ahn (2007) Inference of seasonalcointegration with linear restrictions, Journal of Statistical Computation and Simulation, 77:7,593-603, DOI: 10.1080/10629360600569436

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Journal of Statistical Computation and SimulationVol. 77, No. 7, July 2007, 593–603

Inference of seasonal cointegration with linear restrictions

BYEONGCHAN SEONG†, SINSUP CHO*‡ and SUNG K. AHN†

†Department of Management and Operations, Washington State University Pullman,WA 99164-4736, USA

‡Department of Statistics, Seoul National University, Seoul 151-747, Korea

(Revised 2 April 2005; in final form 25 November 2005)

In this article, we study the statistical inference of seasonal cointegration with joint linear restrictionsamong cointegrating vectors associated with possibly different seasonal unit roots. A Wald-type testand a likelihood ratio test are considered. For the development of the test statistics, we use the Gaussianreduced-rank estimation of Ahn et al. [Ahn, S.K., Cho, S. and Seong, B.C., 2004, Inference of seasonalcointegration: Gaussian reduced rank estimation and tests for various types of cointegration. OxfordBulletin of Economics and Statistics, 66, 261–284], which simultaneously accommodates the coin-tegration corresponding to all seasonal unit roots. We then obtain the asymptotic distributions of thetest statistics. We present methods for accommodating linear restrictions in the Gaussian reduced-rankestimation and obtain the related asymptotic distributions. A Monte Carlo simulation is conducted toinvestigate small-sample properties of the test statistics for some linear restrictions.

Keywords: Hypothesis testing; Cointegrating vectors; Gaussian reduced-rank estimation; Seasonalunit root

JEL Classification: C12; C22; C32

1. Introduction

Since Hylleberg et al. [1] introduced the concept of seasonal cointegration, the statistical infer-ence of seasonal cointegration has been studied extensively. Among others, Lee [2] developedthe maximum likelihood inference of seasonal cointegration for the case with contempo-raneous (or synchronous) cointegration, and Ahn and Reinsel [3] developed the Gaussianreduced-rank (GRR) estimation that covers the case of polynomial (non-synchronous) coin-tegration as well as contemporaneous cointegration. These studies focused on the quarterlyseries, whereas Johansen and Schaumburg [4] developed the maximum likelihood inference forgeneral seasonal series with polynomial cointegration by introducing an error correction model(ECM) with complex-valued coefficient matrices. Cubadda [5] developed the reduced-rankregression analysis using an ECM for complex-valued series that resolves the computationalcomplexities involved in Johansen and Schaumburg [4]. Ahn et al. [6] extended the GRRestimation of Ahn and Reinsel [3] to general seasonal series.

*Corresponding author. Email: sinsup@snu.ac.kr

Journal of Statistical Computation and SimulationISSN 0094-9655 print/ISSN 1563-5163 online © 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/10629360600569436

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594 Byeongchan Seong et al.

While most of the earlier literature on seasonal cointegration considers inference for modelswith no restriction on cointegrating vectors, Johansen and Schaumburg [4] and Cubadda [5]suggested χ2 tests for the linear hypotheses for a cointegrating vector at a given frequencyfor models that have restrictions on cointegrating vectors. Ahn et al. [6] presented tests forcontemporaneous cointegration at a given frequency and common polynomial cointegrationat two different frequencies of seasonal unit roots.

Recently, Cubadda and Omtzigt [7] showed an increase in the efficiency of inference in smallsamples by jointly modeling cointegration restrictions across different frequencies. However,earlier studies, exceptAhn and Reinsel [3] andAhn et al. [6], focused on a single frequency cor-responding to a seasonal root at a time, ignoring any possible cointegration at other frequenciesand regressing out the terms at the other frequencies in the ECM.Although the GRR estimationinAhn and Reinsel [3] andAhn et al. [6] considered cointegration at all frequencies of seasonalunit roots simultaneously, enabling simultaneous tests and estimation with restrictions amongseasonal cointegrating vectors, especially across different frequencies, they did not considerthe tests for cointegrating vectors with linear restrictions across different frequencies.

The reasons for pursuing the simultaneous inference are twofold. First, the efficiency ofestimation can be improved in small samples by imposing true restrictions on the model.For example, Ahn et al. [6] improved the performance of estimation by using the restrictedmodel with true contemporaneous or common polynomial cointegration, and Cubadda andOmtzigt [7] improved the efficiency of inference as mentioned in the previous paragraph.Secondly, empirical studies or some of the economic theories provide motivations for testing(common) restrictions on a subset of cointegrating vectors across different frequencies. Theserestrictions provide simple structures for seasonal cointegrating vectors that are meaningfuland easy to interpret (see [4, p. 320]).

This article is organized as follows. In section 2, we introduce the ECM for seasonal coin-tegration and the GRR estimation in Ahn et al. [6]. In section 3, we consider the tests forhypotheses with generalized linear restrictions on cointegrating vectors at possibly differentfrequencies of seasonal unit roots. In section 4, we study the method of accommodating linearrestrictions in the GRR estimation and obtain the related asymptotic distributions that can beused for inference on the restricted models among seasonal cointegrating vectors. In section 5,we examine finite sample properties of the test statistics for some linear restrictions, using aMonte Carlo simulation.

2. ECM and the GRR estimation

Let yt be an m-vector time series with non-stationary seasonal behavior and period s such that

�(L)yt =(

Im −p∑

k=1

�kLk

)yt = εt , (1)

where εt ’s are independent normal random vectors with mean 0 and variance Ω, and theinitial values y−p+1, . . . , y0 are fixed. Here, for convenience, we consider a model with nodeterministic terms; however, models with deterministic terms can be easily implemented asin Ahn et al. [6], Cubadda [5], and Johansen and Schaumburg [4].

As in Ahn et al. [6], if the series are cointegrated at frequencies θj /2π for distinct seasonalunit roots ωj = exp(iθj ) and θj ∈ [0, π ] for j = 1, . . . , K , where i = √−1 is the imaginarynumber, model (1) may be rewritten in the following ECM,

�∗(L)(1 − Ls)yt =K∑

j=1

{(αjRβ ′jI + αjIβ

′jR)w(j)

t−1 + (αjIβ′jI − αjRβ ′

jR)v(j)

t−1} + εt , (2)

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Inference of seasonal cointegration 595

where y(j)

t−1 = �Kk �=j∇k(L)yt−1, w(j)

t = sin θj y(j)t , and

v(j)t =

{− cos θj y(j)

t for θj = 0 or π

− cos θj y(j)t + y(j)

t−1 for θj ∈ (0, π)

for

∇j (L) ={

1 − cos θjL for θj = 0 or π,

1 − 2 cos θjL + L2 for θj ∈ (0, π),

and

(αjR + iαjR)(βjR + iβjI )′ = −�(ωj )

{ωjf (j)(ωj )} for θj = 0 or π

(αjR + iαjR)(βjR + iβjI )′ = �(ωj )

{iωjf (j)(ωj ) sin θj } for θj ∈ (0, π)

such that αjR , αjI βjR , and βjI are m × rj real-valued matrices with a rank equal to rj ,βjR = [Irj

, β ′0jR]′, and βjI = [Orj

, β ′0jI ]′, where Irj

and Orjare an rj × rj identity matrix

and an rj × rj zero matrix, respectively. Note that αjI and βjI are zero matrices in the caseof θj = 0 or π .

For parameterization, we define η = (← β′jR, β′

jI →, α′)′ with βjR = vec(β0jR), βjI =vec(β0jI ), and α = vec{(← αjR, αjI →, �∗

1, . . . ,�∗p−s)

′}, where vec(A) denotes a vectorformed by stacking the columns of matrix A and ← αjR, αjI →, denotes an arrange-ment of αjR and αjI side by side for all j ’s (in the case of θj = 0 or π , B0jI and αjI

are omitted). Note that βjR, βjI , α, and η are rj dj , rj dj , m{∑θj =0orπ rj + 2∑

θj ∈(0,π) rj +m(p − s)}, and

∑θj =0orπ (rjdj + mrj ) + 2

∑θj ∈(0,π)(rj dj + mrj ) + m2(p − s)-dimensional

vectors, respectively. We further define w(j)

2t and v(j)

2t and denoting the last dj = m − rj

elements of w(j)t and v(j)

t , respectively.The GRR estimator of η, based on T observations y1, . . . , yT , is obtained by the iterative

approximate Newton–Raphson relations

η̂(k+1) = η̂

(k) +(

T∑t=1

Xt−1�−1X′

t−1

)−1 (T∑

t=1

Xt−1�−1εt

)∣∣∣∣∣∣η̂

(k)

, (3)

where η̂(k) is an estimator at the previous iteration,

Xt−1 = [← X′(j)

t−1 →, Im ⊗ X̃′t−1]′, (4)

X(j)

t−1 = [αjI ⊗ w′(j)

2t−1 − αjR ⊗ v′(j)

2t−1, αjR ⊗ w′(j)

2t−1 + αjI ⊗ v′(j)

2t−1]′, (5)

X̃t−1 = [← (β ′jI w(j)

t−1 − β ′jRv(j)

t−1)′, (β ′

jRw(j)

t−1 + β ′jI v(j)

t−1)′ → z′

t−1, . . . , z′t−p+s]′, (6)

and ⊗ is the Kronecker product. We note that in the case of θj = 0 or π , X(j)

t−1 = [−αjR ⊗v′(j)

2t−1]′ and the corresponding component of X̃t−1 is (−β ′jRv(j)

t−1)′. We have the following

results for the GRR estimator, which will be used to obtain the asymptotic distributions of ourtest statistics for generalized linear restrictions.

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596 Byeongchan Seong et al.

LEMMA 1 (Ahn et al. [6]) Let (β̂′jR, β̂′

jI )′ denote the GRR estimator for (β′

jR, β′jI )

′ obtainedfrom equation (3). Then, as T → ∞,

(∑X̂

(j)

t−1�̂−1

X̂′(j)

t−1

)1/2 (β̂jR − βjR

)D−→ N(0, Irj dj

) for θj = 0 or π,

(∑X̂

(j)

t−1�̂−1

X̂′(j)

t−1

)1/2(

β̂jR − βjR

β̂jI − βjI

)D−→ N(0, I2rj dj

) for θj ∈ (0, π),

and these are asymptotically independent for different j = 1, . . . , K , where Ω̂ is a consistentestimator of Ω.

Lemma 1 includes the asymptotic distributions of cointegrating vectors, i.e. (β̂′jR, β̂′

jI )′’s,

only because our study focuses on restrictions with respect to these. In the usual cointegrationanalysis, the parameters of main interest are βjR’s and βjI ’s rather than the stationary parameterα. Restrictions on α can be easily implemented in the usual regression method, using Theorem1 of Ahn et al. [6], and thus we do not pursue the joint restrictions between (β′

jR, β′jI )

′’s andα in this article.

3. Hypothesis test for linear restrictions on seasonal cointegrating vectors

The generalized M different linear restrictions can be considered as follows:

H0: Rβ = r, (7)

where R = (Rij ), β = (← β′jR, β′

jI →)′, and r = (r′1, . . . , r′

M)′ for i = 1, . . . , M and j =1, . . . , K (in the case of θ j = 0 or π , βjI is omitted). Rij is a known matrix of size pi × 2rj dj

for θj ∈ (0, π) or pi × rj dj for θj = 0 or π , representing the particular linear combinationsof β, about which we entertain the hypotheses, and ri is a known pi-dimensional vector,where pi’s are the given constants. This hypothesis is general enough to include the linearhypotheses about seasonal cointegration, which have been considered in the previous literatureon the subject.

For example, contemporaneous cointegration corresponding to one frequency for θk ∈(0, π), i.e. H0: βkI = 0, is the case with M = 1, R1k = [Orkdk

, Irkdk], R1j = 0 for j �= k,

r1 = 0 and p1 = rkdk such that the compact form of hypothesis (7) is

H0: R1k(β′kR, β′

kI )′ = r1

Using this form, we can test the assumption of Lee [12] with respect to frequency 1/4 (θ =π/2). The common polynomial cointegration corresponding to the two frequencies for θk, θl ∈(0, π) with k �= l, i.e. H0: (β′

kR, β′kI )

′ = (β′lR, β′

lI )′, is the case with M = 1, R1k = I2r.d.,

R1l = −I2r.d., R1j = 0 for j �= k and l, and r1 = 0, where r.d. = rkdk = rldl and p1 = 2r·d·such that the compact form is

H0: [R1k, R1l](β′kR, β′

kI , β′lR, β′

lI )′ = r1.

The common polynomial cointegration corresponding to the three frequencies for θk, θl, θm, ∈(0, π) with different k, l, and m, i.e. H0: (β′

kR, β′kI )

′ = (β′IR, β′

lI )′ = (β′

mR, β′mI )

′, is the casewith M = 2, R1k = I2r.d., R1l = −I2r.d., R1j = 0 for j �= l and k, R2l = I2r.d., R2m = −I2r.d.,

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Inference of seasonal cointegration 597

R2j = 0 for j �= l and m, and r1 = 0, and r2 = 0, where, r.d. = rkdk = rldl = rmdm andp1 = p2 = 2r·d· such that the compact form is

H0:[

R1k, R1l , R1m

R2k, R2l , R2m

](β′

kR, β′kI , β

′lR, β′

lI , β′mR, β′

mI )′ =

(r1

r2

).

As an example of a more complicated hypothesis, consider a test to assess whether a linearcombination of cointegrating vectors among different frequencies has a specific value. Forexample,

c1

(βkR

βkI

)+ c2

(βlR

βlI

)+ c3

(βmR

βmI

)= 0 for different k, l, and m, (8)

where c1, c2, and c3 are scalars with non-zero values. This restriction implies that there existsdependency among the long-run equilibrium coefficients at three different frequencies. Thisis the case with M = 1, R1k = c1I2r·d·, R1l = c2I2r·d·, R1m = c3I2r·d·, R1j = 0 for j �= k, l,and m, and r1 = 0, where r.d. = rkdk = rldl = rmdm and p1 = 2r.d..

Some precautions are required while constructing the hypotheses wherein some of thefrequencies belong to 0 or 1/2 (θ = π) and the remaining belong to the seasonal frequenciesother than 0 and 1/2. These include the full cointegration considered by Lee [2] and Engleet al. [8]. More specifically, when we are interested in the common cointegration betweenfrequencies 1/2 (θ = π) and 1/4 (θ = π/2), the form of hypothesis is a joint type, H0: β3I = 0and β2R = β3R , where β2R and (β′

3R, β′3I )

′ denote cointegrating vectors at frequencies 1/2 and1/4, respectively. This can be re-expressed by the following form,

H0:[

R12, R13

R22, R23

](β′

2R, β′3R, β′

3I )′ =

(r1

r2

).

with M = 2, R12 = Or.d., R13 = [Or.d., Ir.d.], R22 = Ir.d., R23 = [−Ir.d., Or.d.], r1 = 0, andr2 = 0, where r.d. = r2d2 = r3d3 and p1 = p2 = r.d.

As a more relevant case for practitioners, hypothesis (7) includes individual restrictions ona subset of the cointegrating vectors. For example, when interested in the zero coefficientsin cointegrating vectors, which is similar to the case in non-seasonal cointegration (see [9]p. 648), we can adjust the matrix R such that Rij (β

′jR, β′

jI )′ = 0, where the elements of Rij

that correspond to the interested subset of the cointegrating vector are ones and the remainingare zeros. Furthermore, as a result of the test for these types of restrictions, if we conclude thatthe kth (k ≤ rj for j = 1, . . . , K) columns in both βjR = [Irj

, β ′0jR]′ and βjI = [Orj

, β ′0jI ]′

matrices are unit and zero vectors, respectively, for all frequencies, this implies that the kthvariable in the cointegrated system is stationary. Hoffmann and MacDonald [10] defined thiscase as a trivial cointegration. Bohl [11], for instance, excluded the interest rate from hisseasonal cointegration analysis to model the relationship between the German money demandand its determinants such as real income and interest rate because the interest rate appearedto be stationary. However, the interest rate could be included in the analysis for testing trivialcointegration at all frequencies and for estimation such that the resulting ECM can allowvariations of the interest rate to play an important role in the short-run dynamics of moneydemand in accordance with the reasoning from economic theory.

In the following theorem, we define a Wald-type test statistic for hypothesis (7) and stateits asymptotic distribution.

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598 Byeongchan Seong et al.

THEOREM 1 Let β̂ = (← β̂′jR, β̂′

jI →)′ denote the GRR estimator obtained from equation (3).For testing hypothesis (7), the Wald test statistic is

W = (Rβ̂ − r)′[RDR′]−1(Rβ̂ − r), (9)

where D = diag(( ∑

X̂(1)t−1�̂

−1X̂

′(1)t−1

)−1, . . . ,

( ∑X̂

(K)t−1�̂

−1X̂

′(K)t−1

)−1), �̂ is a consistent esti-

mator of � and it is asymptotically a χ2 random variable with∑M

i=1 Pi degrees offreedom.

Proof On the basis of the asymptotic independence in Lemma 1, we obtain

⎛⎜⎜⎜⎜⎜⎜⎜⎝

(∑X̂

(1)t−1�̂

−1X̂

′(1)t−1

)1/2(

β̂1R − β1R

β̂1I − β1I

)...(∑

X̂(K)t−1�̂

−1X̂

′(K)t−1

)1/2(

β̂KR − βKR

β̂KI − βKI

)

⎞⎟⎟⎟⎟⎟⎟⎟⎠

D−→ N(0, ),

where = diag(Iv1 , . . . , IvK) and vk is rkdk in the case of θk = 0 or π , or 2rkdk in the case

of θk ∈ (0, π). Given the sample paths of X̂(1)t−1, . . . , X̂

(K)t−1, β̂ is approximately a multivariate

normal random vector with mean β̂ and variance D for large T sample size. Therefore, underthe null H0: Rβ = r, Rβ̂ is approximately a multivariate normal random vector with mean rand variance RDR′, and equation (9) is approximately a χ2-random variable with

∑Mi=1 pi

degrees of freedom. As this asymptotic distribution is independent of the sample paths ofX̂

(1)t−1, . . . , X̂

(K)t−1 it is also an unconditional distribution. �

Next, we consider a likelihood ratio (LR) test statistic for testing hypothesis (7).As a properlystandardized form of β̂ does follow the usual normal theory asymptotics as in Lemma 1, theLR test is asymptotically equivalent to the Wald test [see, for example, 12]; thus, we state thefollowing theorem without proof.

THEOREM 2 For testing hypothesis (7), the LR test statistic is

LR = −T ln|�̂(η̂)||�̂(η̃)| , (10)

where η̂ = (β̂′, α̂′)′ and η̃ = (β̃′, α̃′)′, respectively, denote the unrestricted GRR estimator andthe restricted GRR estimator under restrictions (7), and it is asymptotically equivalent tostatistic (9).

We comment that the test statistic W in Theorem 1 depends only on the GRR estimatorsin the unrestricted model, whereas the LR test statistic in Theorem 2 requires the estimatorsin the restricted model under the null hypothesis as well as in the unrestricted model. TheGRR estimation of Ahn et al. [6] enables us to accommodate linear restrictions on seasonalcointegrating vectors at possibly different frequencies.

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Inference of seasonal cointegration 599

4. Inference on the restricted models

In this section, we study the method of accommodating restricted models with two simplecases of restrictions in the GRR estimation and obtain the related asymptotic distributions.This method can be easily extended to models with various restrictions.

The two simple cases considered are as follows.

Case I. The presence of a contemporaneous cointegration at a given frequency, and

Case II. The presence of a common polynomial cointegration at two different frequencies.For Case I, we are interested in estimating βkR and testing

H0: βkR = b for a known b, (11)

because in this case, βkI = 0 for a frequency with θk ∈ (0, π). For Case II, we have(β′

kR, β′kI )

′ = (β′lR, β′

lI )′ for different frequencies k �= l. As a special case of Case II, common

contemporaneous cointegration at two different frequencies may be considered, i.e. βkR = βlR

and βkI = βlI = 0 for k �= l. In Case II, we are interested in estimating (β′·R, β′

·I )′ as the pooledestimator for (β′

kR, β′kI )

′ = (β′lR, β′

lI )′ and testing

H0: (β′·R, β′

·I )′ = (b′

R, b′I )

′ for a known (b′R, b′

I )′. (12)

For the GRR estimation in the restricted models, equation (5) in the unrestricted model, i.e.X′(j)

t−1 for j = 1, . . . , K , needs to be appropriately modified or combined among the related

frequencies because only through theX′(j)

t−1’s does the structure of the restrictions in the seasonal

cointegrating vectors influence the likelihood function [13]. For the estimator β̂kR in Case I,we use, instead of X

′(k)t−1,

X′∗(k)t−1 = αkI ⊗ w′(k)

2t−1 − αkR ⊗ v′(k)2t−1,

and for the pooled estimator (β̂′·R, β̂′

·I )′ in Case II,

X′(·)t−1 =

[αkI ⊗ w′(k)

2t−1 − αkR ⊗ v′(k)2t−1 + αlI ⊗ w′(l)

2t−1 − αlR ⊗ v′(l)2t−1, αkR ⊗ w′(k)

2t−1

+αkI ⊗ v′(k)2t−1 + αlR ⊗ w′(l)

2t−1 + αlI ⊗ v′(l)2t−1

],

with X′(k)t−1 and X

′(l)t−1 omitted. Further, in the case of the restriction in equation (8) with

c1 = c2 = c3 = 1, we can substitute

X′∗(k)t−1 =

[αkI ⊗ w′(k)

2t−1 − αkR ⊗ v′(k)2t−1 − αmI ⊗ w′(m)

2t−1 + αmR ⊗ v′(m)2t−1, αkR ⊗ w′(k)

2t−1

+αkI ⊗ v′(k)2t−1 − αmR ⊗ w′(m)

2t−1 − αmI ⊗ v′(m)2t−1

],

X′∗(l)t−1 =

[αlI ⊗ w′(l)

2t−1 − αlR ⊗ v′(l)2t−1 − αmI ⊗ w′(m)

2t−1 + αmR ⊗ v′(m)2t−1, αlR ⊗ w′(l)

2t−1

+αlI ⊗ v′(l)2t−1 − αmR ⊗ w′(m)

2t−1 − αmI ⊗ v′(m)2t−1

]for X

′(k)t−1 and X

′(l)t−1, respectively, with X

′(m)t−1 omitted.

When estimating the restricted models with individual restrictions on a subset of thecointegrating vectors, we can use the adjusted matrix X

′∗(j)

t−1 obtained by deleting the kth

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600 Byeongchan Seong et al.

column, which is related to the restrictions, in the corresponding X′(j)

t−1. For example, if thekth (k ≤ rj for j = 1, . . . , K) variable is supposed to have a common trivial cointegrationacross all frequencies, the kth columns of both the first and second block matrices of X

′(j)

t−1, i.e.

[αjI ⊗ w′(j)

2t−1 − αjR ⊗ v′(j)

2t−1] and [αjR ⊗ w′(j)

2t−1 + αjI ⊗ v′(j)

2t−1], respectively, are deleted for

the new X′∗(j)

t−1 . Note that the trivial cointegration implies that the kth columns of both β0jR

and β0jI matrices are zero columns.These modified X

′∗(j)

t−1 ’s and those associated with other restrictions can be obtained fromthe rewritten form of the error term in model (2):

εt =∑

j

X′(j)

t−1(β′jR, β′

jI )′ + {the parts unrelated to (β′

jR, β′jI )

′s}. (13)

For details, see [14, chapter 3]. Using the modified X′∗(j)

t−1 ’s, we adjust the Hessian matrix andscore vector in the Newton–Raphson iteration in order to obtain estimates for the restrictedmodels.

When we estimate the restricted models with more complicated linear restrictionssuch as

R1

(βkR

βkI

)+ R2

(βlR

βlI

)+ R3

(βmR

βmI

)= r for different k, l, and m,

where R1, R2, and R3 are non-zero matrices, the modification of X′(j)

t−1’s among the relatedfrequencies is not easy. In this case, a numerical optimization using, for example, the Lagrangemultiplier is recommended. From this point of view, the Wald test statistic is easier to calculatethan the LR test statistic.

In order to test the hypotheses in the restricted models, we need to find the related asymptoticdistributions. Theorem 3 shows the asymptotic distributions of the GRR estimator in therestricted models of Cases I and II, which can be easily extended to the models with variousrestrictions.

THEOREM 3 In the restricted model of Case I, the asymptotic distribution of β̂kR is, undernull hypothesis (11),(∑

X̂∗(k)t−1 �̂

−1X̂

′∗(k)t−1

)1/2 (β̂kR − b

)D−→ N(0, Irkdk

) for θk ∈ (0, π), (14)

where X̂′∗(k)t−1 = [α̂kl ⊗ w′(k)

2t−1 − α̂kR ⊗ v′(k)t−1] is an m × rkdk matrix and Ω̂ is a consistent esti-

mator of Ω. In the restricted model of Case II, the asymptotic distribution of the pooledestimator (β̂·R, β̂·I ) is, under null hypothesis (12),

(∑X̂

(·)t−1�̂

−1X̂

′(·)t−1

)1/2(

β̂·R − bR

β̂·I − bI

)D−→ N(0, I2r·d·), (15)

where [X̂′(·)t−1 = [α̂kI ⊗ w′(k)

2t−1 − α̂kR ⊗ v′(k)2t−1 + α̂lI ⊗ w′(l)

2t−1 − α̂IR ⊗ v′(l)2t−1, α̂kR ⊗ w′(k)

2t−1 +α̂kI ⊗ v′(k)

2t−1 + α̂IR ⊗ w′(l)2t−1 + α̂lI ⊗ v′(l)

2t−1] is an m × 2r.d. matrix for r·d· = rkdk = rldl , �̂

is an consistent estimator of �.

Proof The asymptotic distribution in Case I is trivial if we regard the corresponding restrictedmodel as the unrestricted model in the second result of Lemma 1 without βjI . For Case II, we

use X̂′(·)t−1 = X̂

′(k)t−1 + X̂

′(l)t−1 for common parameters (β·R, β·I ), and the asymptotic distribution

follows directly from Lemma 1. �

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Inference of seasonal cointegration 601

5. Monte Carlo experiment

In this section, we conduct Monte Carlo experiments to evaluate and compare the performancesof the Wald and LR tests, developed in section 3. We focus on the quarterly series and threetypes of hypotheses:

H0: β3I = 0, (16)

H ′0: β3I = 0 and β2R = β3R, (17)

H ′′0 : β3I = 0 and β1R + β2R + β3R = 0, (18)

where β1R , β2R , and (β′3R, β′

3I )′ denote the cointegrating vectors at frequencies 0, 1/2, and 1/4,

respectively. The form of hypothesis (18) corresponding to that of hypothesis (7) is

H0:[Or·d· , Or·d· , Or·d· , Ir·d·Ir·d· , Ir·d· , Ir·d· , Or·d·

](β′

1R, β′2R, β′

3R, β′3I )

′ = r,

where r·d· = r1d1 = r2d2 = r3d3 and r is a 2r·d·-dimensional zero vector.For the data-generating process, the one used inAhn and Reinsel [3] to cover the uncommon

(polynomial) cointegration at frequencies 1/2 and 1/4 is modified, i.e.

(1 − L4)yt = α1Rβ ′1Rut−1 + α2Rβ ′

2Rvt−1 + (α3Rβ ′3I + α3I β

′3R)wt−1

+ (−α3Rβ ′3R + α3I β

′3I )wt−2 + εt , (19)

where ut−1 = (1 + L)(1 + L2)yt−1, vt−1 = (1 − L)(1 + L2)yt−1, wt−1 = (1 − L2)yt−1,α1R =(0.6, 0.6)′, α2R = (−0.4, 0.6)′, α3R = (0.6, −0.6)′, α3I = (0.4, −0.8)′, β1R = (1, b1)

′ =(1, −0.7)′, β2R = (1, b2) = (1, 0.35)′, β3R = (1, b3) = (1, −b1 − b2)

′, and β3I = (0, b4)′,

with b4 = 0, 0.01. For the variance of εt = (ε1t , ε2t )′, we choose var(ε1t ) = var(ε2t ) = 1

and cov(ε1t , ε2t = 0.3). Note that the first components in β1R, β2R, β3R , and β3I are normaliz-ing constants. When b4 = 0, yt is contemporaneously cointegrated at frequencies 1/4 as wellas 1/2 with a common cointegrating vector (1, 0.35)′. When b4 = 0.01, yt is cointegrated atfrequencies 0, 1/2, and 1/4, whereas it has different cointegrating vectors at each frequency,contrary to the case of b4 = 0. Here, for our two tests, b4 = 0.01 is chosen for the evaluationof powers. When b4 = 0, the roots of the characteristic equation det{�(L)} = 0 are ±1, ±i,−1.1936, 1.2638, and 0.1387 ± 1.3241i; when b4 = 0.01, they are ±1, ±i, −1.1905, 1.2387,and 0.1181 ± 1.3093i.

For the performances of our Wald and LR tests for hypotheses (16), (17), and (18), wegenerate samples of series with length T = 40, 100, 200, and 400, representing quarterly dataover 10, 25, 50, and 100 years, using MATLAB with 1000 replications in each sample length.We calculate the rejection rate at 5% nominal level of our two tests, using the GRR estimation

Table 1. Rejection rates at 5% level of Wald and LR tests for hypotheses (16), (17), and (18),using data-generating process (17), with b4 = 0 (replications =1000).

Sample length T

40 100 200 400

Test type Wald LR Wald LR Wald LR Wald LR

Hypothesis (16) 0.160 0.091 0.078 0.062 0.069 0.060 0.039 0.037Hypothesis (17) 0.193 0.097 0.081 0.059 0.074 0.064 0.051 0.044Hypothesis (18) 0.207 0.094 0.092 0.063 0.072 0.053 0.055 0.052

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602 Byeongchan Seong et al.

Table 2. Rejection rates at 5% level of Wald and LR tests for hypotheses (16), (17), and (18),using data-generating process (17), with b4 = 0.0 (replications =1000).

Sample length T

40 100 200 400

Test type Wald LR Wald LR Wald LR Wald LR

Hypothesis (16) 0.200 0.139 0.570 0.535 0.929 0.921 0.998 0.997Hypothesis (17) 0.227 0.136 0.517 0.464 0.900 0.893 0.998 0.997Hypothesis (18) 0.262 0.135 0.505 0.459 0.901 0.889 0.999 0.999

based on model (1). The approximate distributions used in both tests for hypotheses (16), (17),and (18) are χ2(1), χ2(2), and χ2(2), respectively. Table 1 shows the results for b4 = 0 andtable 2 for b4 = 0.01.

In table 1, it can be seen that in all the sample lengths, with the exception of T = 400,both tests have considerable size distortions, but the Wald test has a larger size distortion.However, several studies possibly anticipate this result for the small-sample performance ofasymptotic tests on the non-seasonal cointegrating vectors [see, for instance, 15]. Johansen [16]considers a correction to the LR test, which depends on the sample size, in order to improvethe approximation to the asymptotic χ2 distribution. For the seasonal case, a correction similarto the one used in the non-seasonal case may improve the performance. Another finding is thatin all cases, the magnitude of the Wald test statistic is larger than that of the LR; this is a well-known fact in the previous literature on the hypothesis testing theory [see, for example, 17].The interesting finding is that the more complex the form (i.e. number of linear restrictions,M) of the hypothesis, the worse the size distortion becomes, particularly in the case of theWald test.

Table 2 contains the empirical powers of both tests. Overall, the Wald test has higher powerthan the LR test. As the sample length becomes larger, both tests show satisfactory powers.

Acknowledgements

The authors thank an anonymous referee for the helpful comments that led to a significantimprovement of this article. Byeongchan Seong was supported by the Post-doctoral FellowshipProgram of Korea Science and Engineering Foundation (KOSEF). The research of Sinsup Choand Sung K. Ahn was supported by the Korea Research Foundation Grant (KRF-2005-070-C00022) funded by the Korean Government (MOEHRD).

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